Merge remote-tracking branch 'upstream/master'

292 files changed, 68149 insertions, 1373 deletions

diff --git a/buch/.gitignore b/.gitignore
index 5d8a46e..cc64005 100644
--- a/buch/.gitignore
+++ b/.gitignore
@@ -14,3 +14,5 @@ buch*.pdf
 .build/
 *.synctex.gz
 *.DS_Store
+
+
diff --git a/buch/Makefile b/buch/Makefile
index 722c177..b83c72a 100755
--- a/buch/Makefile
+++ b/buch/Makefile
@@ -19,7 +19,6 @@ buch.pdf:	buch.tex $(TEXFILES) buch.ind $(BLXFILES)
 	bibtex buch
 
 buch.idx:	buch.tex $(TEXFILES) images
-	touch buch.ind
 	pdflatex buch.tex
 
 buch.ind:	buch.idx
@@ -28,8 +27,19 @@ buch.ind:	buch.idx
 separate:	buch.aux buch.pdf
 	bash splitpapers
 
-numerik.pdf:	
-	pdfjam --outfile numerik.pdf \
+matrizen.pdf:	
+	pdfjam --outfile matrizen.pdf \
 		../cover/front.pdf 1,{} \
 		buch.pdf 1-504 \
 		../cover/back.pdf {},1
+
+tests:	test1.pdf test2.pdf test3.pdf
+
+test1.pdf:	common/test-common.tex common/test1.tex aufgaben1.tex
+	pdflatex common/test1.tex
+
+test2.pdf:	common/test-common.tex common/test1.tex aufgaben2.tex
+	pdflatex common/test2.tex
+
+test3.pdf:	common/test-common.tex common/test1.tex aufgaben3.tex
+	pdflatex common/test3.tex
diff --git a/buch/aufgaben1.tex b/buch/aufgaben1.tex
new file mode 100644
index 0000000..9348019
--- /dev/null
+++ b/buch/aufgaben1.tex
@@ -0,0 +1,13 @@
+%
+% aufgaben1.tex -- Aufgaben für Test 1
+%
+% (c) 2012 Prof. Dr. Andreas Mueller, HSR
+%
+
+\item
+\input chapters/30-endlichekoerper/uebungsaufgaben/3003.tex
+\item
+\input chapters/30-endlichekoerper/uebungsaufgaben/3004.tex
+\item
+\input chapters/30-endlichekoerper/uebungsaufgaben/3005.tex
+
diff --git a/buch/aufgaben2.tex b/buch/aufgaben2.tex
new file mode 100644
index 0000000..dc4fc59
--- /dev/null
+++ b/buch/aufgaben2.tex
@@ -0,0 +1,11 @@
+%
+% aufgaben2.tex -- Aufgaben für Test 2
+%
+% (c) 2021 Prof. Dr. Andreas Mueller, OST
+%
+
+\item
+\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex
+\item
+\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex
+
diff --git a/buch/aufgaben3.tex b/buch/aufgaben3.tex
new file mode 100644
index 0000000..23c9153
--- /dev/null
+++ b/buch/aufgaben3.tex
@@ -0,0 +1,7 @@
+%
+% aufgaben3.tex -- Aufgaben für Test 3
+%
+% (c) 2021 Prof. Dr. Andreas Mueller, OST
+%
+\item
+\input chapters/60-gruppen/uebungsaufgaben/6001.tex
diff --git a/buch/buch.tex b/buch/buch.tex
index 65c2ca7..449bc2a 100644
--- a/buch/buch.tex
+++ b/buch/buch.tex
@@ -44,6 +44,6 @@
 \lhead{Index}
 \rhead{}
 \addcontentsline{toc}{chapter}{\indexname}
-\input{buch.ind}
+\InputIfFileExists{buch.ind}{}{}
 
 \end{document}
diff --git a/buch/chapters/10-vektorenmatrizen/gruppen.tex b/buch/chapters/10-vektorenmatrizen/gruppen.tex
index 9848469..cb37d05 100644
--- a/buch/chapters/10-vektorenmatrizen/gruppen.tex
+++ b/buch/chapters/10-vektorenmatrizen/gruppen.tex
@@ -182,7 +182,7 @@ begegnet, wo wir nur gezeigt haben, dass $AA^{-1}=E$ ist.
 Da aber die invertierbaren Matrizen eine Gruppe
 bilden, folgt jetzt aus dem Satz automatisch, dass auch $A^{-1}A=E$.
 
-\subsubsection{Homomorphismen}
+\subsubsection{Homomorphismen} \label{buch:gruppen:subsection:homomorphismen}
 Lineare Abbildung zwischen Vektorräumen zeichnen sich dadurch aus,
 dass sie die algebraische Struktur des Vektorraumes respektieren.
 Für eine Abbildung zwischen Gruppen heisst dies, dass die Verknüpfung,
@@ -313,14 +313,14 @@ auf einem geeigneten Vektorraum.
 \begin{definition}
 \label{buch:vektorenmatrizen:def:darstellung}
 Eine Darstellung einer Gruppe $G$ ist ein Homomorphismus 
-$G\to\operatorname{GL}_(\mathbb{R})$.
+$G\to\operatorname{GL}_n(\mathbb{R})$.
 \index{Darstellung}
 \end{definition}
 
 \begin{beispiel}
 Die Gruppen $\operatorname{GL}_n(\mathbb{Z})$,
 $\operatorname{SL}_n(\mathbb{Z})$ oder $\operatorname{SO}(n)$ 
-sind alle Teilmengen von $\operatorname{GL}_n(\mathbb{R}$.
+sind alle Teilmengen von $\operatorname{GL}_n(\mathbb{R})$.
 Die Einbettungsabbildung $G\hookrightarrow \operatorname{GL}_n(\mathbb{R})$
 ist damit automatisch eine Darstellung, sie heisst auch die
 {\em reguläre Darstellung} der Gruppe $G$.
diff --git a/buch/chapters/10-vektorenmatrizen/linear.tex b/buch/chapters/10-vektorenmatrizen/linear.tex
index 2fcf199..ac2b85d 100644
--- a/buch/chapters/10-vektorenmatrizen/linear.tex
+++ b/buch/chapters/10-vektorenmatrizen/linear.tex
@@ -839,6 +839,83 @@ die Eigenschaft $A^{-1}A=I$ ganz allgemein gezeigt.
 \subsubsection{Determinante}
 XXX TODO
 
+\begin{beispiel}
+Die Inverse der Matrix
+\begin{equation}
+A=\begin{pmatrix}
+1&a&a\\
+a&1&a\\
+a&a&1
+\end{pmatrix}
+\label{buch:vektoren-und-matrizen:abeispiel:eqn1}
+\end{equation}
+ist mit Hilfe von Determinanten besonders einfach zu invertieren.
+Die Determinante von $A$ ist nach der Sarrus-Formel
+\[
+\det A
+=
+1 + 2a^3  - 3a^2.
+\]
+Die adjungiert Matrix ist
+\begin{align*}
+A^{-1}
+&=
+\frac{1}{\det{A}}
+\begin{pmatrix}
+\det A_{11} & \det A_{21} & \det A_{31} \\
+\det A_{12} & \det A_{22} & \det A_{32} \\
+\det A_{13} & \det A_{23} & \det A_{33} 
+\end{pmatrix}
+\\
+&=
+\frac{1}{2a^3-3a^2+1}
+\renewcommand\arraystretch{1.1}
+\begin{pmatrix*}[r]
+\left|\begin{matrix}1&a\\a&1\end{matrix}\right|
+&
+-\left|\begin{matrix}a&a\\a&1\end{matrix}\right|
+&
+\left|\begin{matrix}a&a\\1&a\end{matrix}\right|
+\\
+-\left|\begin{matrix}a&a\\a&1\end{matrix}\right|
+&
+\left|\begin{matrix}1&a\\a&1\end{matrix}\right|
+&
+-\left|\begin{matrix}1&a\\a&a\end{matrix}\right|
+\\
+\left|\begin{matrix}a&1\\a&a\end{matrix}\right|
+&
+-\left|\begin{matrix}1&a\\a&a\end{matrix}\right|
+&
+\left|\begin{matrix}1&a\\a&1\end{matrix}\right|
+\end{pmatrix*}
+\\
+&=
+\frac{1}{2a^3-3a^2+1}
+\begin{pmatrix}
+1-a^2 & a^2-a & a^2-a\\
+a^2-a & 1-a^2 & a^2-a\\
+a^2-a & a^2-a & 1-a^2
+\end{pmatrix}
+\end{align*}
+Mit $1-a^2=(1+a)(1-a)$ und $a^2-a=a(a-1)$ kann man dies noch etwas
+vereinfachen, indem man den gemeinsamen Faktor $1-a$ ausklammern.
+Man erhält so die Form
+\begin{equation}
+A^{-1}
+=
+\frac{1-a}{2a^3-3a^2+1}
+\begin{pmatrix}
+1+a &  -a &  -a \\
+ -a & 1+a &  -a \\
+ -a &  -a & 1+a
+\end{pmatrix}.
+\label{buch:vektoren-und-matrizen:abeispiel:eqn2}
+\end{equation}
+für die Inverse einer Matrix der Form 
+\eqref{buch:vektoren-und-matrizen:abeispiel:eqn1}.
+\end{beispiel}
+
 %
 % Lineare Abbildungen
 %
@@ -1133,3 +1210,8 @@ n-\operatorname{def}A.
 
 \subsubsection{Quotient}
 TODO: $\operatorname{im} A \simeq \Bbbk^m/\ker A$
+
+
+
+
+
diff --git a/buch/chapters/30-endlichekoerper/galois.tex b/buch/chapters/30-endlichekoerper/galois.tex
index 2f8117e..c7147bf 100644
--- a/buch/chapters/30-endlichekoerper/galois.tex
+++ b/buch/chapters/30-endlichekoerper/galois.tex
@@ -128,6 +128,7 @@ $p_1$ und $p_2$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$.
 Ein Körper kann also nur entstehen, wenn $n$ eine Primzahl ist.
 
 \begin{definition}
+\label{buch:endlichekoerper:def:galois-koerper}
 Ist $p$ eine Primzahl, dann heisst $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$
 der Galois-Körper der Ordnung $p$.
 \end{definition}
diff --git a/buch/chapters/60-gruppen/lie-gruppen.tex b/buch/chapters/60-gruppen/lie-gruppen.tex
index d6fc007..e92c254 100644
--- a/buch/chapters/60-gruppen/lie-gruppen.tex
+++ b/buch/chapters/60-gruppen/lie-gruppen.tex
@@ -29,7 +29,7 @@ wenn es gelingt, eine Karte für eine Umgebung des neutralen Elements
 zu finden.
 Dazu muss gezeigt werden, dass sich aus einer solchen Karte für jedes
 andere Gruppenelement eine Karte für eine Umgebung ableiten lässt.
-Sei also $\varphi_e\colon U_e\mathbb{R}^N$ eine Karte für die Umgebung
+Sei also $\varphi_e\colon U_e \to \mathbb{R}^N$ eine Karte für die Umgebung
 $U_e\subset G$ von $e\in G$.
 Für $g\in G$ ist dann die Abbildung
 \[
diff --git a/buch/chapters/70-graphen/Makefile.inc b/buch/chapters/70-graphen/Makefile.inc
index d8fe742..2a7d9a6 100644
--- a/buch/chapters/70-graphen/Makefile.inc
+++ b/buch/chapters/70-graphen/Makefile.inc
@@ -7,5 +7,6 @@
 CHAPTERFILES = $(CHAPTERFILES)						\
 	chapters/70-graphen/beschreibung.tex				\
 	chapters/70-graphen/spektral.tex				\
+	chapters/70-graphen/waerme.tex					\
 	chapters/70-graphen/wavelets.tex				\
 	chapters/70-graphen/chapter.tex
diff --git a/buch/chapters/70-graphen/beschreibung.tex b/buch/chapters/70-graphen/beschreibung.tex
index 25cfcc0..a0f46da 100644
--- a/buch/chapters/70-graphen/beschreibung.tex
+++ b/buch/chapters/70-graphen/beschreibung.tex
@@ -401,7 +401,7 @@ Sie hat für $i\ne j$ die Einträge
 \\
 &=\text{Anzahl der Kanten, die $i$ mit $j$ verbinden}
 \\
-&=a_{ij}
+&=a_{ij}.
 \end{align*}
 Die Adjazenzmatrix eines Graphen lässt sich also aus der
 Inzidenzmatrix berechnen.
diff --git a/buch/chapters/70-graphen/chapter.tex b/buch/chapters/70-graphen/chapter.tex
index b6e02c9..6def393 100644
--- a/buch/chapters/70-graphen/chapter.tex
+++ b/buch/chapters/70-graphen/chapter.tex
@@ -65,5 +65,6 @@ Basis zur Beschreibung von Funktionen auf dem Graphen.
 
 \input{chapters/70-graphen/beschreibung.tex}
 \input{chapters/70-graphen/spektral.tex}
+\input{chapters/70-graphen/waerme.tex}
 \input{chapters/70-graphen/wavelets.tex}
 
diff --git a/buch/chapters/70-graphen/images/Makefile b/buch/chapters/70-graphen/images/Makefile
index bd77756..5db54c8 100644
--- a/buch/chapters/70-graphen/images/Makefile
+++ b/buch/chapters/70-graphen/images/Makefile
@@ -3,11 +3,14 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	peterson.pdf adjazenzu.pdf adjazenzd.pdf kreis.pdf fundamental.pdf
+all:	peterson.pdf adjazenzu.pdf adjazenzd.pdf kreis.pdf fundamental.pdf \
+	petersonchrind.pdf nine.pdf gh.pdf
 
 peterson.pdf:	peterson.tex
 	pdflatex peterson.tex
 
+petersonchrind.pdf:	petersonchrind.tex
+	pdflatex petersonchrind.tex
 adjazenzu.pdf:	adjazenzu.tex
 	pdflatex adjazenzu.tex
 
@@ -20,3 +23,9 @@ kreis.pdf:	kreis.tex
 fundamental.pdf:	fundamental.tex
 	pdflatex fundamental.tex
 
+nine.pdf:	nine.tex
+	pdflatex nine.tex
+
+gh.pdf:	gh.tex
+	pdflatex gh.tex
+
diff --git a/buch/chapters/70-graphen/images/gh.pdf b/buch/chapters/70-graphen/images/gh.pdf
new file mode 100644
index 0000000..c6e48d7
--- /dev/null
+++ b/buch/chapters/70-graphen/images/gh.pdf
diff --git a/buch/chapters/70-graphen/images/gh.tex b/buch/chapters/70-graphen/images/gh.tex
new file mode 100644
index 0000000..fcceb5f
--- /dev/null
+++ b/buch/chapters/70-graphen/images/gh.tex
@@ -0,0 +1,55 @@
+%
+% gh.tex -- Lokalsierungsfunktionen für Wavelets auf einem Graphen
+%
+% (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}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\kurve#1#2{
+	\draw[color=#2,line width=1.4pt]
+		plot[domain=0:6.3,samples=400]
+			({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+
+\begin{scope}
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\kurve{1}{red}
+\foreach \k in {0,...,4}{
+	\pgfmathparse{0.30*exp(ln(2)*\k)}
+	\xdef\l{\pgfmathresult}
+	\kurve{\l}{blue}
+}
+
+\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$};
+\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$};
+
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+
+\begin{scope}[xshift=7cm]
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\draw[color=darkgreen,line width=1.4pt]
+	plot[domain=0:6.3,samples=100] 
+		({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/images/nine.pdf b/buch/chapters/70-graphen/images/nine.pdf
new file mode 100644
index 0000000..2ae9f68
--- /dev/null
+++ b/buch/chapters/70-graphen/images/nine.pdf
diff --git a/buch/chapters/70-graphen/images/nine.tex b/buch/chapters/70-graphen/images/nine.tex
new file mode 100644
index 0000000..f214c1e
--- /dev/null
+++ b/buch/chapters/70-graphen/images/nine.tex
@@ -0,0 +1,67 @@
+%
+% nine.tex -- Nine node graph to illustrate Wilf's theorem
+%
+% (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}
+\def\skala{1}
+\def\kante#1#2{
+        \draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2);
+}
+\def\knoten#1#2{
+                \fill[color=#2!30] (#1) circle[radius=0.2];
+                \draw[color=#2] (#1) circle[radius=0.2];
+                \draw (#1) circle[radius=0.2];
+}
+\def\R{1.5}
+\definecolor{rot}{rgb}{1,0,0}
+\definecolor{gruen}{rgb}{0,0.6,0}
+\definecolor{blau}{rgb}{0,0,1}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\coordinate (A) at (0:\R);
+\coordinate (B) at (40:\R);
+\coordinate (C) at (80:\R);
+\coordinate (D) at (120:\R);
+\coordinate (E) at (160:\R);
+\coordinate (F) at (200:\R);
+\coordinate (G) at (240:\R);
+\coordinate (H) at (280:\R);
+\coordinate (I) at (320:\R);
+
+\knoten{A}{rot}
+\knoten{B}{blau}
+\knoten{C}{gruen}
+\knoten{D}{blau}
+\knoten{E}{rot}
+\knoten{F}{blau}
+\knoten{G}{rot}
+\knoten{H}{gruen}
+\knoten{I}{blau}
+
+\kante{A}{B}
+\kante{B}{C}
+\kante{C}{D}
+\kante{D}{E}
+\kante{E}{F}
+\kante{F}{G}
+\kante{G}{H}
+\kante{H}{I}
+\kante{I}{A}
+
+\kante{A}{C}
+\kante{A}{D}
+\kante{D}{G}
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/images/petersonchrind.pdf b/buch/chapters/70-graphen/images/petersonchrind.pdf
new file mode 100644
index 0000000..23ef6e9
--- /dev/null
+++ b/buch/chapters/70-graphen/images/petersonchrind.pdf
diff --git a/buch/chapters/70-graphen/images/petersonchrind.tex b/buch/chapters/70-graphen/images/petersonchrind.tex
new file mode 100644
index 0000000..4ae9f39
--- /dev/null
+++ b/buch/chapters/70-graphen/images/petersonchrind.tex
@@ -0,0 +1,142 @@
+%
+% tikztemplate.tex -- template for standalon tikz images
+%
+% (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}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\begin{scope}[xshift=-3.5cm]
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\fill[color=blau] (PA) circle[radius=\r];
+\fill[color=rot] (PB) circle[radius=\r];
+\fill[color=rot] (PC) circle[radius=\r];
+\fill[color=gruen] (PD) circle[radius=\r];
+\fill[color=gruen] (PE) circle[radius=\r];
+
+\fill[color=rot] (QA) circle[radius=\r];
+\fill[color=blau] (QB) circle[radius=\r];
+\fill[color=gruen] (QC) circle[radius=\r];
+\fill[color=rot] (QD) circle[radius=\r];
+\fill[color=blau] (QE) circle[radius=\r];
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius=\r];
+\draw (PC) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius=\r];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\node at (0,{-\Ra}) [below] {$\operatorname{chr}P=3\mathstrut$};
+
+\end{scope}
+
+\begin{scope}[xshift=3.5cm]
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+\definecolor{gelb}{rgb}{0,0,1}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\fill[color=rot] (QA) circle[radius={1.5*\r}];
+\fill[color=rot!40] (QB) circle[radius=\r];
+\fill[color=rot!40] (QE) circle[radius=\r];
+\fill[color=rot!40] (PA) circle[radius=\r];
+
+\fill[color=blau] (PB) circle[radius={1.5*\r}];
+\fill[color=blau!40] (PD) circle[radius=\r];
+\fill[color=blau!40] (PE) circle[radius=\r];
+\fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r];
+
+\fill[color=gruen] (PC) circle[radius={1.5*\r}];
+\fill[color=gruen!40] (QC) circle[radius=\r];
+\fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r];
+\fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r];
+
+\fill[color=gelb] (QD) circle[radius={1.5*\r}];
+\fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r];
+\fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r];
+\fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r];
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius={1.5*\r}];
+\draw (PC) circle[radius={1.5*\r}];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius={1.5*\r}];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius={1.5*\r}];
+\draw (QE) circle[radius=\r];
+
+\node at (0,{-\Ra}) [below] {$\operatorname{ind}P=4\mathstrut$};
+
+\end{scope}
+
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/spektral.tex b/buch/chapters/70-graphen/spektral.tex
index f68c814..5fb3056 100644
--- a/buch/chapters/70-graphen/spektral.tex
+++ b/buch/chapters/70-graphen/spektral.tex
@@ -1,198 +1,465 @@
 %
-% spektral.tex
+% spektral.tex -- spektrale Graphentheorie
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 \section{Spektrale Graphentheorie
 \label{buch:section:spektrale-graphentheorie}}
 \rhead{Spektrale Graphentheorie}
-Die Laplace-Matrix codiert alle wesentliche Information eines
+Die Adjazenz-Matrix, die Grad-Matrix und damit natürlich auch
+die Laplace-Matrix codieren alle wesentliche Information eines
 ungerichteten Graphen.
 Sie operiert auf Vektoren, die für jeden Knoten des Graphen eine
 Komponente haben.
 Dies eröffnet die Möglichkeit, den Graphen über die linearalgebraischen
-Eigenschaften der Laplace-Matrix zu studieren.
-
-\subsection{Grapheigenschaften und Spektrum von $L$
-\label{buch:subsection:grapheigenschaften-und-spektrum-von-l}}
-TODO XXX 
-
-\subsection{Wärmeleitung auf einem Graphen
-\label{buch:subsection:waermeleitung-auf-einem-graphen}}
-Die Vektoren, auf denen die Laplace-Matrix operiert, können betrachtet
-werden als Funktionen, die jedem Knoten einen Wert zuordnen.
-Eine mögliche physikalische Interpretation davon ist die Temperaturverteilung
-auf dem Graphen.
-Die Kanten zwischen den Knoten erlauben der Wärmeenergie, von einem Knoten
-zu einem anderen zu fliessen.
-Je grösser die Temperaturdifferenz zwischen zwei Knoten ist, desto
-grösser ist der Wärmefluss und desto schneller ändert sich die Temperatur
-der beteiligten Knoten.
-Die zeitliche Änderung der Temperatur $T_i$ im Knoten $i$ ist proportional
-\[
-\frac{dT_i}{dt}
-=
-\sum_{\text{$j$ Nachbar von $i$}} \kappa (T_j-T_i)
-=
--
-\kappa
-\biggl(
-d_iT_i
--
-\sum_{\text{$j$ Nachbar von $i$}} T_j
-\biggr)
-\]
-Der Term auf der rechten Seite ist genau die Wirkung der 
-Laplace-Matrix auf dem Vektor $T$ der Temperaturen:
-\begin{equation}
-\frac{dT}{dt}
-=
--\kappa L T.
-\label{buch:graphen:eqn:waermeleitung}
-\end{equation}
-Der Wärmefluss, der durch die
-Wärmeleitungsgleichung~\eqref{buch:graphen:eqn:waermeleitung} beschrieben
-wird, codiert ebenfalls wesentliche Informationen über den Graphen.
-Je mehr Kanten es zwischen verschiedenen Teilen eines Graphen gibt,
-desto schneller findet der Wärmeaustausch zwischen diesen Teilen
-statt.
-Die Lösungen der Wärmeleitungsgleichung liefern also Informationen 
-über den Graphen.
-
-\subsection{Eigenwerte und Eigenvektoren
-\label{buch:subsection:ein-zyklischer-graph}}
-Die Wärmeleitungsgleichung~\eqref{buch:graphen:eqn:waermeleitung} 
-ist eine lineare Differentialgleichung mit konstanten Koeffizienten,
-die mit der Matrixexponentialfunktion gelöst werden.
-Die Lösung ist
-\[
-f(t) = e^{-\kappa Lt}f(0).
-\]
+Eigenschaften dieser Matrizen zu studieren.
+Dieser Abschnitt soll diese Idee an dem ziemlich übersichtlichen Beispiel
+der chromatischen Zahl eines Graphen illustrieren.
+
+\subsection{Chromatische Zahl und Unabhängigkeitszahl
+\label{buch:subsection:chromatische-zahl}}
+Der Grad eines Knotens ist ein mass dafür, wie stark ein Graph
+``vernetzt'' ist.
+Je höher der Grad, desto mehr direkte Verbindungen zwischen Knoten gibt es.
+Noch etwas präziser können diese Idee die beiden mit Hilfe der
+chromatischen zahl und der Unabhängigkeitszahl erfasst werden.
+
+\begin{definition}
+Die {\em chromatische Zahl} $\operatorname{chr}G$ eines Graphen $G$ ist
+die minimale Anzahl von Farben, die Einfärben der Knoten eines Graphen
+nötig sind, sodass benachbarte Knoten verschiedene Farben haben.
+\index{chromatische Zahl}
+\end{definition}
+
+\begin{definition}
+Eine Menge von Knoten eines Graphen heisst {\em unabhängig}, wenn 
+keine zwei Knoten im Graphen verbunden sind.
+Die {\em Unabhängigkeitszahl} $\operatorname{ind}G$ eines Graphen $G$
+ist die maximale Anzahl Knoten einer unabhängigen Menge.
+\index{Unabhängigkeitszahl}
+\end{definition}
 
-Die Berechnung der Lösung mit der Matrixexponentialreihe ist ziemlich
-ineffizient, da grosse Matrizenprodukte berechnet werden müssen.
-Da die Matrix $L$ symmetrisch ist, gibt es eine Basis aus 
-orthonormierten Eigenvektoren und die Eigenwerte sind reell.
-Wir bezeichnen die Eigenvektoren mit $f_1,\dots,f_n$  und die
-zugehörigen Eigenwerte mit $\lambda_i$.
-Die Funktion $f_i(t)= e^{-\kappa\lambda_it}f_i$ ist dann eine Lösung
-der Wärmeleitungsgleichung, denn die beiden Seiten
+Zwischen der chromatischen Zahl und der Unabhängigkeitszahl eines Graphen
+muss es einen Zusammenhang geben.
+Je mehr Verbingungen es im Graphen gibt, desto grösser wird die chromatische
+Zahl.
+Gleichzeitig wird es schwieriger für Mengen von Knoten, unabhängig zu sein.
+
+\begin{satz}
+\label{buch:satz:chrind}
+Ist $G$ ein Graph mit $n$ Knoten, dann gilt
+$\operatorname{chr}G\cdot\operatorname{ind}G\ge n$.
+\end{satz}
+
+\begin{proof}[Beweis]
+Eine minimale Färbung des Graphen mit $\operatorname{chr}G$ Farben
+teilt die Knoten in $\operatorname{chr}G$ Mengen $V_f$ von Knoten mit
+gleicher Farbe $f$ ein.
+Da diese Mengen einfarbig sind, sind sie unabhängig, enthalten also
+höchstens so viele Knoten, wie die Unabhängigkeitszahl erlaubt,
+also $|V_f|\le \operatorname{ind}G$.
+Da die Menge aller Knoten die Vereinigung der Mengen $V_f$ ist,
+ist die Gesamtzahl der Knoten 
 \begin{align*}
-\frac{d}{dt}f_i(t)
+V
 &=
--\kappa\lambda_ie^{-\kappa\lambda_it}f_i
-=
--\kappa\lambda_i f_i(t)
-\\
--\kappa Lf_i(t)
+\bigcup_{\text{$f$ eine Farbe}} V_f
+&&\Rightarrow&
+n
 &=
--\kappa e^{-\kappa\lambda_it} Lf_i
+\sum_{\text{$f$ eine Farbe}} |V_f| 
+\\
+&
+&&&
+&\le
+\sum_{\text{$f$ eine Farbe}} \operatorname{ind}G
 =
--\kappa e^{-\kappa\lambda_it} \lambda_i f_i
+(\text{Anzahl Farben})\cdot \operatorname{ind}G
 =
--\kappa \lambda_i f_i(t)
+\operatorname{chr}G \cdot \operatorname{ind}G.
 \end{align*}
-von \eqref{buch:graphen:eqn:waermeleitung} stimmen überein.
-
-Eine Lösung der Wärmeleitungsgleichung zu einer beliebigen
-Anfangstemperaturverteilung $f$ kann durch Linearkombination aus 
-den Lösungen $f_i(t)$ zusammengesetzt werden.
-Dazu ist nötig, $f$ aus den Vektoren $f_i$ linear zu kombinieren.
-Da aber die $f_i$ orthonormiert sind, ist dies besonders einfach,
-die Koeffizienten sind die Skalarprodukte mit den Eigenvektoren:
-\[
-f=\sum_{i=1}^n \langle f_i,f\rangle f_i.
-\]
-Daraus kann man die allgmeine Lösungsformel
+Damit ist $n\le \operatorname{chr}G\cdot\operatorname{ind}G$ gezeigt.
+\qedhere
+\end{proof}
+
+\begin{beispiel}
+In einem vollständigen Graphen ist jeder Knoten mit jedem anderen verbunden.
+Jede Menge mit zwei oder mehr Knoten kann daher nicht unabhängig sein, die
+Unabhängigkeitszahl ist daher $\operatorname{ind}G=1$.
+Andererseits ist für jeden Knoten eine eigene Farbe nötig, daher ist die
+chromatische Zahl $\operatorname{chr}G=n$.
+Die Ungleichung von Satz~\ref{buch:satz:chrind} ist erfüllt, sogar mit
+Gleichheit.
+Das Beispiel zeigt, dass die Ungleichung nicht ohne zusätzliche Annahmen
+verbessert werden kann.
+\end{beispiel}
+
+\begin{figure}
+\centering
+\includegraphics{chapters/70-graphen/images/petersonchrind.pdf}
+\caption{Chromatische Zahl und Unabhängigkeitszahl des Peterson-Graphen.
+Die chromatische Zahl ist $3$, da der Graph sich mit drei Farben einfärben
+lässt (links).
+Die Unabhängigkeitszahl ist $4$, die vier grösseren Knoten im rechten
+Graphen sind unabhängig.
+Die Farben der kleinen Knoten sind die additive Mischung der Farben
+der grossen Knoten, mit denen sie verbunden sind.
+\label{buch:graphen:fig:chrindpeterson}}
+\end{figure}
+
+\begin{beispiel}
+Der Peterson-Graph $P$ von Abbildung~\ref{buch:graphen:fig:chrindpeterson}
+hat chromatische Zahl $\operatorname{chr}P=3$ und unabhängigkeitszahl
+$\operatorname{ind}P=4$.
+Die Ungleichung von Satz~\ref{buch:satz:chrind} ist erfüllt, sogar als
+Ungleichung: $\operatorname{chr}P\cdot\operatorname{ind}P=3\cdot 4=12>10=n$.
+\end{beispiel}
+
+Nach Definition ist Unabhängigkeitszahl ein Mass für die Grösse einer
+unabhängigen Menge von Punkten.
+Der Beweis von Satz~\ref{buch:satz:chrind} zeigt, dass man sich die
+chromatische Zahl als ein Mass dafür, wieviele solche anabhängige 
+Mengen in einem Graphen untergebracht werden können.
+
+%
+% Chromatische Zahl und maximaler Grad
+%
+\subsection{Chromatische Zahl und maximaler Grad
+\label{buch:subsection:chr-und-maximaler-grad}}
+Wenn kein Knoten mehr als $d$ Nachbarn hat, dann reichen
+$d+1$ Farben immer, um diesen Knoten und seine Nachbarn einzufärben.
+Das heisst aber noch nicht, dass dann auch $d+1$ Farben zur
+Einfärbung des ganzen Graphen reichen.
+Genau dies garantiert jedoch der folgende Satz.
+
+\begin{definition}
+Der maximale Grad
+\(
+\max_{v\in V} \deg(v)
+\)
+wird mit $d$ bezeichnet.
+\end{definition}
+
+\begin{satz}
+\label{buch:graphen:satz:chrmaxgrad}
+Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt 
+$\operatorname{chr}G \le d+1$.
+\end{satz}
+
+\begin{proof}[Beweis]
+Wir führen den Beweis mit Hilfe von vollständiger Induktion nach der
+Anzahl Knoten eines Graphen.
+Ein Graph mit nur einem Knoten hat keine Kanten, der maximale Grad ist
+daher $0$ und $d+1=1$ Farbe reicht auch tatsächlich zur Einfärbung des
+einen Knotens.
+
+Wir nehmen jetzt an, die Behaupt sei für Graphen mit $n-1$ Knoten bereits
+bewiesen, ein Graph $G'$ mit $n-1$ Knoten und maximalem Grad $d'$ erfüllt
+also die Ungleichung $\operatorname{chr}G'\le d'+1$.
+
+Wir wählen jetzt einen beleibigen Knoten $v$ des Graphen $G$ und bilden
+den Graphen $G'$, der aus $G$ entsteht, indem man den Knoten $v$
+entfernt: $G'=G\setminus\{v\}$.
+Der maximale Grad $d'$ von $G'$ kann dabei nicht grösser werden, es ist
+also $d'\le d$.
+Da $G'$ genau $n-1$ Knoten hat, lässt er sich mit höchstens $d'+1\le d+1$
+Farben einfärben.
+Es muss jetzt also nur noch eine Farbe für den Knoten $v$ gefunden werden.
+Da $d$ der maximale Grad ist, hat $v$ höchstens $d$ Nachbarn, die höchstens
+$d$ verschiedene Farben haben können.
+Von den $d+1$ zur Verfügung stehenden Farben bleibt also mindestens eine
+übrig, mit der man den Knoten $v$ einfärben kann.
+Damit ist der Induktionsschritt gelungen und somit der Satz bewiesen.
+\end{proof}
+
+Das Argument im Beweis von Satz~\ref{buch:graphen:satz:chrmaxgrad}
+ist für alle Begriffe anwendbar, die sich bei der Bildung eines 
+Untergraphen auf ``monotone'' Art ändern.
+Die chromatische Zahl eines Untergraphen ist höchstens so gross wie die
+des ganzen Graphen. 
+Dann kann man eine Ungleichung für grosse Graphen schrittweise aus
+entsprechenden Ungleichungen für die kleineren Teilgraphen gewinnen.
+Ziel der folgenden Abschnitte ist zu zeigen, dass sich eine Grösse
+mit ähnlichen Eigenschaften aus dem Eigenwertspektrum der Adjazenzmatrix
+ablesen lässt.
+Daraus ergibt sich dann eine bessere Abschätzung der chromatischen Zahl
+eines Graphen.
+
+%
+% maximaler Eigenwert und maximaler Grad
+%
+\subsection{Maximaler Eigenwert von $A(G)$ und maximaler Grad
+\label{buch:subsection:maximaler-eigenwert}}
+Die Adjazenzmatrix $A(G)$ eines Graphen $G$  mit $n$ Knoten enthält unter
+anderem auch die Information über den Grad eines Knotens.
+Die Summe der Elemente einer Zeile oder einer Spalte ergibt einen Vektor,
+der die Grade der Knoten als Komponenten enthält.
+Ist $U$ ein $n$-dimensionaler Vektor aus lauter Einsen, dann ist
+ist $A(G)U$ ein Spaltenvektor bestehend aus den Zeilensummen der Matrix 
+$A(G)$ und
+$U^tA(G)$ ein Zeilenvektor bestehend aus den Spaltensummen.
+$A(G)U$ ist also der Vektor der Grade der Knoten.
+
+Das Skalarprodukt von $A(G)U$ mit $U$ ist die Summe der Grade.
+Somit ist
 \begin{equation}
-f(t)
+\frac{\langle A(G)U,U\rangle}{\langle U,U\rangle}
 =
-\sum_{i=1}^n \langle f_i,f\rangle f_i(t)
+\frac{1}{\langle U,U\rangle}\sum_{v\in V}\deg(v)
 =
-\sum_{i=1}^n \langle f_i,f\rangle e^{-\kappa\lambda_i t}f_i
-\label{buch:graphen:eqn:eigloesung}
+\frac{1}{n}(d_1+\dots+d_n)
+\label{buch:graphen:eqn:AUdavg}
 \end{equation}
-ableiten.
+der mittlere Grad, der mit $\overline{d}$ bezeichnet werden soll.
 
-\subsection{Beispiel: Ein zyklischer Graph}
-\begin{figure}
-\centering
-\includegraphics{chapters/70-graphen/images/kreis.pdf}
-\caption{Beispiel Graph zur Illustration der verschiedenen Basen auf einem
-Graphen.
-\label{buch:graphen:fig:kreis}}
-\end{figure}
-Wir illustrieren die im folgenden entwickelte Theorie an dem Beispielgraphen
-von Abbildung~\ref{buch:graphen:fig:kreis}.
-Besonders interessant sind die folgenden Funktionen:
+Da $A(G)$ eine symmetrische Matrix ist, ist $A(G)$ diagonalisierbar,
+die Eigenwerte sind also alle reell.
+Es ist ausserdem bekannt, dass der Eigenvektor $f$ zum grössten Eigenwert
+$\alpha_{\text{max}}$ von $A(G)$
+den Bruch
 \[
-\left.
+\frac{\langle A(G)f,f\rangle}{\langle f,f\rangle}
+\]
+für Vektoren $f\ne 0$ maximiert.
+Aus~\eqref{buch:graphen:eqn:AUdavg} folgt damit, dass
+\begin{equation}
+\overline{d}
+\le
+\alpha_{\text{max}}
+\label{buch:graphen:eqn:dqueramax}
+\end{equation}
+ist.
+
+In Abschnitt~\ref{buch:section:positive-vektoren-und-matrizen}
+des nächsten Kapitels wird die Perron-Frobenius-Theorie positiver
+Matrizen vorgestellt, welche einer Reihe interessanter Aussagen
+über den betragsgrössten Eigenwert und den zugehörigen Eigenvektor
+macht.
+Die Adjazenz-Matrix ist eine nichtnegative Matrix und $\alpha_{\text{max}}$
+ist der grösste Eigenwert, also genau die Grösse, auf die die
+Sätze~\ref{buch:wahrscheinlichkeit:satz:perron-frobenius}
+und \label{buch:wahrscheinlichkeit:satz:perron-frobenius2}
+anwendbar sind.
+Dazu muss die Matrix allerdings primitiv sein, was gleichbedeutend
+ist damit, dass der Graph zusammenhängend ist.
+Im folgenden soll dies daher jeweils angenommen werden.
+
+\begin{satz}
+Ist $G$ ein zusammenhänger Graph mit $n$ Knoten und maximalem Grad $d$,
+dann gilt
+\[
+\frac1n\sum_{v\in V} \deg(v) 
+=
+\overline{d}
+\le \alpha_{\text{max}} \le d.
+\]
+\end{satz}
+
+\begin{proof}[Beweis]
+Wir wissen aus \eqref{buch:graphen:eqn:dqueramax} bereits, dass
+$\overline{d}\le\alpha_{\text{max}}$ gilt, es bleibt also nur noch
+$\alpha_{\text{max}}\le d$ zu beweisen.
+
+Sei $f$ der Eigenvektor zum Eigenwert $\alpha_{\text{max}}$.
+Nach Satz~\label{buch:wahrscheinlichkeit:satz:perron-frobenius2}
+ist $f$ ein positiver Vektor mit der Eigenschaft $A(G)f=\alpha_{\text{max}}f$.
+Der Eigenvektor $f$ ist eine Funktion auf den Knoten des Graphen,
+die $v$-Komponente des Vektors $f$ für einen Vertex $v\in V$ ist $f(v)$.
+Die Eigenvektoreigenschaft bedeutet $(A(G)f)(v)=\alpha_{\text{max}} f(v)$.
+Die Adjazenzmatrix $A(G)$ enthält in Zeile $v$ Einsen genau für diejenigen
+Knoten $u\in V$, die zu $v$ benachbart sind.
+Schreiben wir $u\sim v$ für die Nachbarschaftsrelation, dann ist 
+\[
+(A(G)f)(v)
+=
+\sum_{u\sim v} f(u).
+\]
+Die Summe der Komponenten $A(G)f$ kann man erhalten durch Multiplikation
+von $A(G)f$ mit einem Zeilenvektor $U^t$ aus lauter Einsen, also
+\begin{equation}
 \begin{aligned}
-s_m(k)
+\sum_{v\in V}\sum_{u\sim v}f(v)
 &=
-\sin\frac{2\pi mk}{n}
+U^tA(G)f
+=
+(U^tA(G))f
+=
+\begin{pmatrix}d_1&d_2&\dots&d_n\end{pmatrix} f
 \\
-c_m(k)
 &=
-\cos\frac{2\pi mk}{n}
+\sum_{v\in V}\deg (v) f(v)
+\le
+\sum_{v\in V}df(v)
+=
+d
+\sum_{v\in V}f(v).
 \end{aligned}
-\;
-\right\}
-\quad
-\Rightarrow
-\quad
-e_m(k)
+\label{buch:graphen:eqn:sumkomp}
+\end{equation}
+Andererseits ist $A(G)f=\alpha_{\text{max}}f$, die linke Seite
+von~\eqref{buch:graphen:eqn:sumkomp} ist daher
+\begin{equation}
+\sum_{v\in V}\sum_{u\sim v}f(v)
 =
-e^{2\pi imk/n}
+U^tA(G)f
 =
-c_m(k) + is_m(k).
+\alpha_{\text{max}}
+U^tf
+=
+\alpha_{\text{max}} \sum_{v\in V}f(v).
+\label{buch:graphen:eqn:sumkomp2}
+\end{equation}
+Die Ungleichung~\eqref{buch:graphen:eqn:sumkomp}
+und die Gleichung~\eqref{buch:graphen:eqn:sumkomp2} ergeben zusammen
+die Ungleichung
+\[
+\alpha_{\text{max}} \sum_{v\in V}f(v)
+\le d\sum_{v\in V}f(v)
+\qquad\Rightarrow\qquad
+\alpha_{\text{max}} \le d,
 \]
-Das Skalarprodukt dieser Funktionen ist
+da die Summe der Komponenten des positiven Vektors $f$ nicht verschwinden
+kann.
+Damit ist die Ungleichung bewiesen.
+\end{proof}
+
+%
+% alpha_max eines Untergraphen
+%
+\subsection{$\alpha_{\text{max}}$ eines Untergraphen
+\label{buch:subsection:alphamax-eines-untergraphen}}
+Der grösste Eigenwert $\alpha_{\text{max}}$ ist ein potentieller 
+Anwärter für eine bessere Abschätzung der chromatischen Zahl.
+Bereits früher wurde bemerkt, dass dies auch bedeutet, dass man 
+das Verhalten des grössten Eigenwerts bei einem Übergang zu einem
+Untergraphen verstehen muss.
+
+\begin{satz}
+\label{buch:graphen:satz:amaxuntergraph}
+Sei $G'$ ein echter Untergraph von $G$ mit Adjazenzmatrix $A(G')$ und
+grösstem Eigenwert $\alpha_{\text{max}}'=\varrho(A(G'))$, dann ist
+$\alpha_{\text{max}}' \le \alpha_{\text{max}}$.
+\end{satz}
+
+\begin{proof}[Beweis]
+Sei $f'$ der positive Eigenvektor zum Eigenwert $\alpha_{\text{max}}'$
+der Matrix $A(G')$.
+$f'$ ist definiert auf der Menge $V'$ der Knoten von $G'$.
+Aus $f'$ lässt sich ein Vektor $g$ mit den Werten
 \[
-\langle e_m, e_{m'}\rangle
+g(v)
 =
-\frac1n
-\sum_{k=1}^n
-\overline{e^{2\pi i km/n}}
-e^{2\pi ikm'/n}
-=
-\frac1n
-\sum_{k=1}^n
-e^{\frac{2\pi i}{n}(m'-m)k}
-=
-\delta_{mm'}
+\begin{cases}
+f'(v)&\qquad v\in V'\\
+    0&\qquad\text{sonst}
+\end{cases}
 \]
-Die Funktionen bilden daher eine Orthonormalbasis des Raums der
-Funktionen auf $G$.
-Wegen $\overline{e_m} = e_{-m}$ folgt, dass für gerade $n$
-die Funktionen
+konstruieren, der auf ganz $V$ definiert ist.
+
+Die Vektoren $f'$ und $g$ haben die gleichen Komponenten, also ist auch
+$\langle f',f'\rangle = \langle g,g\rangle$.
+Die Matrixelemente von $A(G')$ und $A(G)$ auf gemeinsamen Knoten $u,v\in V'$ 
+erfüllen $A(G')_{uv}\le A(G)_{uv}$, da jede Kante von $G'$ auch in $G$ ist.
+Daher gilt
 \[
-c_0, c_1,s_1,c_2,s_2,\dots c_{\frac{n}2-1},c_{\frac{n}2-1},c_{\frac{n}2}
+\langle A(G')f',f'\rangle
+\le
+\langle A(G)g,g\rangle,
 \]
-eine orthonormierte Basis.
+woraus sich die Ungleichung
+\[
+\alpha_{\text{max}}'
+=
+\frac{\langle A(G')f',f'\rangle}{\langle f',f'\rangle}
+=
+\frac{\langle A(G)g,g\rangle}{\langle g,g\rangle}
+\le
+\alpha_{\text{max}}
+\]
+ergibt, da $\alpha_{\text{max}}$ das Maximum von
+$\langle A(G)h,h\rangle/\langle h,h\rangle$ für alle Vektoren $h\ne 0$ ist.
+\end{proof}
 
+%
+% Der Satz von Wilf
+%
+\subsection{Chromatische Zahl und $\alpha_{\text{max}}$: Der Satz von Wilf
+\label{buch:subsection:chr-und-alpha-max}}
+Die in Satz~\ref{buch:graphen:satz:amaxuntergraph} beschriebene
+Eigenschaft von $\alpha_{\text{max}}$ beim Übergang zu einem Untergraphen
+ermöglich jetzt, eine besser Abschätzung für die chromatische Zahl
+zu finden.
 
-Die Laplace-Matrix kann mit der folgenden Definition zu einer linearen
-Abbildung auf Funktionen auf dem Graphen gemacht werden.
-Sei $f\colon V\to \mathbb{R}$ und $L$ die Laplace-Matrix mit
-Matrixelementen $l_{vv'}$ wobei $v,v'\in V$ ist.
-Dann definieren wir die Funktion $Lf$ durch
+\begin{satz}[Wilf]
+\label{buch:graphen:satz:wilf}
+Sie $G$ ein zusammenhängder Graph und $\alpha_{\text{max}}$ der grösste
+Eigenwert seiner Adjazenzmatrix. Dann gilt
 \[
-(Lf)(v)
-=
-\sum_{v'\in V} l_{vv'}f(v').
+\operatorname{chr}G\le \alpha_{\text{max}}+1.
 \]
+\end{satz}
 
-\subsection{Standardbasis und Eigenbasis
-\label{buch:subsection:standardbasis-und-eigenbasis}}
-Die einfachste Basis, aus der siche Funktionen auf dem Graphen linear
-kombinieren lassen, ist die Standardbasis.
-Sie hat für jeden Knoten $v$ des Graphen eine Basisfunktion mit den Werten
+\begin{proof}[Beweis]
+Wie der Satz~\ref{buch:graphen:satz:chrmaxgrad} kann auch der Satz von Wilf
+mit Hilfe von vollständiger Induktion über die Anzahl $n$ der Knoten
+bewiesen werden.
+
+Ein Graph mit nur einem Knoten hat die $0$-Matrix als Adjazenzmatrix,
+der maximale Eigenwert ist $\alpha_{\text{max}}=0$, und tatsächlich reicht
+$\alpha_{\text{max}}+1=1$ Farbe, um den einen Knoten einzufärben.
+
+Wir nehmen jetzt an, der Satz sei für Graphen mit $n-1$ Knoten bereits
+beweisen.
+Wir müssen dann zeigen, dass der Satz dann auch für Graphen mit $n$ Knoten
+gilt.
+
+Sei $v\in V$ ein Knoten minimalen Grades und $G'=G\setminus{v}$ der
+Untergraph, der entsteht, wenn der Knoten $v$ entfernt wird.
+Da $G'$ genau $n-1$ Knoten hat, gilt der Satz von Wilf für $G'$
+und daher kann $G'$ mit höchstens
 \[
-e_v\colon V\to\mathbb R:v'\mapsto \begin{cases}
-1\qquad&v=v'\\
-0\qquad&\text{sonst.}
-\end{cases}
+\operatorname{chr}G' \le 1 + \alpha_{\text{max}}'
 \]
+Farben eingefärbt werden.
+Nach Satz~\ref{buch:graphen:satz:amaxuntergraph} ist
+$\alpha_{\text{max}}'\le \alpha_{\text{max}}$,
+Also kann $G'$ mit höchstens $\alpha_{\text{max}}+1$ Farben eingefärbt werden.
+
+Da $v$ ein Knoten minimalen Grades ist, ist sein Grad
+$d(v)\le \overline{d}\le \alpha_{\text{max}}$.
+Die Nachbarn von $v$ haben also hächstens $\alpha_{\text{max}}$ verschiedene
+Farben, mit einer weiteren Farbe lässt sich also auch $G$ einfärben.
+Daraus folgt $\operatorname{chr}G\le \alpha_{\text{max}}+1$.
+\end{proof}
+
+\begin{figure}
+\centering
+\includegraphics{chapters/70-graphen/images/nine.pdf}
+\caption{Beispiel für einen Graphen, für den der
+Satz~\ref{buch:graphen:satz:wilf} von Wilf die bessere
+Abschätzung für die chromatische Zahl eines Graphen gibt als der
+maximale Grad.
+\label{buch:graphen:fig:wilfexample}}
+\end{figure}
+
+\begin{beispiel}
+Der Graph in Abbildung~\ref{buch:graphen:fig:wilfexample} 12 Kanten und 9
+Knoten, daher ist $\overline{d}\le \frac{24}{9}$.
+Der maximale Grad ist $4$ und durch explizite Rechnung mit Hilfe zum Beispiel
+von Octave ergibt, dass $\alpha_{\text{max}}\approx 2.9565$.
+Aus dem Satz von Wilf folgt, dass
+$\operatorname{chr}G\le \alpha_{\text{max}}+1$, und daraus ergibt sich
+$\operatorname{chr}G\le 3$.
+Tatsächlich ist die chromatische Zahl $\operatorname{chr}G=3$, da 
+der Graph mindestens ein Dreieck enthält.
+Der maximale Grad ist 4, somit gibt der
+Satz~\ref{buch:graphen:satz:chrmaxgrad}
+die Schranke 
+$\operatorname{chr}G\le 4+1=5$
+für die chromatische Zahl.
+Der Satz von Wilf ist also eine wesentliche Verbesserung, er liefert in
+diesem Fall den exakten Wert der chromatischen Zahl.
+\end{beispiel}
+
 
 
diff --git a/buch/chapters/70-graphen/waerme.tex b/buch/chapters/70-graphen/waerme.tex
new file mode 100644
index 0000000..e7fc023
--- /dev/null
+++ b/buch/chapters/70-graphen/waerme.tex
@@ -0,0 +1,184 @@
+%
+% waerme.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Wärmeleitung auf einem Graphen
+\label{buch:section:waermeleitung-auf-einem-graphen}}
+Die Vektoren, auf denen die Laplace-Matrix operiert, können betrachtet
+werden als Funktionen, die jedem Knoten einen Wert zuordnen.
+Eine mögliche physikalische Interpretation davon ist die Temperaturverteilung
+auf dem Graphen.
+Die Kanten zwischen den Knoten erlauben der Wärmeenergie, von einem Knoten
+zu einem anderen zu fliessen.
+Je grösser die Temperaturdifferenz zwischen zwei Knoten ist, desto
+grösser ist der Wärmefluss und desto schneller ändert sich die Temperatur
+der beteiligten Knoten.
+Die zeitliche Änderung der Temperatur $T_i$ im Knoten $i$ ist proportional
+\[
+\frac{dT_i}{dt}
+=
+\sum_{\text{$j$ Nachbar von $i$}} \kappa (T_j-T_i)
+=
+-
+\kappa
+\biggl(
+d_iT_i
+-
+\sum_{\text{$j$ Nachbar von $i$}} T_j
+\biggr)
+\]
+Der Term auf der rechten Seite ist genau die Wirkung der 
+Laplace-Matrix auf dem Vektor $T$ der Temperaturen:
+\begin{equation}
+\frac{dT}{dt}
+=
+-\kappa L T.
+\label{buch:graphen:eqn:waermeleitung}
+\end{equation}
+Der Wärmefluss, der durch die
+Wärmeleitungsgleichung~\eqref{buch:graphen:eqn:waermeleitung} beschrieben
+wird, codiert ebenfalls wesentliche Informationen über den Graphen.
+Je mehr Kanten es zwischen verschiedenen Teilen eines Graphen gibt,
+desto schneller findet der Wärmeaustausch zwischen diesen Teilen
+statt.
+Die Lösungen der Wärmeleitungsgleichung liefern also Informationen 
+über den Graphen.
+
+\subsection{Eigenwerte und Eigenvektoren
+\label{buch:subsection:ein-zyklischer-graph}}
+Die Wärmeleitungsgleichung~\eqref{buch:graphen:eqn:waermeleitung} 
+ist eine lineare Differentialgleichung mit konstanten Koeffizienten,
+die mit der Matrixexponentialfunktion gelöst werden.
+Die Lösung ist
+\[
+f(t) = e^{-\kappa Lt}f(0).
+\]
+
+Die Berechnung der Lösung mit der Matrixexponentialreihe ist ziemlich
+ineffizient, da grosse Matrizenprodukte berechnet werden müssen.
+Da die Matrix $L$ symmetrisch ist, gibt es eine Basis aus 
+orthonormierten Eigenvektoren und die Eigenwerte sind reell.
+Wir bezeichnen die Eigenvektoren mit $f_1,\dots,f_n$  und die
+zugehörigen Eigenwerte mit $\lambda_i$.
+Die Funktion $f_i(t)= e^{-\kappa\lambda_it}f_i$ ist dann eine Lösung
+der Wärmeleitungsgleichung, denn die beiden Seiten
+\begin{align*}
+\frac{d}{dt}f_i(t)
+&=
+-\kappa\lambda_ie^{-\kappa\lambda_it}f_i
+=
+-\kappa\lambda_i f_i(t)
+\\
+-\kappa Lf_i(t)
+&=
+-\kappa e^{-\kappa\lambda_it} Lf_i
+=
+-\kappa e^{-\kappa\lambda_it} \lambda_i f_i
+=
+-\kappa \lambda_i f_i(t)
+\end{align*}
+von \eqref{buch:graphen:eqn:waermeleitung} stimmen überein.
+
+Eine Lösung der Wärmeleitungsgleichung zu einer beliebigen
+Anfangstemperaturverteilung $f$ kann durch Linearkombination aus 
+den Lösungen $f_i(t)$ zusammengesetzt werden.
+Dazu ist nötig, $f$ aus den Vektoren $f_i$ linear zu kombinieren.
+Da aber die $f_i$ orthonormiert sind, ist dies besonders einfach,
+die Koeffizienten sind die Skalarprodukte mit den Eigenvektoren:
+\[
+f=\sum_{i=1}^n \langle f_i,f\rangle f_i.
+\]
+Daraus kann man die allgmeine Lösungsformel
+\begin{equation}
+f(t)
+=
+\sum_{i=1}^n \langle f_i,f\rangle f_i(t)
+=
+\sum_{i=1}^n \langle f_i,f\rangle e^{-\kappa\lambda_i t}f_i
+\label{buch:graphen:eqn:eigloesung}
+\end{equation}
+ableiten.
+
+\subsection{Beispiel: Ein zyklischer Graph}
+\begin{figure}
+\centering
+\includegraphics{chapters/70-graphen/images/kreis.pdf}
+\caption{Beispiel Graph zur Illustration der verschiedenen Basen auf einem
+Graphen.
+\label{buch:graphen:fig:kreis}}
+\end{figure}
+Wir illustrieren die im folgenden entwickelte Theorie an dem Beispielgraphen
+von Abbildung~\ref{buch:graphen:fig:kreis}.
+Besonders interessant sind die folgenden Funktionen:
+\[
+\left.
+\begin{aligned}
+s_m(k)
+&=
+\sin\frac{2\pi mk}{n}
+\\
+c_m(k)
+&=
+\cos\frac{2\pi mk}{n}
+\end{aligned}
+\;
+\right\}
+\quad
+\Rightarrow
+\quad
+e_m(k)
+=
+e^{2\pi imk/n}
+=
+c_m(k) + is_m(k).
+\]
+Das Skalarprodukt dieser Funktionen ist
+\[
+\langle e_m, e_{m'}\rangle
+=
+\frac1n
+\sum_{k=1}^n
+\overline{e^{2\pi i km/n}}
+e^{2\pi ikm'/n}
+=
+\frac1n
+\sum_{k=1}^n
+e^{\frac{2\pi i}{n}(m'-m)k}
+=
+\delta_{mm'}
+\]
+Die Funktionen bilden daher eine Orthonormalbasis des Raums der
+Funktionen auf $G$.
+Wegen $\overline{e_m} = e_{-m}$ folgt, dass für gerade $n$
+die Funktionen
+\[
+c_0, c_1,s_1,c_2,s_2,\dots c_{\frac{n}2-1},c_{\frac{n}2-1},c_{\frac{n}2}
+\]
+eine orthonormierte Basis.
+
+
+Die Laplace-Matrix kann mit der folgenden Definition zu einer linearen
+Abbildung auf Funktionen auf dem Graphen gemacht werden.
+Sei $f\colon V\to \mathbb{R}$ und $L$ die Laplace-Matrix mit
+Matrixelementen $l_{vv'}$ wobei $v,v'\in V$ ist.
+Dann definieren wir die Funktion $Lf$ durch
+\[
+(Lf)(v)
+=
+\sum_{v'\in V} l_{vv'}f(v').
+\]
+
+\subsection{Standardbasis und Eigenbasis
+\label{buch:subsection:standardbasis-und-eigenbasis}}
+Die einfachste Basis, aus der siche Funktionen auf dem Graphen linear
+kombinieren lassen, ist die Standardbasis.
+Sie hat für jeden Knoten $v$ des Graphen eine Basisfunktion mit den Werten
+\[
+e_v\colon V\to\mathbb R:v'\mapsto \begin{cases}
+1\qquad&v=v'\\
+0\qquad&\text{sonst.}
+\end{cases}
+\]
+
+
diff --git a/buch/chapters/70-graphen/wavelets.tex b/buch/chapters/70-graphen/wavelets.tex
index 9c88c08..ef1520e 100644
--- a/buch/chapters/70-graphen/wavelets.tex
+++ b/buch/chapters/70-graphen/wavelets.tex
@@ -103,22 +103,230 @@ aus der sich alle Vektoren linear kombinieren lassen, in der aber
 auch auf die für die Anwendung interessante Längenskala angepasste
 Funktionen gefunden werden können.
 
-\subsection{Wavelets und Frequenzspektrum}
-Eine Wavelet-Basis der Funktionen auf $\mathbb{R}$ zerlegt
+\subsection{Wavelets auf einem Graphen}
+Die Fourier-Theorie analysiert Funktionen nach Frequenzen, wobei die 
+zeitliche Position von interessanten Stellen der Funktion in der Phase
+der einzelnen Komponenten verschwindet.
+Die Lokalisierung geht also für viele praktische Zwecke verloren.
+Umgekehrt haben einzelne Ereignisse wie eine $\delta$-Funktion keine
+charakteristische Frequenz, sie sind daher im Frequenzraum überhaupt 
+nicht lokalisierbar.
+Die Darstellung im Frequenzraum und in der Zeit sind also extreme
+Darstellungen, entweder Frequenzlokalisierung oder zeitliche Lokalisierung
+ermöglichen, sich aber gegenseitig ausschliessen.
 
+\subsubsection{Dilatation}
+Eine Wavelet-Basis für die $L^2$-Funktionen auf $\mathbb{R}$ erlaubt
+eine Funktion auf $\mathbb{R}$ auf eine Art zu analysieren, die eine
+ungenaue zeitliche Lokalisierung bei entsprechend ungenauer
+Frequenzbestimmung ermöglicht.
+Ausserdem entstehen die Wavelet-Funktionen aus einer einzigen Funktion
+$\psi(t)$ durch Translation um $b$ und Dilatation mit dem Faktor $a$:
+\[
+\psi_{a,b}(t)
+=
+\frac{1}{\sqrt{|a|}} \psi\biggl(\frac{t-b}a\biggr)
+=
+T_bD_a\psi(t)
+\]
+in der Notation von \cite{buch:mathsem-wavelets}.
+Auf einem Graphen ist so eine Konstruktion grundsätzlich nicht möglich,
+da es darauf weder eine Translations- noch eine Streckungsoperation gibt.
+
+In der Theorie der diskreten Wavelet-Transformation ist es üblich, sich
+auf Zweierpotenzen als Streckungsfaktoren zu beschränken.
+Ein Gitter wird dadurch auf sich selbst abgebildet, aber auf einem
+Graphen gibt es keine Rechtfertigung für diese spezielle Wahl von
+Streckungsfaktoren mehr.
+Es stellt sich daher die Frage, ob man für eine beliebige Menge
+\(
+T= \{ t_1,t_2,\dots\} \}
+\)
+von Streckungsfaktoren eine Familie von Funktionen $\chi_j$ zu finden
+derart, dass man sich die $\chi_j$ in einem gewissen Sinn als aus
+$\chi_0$ durch Dilatation entstanden vorstellen kann.
 
-\subsection{Frequenzspektrum
-\label{buch:subsection:frequenzspektrum}}
-Die Fundamentallösung der Wärmeleitunsgleichung haben ein Spektrum, welches
-wie $e^{-k^2}$ gegen $0$ geht.
+Die Dilatation kann natürlich nicht von einer echten
+Dilatation im Ortsraum herstammen, aber man kann wenigstens versuchen, die
+Dilatation im Frequenzraum nachzubilden.
+Für Funktionen in $L^2(\mathbb{R})$ entspricht die Dilatation mit dem
+Faktor $a$ im Ortsraum der Dilatation mit dem Faktor $1/a$ im Frequenzraum:
+\[
+\widehat{D_af}(\omega) = D_{1/a}\hat{f}(\omega).
+\]
+\cite[Satz~3.14]{buch:mathsem-wavelets}.
+Es bleibt aber das Problem, dass sich auch die Skalierung im Frequenzraum
+nicht durchführen lässt, da auch das Frequenzspektrum des Graphen nur eine
+Menge von reellen Zahlen ohne innere algebraische Struktur ist.
+
+\subsubsection{Mutterwavelets}
+\begin{figure}
+\centering
+\includegraphics{chapters/70-graphen/images/gh.pdf}
+\caption{Lokalisierungsfunktion $g(\lambda)$ für die Dilatation (links).
+Die Dilatierten Funktionen $g_i=\tilde{D}_{1/a_i}g$ lokalisieren
+die Frequenzen jeweils um die Frequenzen $a_i$ im Frequenzraum.
+Der Konstante Vektor ist vollständig delokalisiert, die Funktion $h$
+in der rechten Abbildung entfernt die hohen Frequenzen und liefert Funktionen,
+die in der Umgebung eines Knotens wie die Konstante Funktion aussehen.
+\label{buch:graphs:fig:lokalisierung}}
+\end{figure}
+Das Mutter-Wavelet einer Wavelet-Analyse zeichnet definiert, in welchem Mass
+sich Funktionen im Orts- und im Frequenzraum lokalisieren lassen.
+Die Standardbasis der Funktionen auf einem Graphen repräsentieren die
+perfekte örtliche Lokalisierung, Eigenbasis der Laplace-Matrix $L$ repräsentiert
+die perfekte Lokalisierung im Frequenzraum.
+Sei $g(\lambda)\ge 0$ eine Funktion im Frequenzraum, die für  $\lambda\to0$ und
+$\lambda\to\infty$ rasch abfällt mit einem Maximum irgendwo dazwischen
+(Abbildung~\ref{buch:graphs:fig:lokalisierung}).
+Sie kann als eine Lokalisierungsfunktion im Frequenzraum betrachtet werden.
 
-Die Fundamentallösung entsteht dadurch, dass die hohen Frequenzen
-schneller dämpft als die tiefen Frequenzen.
+Die Matrix $g(L)$ bildet entfernt aus einer Funktion die ganz hohen und 
+die ganz tiefen Frequenz, lokalisiert also die Funktionen im Frequenzraum.
+Die Standardbasisvektoren werden dabei zu Funktionen, die nicht mehr nur
+auf einem Knoten von $0$ verschieden sind, aber immer noch einigermassen
+auf dem Graphen lokalisiert sind.
+Natürlich sind vor allem die Werte auf den Eigenwerten
+$\lambda_0 < \lambda_1\le \dots\le \lambda_n$ der Laplace-Matrix
+von Interesse.
 
+Die Matrix $g(L)$ kann mit Hilfe der Spektraltheorie berechnet werden,
+was im vorliegenden Fall naheliegend ist, weil ja die Eigenvektoren von
+der Laplace-Matrix bereits bekannt sind.
+Die Matrix $\chi^t$ bildet die Standardbasisvektoren in die
+Eigenbasis-Vektoren ab, also in eine Zerlegung im Frequenzraum ab,
+$\chi$ vermittelt die Umkehrabbildung.
+Mit der Spektraltheorie findet man für die Abbildung $g(L)$ die Matrix
+\begin{equation}
+g(L)
+=
+\chi
+\begin{pmatrix}
+g(\lambda_0)&0&\dots&0\\
+0&g(\lambda_1)&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+0&0&\dots&g(\lambda_n)
+\end{pmatrix}
+\chi^t.
+\label{buch:graphen:eqn:mutterwavelet}
+\end{equation}
 
-\subsection{Wavelet-Basen
-\label{buch:subsection:}}
+\subsubsection{Dilatation}
+Die Dilatation um $a$ im Ortsraum wird zu einer Dilatation um $1/a$ im
+Frequenzraum.
+Statt also nach einer echten Dilatation der Spaltenvektoren in $g(L)$
+zu suchen, kann man sich darauf verlegen, Funktionen zu finden, deren
+Spektrum von einer Funktionen lokalisiert worden ist, die eine Dilatation
+von $g$ ist.
+Man wählt daher eine ansteigende Folge $A=(a_1,\dots)$ von Streckungsfaktoren
+und betrachtet anstelle von $g$ die dilatierten Funktionen
+$g_i=\tilde{D}_{1/a_i}g$.
+Die zugehörigen Wavelet-Funktionen auf dem Graphen können wieder mit
+der Formel~\eqref{buch:graphen:eqn:mutterwavelet} berechnet werden,
+man erhält
+\begin{equation}
+\tilde{D}_{1/a_i}g(L)
+=
+g_i(L)
+=
+\chi
+\begin{pmatrix}
+g(a_i\lambda_0)&0&\dots&0\\
+0&g(a_i\lambda_1)&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+0&0&\dots&g(a_i\lambda_n)
+\end{pmatrix}
+\chi^t .
+\end{equation}
+Die Spalten von $g_i(L)$ bilden wieder eine Menge von Funktionen, die
+eine gemäss $g_i$ lokalisiertes Spektrum haben.
 
+\subsubsection{Vater-Wavelet}
+Wegen $g(0)=0$ wird die konstante Funktion, die Eigenvektor zum Eigenwert
+$\lambda_0=0$ ist, von den Abbildungen $g_i(L)$ auf $0$ abgebildet.
+Andererseits ist diese Funktion nicht lokalisiert, man möchte Sie also
+für die Analyse nicht unbedingt verwenden.
+Man wählt daher eine Funktion $h(\lambda)$ mit $h(0)=1$ so, dass
+für $\lambda\to \infty$ der Wert $h(\lambda)$ genügend rasch gegen $0$
+geht.
+Die Matrix $h(L)$ bildet daher den konstanten Vektor nicht auf $0$ ab,
+sondern lokalisiert ihn im Ortsraum.
+Wir erhalten daher in den Spalten von $h(L)$ Vektoren, die um die
+einzelnen Knoten lokalisiert sind.
+
+\subsubsection{Rekonstruktion}
+Die Operatoren $h(L)$ und $g_i(L)$ erzeugen analysieren eine Funktion
+nach den verschiedenen Frequenzen mit den Skalierungsfaktoren $a_i$,
+aber die Rekonstruktion ist noch nicht klar.
+Diese wäre einfacher, wenn die Operatoren zusammen die identische
+Abbildung ergäben, wenn also
+\[
+h(L) + \sum_{i}g_i(L)=I
+\]
+gelten würde.
+Nach der Spektraltheorie gilt das nur, wenn für alle Eigenwerte
+$\lambda_k$, $k=1,\dots,n$
+\[
+h(\lambda_k) + \sum_ig(a_i\lambda_k)=1
+\]
+gilt.
+Für beleibige Funktionen $g$ und $h$ kann man nicht davon ausgehen,
+aber man kann erwarten.
+Man muss daher zusätzlich verlangen, dass
+\[
+h(\lambda_k) + \sum_{i} g(a_i\lambda_k) > 0
+\]
+ist für alle Eigenwerte $\lambda_k$.
+
+\subsubsection{Frame}
+Die Menge von Vektoren, die in der vorangegangenen Konstruktion gefunden
+wurden, ist zu gross, um eine Basis zu sein.
+Vektoren lassen sich darin auf verschiedene Art darstellen.
+Wir verlangen aber auch keine eindeutige Darstellung, nur eine 
+Darstellung, in der wir die ``dominierenden'' Komponenten in jeder
+Frequenzskala identifizieren können.
+
+\begin{definition}
+\label{buch:graphen:def:frame}
+Ein Frame des Vektorraumes $\mathbb{R}^n$ ist eine Menge
+$F=\{e_k\;|\; k=1,\dots,N\}$ von Vektoren mit der Eigenschaft
+\begin{equation}
+A\|v\|^2
+\le
+\sum_{k=1}^N  |\langle v,e_k\rangle|^2
+\le
+B\|v\|^2
+\label{buch:graphen:eqn:frame}
+\end{equation}
+Die Zahlen $A$  und $B$ heissen die {\em Frame-Konstanten} des Frames.
+\end{definition}
+
+Die oben gefundenen Vektoren, die Spalten Vektoren von $h(L)$ und $g_i(L)$
+bilden daher ein Frame.
+Die Frame-Konstanten kann man unmittelbar ausrechnen.
+Der mittlere Term von \eqref{buch:graphen:eqn:frame} ist 
+\[
+\|h(L) v\|^2
++
+\sum_{i} \|g_i(L)v\|^2,
+\]
+die durch die Funktion
+\[
+f(\lambda)
+=
+h(\lambda)^2 + \sum_i g_i(\lambda)^2
+\]
+abgeschätzt werden kann.
+Die Frame-Konstanten sind daher
+\begin{align*}
+A&=\min_{k} f(\lambda_k)
+&
+&\text{und}&
+B&=\max_{k} f(\lambda_k).
+\end{align*}
+Die Konstruktion hat also ein Frame für die Funktionen auf dem Graphen
+etabliert, die viele Eigenschaften einer Multiskalenanalyse in diese
+wesentlich weniger symmetrische Situation rettet.
 
 
 
diff --git a/buch/chapters/80-wahrscheinlichkeit/parrondo.tex b/buch/chapters/80-wahrscheinlichkeit/parrondo.tex
index a62d813..50e7fda 100644
--- a/buch/chapters/80-wahrscheinlichkeit/parrondo.tex
+++ b/buch/chapters/80-wahrscheinlichkeit/parrondo.tex
@@ -24,15 +24,15 @@ Je nach Ausgang gewinnt oder verliert der Spieler eine Einheit.
 Sei $X$ die Zufallsvariable, die den gewonnen Betrag beschreibt.
 Für eine faire Münze ist die Gewinnerwartung in diesem Spiel natürlich
 $E(X)=0$.
-Wenn die Wahrscheinlichkeit für einen Gewinn $1+e$ ist, dann muss
-die Wahrscheinlichkeit für einen Verlust $1-e$ sein, und die 
+Wenn die Wahrscheinlichkeit für einen Gewinn $\frac12+e$ ist, dann muss
+die Wahrscheinlichkeit für einen Verlust $\frac12-e$ sein, und die 
 Gewinnerwartung ist
 \(
 E(X)
 =
 1\cdot P(X=1) + (-1)\cdot P(X=-1)
 =
-1+e + (-1)(1-e)
+\frac12+e + (-1)\biggl(\frac12-e\biggr)
 =
 2e.
 \)
@@ -763,7 +763,7 @@ Eigenwert $1$ finden, die Rechnung mit dem Gauss-Algorithmus liefert
 p=
 \frac{1}{709}
 \begin{pmatrix}
-245\\180\\84
+245\\180\\284
 \end{pmatrix}.
 \]
 Damit kann man jetzt die Gewinnwahrscheinlichkeit im iterierten Spiel
diff --git a/buch/chapters/references.bib b/buch/chapters/references.bib
index a4579e7..a5d0201 100644
--- a/buch/chapters/references.bib
+++ b/buch/chapters/references.bib
@@ -21,6 +21,12 @@ abstract = "In this paper, we present Google, a prototype of a large-scale searc
 }
 
 
+@book{buch:mathsem-wavelets,
+	title = {Mathematisches Seminar Wavelets},
+	author = { Andreas M"uller and others },
+	year = {2019},
+}
+
 @book{buch:mathsem-dgl,
 	title = {Mathematisches Seminar Differentialgleichungen},
 	author = { Andreas M"uller and others },
diff --git a/buch/test3.tex b/buch/common/test-common.tex
index 71b1529..289e59c 100644
--- a/buch/test3.tex
+++ b/buch/common/test-common.tex
@@ -1,9 +1,8 @@
 %
-% test3.tex -- Test 3
+% test.tex -- Gemeinsamer Rahmen für Kurztests
 %
-% (c) 2021 Prof. Dr. Andreas Mueller, OST
+% (c) 2021 Prof. Dr. Andreas Mueller, OST Ostschweizer Fachhochschule
 %
-%\documentclass[a4paper,12pt]{book}
 \documentclass[a4paper,12pt]{article}
 \usepackage{geometry}
 \geometry{papersize={210mm,297mm},total={165mm,260mm}}
@@ -72,20 +71,3 @@
 \renewcommand{\qedsymbol}{}
 \begin{proof}[Hinweis]}{\end{proof}}
 
-\begin{document}
-{\parindent0pt\hbox to\hsize{%
-Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
-\vspace{0.5cm}
-
-\section*{Kurztest 3}
-
-\begin{uebungsaufgaben}
-
-\item
-\input chapters/60-gruppen/uebungsaufgaben/6001.tex
-%\item
-%\input chapters/60-gruppen/uebungsaufgaben/6002.tex
-
-\end{uebungsaufgaben}
-
-\end{document}
diff --git a/buch/common/test1.tex b/buch/common/test1.tex
new file mode 100644
index 0000000..1f5a155
--- /dev/null
+++ b/buch/common/test1.tex
@@ -0,0 +1,21 @@
+%
+% test1.tex -- Test 1
+%
+% (c) 2021 Prof. Dr. Andreas Mueller, OST
+%
+\input{common/test-common.tex}
+
+\begin{document}
+{\parindent0pt\hbox to\hsize{%
+Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
+\vspace{0.5cm}
+
+\section*{Kurztest 1}
+
+\begin{uebungsaufgaben}
+
+\input{aufgaben1.tex}
+
+\end{uebungsaufgaben}
+
+\end{document}
diff --git a/buch/common/test2.tex b/buch/common/test2.tex
new file mode 100644
index 0000000..0980e44
--- /dev/null
+++ b/buch/common/test2.tex
@@ -0,0 +1,21 @@
+%
+% test2.tex -- Test 2
+%
+% (c) 2012 Prof. Dr. Andreas Mueller, OST
+%
+\input{common/test-common.tex}
+
+\begin{document}
+{\parindent0pt\hbox to\hsize{%
+Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
+\vspace{0.5cm}
+
+\section*{Kurztest 2}
+
+\begin{uebungsaufgaben}
+
+\input{aufgaben2.tex}
+
+\end{uebungsaufgaben}
+
+\end{document}
diff --git a/buch/common/test3.tex b/buch/common/test3.tex
new file mode 100644
index 0000000..8b24262
--- /dev/null
+++ b/buch/common/test3.tex
@@ -0,0 +1,21 @@
+%
+% test3.tex -- Test 3
+%
+% (c) 2021 Prof. Dr. Andreas Mueller, OST
+%
+\input{common/test-common.tex}
+
+\begin{document}
+{\parindent0pt\hbox to\hsize{%
+Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
+\vspace{0.5cm}
+
+\section*{Kurztest 3}
+
+\begin{uebungsaufgaben}
+
+\input{aufgaben3.tex}
+
+\end{uebungsaufgaben}
+
+\end{document}
diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex
new file mode 100644
index 0000000..0db5617
--- /dev/null
+++ b/buch/papers/clifford/0_ElevatorPitch.tex
@@ -0,0 +1,2 @@
+TODO...
+GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains.
\ No newline at end of file
diff --git a/buch/papers/clifford/10_Quaternionen.tex b/buch/papers/clifford/10_Quaternionen.tex
new file mode 100644
index 0000000..8945ba8
--- /dev/null
+++ b/buch/papers/clifford/10_Quaternionen.tex
@@ -0,0 +1,61 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Quaternionen}
+\rhead{Quaternionen}
+Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den 3 dimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine dreh-streckende Eigenschaft.
+Sie finden beispielsweise in der Computergraphik und in der Robotik Anwendung.
+Die Quaternionen werden so definiert.
+\begin{align}
+	q = w + xi + yj + zk; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{H}
+\end{align}
+Eine Drehstreckung wird dabei mit dieser Formel erreicht. 
+\begin{align} \label{QuatRot}
+	\begin{split} 
+		&v'' = qvq^{-1};\quad q,v,q^{-1} \in \mathbb{H}\\
+		&Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1)
+	\end{split}
+\end{align}
+Die Quaternionen besitzen im Gegensatz zu dem komplexen Zahlen 3 imaginäre Einheiten $i,j,k$. Wieso 3? Weil es in der dritten Dimension 3 Drehachsen gibt, anstatt nur eine. Nun haben wir ein kleines Problem. Wie sollen wir die Quaternionen darstellen? Wir bräuchten 4 Achsen für die 3 Imaginären Einheiten und die eine reelle Einheit. Ein weiterer Nachteil in visueller Hinsicht entsteht beim Anwenden eines Quaternion auf einen Vektor. Sie befinden sich nicht im gleichen Raum und müssen zuerst ineinander umgewandelt werden, um damit zu rechnen, wie man bei $v$ in der Formel (\ref{QuatRot}) sieht.
+
+\subsection{geometrischen Algebra}
+Die geometrische Algebra besitzt die Fähigkeit beide Probleme zu lösen. Die Quaternionen können, wie schon im 2 dimensionalen Fall durch die gerade Grade $\mathbb{G}_3^+ \cong \mathbb{H}$ dargestellt werden. Da wir uns jetzt aber in $\mathbb{G}_3$ befinden haben wir 3 Basisvektoren $e_1, e_2, e_3$ und können somit 3 Bivektoren bilden $e_{12}, e_{23}, e_{31}$.
+\begin{align}
+	\mathbf{q} = w + x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}}; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{G}_3^+
+\end{align}
+Die Probleme werden dadurch gelöst, da wir die Bivektoren im Raum nicht durch einzelne Achsen darstellen müssen, sondern sie als eine orientiere Fläche darstellen können. Anstatt die Vektoren in Quaternionen umzurechnen, können wir jetzt die Vektoren separat im gleichen Raum darstellen. 
+\\BILD VEKTOR, QUATERNION IN G3\\
+Wie schon im 2 dimensionalen Fall beschreibt ein Bivektor, um wie viel der um 90 grad gedrehte orginale Vektor gestreckt wird. Dabei dreht jeder Bivektor den Vektor um eine andere Achse.
+\\BILD?\\
+In der Computergraphik und Robotik macht eine Drehstreckung aber nicht viel Sinn. Wieso sollte ein Objekt bei einer Drehung zusätzlich noch grösser werden? Darum verwendet man sogenannte Einheitsquaternion, welche den Betrag $|q|=1$ haben. Sie rotieren die Objekte bzw. Vektoren lediglich.
+\begin{align}
+	\mathbf{q} = \cos(\alpha) + sin(\alpha)(x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}})
+\end{align}
+wobei definiert ist, dass $x^2+y^2+z^2=1$. Somit beträgt der Betrag immer 1.
+\begin{align}
+	|q| = \sqrt{cos(\alpha)^2 + sin(\alpha)^2(x^2+y^2+z^2) } = \sqrt{cos(\alpha)^2 + sin(\alpha)^2} = 1
+\end{align}
+Man verwendet um einen Vektor zu drehen wieder die gleiche Formel, wie auch schon im 2 dimensionalen Fall.
+\begin{align} \label{QuatRot}
+	\begin{split} 
+		&v'' = qvq^{-1}\\
+		&Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1)
+	\end{split}
+\end{align}
+Es ist wichtig bei Quaternionen für eine reine Drehstreckung mit $q$ und $q^{-1}$ beidseitig zu multiplizieren, sonst werden die senkrechten Anteile zu den Bivektorebenen ebenfalls beeinflusst, wie man im Kapitel Rotation bei der Formel (\ref{RotAufPerpPar}) sehen kann
+
+\subsection{Gimbal-Lock und Interpolation}
+
+\subsection{Fazit}
+andere Darstellungsweise. Besser für Verständnis => komplexe Zahlen erscheinen ähnlicher zu Quaternionen? Eine Sprache für alle Geometrische Probleme
+
+
+\begin{tikzpicture}
+	\draw[thin,gray!40] (-3,-3) grid (3,3);
+	\draw[<->] (-3,0)--(3,0) node[right]{$x$};
+	\draw[<->] (0,-3)--(0,3) node[above]{$y$};
+	\draw[line width=2pt,blue,-stealth](0,0)--(1,1) node[anchor=south west]{$\boldsymbol{u}$};
+	\draw[line width=2pt,red,-stealth](0,0)--(-1,-1) node[anchor=north east]{$\boldsymbol{-u}$};
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex
new file mode 100644
index 0000000..88a5789
--- /dev/null
+++ b/buch/papers/clifford/1_Vektordarstellung.tex
@@ -0,0 +1,71 @@
+\section{Vektoroperationen\label{clifford:section:Vektoroperationen}}
+\rhead{Vektoroperationen}
+\subsection{Vektordarstellung\label{clifford:section:Vektordarstellung}}
+Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden
+\begin{equation}
+    \begin{split}
+    \textbf{a} 
+    &=
+    \begin{pmatrix} 
+    a_1 \\ a_2 \\ \vdots \\ a_n   
+    \end{pmatrix} 
+    =
+    a_1 \begin{pmatrix}
+    1 \\ 0 \\ \vdots \\ 0  
+    \end{pmatrix} 
+    + 
+    a_2\begin{pmatrix} 
+    0 \\ 1 \\ \vdots \\ 0  
+    \end{pmatrix} + \dots 
+    + 
+    a_n\begin{pmatrix}
+    0 \\ 0 \\ \vdots \\ 1  
+    \end{pmatrix} \\\ 
+    &= 
+    a_1\textbf{e}_1 
+    +
+    a_2\textbf{e}_2
+    + 
+    \dots + a_n\textbf{e}_n
+    = 
+    \sum_{i=1}^{n} a_i \textbf{e}_i
+    \qquad
+    a_i \in \mathbb{R}
+    , \textbf{e}_i \in \mathbb{R}^n.
+    \end{split}
+\end{equation}
+Diese Basisvektoren sollen orthonormal sein und um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, ...$ ersetzt.
+\begin{beispiel}
+Linearkombination von Basisvektoren in $\mathbb{R}^4$
+    \begin{equation}
+        \begin{pmatrix} 
+        42 \\ 2 \\ 1291 \\ 4    
+        \end{pmatrix} 
+        = 
+        42 \begin{pmatrix}
+        1 \\ 0 \\ 0 \\ 0 
+        \end{pmatrix} 
+        +
+        2 \begin{pmatrix} 
+        0 \\ 1 \\ 0 \\ 0 
+        \end{pmatrix}
+        +
+        1291 
+        \begin{pmatrix} 
+        0 \\ 0 \\ 1 \\ 0 
+        \end{pmatrix} 
+        +
+        4 \begin{pmatrix} 
+        0 \\ 0 \\ 0 \\ 1 
+        \end{pmatrix} 
+        = 
+        42\textbf{e}_1
+        + 
+        2\textbf{e}_2
+        + 
+        1291\textbf{e}_3
+        + 
+        4\textbf{e}_4
+    \end{equation}
+\end{beispiel}
+Wobei Beispiel für einen vier dimensionalen Vektor ist, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden.
\ No newline at end of file
diff --git a/buch/papers/clifford/2_QuadratVektoren.tex b/buch/papers/clifford/2_QuadratVektoren.tex
new file mode 100644
index 0000000..cfb05d6
--- /dev/null
+++ b/buch/papers/clifford/2_QuadratVektoren.tex
@@ -0,0 +1,110 @@
+\subsection{Quadrat von Vektoren}
+Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. 
+Was soll es schon heissen zwei Vektoren miteinander zu multiplizieren? 
+\newline
+Im Folgenden werden wir versuchen diese Operation ähnlich intuitiv darzustellen.
+\newline
+Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation ist. 
+Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen, auch so definiert ist.  
+\newline 
+Zusätzlich wollen wir auch das Assoziativgesetz und das Kommutativgesetz für Skalare beibehalten. Wobei das Kommutativgesetz leider, oder wie man sehen wird zum Glück, in der geometrischen Algebra im generellen nicht mehr gilt. Das heisst wir dürfen ausklammern \ref{eq:assoziativ} und die Position von Skalaren im Produkt ändern \ref{eq:kommSkalar}, allerdings nicht die Position der Vektoren \ref{eq:kommVector}.
+\begin{equation}
+    \label{eq:assoziativ}
+    \textbf{e}_i(\textbf{e}_j + \textbf{e}_k) 
+    = 
+    \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k 
+\end{equation}
+\begin{equation}
+    \label{eq:kommSkalar}
+    a\textbf{e}_ib\textbf{e}_j 
+    = 
+    ab\textbf{e}_i\textbf{e}_j
+\end{equation}
+\begin{equation}
+    \label{eq:kommVector}
+    \textbf{e}_i\textbf{e}_j 
+    \neq 
+    \textbf{e}_j\textbf{e}_i
+\end{equation}
+Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors.
+\begin{align}
+    \textbf{a}^2 &= 
+    \left ( 
+    \sum_{i=1}^{n} a_i \textbf{e}_i 
+    \right ) 
+    \left ( 
+    \sum_{i=1}^{n} a_i \textbf{e}_i 
+    \right )
+    \label{eq:quad_a_1}
+    \\
+    &= 
+    \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} 
+    + 
+    \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  a_ia_j\textbf{e}_i\textbf{e}_j } 
+    \label{eq:quad_a_2}
+    \\
+    &= \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  a_ia_j\textbf{e}_i\textbf{e}_j}.
+    \label{eq:quad_a_3}
+\end{align}
+
+\begin{beispiel}
+Quadrat eines Vektors in $\mathbb{R}^2$
+\begin{equation}
+    \begin{split}
+    \textbf{a}^2 
+    &= (a_1\textbf{e}_1+a_2\textbf{e}_2)(a_1\textbf{e}_1+a_2\textbf{e}_2) \\\
+    &= \textcolor{red}{a_1^2\textbf{e}_1^2 + a_2^2\textbf{e}_2^2} 
+    + \textcolor{blue}{a_1\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2}   \\\
+    & = \textcolor{cyan}{a_1^2 + a_2^2} + \textcolor{orange}{a_1b\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2}
+    \end{split}
+\end{equation}
+
+\end{beispiel}
+Der Vektor wird in \ref{eq:quad_a_1} als Linearkombination geschrieben.
+Das Quadrat kann, wie in \ref{eq:quad_a_2} gezeigt, in zwei Summen aufteilen werden , wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. 
+\newline
+Da $\textbf{e}_i^2 = 1$ gilt, da zuvor vorausgesetzt wurde, dass man mit orthonormalen Einheitsvektoren arbeitet, wird dies nun eingesetzt ergibt sich \ref{eq:quad_a_3}
+\newline
+Die hellblaue Teil ist nun bereits Länge im Quadrat eines Vektors, also das Ziel der Multiplikation. 
+Daher muss der restliche Teil dieser Gleichung null ergeben. 
+Aus dieser Erkenntnis leiten wir in \ref{eq:Mischterme_Null} weitere Eigenschaften für die Multiplikation her.
+\begin{equation}
+    \label{eq:Mischterme_Null}
+    \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  a_ia_j\textbf{e}_i\textbf{e}_j  = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots =  0
+\end{equation}
+Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_2$ gleich null gesetzt. Somit fallen alle Terme bis auf den blauen weg. Wird dies weiter vereinfacht ergibt sich
+\begin{equation}
+\begin{split}
+    a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1) &= 0 \\
+    a_1a_2\textbf{e}_1\textbf{e}_2 &= -a_1a_2\textbf{e}_2\textbf{e}_1 \\
+    \textbf{e}_1\textbf{e}_2 &= -\textbf{e}_2\textbf{e}_1.
+\end{split}
+\end{equation}
+\begin{satz}
+    Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind.
+    \begin{equation}
+        \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \qquad \textbf{e}_i \perp \textbf{e}_j
+    \end{equation}
+\end{satz}
+Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in \ref{tab:multip_vec} gemacht wurde.
+\begin{table}
+\caption{Multiplikationstabelle für Vektoren}
+\label{tab:multip_vec}
+\begin{center}
+\begin{tabular}{ |c|c|c|c|c|c| } 
+ \hline
+  & $\textbf{e}_1$ & $\textbf{e}_2$ & $\dots$ & $\textbf{e}_{n-1}$ & $\textbf{e}_{n}$ \\
+  \hline
+ $\textbf{e}_1$ & 1 & $\textbf{e}_1\textbf{e}_2$ & $\dots$ & $\textbf{e}_1\textbf{e}_{n-1}$ & $\textbf{e}_1\textbf{e}_{n}$ \\
+ \hline
+ $\textbf{e}_2$ & $-\textbf{e}_1\textbf{e}_2$ & 1 & $\dots$ & $\textbf{e}_2\textbf{e}_{n-1}$ & $\textbf{e}_2\textbf{e}_{n}$ \\
+ \hline
+ $\vdots$ & $\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ & $\vdots$ \\
+ \hline
+ $\textbf{e}_{n-1}$ & $-\textbf{e}_1\textbf{e}_{n-1}$ & $-\textbf{e}_2\textbf{e}_{n-1}$  & $\dots$ & $1$ & $\textbf{e}_{n-1}\textbf{e}_{n}$ \\
+ \hline
+ $\textbf{e}_{n}$ & $-\textbf{e}_1\textbf{e}_{n}$ & $-\textbf{e}_2\textbf{e}_{n}$  & $\dots$ & $-\textbf{e}_{n-1}\textbf{e}_{n}$ & 1 \\
+ \hline
+\end{tabular}
+\end{center}
+\end{table}
\ No newline at end of file
diff --git a/buch/papers/clifford/3_MultiplikationVektoren.tex b/buch/papers/clifford/3_MultiplikationVektoren.tex
new file mode 100644
index 0000000..841dde4
--- /dev/null
+++ b/buch/papers/clifford/3_MultiplikationVektoren.tex
@@ -0,0 +1,175 @@
+\subsection{Multiplikation von Vektoren}
+Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipliziert werden?
+\begin{equation}
+    \textbf{u} = 
+    \sum_{i=1}^{n} u_i \textbf{e}_i 
+    \qquad 
+    \textbf{v} = \sum_{i=1}^{n} v_i \textbf{e}_i
+\end{equation}
+\begin{equation}
+    \begin{split}
+        \textbf{u}\textbf{v} 
+        =
+        \left ( 
+        \sum_{i=1}^{n} u_i \textbf{e}_i
+        \right ) 
+        \left ( 
+        \sum_{i=1}^{n} v_i \textbf{e}_i
+        \right) 
+        = 
+        \sum_{i=1}^n u_iv_i\underbrace{\textbf{e}_i^2}_{1} 
+        + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  u_iv_j\textbf{e}_i\textbf{e}_j 
+    \end{split}
+\end{equation}
+\begin{beispiel}
+    Multiplikation von Vektoren in $\mathbb{R}^2$
+\end{beispiel}
+\begin{equation}
+    \begin{split}
+        \textbf{u}\textbf{v} 
+        &= 
+        (u_1\textbf{e}_1 + u_2\textbf{e}_2)(v_1\textbf{e}_1 + v_2\textbf{e}_2) 
+        = 
+        u_1v_1\textbf{e}_1^2
+        + 
+        u_2v_2\textbf{e}_2^2 
+        + 
+        u_1v_2\textbf{e}_1\textbf{e}_2 
+        +  
+        u_2v_1\underbrace{\textbf{e}_2\textbf{e}_1}_{-\textbf{e}_1\textbf{e}_2}
+        \\\ 
+        &=  
+        \underbrace{(u_1v_1 + u_2v_2)}_{\text{Skalarprodukt}} 
+        + 
+        \underbrace{(u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2}_{\text{Äusseres Produkt}}
+    \end{split}
+\end{equation}
+Der linke Teil dieser Multiplikation ergibt das Skalarprodukt der zwei Vektoren, der rechte Term ergibt etwas neues das sich das äussere Produkt der zwei Vektoren nennt.
+\subsubsection{Äusseres Produkt}
+Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt
+\begin{equation}
+    \textbf{u}\wedge \textbf{v} 
+    = 
+    \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  u_iv_j\textbf{e}_i\textbf{e}_j 
+\end{equation}
+\begin{beispiel}
+Äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$
+\end{beispiel}
+\begin{equation}
+    \begin{split}
+        u \wedge v 
+        &= 
+        u_1v_2\textbf{e}_1\textbf{e}_2 
+        + 
+        u_1v_3\textbf{e}_1\textbf{e}_3 
+        + 
+        u_2v_2\textbf{e}_2\textbf{e}_3 
+        + 
+        u_2v_1\textbf{e}_2\textbf{e}_1 
+        + 
+        u_3v_1\textbf{e}_3\textbf{e}_1 
+        +
+        u_3v_2\textbf{e}_3\textbf{e}_2 \\\ 
+        &= 
+        (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 
+        + 
+        (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 
+        + 
+        (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3
+    \end{split}
+\end{equation}
+Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \ref{eq:u_wedge_v}-\ref{eq:u_wedge_v_5} hergeleitet.
+\begin{align}
+        \textbf{u}\wedge \textbf{v}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n  
+        u_iv_j\textbf{e}_i\textbf{e}_j 
+        \label{eq:u_wedge_v}
+        \\
+        \label{eq:u_wedge_v_1}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j 
+        + 
+        \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j 
+        \\
+        \label{eq:u_wedge_v_2}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j 
+        + 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i
+        \\
+        \label{eq:u_wedge_v_3}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j 
+        - 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j
+        \\
+        \label{eq:u_wedge_v_4}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j
+        \\
+        \label{eq:u_wedge_v_5}
+        &= 
+        \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n \begin{vmatrix} 
+        u_i & v_i \\
+        u_j & v_j
+    \end{vmatrix}\textbf{e}_i\textbf{e}_j
+\end{align}
+Die Summe aus \ref{eq:u_wedge_v_1} wird in \ref{eq:u_wedge_v} in zwei verschiedene Summen aufgeteilt. 
+Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat. 
+\newline
+Bei \ref{eq:u_wedge_v_2} werden die Indexe der zweiten Summe vertauscht, damit man nun bei beiden Teilen die gleiche Summe hat.
+Danach werden in \ref{eq:u_wedge_v_3}, mit Hilfe der Antikommutativität, die Einheitsvektoren der zweiten Summe vertauscht.
+\newline
+Nun können die Summen, wie in \ref{eq:u_wedge_v_4} wieder in eine Summe zusammengefasst werden.
+\newline
+Der Term in der Klammer in \ref{eq:u_wedge_v_4} kann auch als Determinante einer 2x2 Matrix dargestellt werden, was in \ref{eq:u_wedge_v_5} gemacht wird.
+\newline
+Die Determinante einer Matrix beschreibt welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt.
+\begin{figure}
+\centering
+\begin{tikzpicture}
+  \draw[thin,gray!40] (0,0) grid (4,4);
+  \draw[<->] (0,0)--(4,0) ;
+  \draw[<->] (0,0)--(0,4) ;
+  \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2);
+  \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north
+  west]{$\boldsymbol{u}$};
+  \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$};
+  \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$\begin{vmatrix} 
+        u_i & v_i \\
+        u_j & v_j
+    \end{vmatrix} = u_iv_j - v_iu_j$};
+\end{tikzpicture}
+\caption{Geometrische Interpretation der Determinante einer 2x2 Matrix\label{figure:det}}
+\end{figure}
+\newline
+Das äussere Produkt besteht nun also aus der Summe 
+    $\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n$
+    von Flächen 
+    $\begin{vmatrix} 
+        u_i & v_i \\
+        u_j & v_j
+    \end{vmatrix}$, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. 
+Dieses Produkt $\textbf{e}_i\textbf{e}_j$ der Basisvektoren interpretiert man als Umlaufrichtung.
+Wobei die gebildete Fläche in Richtung des ersten Vektors umschritten wird. 
+Dies ist in \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. 
+Diese Fläche mit einer Richtung nennt man in der geometrischen Algebra einen Bivektor, da er eine Art zwei dimensionaler Vektor ist. 
+\begin{figure}
+\centering
+\begin{tikzpicture}
+  \draw[thin,gray!40] (0,0) grid (4,4);
+  \draw[<->] (0,0)--(4,0) node[right]{$x$};
+  \draw[<->] (0,0)--(0,4) node[above]{$y$};
+  \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2);
+  \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north
+  west]{$\boldsymbol{u}$};
+  \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$};
+ \draw[->] (2.15,1.5) arc (0:310:0.3);
+  \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$u\wedge v = \begin{vmatrix} 
+        u_i & v_i \\
+        u_j & v_j
+    \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$};
+\end{tikzpicture}
+\caption{Geometrische Interpretation des äusseren Produkt in $\mathbb{R}^2$\label{figure:wedge}}
+\end{figure}
\ No newline at end of file
diff --git a/buch/papers/clifford/4_GeometrischesProdukt.tex b/buch/papers/clifford/4_GeometrischesProdukt.tex
new file mode 100644
index 0000000..a19e983
--- /dev/null
+++ b/buch/papers/clifford/4_GeometrischesProdukt.tex
@@ -0,0 +1,59 @@
+\subsection{Geometrisches Produkt}
+Die Multiplikation von zwei Vektoren nennt man in der Clifford Algebra das geometrische Produkt, dieses können wir nun als Summe aus dem Skalar- und dem äusseren Produkt darstellen
+\begin{equation}
+    \textbf{u}\textbf{v} = \textbf{u}\cdot \textbf{v} + \textbf{u} \wedge \textbf{v}.
+\end{equation}
+Dieses Additionszeichen zwischen diesen zwei Produkten mag vielleicht ein wenig eigenartig wirken, da uns das Skalarprodukt ein Skalar und das äussere Produkt einen Bivektor zurück gibt. Was bedeutet es nun also diese beiden Elemente zu addieren?
+Man kann sich die Addition wie bei den komplexen Zahlen vorstellen, wobei die imaginäre Einheit auch nicht explizit zu dem reelen Teil addiert werden kann, sondern die zwei Teile zusammen ein Objekt, eine komplexe Zahl bilden. 
+Dieses Objekt, also die Summe von verschiedenen Elemente der Clifford Algebra, wird Multivektor genannt.
+\begin{definition}
+Ein Multivektor besteht aus den verschiedenen Bauteilen, wie zum Beispiel Vektoren, Bivektoren oder Trivektoren (Volumen mit einer Richtung), der Clifford Algebra.
+\begin{equation}
+    M = \sum \left ( \prod a_i\textbf{e}_j \right)
+\end{equation}
+\end{definition}
+Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $\mathbb{G}^n$ beschrieben. 
+\begin{beispiel}
+Allgemeiner Multivektor in $\mathbb{G}^3$
+\begin{equation}
+    M = a 
+    + 
+    \underbrace{b\textbf{e}_1 + c\textbf{e}_2 + d\textbf{e}_3}_{\text{Vektorteil}} 
+    +
+    \underbrace{f\textbf{e}_1\textbf{e}_2 + g\textbf{e}_1\textbf{e}_3 + h\textbf{e}_2\textbf{e}_3 }_{\text{Bivektorteil}} 
+    +   
+    \underbrace{k\textbf{e}_1\textbf{e}_2\textbf{e}_3}_{\text{Trivektorteil}}
+\end{equation}
+\end{beispiel}
+\begin{definition}
+Um das Produkt von Basisvektoren in Zukunft darzustellen wird folgende Notation definiert
+    \begin{equation}
+        e_ie_j = e_{ij}
+    \end{equation}
+\end{definition}
+Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $\mathbb{G}^n$ erstellt werden. In \ref{tab:multip} ist dies für  $\mathbb{G}^3$ gemacht.
+\begin{table}
+    \caption{Multiplikationstabelle für $\mathbb{G^3}$}
+    \label{tab:multip}
+    \begin{center}
+    \begin{tabular}{ |c|c|c|c|c|c|c|c| } 
+     \hline
+     1 & $\textbf{e}_1$ & $\textbf{e}_2$ &$\textbf{e}_3$ & $\textbf{e}_{12}$ & $\textbf{e}_{13}$ & $\textbf{e}_{23}$ & $\textbf{e}_{123}$\\
+     \hline
+     $\textbf{e}_1$ & 1 & $\textbf{e}_{12}$ & $\textbf{e}_{12}$ & $\textbf{e}_2$ & $\textbf{e}_3$ & $\textbf{e}_{123}$ & $\textbf{e}_{23}$\\
+     \hline
+     $\textbf{e}_2$ & $-\textbf{e}_{12}$ & 1 & $\textbf{e}_{23}$ & $-\textbf{e}_1$ & $-\textbf{e}_{123}$ & $\textbf{e}_3$ & $-\textbf{e}_{13}$\\
+     \hline
+     $\textbf{e}_3$ & $-\textbf{e}_{13}$ & $-\textbf{e}_{23}$ & 1 & $\textbf{e}_{123}$ & $-\textbf{e}_1$ & $-\textbf{e}_2$ & $\textbf{e}_{12}$\\
+     \hline
+     $\textbf{e}_{12}$ & -$\textbf{e}_2$ & $\textbf{e}_1$& $\textbf{e}_{123}$ & -1 & $-\textbf{e}_{23}$ & $\textbf{e}_{13}$ &  $-\textbf{e}_{3}$\\
+     \hline
+     $\textbf{e}_{13}$ & $-\textbf{e}_{3}$ & $-\textbf{e}_{123}$ & $\textbf{e}_{1}$ & $\textbf{e}_{23}$ & -1 & $-\textbf{e}_{12}$ &  $\textbf{e}_{2}$\\
+     \hline
+     $\textbf{e}_{23}$ &  $\textbf{e}_{123}$ & $-\textbf{e}_{3}$ & $\textbf{e}_{2}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & -1 & $-\textbf{e}_{1}$ \\
+     \hline
+     $\textbf{e}_{123}$ & $\textbf{e}_{23}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & $-\textbf{e}_{3}$& $\textbf{e}_{2}$ & $-\textbf{e}_{1}$ & -1 \\
+     \hline
+    \end{tabular}
+    \end{center}
+\end{table}
diff --git a/buch/papers/clifford/5_PolareDarstellung.tex b/buch/papers/clifford/5_PolareDarstellung.tex
new file mode 100644
index 0000000..80fb49f
--- /dev/null
+++ b/buch/papers/clifford/5_PolareDarstellung.tex
@@ -0,0 +1,29 @@
+\subsection{Polare Darstellung des geometrischen Produktes}
+Beide Teile des geometrischen Produktes lassen sich durch trigonometrische Terme beschreiben. Das Skalarprodukt kann als 
+\begin{equation}
+    \textbf{u}\cdot \textbf{v} = |\textbf{u}||\textbf{v}|\cos{\alpha}
+\end{equation}
+beschrieben werden. Wobei $\alpha$ den Winkel zwischen den beiden Vektoren beschreibt.
+\newline
+Beim äusseren Produkt wurde bereits erwähnt, dass es aus dem Produkt der Fläche des von den zwei Vektoren aufgespannten Parallelogram und einer Umlaufrichtung beschrieben wird. Die Fläche eines Parallelograms lässt sich auch mit einen Sinus Term beschreiben
+\begin{equation}
+    \textbf{u} \wedge \textbf{v}
+    = 
+    \begin{vmatrix} 
+        u_i & v_i \\
+        u_j & v_j
+    \end{vmatrix}\textbf{e}_i\textbf{e}_j  
+    = 
+    \underbrace{|u||v|\sin{\alpha}}_{\text{Fläche}}\textbf{e}_i\textbf{e}_j
+\end{equation}
+Wobei die Fläche des Parallelogram auf der von $\textbf{e}_i$ und $\textbf{e}_j$ aufgespannten Ebene liegen.\newline
+Nun kann man diese Terme wieder zum geometrischen Produkt vereinen
+\begin{equation}
+    \textbf{u}\textbf{v}
+    = 
+    |\textbf{u}||\textbf{v}|\cos{(\alpha)} 
+    + 
+    |\textbf{u}||\textbf{v}|\sin{(\alpha)} \textbf{e}_i\textbf{e}_j
+    = 
+    |\textbf{u}||\textbf{v}|(\cos{(\alpha)} + \sin{(\alpha)}\textbf{e}_i\textbf{e}_j)
+\end{equation}
\ No newline at end of file
diff --git a/buch/papers/clifford/6_Dirac-Matrizen.tex b/buch/papers/clifford/6_Dirac-Matrizen.tex
new file mode 100644
index 0000000..6417bb3
--- /dev/null
+++ b/buch/papers/clifford/6_Dirac-Matrizen.tex
@@ -0,0 +1,7 @@
+%
+% einleitung.tex -- Beispiel-File für die Einleitung
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Dirac-Matrizen}
+\rhead{Dirac-Matrizen}
diff --git a/buch/papers/clifford/7_Reflektion.tex b/buch/papers/clifford/7_Reflektion.tex
new file mode 100644
index 0000000..d4942e0
--- /dev/null
+++ b/buch/papers/clifford/7_Reflektion.tex
@@ -0,0 +1,33 @@
+%
+% teil1.tex -- Beispiel-File für das Paper
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Reflektion/ Spiegelung}
+\rhead{Reflektion/ Spiegelung}
+Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die Geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Reflektion auch eine einfache, praktische Formulierung besitzen. \\HIER BILD
+\subsection{linearen Algebra}
+Aus der linearen Algebra ist bekannt, dass man eine Reflektion wie folgt beschreiben kann.
+\begin{align} \label{RefLinAlg}
+	\mathbf{v^{'}} = \mathbf{v} - 2 \cdot \mathbf{v_{\perp u}}
+\end{align}
+Dabei stellt $\mathbf{u}$ die Spiegelachse dar.
+Es scheint für diese Formel aber umständlich zu sein, weitere Reflektionen, mit weiteren Spiegelachsen, anzufügen. Man kann die Abbildung des Vektors auf den Reflektierten Vektor auch als Matrix schreiben, welche aus den Komponenten des zu der Spiegelachse orthonormalen Vektors $\mathbf{\hat{n}}$ besteht.
+\\MATRIZEN O(2) und O(3) zeigen\\
+Diese Matrizen gehören der Matrizengruppe $O(n)$ an....
+\subsection{geometrischen Algebra}
+Die Geometrische Algebra leitet aus der obigen Formel (\ref{RefLinAlg}) eine einfache und intuitive Form her, welche auch für weitere Operationen einfach erweitert werden kann.
+\begin{align}
+	\mathbf{v'} = \mathbf{uvu^{-1}}
+\end{align}
+wobei die Inverse eines Vektors so definiert ist, dass multipliziert mit sich selbst das neutrale Element 1 ergibt.
+\begin{align}
+	u^{-1} = \dfrac{u}{|u|^2} \Rightarrow uu^{-1} = 1
+\end{align}
+verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird somit die Formel reduziert zu einer beidseitigen Multiplikation von $\mathbf{\hat{u}}$.
+\begin{align}
+	\mathbf{v'} = \mathbf{\hat{u}v\hat{u}}
+\end{align}
+Im Gegensatz zu den Abbildungen in der linearen Algebra, welche in jeder anderen Dimension durch andere Matrizen beschrieben werden müssen, ist es in der geometrischen Algebra immer der gleiche Vorgehensweise.
+Zudem ist diese kompakte Schreibweise in der linearen Algebra nicht möglich, da keine Multiplikation von Vektoren definiert ist. 
+\\BEISPIEL?
\ No newline at end of file
diff --git a/buch/papers/clifford/8_Rotation.tex b/buch/papers/clifford/8_Rotation.tex
new file mode 100644
index 0000000..c2928bf
--- /dev/null
+++ b/buch/papers/clifford/8_Rotation.tex
@@ -0,0 +1,100 @@
+%
+% teil2.tex -- Beispiel-File für teil2 
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Rotation}
+\rhead{Rotation}
+Eine Rotation kann man aus zwei, aufeinanderfolgende Reflektionen bilden. Das war für mich zuerst eine verwirrende Aussage, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Reflektion schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Reflektion, wie bei einer Rotation, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Reflektion invertiert wird. Stellt man sich beispielsweise ein Objekt in 3D vor und spiegelt dieses an einer Fläche, dann ist es unmöglich nur durch eine Rotation (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zwei mal invertiert wurde.
+\\BILD
+
+\subsection{linearen Algebra}
+In der linearen Algebra haben wir Drehungen durch die Matrizen der Gruppe $SO(n)$ beschrieben. Die SO(2) werden beispielsweise auf diese Weise gebildet.
+\begin{align}
+	D = 
+	\begin{pmatrix}
+		cos(\alpha) & sin(\alpha) \\
+		-sin(\alpha) & cos(\alpha) 
+	\end{pmatrix}
+\end{align}
+
+\subsection{geometrischen Algebra}
+Da wir jetzt aus der Geometrie wissen, dass eine Rotation durch zwei Reflektionen gebildet werden kann, können wir die Rotation einfach herleiten.
+\begin{align} \label{rotGA}
+	v'' = wv'w^{-1} = w(uvu^{-1})w^{-1} 
+\end{align}
+Die Vektoren $\mathbf{w}$ und $\mathbf{u}$ bilden hier wiederum die Spiegelachsen. Diese versuchen wir jetzt noch zu verbessern. Dazu leiten wir zuerst die bekannte Polarform her. (Anmerkung: Hier wird eine Rotation auf der $\mathbf{e_{12}}$ Ebene hergeleitet. Weitere Drehungen können in höheren Dimensionen durch Linearkombinationen von Drehungen in den $\mathbf{e_{ij}}, i\not=j$ Ebenen erreicht werden)
+\begin{align}
+	\mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_2\right]
+\end{align}
+Dabei können wir ausnützen, dass $e_1^2 = 1$ ist. Was nichts ändert wenn wir es einfügen. Zudem klammern wir dann $e_1$ aus. 
+\begin{align}
+	\mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_1e_1e_2\right] 
+\end{align}
+\begin{align} \label{e1ausklammern}
+	\mathbf{w} = |w|e_1\left[\cos(\theta_w)+ \sin(\theta_w) e_{12}\right]
+\end{align}
+Durch die Reihenentwicklung ist es uns jetzt möglich den Term in eckigen Klammern mit der e-Funktion zu schreiben.
+\begin{align}
+	\mathbf{w} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}}
+\end{align}
+Man kann es so interpretieren, dass der Einheitsvektor $e_1$ um die Länge w gestreckt und um $theta_w$ gedreht wird.
+Nun werden wir den Effekt von zwei aneinandergereihten Vektoren $(wu)$ betrachten.
+\begin{align}
+	\mathbf{wu} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}}||u||\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}}
+\end{align}
+Um die beiden $\mathbf{e_1}$ zu kürzen, können wir die Reihenfolge des exponential Terms mit $\mathbf{e_1}$ wechseln, indem man bei der Gleichung (\ref{e1ausklammern}), anstatt mit $\mathbf{e_1e_1e_2}$ mit $\mathbf{e_2e_1e_1}$ erweitert. 
+\begin{align} 
+	\mathbf{w} = |w|\left[\cos(\theta_w)+ \sin(\theta_w) \mathbf{e_2e_1}\right]\mathbf{e_1}
+\end{align}
+Da $\mathbf{e_2e_1 = -e_{12}}$ können wir einfach den Winkel negieren.
+Jetzt können wir wieder $e_1e_1 = 1$ kürzen. Die Längen können als Skalare beliebig verschoben werden und die exponential Terme zusammengefasst werden.
+\begin{align}
+	\mathbf{wu} = |w||u|e^{-\theta_w \mathbf{e_{12}}}\mathbf{e_1}\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}}
+\end{align}
+\begin{align}
+	\mathbf{wu} = |w||u|e^{(\theta_u-\theta_w) \mathbf{e_{12}}}
+\end{align}
+der Term $\mathbf{u^{-1}w^{-1}}$ kann durch die selbe Methode zusammengefasst werden. 
+\begin{align}
+	\mathbf{u^{-1}w^{-1}} = \dfrac{1}{|w||u|}e^{(\theta_w-\theta_u) \mathbf{e_{12}}}
+\end{align}
+Dabei definieren wir den Winkel zwischen den Vektoren  $\mathbf{w}$ und $\mathbf{u}$ als $\theta = \theta_w - \theta_u$. Setzten wir nun unsere neuen Erkenntnisse in die Gleichung (\ref{rotGA}) ein.
+\begin{align}
+	\mathbf{v''} = |w||u|e^{-\theta \mathbf{e_{12}}} v \dfrac{1}{|w||u|}e^{\theta \mathbf{e_{12}}}
+\end{align}
+HIER DEFINITION/IST WICHTIGE FORMEL
+\begin{align}
+	\mathbf{v''} = e^{-\theta \mathbf{e_{12}}} v e^{\theta \mathbf{e_{12}}}
+\end{align}
+Wir wissen nun, dass das diese beidseitige Multiplikation die Länge von $\mathbf{v}$ nicht verändert, da sich die Längen von $\mathbf{w}$ und $\mathbf{u}$ kürzen. Betrachten wir nun den Effekt der Exponentialterme auf $\mathbf{v}$. Dabei Teilen wir den Vektor $\mathbf{v}$ auf in einen Anteil $\mathbf{v_\parallel}$, welcher auf der Ebene $\mathbf{e_{12}}$ liegt, und einen Anteil $\mathbf{v_\perp}$, welcher senkrecht zu der Ebene steht.
+\begin{align} \label{RotAufPerpPar}
+	\mathbf{v''} = e^{-\theta \mathbf{e_{12}}} (\mathbf{v_\perp + v_\parallel}) e^{\theta \mathbf{e_{12}}}
+\end{align}
+\begin{align}
+	\mathbf{v''} = e^{-\theta \mathbf{e_{12}}} \mathbf{v_\perp} e^{\theta \mathbf{e_{12}}} + e^{-\theta \mathbf{e_{12}}} \mathbf{v_\parallel} e^{\theta \mathbf{e_{12}}}
+\end{align}
+Auf eine allgemeine Herleitung wird hier zwar verzichtet, aber man kann zeigen, dass die Reihenfolge so vertauscht werden kann. Der Winkel wird dabei beim parallelen Term negiert.
+\begin{align}
+	\mathbf{v''} = \mathbf{v_\perp} e^{-\theta \mathbf{e_{12}}}  e^{\theta \mathbf{e_{12}}} +  \mathbf{v_\parallel} e^{-(-\theta) \mathbf{e_{12}}} e^{\theta \mathbf{e_{12}}}
+\end{align}
+\begin{align}
+	\mathbf{v''} = \mathbf{v_\perp} +  \mathbf{v_\parallel} e^{2\theta \mathbf{e_{12}}}
+\end{align}
+Man kann an dieser Gleichung sehen, dass nur der parallele Anteil des Vektors $\mathbf{v}$ auf der Ebene $\mathbf{e_{12}}$ um $2\theta$ gedreht wird. Der senkrechte Anteil bleibt gleich. Wichtig dabei zu sehen ist, dass nur der Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ von Bedeutung ist. Die Länge und Richtung der einzelnen Vektoren spielt keine Rolle. 
+\\BEISPIEL
+\begin{align}
+	\begin{split}
+		&\mathbf{v} = 1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}\quad\Rightarrow\quad \mathbf{v_\parallel} = 1\mathbf{e_1} + 2\mathbf{e_2}; \quad \mathbf{v_\perp} = 3\mathbf{e_3}\\ &\mathbf{wu} = 1e^{(-\pi/2) \mathbf{e_{12}}} = 1[\cos(-\pi/2)\mathbf{e_1}+\sin(-\pi/2)\mathbf{e_2}] = -\mathbf{e_2}; \\ &\mathbf{u^{-1}w^{-1}} = 1e^{(\pi/2) \mathbf{e_{12}}} = \mathbf{e_2}
+	\end{split}
+\end{align}
+\begin{align}
+	\begin{split}
+		\mathbf{v''} = &\mathbf{(wu)v(u^{-1}w^{-1})} \\ 
+		&-\mathbf{e_2} (1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}) \mathbf{e_2} \\
+		& -1\mathbf{e_2e_1e_2} - 2\mathbf{e_2e_2e_2} - 3\mathbf{e_2e_3e_2} \\
+		& 1\mathbf{e_2e_2e_1} - 2\mathbf{e_2} + 3\mathbf{e_2e_2e_3} \\
+		& 1\mathbf{e_1} - 2\mathbf{e_2} + 3\mathbf{e_3}
+	\end{split}
+\end{align}
+Man sieht, dass sich der Vektor $\mathbf{v_\parallel}$ sich um $2\cdot90^\circ$ gedreht hat und der Vektor $\mathbf{v_\perp}$ unverändert blieb.
\ No newline at end of file
diff --git a/buch/papers/clifford/9_KomplexeZahlen.tex b/buch/papers/clifford/9_KomplexeZahlen.tex
new file mode 100644
index 0000000..4dbab2c
--- /dev/null
+++ b/buch/papers/clifford/9_KomplexeZahlen.tex
@@ -0,0 +1,28 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{komplexe Zahlen}
+\rhead{komplexe Zahlen}
+Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Rotationen und Schwingungen gut beschreiben können. Nachdem vorherigen Kapitel überrascht es wahrscheinlich nicht viele, dass es möglich ist Komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der 2 Dimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbb{G}_2^+ \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl kann durch ein Skalar (Grade 0) und einem Bivektor (Grade 2) dargestellt werden. Als Abkürzung nehme ich die Bezeichnung $g_n \in  \mathbb{G}_2^+$.
+\begin{align}
+	a_0 + a_1 j \cong a_0 + a_1 e_{12} = g_n;\quad a_0, a_1 \in \mathbb{R}
+\end{align}
+oder in Polarform.
+\begin{align}
+	|r|e^{\theta j} \cong |r|e^{\theta e_{12}} = g_n; \quad r, \theta \in \mathbb{R}
+\end{align}
+Man beachte, dass wenn wir, wie bei den komplexen Zahlen, Elemente von $\mathbb{G}_2^+$ miteinander Multiplizieren, ist es nicht, wie im Kapitel Rotation bei der Formel (\ref{rotGA})beschrieben, eine Multiplikation von zwei $g_n$ mit einem Vektor. Im 2 dimensionalen bewirken beide Multiplikationen grundsätzlich das Gleiche (eine Drehstreckung), aber die Multiplikation von mehreren $g_n$ ist kommutativ, wie wir es von den komplexen zahlen kennen.
+\begin{align}
+	\begin{split}
+		&(a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}}) = (c + d \mathbf{e_{12}})(a + b \mathbf{e_{12}})\\
+		&(a + b \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2})(c + d \mathbf{e_{12}}) \not= (a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2})
+	\end{split}
+\end{align}
+Um später die Auswirkung der Quaternionen besser zu verstehen, möchte ich kurz darauf eingehen, was ein $g_n$ für eine Auswirkung auf einen Vektor hat.
+Wir kennen diesen Effekt schon von den komplexen Zahlen. Wenn eine komplexe Zahl $c_1=a+bj$ mit einer zweiten $c_2=c+dj$ multipliziert wird, dann kann man diese so aufteilen.
+\begin{align}
+	c = (a + bj)(c + dj) = c\cdot(a+bj) + dj\cdot(a+bj)
+\end{align}
+Wobei $c\cdot(a+bj)$ die jetzige komplexe Zahl $c_1$ um den Faktor $c$ steckt und $dj\cdot(a+bj)$ die um 90° im gegenuhrzeigersinn gedrehte Zahl $c_1$ um den Faktor $d$ streckt. Diese Anteile addiert ergeben, dann den um $c_2$ drehgestreckten Vektor $c_1$. Die wirklichen Vorteile der geometrischen Algebra werden sich aber erst bei den Quaternionen zeigen.
diff --git a/buch/papers/clifford/Makefile.inc b/buch/papers/clifford/Makefile.inc
index 7b941b3..8cdd02e 100644
--- a/buch/papers/clifford/Makefile.inc
+++ b/buch/papers/clifford/Makefile.inc
@@ -3,12 +3,18 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-dependencies-clifford =						\
+dependencies-clifford =							\
 	papers/clifford/packages.tex					\
 	papers/clifford/main.tex					\
-	papers/clifford/references.bib				\
-	papers/clifford/teil0.tex					\
-	papers/clifford/teil1.tex					\
-	papers/clifford/teil2.tex					\
-	papers/clifford/teil3.tex
-
+	papers/clifford/references.bib					\
+	papers/clifford/0_ElevatorPitch.tex				\
+	papers/clifford/1_Vektordarstellung.tex				\
+	papers/clifford/2_QuadratVektoren.tex				\
+	papers/clifford/3_MultiplikationVektoren.tex			\
+	papers/clifford/4_GeometrischesProdukt.tex			\
+	papers/clifford/5_PolareDarstellung.tex				\
+	papers/clifford/6_Dirac-Matrizen.tex				\
+	papers/clifford/7_Reflektion.tex				\
+	papers/clifford/8_Rotation.tex					\
+	papers/clifford/9_KomplexeZahlen.tex				\
+	papers/clifford/10_Quaternionen.tex
diff --git a/buch/papers/clifford/main.tex b/buch/papers/clifford/main.tex
index 5533c55..46d04bd 100644
--- a/buch/papers/clifford/main.tex
+++ b/buch/papers/clifford/main.tex
@@ -3,34 +3,23 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:clifford}}
-\lhead{Thema}
+\chapter{Clifford Algebra\label{chapter:clifford}}
+\lhead{Clifford Algebra}
 \begin{refsection}
-\chapterauthor{Hans Muster}
+\chapterauthor{Thierry Schwaller, Marius Baumann}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
 
-\input{papers/clifford/teil0.tex}
-\input{papers/clifford/teil1.tex}
-\input{papers/clifford/teil2.tex}
-\input{papers/clifford/teil3.tex}
+\input{papers/clifford/0_ElevatorPitch.tex}
+\input{papers/clifford/1_Vektordarstellung.tex}
+\input{papers/clifford/2_QuadratVektoren.tex}
+\input{papers/clifford/3_MultiplikationVektoren.tex}
+\input{papers/clifford/4_GeometrischesProdukt.tex}
+\input{papers/clifford/5_PolareDarstellung.tex}
+\input{papers/clifford/6_Dirac-Matrizen.tex}
+\input{papers/clifford/7_Reflektion.tex}
+\input{papers/clifford/8_Rotation.tex}
+\input{papers/clifford/9_KomplexeZahlen.tex}
+\input{papers/clifford/10_Quaternionen.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/clifford/packages.tex b/buch/papers/clifford/packages.tex
index 8abcef1..8fb4bd9 100644
--- a/buch/papers/clifford/packages.tex
+++ b/buch/papers/clifford/packages.tex
@@ -7,4 +7,3 @@
 % if your paper needs special packages, add package commands as in the
 % following example
 %\usepackage{packagename}
-
diff --git a/buch/papers/clifford/papers/clifford/teil0.tex b/buch/papers/clifford/papers/clifford/teil0.tex
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/clifford/papers/clifford/teil0.tex
diff --git a/buch/papers/clifford/teil0.tex b/buch/papers/clifford/teil0.tex
deleted file mode 100644
index ac943f4..0000000
--- a/buch/papers/clifford/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{clifford:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{clifford:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/clifford/teil1.tex b/buch/papers/clifford/teil1.tex
deleted file mode 100644
index 0674afb..0000000
--- a/buch/papers/clifford/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{clifford:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{clifford:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{clifford:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{clifford:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{clifford:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/clifford/teil2.tex b/buch/papers/clifford/teil2.tex
deleted file mode 100644
index bbcefb0..0000000
--- a/buch/papers/clifford/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{clifford:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{clifford:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/clifford/teil3.tex b/buch/papers/clifford/teil3.tex
deleted file mode 100644
index f50d42d..0000000
--- a/buch/papers/clifford/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{clifford:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{clifford:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/ifs/images/FIC.pdf b/buch/papers/ifs/images/FIC.pdf
new file mode 100644
index 0000000..1c76dfe
--- /dev/null
+++ b/buch/papers/ifs/images/FIC.pdf
@@ -0,0 +1,2003 @@
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+  /UnderlineThickness get Ts mul /Tt exch def
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+} bd
+/OLE {
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+  cf findfont cs scalefont dup
+  /FontMatrix get 0 get /Ts exch def /FontInfo get dup
+  /UnderlinePosition get Ts mul /To exch def
+  /UnderlineThickness get Ts mul /Tt exch def
+  ux uy To add cs add moveto Tcx uy To add cs add lineto
+  Tt setlinewidth stroke
+  grestore
+} bd
+/SOE {
+  /Tcx currentpoint pop def
+  gsave
+  newpath
+  cf findfont cs scalefont dup
+  /FontMatrix get 0 get /Ts exch def /FontInfo get dup
+  /UnderlinePosition get Ts mul /To exch def
+  /UnderlineThickness get Ts mul /Tt exch def
+  ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto
+  Tt setlinewidth stroke
+  grestore
+} bd
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+currentpoint
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+} bd
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+} ifelse
+} if
+} ifelse
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+} if
+} if
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+} if
+} bd
+/RE { % /NewFontName [NewEncodingArray] /FontName RE -
+  findfont dup length dict begin
+  {
+    1 index /FID ne
+    {def} {pop pop} ifelse
+  } forall
+  /Encoding exch def
+  /FontName 1 index def
+  currentdict definefont pop
+  end
+} bind def
+%%EndResource
+%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0
+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
+/BeginEPSF { %def
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+/dict_count countdictstack def % Count objects on dict stack
+/op_count count 1 sub def      % Count objects on operand stack
+userdict begin                 % Push userdict on dict stack
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+0 setgray 0 setlinecap         % Prepare graphics state
+1 setlinewidth 0 setlinejoin
+10 setmiterlimit [ ] 0 setdash newpath
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+{pop languagelevel             % set strokeadjust and
+1 ne                           % overprint to their defaults.
+{false setstrokeadjust false setoverprint
+} if
+} if
+} bd
+/EndEPSF { %def
+count op_count sub {pop} repeat            % Clean up stacks
+countdictstack dict_count sub {end} repeat
+b4_Inc_state restore
+} bd
+%%EndResource
+%%EndProlog
+%%Page: 1 1
+%%PageBoundingBox: 0 0 1152 562
+%%BeginPageSetup
+[1 0 0 -1 0 562] CT
+%%EndPageSetup
+GS
+[0.6 0 0 0.6 0 0.39996] CT
+1 GC
+N
+0 0 1920 936 re
+f
+GR
+GS
+[0.48 0 0 0.48 0 112.71998] CT
+[1 0 0 1 0 0] CT
+N
+0 -234 M
+2400 -234 L
+2400 936 L
+0 936 L
+0 -234 L
+cp
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+GS
+0 0 translate
+1920 936 scale
+%AXGBeginBitmap: java.awt.image.BufferedImage
+{{
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+/DeviceRGB setcolorspace
+<<
+  /ImageType 1
+  /Decode [0 1 0 1 0 1]
+  /DataSource Data
+  /Height 936
+  /ImageMatrix [1920 0 0 936 0 0]
+  /Width 1920
+  /BitsPerComponent 8
+>> image
+} stopped {handleerror} if
+  RawData flushfile
+} exec
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+{{
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+<<
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+  /DataSource Data
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+  /Width 480
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+} stopped {handleerror} if
+  RawData flushfile
+} exec
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+>> image
+} stopped {handleerror} if
+  RawData flushfile
+} exec
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+} stopped {handleerror} if
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+<<
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+  /Decode [0 1 0 1 0 1]
+  /DataSource Data
+  /Height 234
+  /ImageMatrix [1920 0 0 234 0 0]
+  /Width 1920
+  /BitsPerComponent 8
+>> image
+} stopped {handleerror} if
+  RawData flushfile
+} exec
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+} stopped {handleerror} if
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+} exec
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diff --git a/buch/papers/ifs/images/original.png b/buch/papers/ifs/images/original.png
new file mode 100644
index 0000000..2932af1
--- /dev/null
+++ b/buch/papers/ifs/images/original.png
diff --git a/buch/papers/ifs/images/rapperswil.png b/buch/papers/ifs/images/rapperswil.png
new file mode 100644
index 0000000..62a3564
--- /dev/null
+++ b/buch/papers/ifs/images/rapperswil.png
diff --git a/buch/papers/ifs/images/rapperswil001.PNG b/buch/papers/ifs/images/rapperswil001.PNG
new file mode 100644
index 0000000..7015fe4
--- /dev/null
+++ b/buch/papers/ifs/images/rapperswil001.PNG
diff --git a/buch/papers/ifs/images/rapperswil01.PNG b/buch/papers/ifs/images/rapperswil01.PNG
new file mode 100644
index 0000000..067b08a
--- /dev/null
+++ b/buch/papers/ifs/images/rapperswil01.PNG
diff --git a/buch/papers/ifs/images/rapperswil04.PNG b/buch/papers/ifs/images/rapperswil04.PNG
new file mode 100644
index 0000000..7e946fa
--- /dev/null
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diff --git a/buch/papers/ifs/images/sierpinski.PNG b/buch/papers/ifs/images/sierpinski.PNG
new file mode 100644
index 0000000..1e57bf1
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new file mode 100644
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+See the LaTeX manual or LaTeX Companion for explanation.
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+ ...                                              
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+ ...                                              
+                                                  
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+Missing character: There is no u in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no O in font nullfont!
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+Missing character: There is no t in font nullfont!
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+Missing character: There is no . in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no S in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no V in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no w in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no B in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no h in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no z in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no w in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no . in font nullfont!
+
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+Missing character: There is no B in font nullfont!
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+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no S in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no S in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no z in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no D in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no m in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no w in font nullfont!
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+
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+\/cmr/m/n/10 ^^?o
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+
+
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+[]\/cmr/m/n/10 A
+ []
+
+
+Overfull \hbox (5.00002pt too wide) in paragraph at lines 21--26
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+Missing character: There is no B in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no f in font nullfont!
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+Missing character: There is no F in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no h in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no n in font nullfont!
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+
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+\/cmr/m/n/10 ^^?u
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+
+
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+[]\/cmr/m/n/10 U
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+
+(./teil0.tex
+! Undefined control sequence.
+l.6 \section
+            {Teil 0\label{ifs:section:teil0}}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no T in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no 0 in font nullfont!
+! Undefined control sequence.
+l.7 \rhead
+          {Teil 0}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no T in font nullfont!
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+Missing character: There is no 0 in font nullfont!
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+Missing character: There is no t in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no d in font nullfont!
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+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no [ in font nullfont!
+\citation{ifs:bibtex}
+! Undefined control sequence.
+<write> ...: Citation `\@citeb ' on page \thepage 
+                                                  \space undefined\on@line .
+l.10 erat, sed diam voluptua \cite{ifs:bibtex}
+                                              .
+The control sequence at the end of the top line
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+! Undefined control sequence.
+l.34 \subsection
+                {De finibus bonorum et malorum
+The control sequence at the end of the top line
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+! Undefined control sequence.
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+                                                 {000tempmlate:equation1}.
+The control sequence at the end of the top line
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+! Undefined control sequence.
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+l.43 \ref{ifs:section:loesung}
+                              .
+The control sequence at the end of the top line
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+
+
+LaTeX Warning: Reference `ifs:section:loesung' on page  undefined on input line
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+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no N in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no N in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no U in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no ? in font nullfont!
+Missing character: There is no Q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
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+[]
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+
+! Undefined control sequence.
+l.24 \subsection
+                {De finibus bonorum et malorum
+The control sequence at the end of the top line
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+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
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+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no I in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 24--39
+[]
+ []
+
+) (./teil3.tex
+! Undefined control sequence.
+l.6 \section
+            {Teil 3
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no T in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no 3 in font nullfont!
+! Undefined control sequence.
+l.8 \rhead
+          {Teil 3}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no T in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no 3 in font nullfont!
+Missing character: There is no S in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no N in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no N in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no U in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no m in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no ? in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23
+[]
+ []
+
+! Undefined control sequence.
+l.24 \subsection
+                {De finibus bonorum et malorum
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no N in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no T in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no r in font nullfont!
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+
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+)
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+                       [heading=subbibliography]
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+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
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+
+! LaTeX Error: \begin{document} ended by \end{refsection}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.36 \end{refsection}
+                     
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+)
+! Emergency stop.
+<*> main.tex
+            
+*** (job aborted, no legal \end found)
+
+ 
+Here is how much of TeX's memory you used:
+ 37 strings out of 492609
+ 427 string characters out of 6132818
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+ 1141 hyphenation exceptions out of 8191
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+!  ==> Fatal error occurred, no output PDF file produced!
diff --git a/buch/papers/ifs/main.tex b/buch/papers/ifs/main.tex
index 8d70951..cceaf87 100644
--- a/buch/papers/ifs/main.tex
+++ b/buch/papers/ifs/main.tex
@@ -3,34 +3,16 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:ifs}}
-\lhead{Thema}
+\chapter{Iterierte Funktionsschemata\label{chapter:ifs}}
+\lhead{Iterierte Funktionschemata und ihre Anwendungen}
 \begin{refsection}
-\chapterauthor{Hans Muster}
-
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
+\chapterauthor{Alain Keller}
 
 \input{papers/ifs/teil0.tex}
 \input{papers/ifs/teil1.tex}
 \input{papers/ifs/teil2.tex}
 \input{papers/ifs/teil3.tex}
 
+
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib
index 716857f..817c5a4 100644
--- a/buch/papers/ifs/references.bib
+++ b/buch/papers/ifs/references.bib
@@ -4,32 +4,59 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{ifs:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
+@online{ifs:chaos,
+	title = {Chaosspiel},
+	url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel},
+	date = {20201-06-13},
+	year = {2021},
+	month = {6},
+	day = {13}
 }
 
-@book{ifs:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
+@online{ifs:barnsleyfern,
+	title = {Barnsley fern},
+	url = {https://en.wikipedia.org/wiki/Barnsley_fern},
+	date = {20201-06-13},
+	year = {2021},
+	month = {6},
+	day = {13}
+}
+@book{ifs:fractal-geometry,
+	title = {Fractal Geometry},
+	author = {Kenneth Falconer},
+	publisher = {John Wiley \& Sons},
+	year = {1900},
+	isbn = {0-471-92287-0},
+}
+
+@Inbook{ifs:Rousseau2012,
+	author= {Rousseau, Christiane
+	and Saint-Aubin, Yvan
+	and Stern, Manfred},
+	title={Bildkompression: Iterierte Funktionensysteme},
+	bookTitle={Mathematik und Technologie},
+	year={2012},
+	publisher={Springer Berlin Heidelberg},
+	address={Berlin, Heidelberg},
+	pages={341--386},
+	abstract={Dieses Kapitel kann in ein bis zwei Wochen Vorlesungen behandelt werden. Steht nur eine Woche zur Verfügung, dann können Sie kurz die Einführung behandeln (Abschnitt 11.1) und anschlie{\ss}end ausf{\"u}hrlich den Begriff des Attraktors eines iterierten Funktionensystems betrachten (Abschnitt 11.3), wobei Sie sich auf das Sierpi{\'{n}}ski- Dreieck (Beispiel 11.5) konzentrieren. Beweisen Sie den Satz {\"u}ber die Konstruktion von affinen Transformationen, die drei Punkte der Ebene auf drei Punkte der Ebene abbilden und diskutieren Sie die speziellen affinen Transformationen, die h{\"a}ufig bei iterierten Funktionensystemen verwendet werden (Abschnitt 11.2).},
+	isbn={978-3-642-30092-9},
+	doi={10.1007/978-3-642-30092-9_11},
+	url={https://doi.org/10.1007/978-3-642-30092-9_11}
 }
 
-@article{ifs:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@article{ifs:pifs,
+	title = {Applications of Partitioned Iterated Function Systems in Image and Video Compression},
+	journal = {Journal of Visual Communication and Image Representation},
+	volume = 7,
+	number = {2},
+	pages = {144-154},
+	year = 1996,
+	issn = {1047-3203},
+	doi = {https://doi.org/10.1006/jvci.1996.0014},
+	url = {https://www.sciencedirect.com/science/article/pii/S1047320396900140},
+	author = {Guojun Lu and Toon Lin Yew},
+	abstract = {Iterated function systems (IFS) have been used to compress image data. Because of difficulty in finding IFS in natural images, a technique based on partitioned IFS (PIFS) has been proposed for image compression. In this technique, an image to be compressed is divided into nonoverlapping blocks. For each block an affine transformation is found in the image. This set of affine transformations (called PIFS) corresponds to a unique image. In the simplest case, images are partitioned into fixed size blocks. In this paper, we investigate image and video compression techniques using variable block sizes based on the quadtree partition. One property of images generated using PIFS is scalability: they have fine detail in any scale. We exploit this property to reduce required compression time and improve compression performance. There are large amounts of temporal redundancy between fames of a video sequence. We describe a method to remove temporal redundancies effectively using a quadtree partitioning technique. We have implemented the above schemes to compress image and video sequences and will report our experimental results.}
 }
 
+
diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex
index b605bfe..833748c 100644
--- a/buch/papers/ifs/teil0.tex
+++ b/buch/papers/ifs/teil0.tex
@@ -3,20 +3,10 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 0\label{ifs:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{ifs:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
+\section{Einleitung \label{ifs:section:teil0}}
+\rhead{Was ist ein Iteriertes Funktionsschema}
+Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden.
+In der Regel sind diese Bilder Fraktale.
+Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen.
 
 
diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex
index c824cb4..a75b529 100644
--- a/buch/papers/ifs/teil1.tex
+++ b/buch/papers/ifs/teil1.tex
@@ -3,53 +3,132 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 1
+\section{Fraktale
 \label{ifs:section:teil1}}
 \rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{ifs:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{ifs:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{ifs:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{ifs:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
+Bevor wir die IFS ansehen, schauen wir uns Fraktale genauer an.
 
 
+Über die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig. 
+In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt.
+Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: 
+\begin{enumerate}
+	\item $F$ hat eine unendlich feine Struktur
+	\item $F$ kann nicht mit der klassischen Geometrie beschrieben werden.
+	\item Oftmals hat $F$ eine Form von Selbstähnlichkeit.
+	\item Die 'fraktale Dimension' ist grösser als die topologische Dimension
+	\item Viele Fraktale lassen sich auf eine simple Art definieren. Es genügen zum Beispiel nur wenige Funktionen, welche rekursiv ausgeführt werden, um ein Fraktal zu definieren.  
+\end{enumerate}
+\subsection{Koch Kurve
+	\label{ifs:subsection:lilkoch}}
+Diese Eigenschaften möchten wir nun am Beispiel der Koch Kurve näher anschauen.
+In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht aus lauter kleineren Kopien von sich selber. 
+Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt.
+Gestartet wird mit einer einzelnen Strecke der Länge $a$.
+Diese wird in ersten Schritt durch vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt.
+In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. 
+Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt.
+Die Kurve besteht also aus vier kleineren Kopien der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist.
+Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind.
+
+
+\begin{figure}
+	\centering
+	\includegraphics{papers/ifs/images/koch8}
+	\caption{Koch Kurve}
+	\label{ifs:kochkurve8}
+\end{figure}
+
+\begin{figure}
+	\centering
+	\subfigure[]{
+		\label{ifs:kochconsta}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/koch0}}
+	\subfigure[]{
+		\label{ifs:kochconstb}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/koch1}} 
+	\subfigure[]{
+		\label{kochconstc}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/koch2}}
+	\caption{(a) Start (b) 1. Iteration (c) 2. Iteration}
+	\label{ifs:kochconst}
+\end{figure}
+
+Die resultierende Kurve hat ein paar interessante Eigenschaften.
+Die Länge der Kurve der jeweiligen Iteration lässt sich mit 
+\begin{align*}
+	l_0 = a ,\quad l_1 = a  \frac{4}{3} ,\quad l_2 = a  \left( \frac{4}{3}\right)^2 , \quad \cdots , \quad
+	l_n = a \cdot \left( \frac{4}{3}\right)^n \quad
+	\Rightarrow \quad
+	\lim_{n\to\infty} a  \left( \frac{4}{3}\right)^n = \infty
+\end{align*}
+berechnen.
+In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert. 
+
+
+Die Fläche unter der Kurve lässt sich folgendermassen berechnen
+\begin{align*}
+	A_0 &= 0 \\
+	A_1 &= \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\
+	A_2 &= A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ 
+	A_3 &= A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1.
+\end{align*}
+Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist.
+Die Gesamtfläche ist daher gegeben durch die konvergierende geometrische Reihe,
+\begin{align*}
+	A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n =  a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\
+\end{align*}
+mit dem Grenzwert
+\begin{align*}
+	\lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2. 
+\end{align*}
+Wie wir sehen ist die Koch-Kurve ein Objekt mit endlicher Fläche, aber unendlichem Umfang.
+
+
+Zu guter Letzt bestimmen wir die Dimension der Kurve. 
+Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern.
+Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten.
+In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}.
+Die Ähnlichkeits-Dimension $D$ ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$
+  
+\begin{align*}
+	D = - \frac{\log N}{\log \epsilon }.
+\end{align*}
+Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen.
+Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}.
+Somit hat das Dreieck die Dimension $D = 2$.
+Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$.
+Ihre  Ähnlichkeits-Dimension ist somit
+\begin{align*}
+	D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619.
+\end{align*}
+Wie wir nun sehen besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen. 
+Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten.
+\begin{figure}
+	\centering
+	\begin{tikzpicture}
+		
+		% draw the background
+		\draw [line width=1.5pt, fill=gray!2] (0,0) -- (60:4) -- (4,0) -- cycle;
+		
+		\coordinate[label=left:$A$]  (A) at (0,0);
+		\coordinate[label=right:$B$] (B) at (4,0);
+		\coordinate[label=above:$C$] (C) at (2,3.464);
+		
+		\coordinate[label=below:$l$](c) at ($ (A)!.5!(B) $);
+		\coordinate[label=left:$l$] (b) at ($ (A)!.5!(C) $);
+		\coordinate[label=right:$l$](a) at ($ (B)!.5!(C) $);
+		
+		\coordinate[label=below:$l/2$](d) at ($ (b)!.5!(a)$);
+		
+		% the triangle
+		\draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle;
+		\draw [line width=0.5pt] (a) -- (b);
+		\draw [line width=0.5pt] (a) -- (c);
+		\draw [line width=0.5pt] (c) -- (b);
+		
+	\end{tikzpicture}
+	\caption{Selbstähnlichkeit eines gleichseitigen Dreiecks}
+	\label{ifs:trinagle}
+\end{figure}
+
diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex
index bfd1684..fd10634 100644
--- a/buch/papers/ifs/teil2.tex
+++ b/buch/papers/ifs/teil2.tex
@@ -3,38 +3,254 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 2 
+\section{Fraktale mit IFS 
 \label{ifs:section:teil2}}
 \rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{ifs:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
+Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann.
+Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck.
+\begin{figure}
+	\centering
+	\includegraphics[width=0.5\textwidth]{papers/ifs/images/sierpinski}
+	\caption{Sierpinski-Dreieck}
+	\label{ifs:sierpinski10}
+\end{figure}
+Es besteht aus drei kleineren Kopien von sich selbst.
+Es ist also ein Selbstähnliches Gebilde.
+Diese Eigenschaft wollen wir uns zunutze machen.
 
 
+Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$.
+Ausserdem bestimmen wir drei Funktionen
+\begin{align*}
+	f_1(x,y)
+	= 
+	\begin{pmatrix}
+		\frac{1}{2} & 0 \\
+		0 & \frac{1}{2} \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	,\quad
+	f_2(x,y)
+	= 
+	\begin{pmatrix}
+		\frac{1}{2} & 0 \\
+		0 & \frac{1}{2} \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	+
+	\begin{pmatrix}
+		\frac{1}{2} \\
+		0
+	\end{pmatrix}
+	, \quad
+	f_3(x,y)
+	= 
+	\begin{pmatrix}
+		\frac{1}{2} & 0 \\
+		0 & \frac{1}{2} \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	+
+	\begin{pmatrix}
+		\frac{1}{4} \\
+		\frac{1}{2}
+	\end{pmatrix},
+\end{align*}
+welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet.
+$f_1$ bildet das Dreieck auf das Teilstück unten links ab, $f_2$ auf das Teilstück unten rechts und $f_3$ auf das obere Teilstück.
+Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an
+\begin{align*}
+	X = \bigcup\limits_{i = 1}^{3} f_i(X),
+\end{align*}
+entsteht also wieder ein Sierpinski-Dreieck.
+Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck.
+\begin{figure}	
+	\centering
+	\subfigure[]{
+		\label{ifs:sierpconsta}
+		\includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski1}}
+	\subfigure[]{
+		\label{ifs:sierpconstb}
+		\includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski2}} 
+	\subfigure[]{
+		\label{ifs:sierpconstc}
+		\includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski3}}
+	\subfigure[]{
+		\label{ifs:sierpconstd}
+		\includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski6}}
+	\caption{Konstruktion eines Sierpinski-Dreiecks mit einem Schwarzen Quadrat als Start\\
+		(a) 1. Iteration (b) 2. Iteration (c) 3. Iteration (d) 5. Iteration}
+	\label{ifs:sierpconst}
+\end{figure}
+Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird.
+Der Abstand zum Original wird immer kleiner, und konvergiert gegen null.
+
+\subsection{Iterierte Funktionensysteme
+\label{ifs:subsection:IteratedFunktionensysteme}}
+In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern.
+
+
+$S_1,\dots,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt
+\begin{align}
+	|S_i(x) - S_i(y)| \leq c_i|x - y|
+\end{align}
+für jedes i mit einem $c_i < 1$.
+Der Banachsche Fixpunktsatz besagt, dass für solche Kontraktionen ein Eindeutiges $A$ existiert, für das $S(A) = A$ gilt.
+Den Beweis kann man in \cite{ifs:Rousseau2012} nachlesen.
+Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeutige kompakte Menge $F$ für die gilt
+\begin{equation}
+	F = \bigcup\limits_{i = 1}^{m} S_i(F).
+\end{equation}
+Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leere Menge
+\begin{equation}
+	S(E) = \bigcup\limits_{i = 1}^m S_i(E).
+	\label{ifs:transformation}
+\end{equation}
+Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt
+\begin{equation}
+	F = \bigcap\limits_{k = 1}^{\infty} S^k(E).
+	\label{ifs:ifsForm}
+\end{equation}
+In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert.
+Diese Menge ist auch als Attraktor eines IFS bekannt.
+Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. 
+
+\subsection{Beispiel: Barnsley-Farn}
+Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann.
+Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben.
+Die vier affinen Transformationen
+\begin{align}
+	& {S_1(x,y)}
+	= 
+	\begin{pmatrix}
+		0 & 0 \\
+		0 & 0.16 \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix}, \quad &
+	{S_2(x,y)}
+	&= 
+	\begin{pmatrix}
+		0.85 & 0.04 \\
+		-0.04 & 0.85 \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	+
+	\begin{pmatrix}
+		0 \\
+		1.6
+	\end{pmatrix}\\
+	& {S_3(x,y)}
+	= 
+	\begin{pmatrix}
+		0.2 & -0.26 \\
+		0.23 & 0.22 \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	+
+	\begin{pmatrix}
+		0 \\
+		1.6
+	\end{pmatrix}, \quad &
+	{S_4(x,y)}
+	&= 
+	\begin{pmatrix}
+		-0.15 & 0.28 \\
+		0.26 & 0.24 \\
+	\end{pmatrix}
+	\begin{pmatrix}
+		x\\
+		y\\
+	\end{pmatrix} 
+	+
+	\begin{pmatrix}
+		0 \\
+		0.44
+	\end{pmatrix}\\
+	\label{ifs:farnFormel}
+\end{align}
+, welche für die konstruktion des Farns benötigt werden sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt.
+Das gesamte Farnblatt ist in der schwarzen Box.
+Auf diese werden die Transformationen angewendet
+$S_1$ erstellt den Stiel des Farnblattes (rot).
+Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab.
+$S_2$ (grün) erstellt den Hauptteil des Farnes. 
+Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$.
+$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab.
+$S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab.  
+\subsection{Erzeugung eines Bildes zu einem IFS}
+Es gibt zwei verschiedene Methoden um das Bild zu einem IFS zu erzeugen.
+Die erste Methode ist wahrscheinlich die intuitivste. 
+Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab.
+Das neue Bild, dass entsteht, ist die nächste Iterierte.
+Dieses wird wieder mit den Transformationen abgebildet.
+Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind.
+
+Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht.
+In Abbildung \ref{ifs:sierpinski10} ist die zehnte Iterierte zu sehen.
+Weitere Iterationen hätten in dieser Darstellungsgrösse kaum mehr einen Unterschied gemacht.
+
+
+Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}. 
+Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden.
+Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist die erste Methode ziemlich rechenintensiv.
+Beim Chaosspiel werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt.
+Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein.
+Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet.
+Da, wie wir beim Barnsley-Farn gut sehen, nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet.
+Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es um die resultierende Teilabbildung darzustellen.
+Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt.
+Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$.
+
+In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet.
+Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn nicht so gut abgebildet wird.
+
+Am besten sieht man den Effekt einer schlechten Gewichtung in Abbildung \ref{ifs:farnrightWeight}.
+Hier wurde $S_4$, welches für das rechte untere Teilblatt zuständig ist, mit nur $1\%$ statt $7\%$ gewichtet.
+Man sieht, wie sich der Mangel an Punkten auf die anderen Abbildungen das Farnblattes auswirkt.
+In jeder Kopie des ganzen Farns fehlen die Punkte für dieses rechte untere Teilblatt.
+
+
+
+\begin{figure}	
+	\centering
+	\makebox[\textwidth][c]{
+		\includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}}
+	\caption{Barnsley-Farn}
+	\label{ifs:farn}
+\end{figure}
+\begin{figure}
+	\centering
+	\includegraphics[width=\textwidth]{papers/ifs/images/farncolor2}
+	\caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben}
+	\label{ifs:farncolor}
+\end{figure}
+
+\begin{figure}	
+	\centering
+	\subfigure[]{
+		\label{ifs:farnNoWeight}
+		\includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}}
+	\subfigure[]{
+		\label{ifs:farnrightWeight}
+		\includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}}
+	\caption{(a) Chaosspiel ohne Gewichtung (b) $S_4$ zu wenig gewichtet}
+	\label{ifs:farnweight}
+\end{figure}
diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex
index 23fabbc..78fb935 100644
--- a/buch/papers/ifs/teil3.tex
+++ b/buch/papers/ifs/teil3.tex
@@ -3,38 +3,183 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 3
+\section{Fraktale Bildkomprimierung
 \label{ifs:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
+\rhead{Fraktale Bildkomprimierung}
+Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren.
+Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat.
+Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat.
+In diesem Unterkapitel wollen wir eine Methode dafür anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist.
+
+Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind.
+Ein IFS, wie wir es in \ref{ifs:subsection:IteratedFunktionensysteme} definiert haben, wird uns also nicht weiter helfen.
+Die Lösung dazu sind Partitionierte IFS (PIFS) \cite{ifs:pifs}.
+In \ref{ifs:transformation} wurde definiert, dass die Kontraktionen $S_i$ bei IFS auf die gesamte Menge $E$ angewendet werden.
+Bei einem PIFS wird der Attraktor in disjunkte Teilmengen aufgeteilt. 
+Für jede dieser Teilmengen $R_i$ braucht es dann eine grössere Teilmenge, welche mit einer affinen Transformation eine zu $R_i$ ähnliche Menge bildet.
+Wir müssen nicht mehr Ähnlichkeiten zum ganzen Bild finden, sondern zwischen Teilen des Bildes.
+Doch wie finden wir das PIFS, welches das Bild als Attraktor hat?
+
+\subsection{das Kompressionsverfahren
 \label{ifs:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
+Wir beschränken das Verfahren für Graustufenbilder. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert.
+Ein Graustufenbild kann man als Pixelraster mit einer x und y Achse verstehen.
+Jedem dieser Pixel wird ein Grauwert zugeordnet.
+Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist
+\begin{align*}
+	z = f(x,y).
+\end{align*} 
+
+Wir suchen ein PIFS welches das zu komprimierende Bild als Attraktor hat.
+In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$
+Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. 
+Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist.
+Zwei Beispiele wie solche Domain-, und Range-Block Paare aussehen können, sehen wir in Abbildung \ref{ifs:FIC}
+
+\subsubsection{Finden des ähnlichsten $D_j$}
+Zuerst brauchen wir die Transformation 
+\begin{align*}
+	T_i(x,y,z) = 
+	\begin{pmatrix}
+		a_i & b_i & 0 \\
+		c_i & d_i & 0 \\
+		0 & 0 & s_i
+	\end{pmatrix}
+	\begin{pmatrix}
+		x \\
+		y \\
+		z
+	\end{pmatrix}
+	+
+	\begin{pmatrix}
+		\alpha_i \\
+		\beta_i \\
+		g_i
+	\end{pmatrix}
+\end{align*}
+um ein Element aus $D$ auf ein Element von $R$ Abzubilden.
+Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung
+\begin{align}
+	t_i(x,y) = 	
+	\begin{pmatrix}
+		a_i & b_i \\
+		c_i & d_i
+	\end{pmatrix}
+	\begin{pmatrix}
+		x \\
+		y
+	\end{pmatrix}
+	+
+	\begin{pmatrix}
+		\alpha_i \\
+		\beta_i
+	\end{pmatrix}.
+\label{ifs:affTrans}
+\end{align}
+Da wir mit Pixeln arbeiten, ist die Auswahl der möglichen Abbildungen begrenzt.
+Wir sind auf folgende acht Abbildungen beschränkt:
+\begin{itemize}
+	\item Identische Transformation, keine Änderung
+	\item Drehung um 90, 180 oder 270 Grad.
+	\item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen.
+\end{itemize}
+Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren.
+Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen.
+
+
+Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion
+\begin{align*}
+	z' = s_i z + g_i.
+\end{align*}
+Für die Bestimmung dieser Parameter führen wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein.
+$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$.
+
+Wir suchen $s_i$ und $g_i$ so das
+\begin{align*}
+	f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}.
+\end{align*}
+Die Parameter lassen sich mit
+\begin{align*}	
+	s = \frac{\operatorname{cov}(f_{R_i}), f(\tilde{f_{R_i}}))}{\operatorname{var}(\tilde{f_{R_i}})} \\
+	g = E(f_{R_i}) - s E(f(\tilde{f_{R_i}}))
+\end{align*}
+berechnen.
+Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt.
+Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Abstand
+\begin{align*}
+	e = d(f_{R_i}, \bar{f_{R_i}}).
+\end{align*}
+Dieser Abstand sollte so klein wie möglich sein.
+
+Wir bestimmen die Parameter $s$ und $g$ für jede der acht möglichen affinen Abbildungen und das mit jedem Domain-Block.
+Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist die beste.
+
+Diese Schritte führen wir für jeden Range-Block $R_i$ aus.
+Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden.
+
+\begin{figure}	
+	\centering
+	\includegraphics[width=\textwidth]{papers/ifs/images/FIC}
+	\caption{Domain-, und Range-Block Paare in Grün und Rot}
+	\label{ifs:FIC}
+\end{figure}
+
+\subsubsection{Rekonstruktion des Bildes}
+Mit den gefundenen Abbildungen lässt sich das Bild generieren.
+Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge.
+In unserem Fall ist dieses ein Bild  $f_0$ derselben Grösse.
+Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$.
+Dies wird verkürzt als Operator $W$ geschrieben.
+So erhalten wir ein neues Bild $f_1 = W(f_0)$.
+Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen Unterschied feststellen. Die Iteration hat nun ihren Attraktor, das Bild, erreicht.
+
+\subsubsection{Farbbilder}
+Dieses Verfahren mit Graustufenbilder lässt sich ganz einfach auf Farbbilder erweitern.
+Jeder Pixel eines Farbbildes besteht aus einem Rot, Grün und Blauwert (RGB).
+Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein Farbwert benutzt, erhält man drei Bilder, welche wie ein Graustufenbild sind.
+Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen. 
+
+\subsubsection{Performance des Verfahren}
+Dieser Grundalgorithmus der fraktalen Bildkompression ist recht langsam und skaliert auch schlecht für grössere Bilder.
+Dies resultiert aus eigenen Experimenten.
+Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nicht so schnell wie zum Beispiel das JPEG-Verfahren.
+Es wurden bessere Algorithmen der fraktalen Bildkompression entwickelt, doch auch diese können, vor allem in der Laufzeit, noch nicht mit herkömmlichen Komprimierungsverfahren mithalten.
 
+\subsection{Beispiel}
+Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert.
+Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren.
+Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$.
+Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht.
+Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}.
+Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}.
+Es kann jedoch ein beliebiges Startbild
+Nun lassen wir das PIFS laufen.
+Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar.
+Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden.
+\begin{figure}	
+	\centering
+	\includegraphics[width=0.4\textwidth]{papers/ifs/images/original}
+	\caption{Original Bild von Rapperswil}
+	\label{ifs:original}
+\end{figure}
+\begin{figure}
+	\centering
+	\includegraphics[width=0.4\textwidth]{papers/ifs/images/rapperswil}
+	\caption{Startbild}
+	\label{ifs:bild0}
+\end{figure}
 
+\begin{figure}
+	\centering
+	\subfigure[]{
+		\label{ifs:rappirecoa}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil01}}
+	\subfigure[]{
+		\label{ifs:rappirecob}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil001}} 
+	\subfigure[]{
+		\label{ifs:rappirecoc}
+		\includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil04}}
+	\caption{(a) 1. Iteration (b) 2. Iteration (c) 5. Iteration}
+		\label{ifs:rappireco}
+\end{figure}
diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc
index 7c6e70d..b6a76c1 100644
--- a/buch/papers/punktgruppen/Makefile.inc
+++ b/buch/papers/punktgruppen/Makefile.inc
@@ -3,12 +3,12 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-dependencies-punktgruppen =						\
-	papers/punktgruppen/packages.tex					\
-	papers/punktgruppen/main.tex					\
-	papers/punktgruppen/references.bib				\
-	papers/punktgruppen/teil0.tex					\
-	papers/punktgruppen/teil1.tex					\
-	papers/punktgruppen/teil2.tex					\
-	papers/punktgruppen/teil3.tex
+dependencies-punktgruppen =	\
+	papers/punktgruppen/packages.tex \
+	papers/punktgruppen/main.tex \
+	papers/punktgruppen/intro.tex \
+	papers/punktgruppen/symmetry.tex \
+	papers/punktgruppen/crystals.tex \
+	papers/punktgruppen/piezo.tex \
+	papers/punktgruppen/references.bib
 
diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex
new file mode 100644
index 0000000..6de2bca
--- /dev/null
+++ b/buch/papers/punktgruppen/crystals.tex
@@ -0,0 +1,16 @@
+\section{Kristalle}
+Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. 
+Wir werden uns aber nicht auf sein Äusseres fokussieren, sondern was ihn im Inneren ausmacht.
+Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert.
+\begin{definition}[Kristall]
+    Ein Kristall besteht aus Atomen, welche sich in einem Muster arrangieren, welches sich in drei Dimensionen periodisch wiederholt.
+\end{definition}
+
+
+Ein Zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattce-grid}.
+Für die Überschaubarkeit haben wir ein simples Muster eines einzelnen XgrauenX Punktes gewählt in nur Zwei Dimensionen.
+Die eingezeichneten Vektoren a und b sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt.
+Dadurch können von einem einzelnen XGrauenX Gitterpunkt in \ref{fig:punktgruppen:lattce-grid} können mit einer ganzzahligen Linearkombination von a und b alle anderen Gitterpunkte des Kristalles erreicht werden.
+Ein Kristallgitter kann eindeutig mit a und b und deren winkeln beschrieben werden weswegen a und b auch Gitterparameter genannt werden.
+Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor also FRMEL FÜR TRANSLATIONSVEKTOR  erreicht werden.
+Da sich das Ganze Kristallgitter wiederholt, wiederholen sich auch die Eigenschaften eines Gitterpunktes Periodisch mit eiem
diff --git a/buch/papers/punktgruppen/intro.tex b/buch/papers/punktgruppen/intro.tex
new file mode 100644
index 0000000..10dea79
--- /dev/null
+++ b/buch/papers/punktgruppen/intro.tex
@@ -0,0 +1,10 @@
+\section{Einleitung}
+Es gibt viele möglichkeiten sich in Kristallen zu verlieren.
+Auch wen man nur die Mathematischen möglichkeiten in betracht zieht, hat man noch viel zu viele Möglichkeiten sich mit kristallen zu beschäftigen.
+In diesem Articel ist daher der Fokus "nur" auf die Symmetrie gelegt.
+Im Abschitt über Symmetrien werden wir sehen, wie eine Symmetrie eines Objektes weit 
+2.ter versuch:
+Die Kristallographie ist ein grosses Thema, Symmetrien auch. 
+Für beide bestehen schon bewährte Mathematische Modelle und Definitionen.
+Die
+
diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex
index fc91913..d88e221 100644
--- a/buch/papers/punktgruppen/main.tex
+++ b/buch/papers/punktgruppen/main.tex
@@ -3,34 +3,19 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:punktgruppen}}
-\lhead{Thema}
+\chapter{Crystal M\rotatebox[origin=c]{180}{a}th\label{chapter:punktgruppen}}
+\lhead{Crystal M\rotatebox[origin=c]{180}{a}th}
 \begin{refsection}
-\chapterauthor{Hans Muster}
+\chapterauthor{Tim T\"onz, Naoki Pross}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
+\input{papers/punktgruppen/intro}
+\input{papers/punktgruppen/symmetry}
+\input{papers/punktgruppen/crystals}
+\input{papers/punktgruppen/piezo}
 
-\input{papers/punktgruppen/teil0.tex}
-\input{papers/punktgruppen/teil1.tex}
-\input{papers/punktgruppen/teil2.tex}
-\input{papers/punktgruppen/teil3.tex}
+\nocite{punktgruppen:pinter-algebra}
+\nocite{punktgruppen:sands-crystal}
+\nocite{punktgruppen:lang-elt2}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/punktgruppen/packages.tex b/buch/papers/punktgruppen/packages.tex
index 971bcfe..a6efdbf 100644
--- a/buch/papers/punktgruppen/packages.tex
+++ b/buch/papers/punktgruppen/packages.tex
@@ -4,7 +4,4 @@
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 
-% if your paper needs special packages, add package commands as in the
-% following example
-%\usepackage{packagename}
-
+\usepackage{dsfont}
diff --git a/buch/papers/punktgruppen/piezo.tex b/buch/papers/punktgruppen/piezo.tex
new file mode 100644
index 0000000..7ee4174
--- /dev/null
+++ b/buch/papers/punktgruppen/piezo.tex
@@ -0,0 +1 @@
+\section{Piezoelektrizit\"at}
diff --git a/buch/papers/punktgruppen/references.bib b/buch/papers/punktgruppen/references.bib
index aa7eb14..9edb8bd 100644
--- a/buch/papers/punktgruppen/references.bib
+++ b/buch/papers/punktgruppen/references.bib
@@ -4,32 +4,32 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{punktgruppen:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
+@book{punktgruppen:pinter-algebra,
+	title = {A Book of Abstract Algebra},
+	author = {Charles C. Pinter},
+	publisher = {Dover Publications Inc.; 2. Edition},
+	year = {2010},
+	month = {1},
+	day = {10},
+	isbn = {978-0-486-47417-5},
+	inseries = {Dover Books on Mathematics},
 }
 
-@book{punktgruppen:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
+@book{punktgruppen:sands-crystal,
+	title = {Introduction to Crystallography},
+	author = {Donald E. Sands},
+	publisher = {Dover Publications Inc.},
+	year = {1993},
+	isbn = {978-0-486-67839-9},
+	inseries = {Dover Books on Science},
 }
 
-@article{punktgruppen:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@book{punktgruppen:lang-elt2,
+	title = {Elektrotechnik 2},
+	author = {Hans-Dieter Lang},
+	publisher = {Fachhochschule Ostschweiz Rapperswil},
+	year = {2020},
+	month = {2},
+	inseries = {Vorlesungsskript zum Modul ELT},
 }
 
diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex
new file mode 100644
index 0000000..db05ff5
--- /dev/null
+++ b/buch/papers/punktgruppen/symmetry.tex
@@ -0,0 +1,182 @@
+\section{Symmetrie}
+Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem
+ursprünglichen griechischen Wort
+\(\mathrm{\sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\)
+\footnote{\emph{Simmetr\'ia}: ein gemeinsames Mass habend, gleichmässig,
+verhältnismässig} fast nicht verändert. In der Alltagssprache mag es ein
+locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr
+präzise Bedeutung.
+\begin{definition}[Symmetrie]
+	Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer
+	bestimmten Operation invariant ist.
+\end{definition}
+
+Wenn der Leser noch nicht mit der Gruppentheorie in Berührung gekommen ist, ist
+vielleicht nicht ganz klar, was eine Operation ist, aber die Definition sollte
+trotzdem Sinn machen. Die Formalisierung dieser Idee wird bald kommen, aber
+zunächst wollen wir eine Intuition aufbauen.
+
+\begin{figure}[h]
+	\centering
+	\begin{tikzpicture}[
+			node distance = 2cm,
+			shapetheme/.style = {
+				very thick, draw = black, fill = magenta!20!white,
+				minimum size = 2cm,
+			},
+			line/.style = {thick, draw = darkgray},
+			axis/.style = {line, dashed},
+			dot/.style = {
+				circle, draw = darkgray, fill = darkgray,
+				minimum size = 1mm, inner sep = 0, outer sep = 0,
+			},
+		]
+
+		\node[
+			shapetheme,
+			rectangle
+		] (R) {};
+		\node[dot] at (R) {};
+		\draw[axis] (R) ++(-1.5, 0) to ++(3, 0) node[right] {\(\sigma\)};
+
+		\node[
+			shapetheme,
+			regular polygon,
+			regular polygon sides = 5,
+			right = of R,
+		] (Ps) {};
+		\node[dot] (P) at (Ps) {};
+		\draw[line, dotted] (P) to ++(18:1.5);
+		\draw[line, dotted] (P) to ++(90:1.5);
+		\draw[line, ->] (P) ++(18:1.2) 
+			arc (18:90:1.2) node[midway, above right] {\(r, 72^\circ\)};
+
+		\node[
+			shapetheme,
+			circle, right = of P
+		] (Cs) {};
+		\node[dot] (C) at (Cs) {};
+		\draw[line, dotted] (C) to ++(1.5,0);
+		\draw[line, dotted] (C) to ++(60:1.5);
+		\draw[line, ->] (C) ++(1.2,0)
+			arc (0:60:1.2) node[midway, above right] {\(r, \alpha\)};
+
+	\end{tikzpicture}
+	\caption{
+		Beispiele für geometrisch symmetrische Formen.
+		\label{fig:punktgruppen:geometry-example}
+	}
+\end{figure}
+
+Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit
+einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden,
+ist das Konzept der Symmetrie eigentlich viel allgemeiner.  In Abbildung
+\ref{fig:punktgruppen:geometry-example} haben wir einige Formen, die
+offensichtlich symmetrisch sind.  Zum Beispiel hat ein Quadrat viele Achsen, um
+die es gedreht werden kann, ohne sein Aussehen zu verändern.  Regelmässige
+Polygone mit \(n\) Seiten sind gute Beispiele, um eine diskrete
+Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um
+einen Punkt um einen bestimmten Winkel \(360^\circ/n\) sie unverändert lässt.
+Das letzte Beispiel auf der rechten Seite ist eine unendliche
+Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für
+\(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen.  Dies ist
+hoffentlich ausreichend, um die Bedeutung hinter der Notation zu verstehen, die
+nun eingeführt wird.
+
+\begin{definition}[Symmetriegruppe]
+	Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt.
+	Bei einer anderen Operation \(h\) definieren wir die Komposition \(h\circ g\)
+	als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter
+	Komposition eine Gruppe, die Symmetriegruppe genannt wird.
+\end{definition}
+
+Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. Wenn wir
+\(r\) eine Drehung von \(2\pi/n\) sein lassen, gibt es eine wohlbekannte Symmetriegruppe
+\[
+	C_n = \langle r \rangle
+		= \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\}
+		= \mathbb{Z}/n\mathbb{Z},
+\]
+die Zyklische Gruppe heisst. Hier die Potenzen von \(r\) sind als wiederholte
+Komposition gemeint, d.h. \(r^n = r\circ r \circ \cdots r\circ r\).  Die
+Schreibweise mit den spitzen Klammern wird als Erzeugendensystem bezeichnet.
+Das liegt daran, dass alle Elemente der Symmetriegruppe aus Kombinationen einer
+Teilmenge erzeugt werden, die als erzeugende Elemente bezeichnet werden.  Die
+Reflexionssymmetriegruppe ist nicht so interessant, da sie nur
+\(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit
+der Rotation, erhält man die so genannte Diedergruppe
+\[
+	D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle
+		= \left\{
+				\mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1}
+		\right\}.
+\]
+Diesmal muss die Generator-Notation die Beziehungen zwischen den beiden
+Operationen beinhalten. Die ersten beiden sind leicht zu erkennen, für die
+letzte empfehlen wir, sie an einem 2D-Quadrat auszuprobieren.
+
+Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich
+möglich ist, eine nicht kommutative Algebra zu erstellen. Die naheliegende
+Frage ist dann, könnte es sein, dass wir bereits etwas haben, das dasselbe tut?
+Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein.
+\begin{definition}[Darstellung einer Gruppe, Gruppenhomomorphismus]
+	Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\)
+	bzw.  \(\star\). Ein Homomorphismus\footnote{ Für eine ausführlichere
+	Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist
+	eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt
+	\(f(a\diamond b) = f(a) \star f(b)\).  Man sagt, dass der Homomorphismus
+	\(f\) \(G\) in \(H\) transformiert, oder dass \(H\) eine Darstellung von
+	\(G\) ist.
+\end{definition}
+\begin{beispiel}
+	Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine
+	Drehung von \(2\pi k/n\) um den Ursprung dar. Die mit der Matrix 
+	\[
+		\Phi(r^k) = \begin{pmatrix}
+			\cos(2\pi k/n) & -\sin(2\pi k/n) \\
+			\sin(2\pi k/n) &  \cos(2\pi k/n)
+		\end{pmatrix}
+	\]
+	definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von
+	\(C_n\). In diesem Fall ist die erste Gruppenoperation die Komposition und
+	die zweite die Matrixmultiplikation. Man kann überprüfen, dass \(\Phi(r^2
+	\circ r) = \Phi(r^2)\Phi(r)\).
+\end{beispiel}
+\begin{beispiel}
+	Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen
+	Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem
+	komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\)
+	ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben.
+\end{beispiel}
+
+Die Symmetrien, die wir bis jetzt besprochen haben, haben immer mindestens
+einen Punkt unbesetzt gelassen. Im Fall der Rotation war es der Drehpunkt, bei
+der Spiegelung die Achse. Dies ist jedoch keine Voraussetzung für eine
+Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt
+verschieben können. Ein aufmerksamer Leser wird bemerken, dass die
+unveränderten Punkte zum Eigenraum\footnote{Zur Erinnerung \(E_\lambda =
+\mathrm{null}(\Phi - \lambda I)\), \(\vec{v}\in E_\lambda \implies \Phi \vec{v}
+= \lambda\vec{v}\)} der Matrixdarstellung der Symmetrieoperation gehören.
+Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man
+Punktsymmetrie.
+\begin{definition}[Punktgruppe]
+	Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens
+	einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine
+	Punktgruppe ist.
+\end{definition}
+Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren:
+eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr
+nützlich ist, nämlich die Translationssymmetrie.  Von einem mathematischen
+Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\)
+hat, wenn es die Gleichung 
+\[
+	U(x) = U(Q(x)) = U(x + a),
+\]
+für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche
+Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine
+zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\)
+dann \(\vec{v}(t) = \vec{v}(t + \tau)\).
+
+% \subsection{Sch\"onflies notation} 
+
+% vim:ts=2 sw=2 spell spelllang=de:
diff --git a/buch/papers/punktgruppen/teil0.tex b/buch/papers/punktgruppen/teil0.tex
deleted file mode 100644
index 5a8278e..0000000
--- a/buch/papers/punktgruppen/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{punktgruppen:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{punktgruppen:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/punktgruppen/teil1.tex b/buch/papers/punktgruppen/teil1.tex
deleted file mode 100644
index 228af33..0000000
--- a/buch/papers/punktgruppen/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{punktgruppen:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{punktgruppen:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{punktgruppen:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{punktgruppen:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{punktgruppen:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/punktgruppen/teil2.tex b/buch/papers/punktgruppen/teil2.tex
deleted file mode 100644
index b48e785..0000000
--- a/buch/papers/punktgruppen/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{punktgruppen:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{punktgruppen:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/punktgruppen/teil3.tex b/buch/papers/punktgruppen/teil3.tex
deleted file mode 100644
index 94abd74..0000000
--- a/buch/papers/punktgruppen/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{punktgruppen:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{punktgruppen:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/reedsolomon/.gitignor b/buch/papers/reedsolomon/.gitignor
new file mode 100644
index 0000000..52a02ac
--- /dev/null
+++ b/buch/papers/reedsolomon/.gitignor
@@ -0,0 +1,24 @@
+RS.aux
+RS.bbl
+RS.bib
+RS.blg
+RS.idx
+RS.ilg
+RS.ind
+RS.log
+RS.out
+RS.pdf
+RS.run.xml
+RS.toc
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
diff --git a/buch/papers/reedsolomon/RS presentation/README.txt b/buch/papers/reedsolomon/RS presentation/README.txt
new file mode 100644
index 0000000..4d0620f
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/README.txt
@@ -0,0 +1 @@
+Dies ist die Presentation des Reed-Solomon-Code
\ No newline at end of file
diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux b/buch/papers/reedsolomon/RS presentation/RS.aux
new file mode 100644
index 0000000..065ba66
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/RS.aux
@@ -0,0 +1,146 @@
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new file mode 100644
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diff --git a/buch/papers/reedsolomon/RS presentation/RS.out b/buch/papers/reedsolomon/RS presentation/RS.out
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+\BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1
+\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2
+\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3
+\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4
+\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5
+\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6
+\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7
+\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8
diff --git a/buch/papers/reedsolomon/RS presentation/RS.pdf b/buch/papers/reedsolomon/RS presentation/RS.pdf
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diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz
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diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex
new file mode 100644
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--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/RS.tex
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+\documentclass[11pt,aspectratio=169]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[ngerman]{babel}
+\usepackage{tikz}
+\usetheme{Hannover}
+
+\begin{document}
+	\author{Joshua Bär und Michael Steiner}
+	\title{Reed-Solomon-Code}
+	\subtitle{}
+	\logo{}
+	\institute{OST Ostschweizer Fachhochschule}
+	\date{26.04.2021}
+	\subject{Mathematisches Seminar}
+	%\setbeamercovered{transparent}
+	\setbeamercovered{invisible}
+	\setbeamertemplate{navigation symbols}{}
+	\begin{frame}[plain]
+		\maketitle
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Einführung}
+	\begin{frame}
+		\frametitle{Reed-Solomon-Code:}
+		\begin{itemize}
+		\visible<1->{\item Für Übertragung von Daten}
+		\visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler}
+		\visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.}
+		\end{itemize}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Polynom Ansatz}
+	\begin{frame}
+		\begin{itemize}
+			\item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern
+		\end{itemize}
+	\end{frame}
+	\begin{frame}
+		\frametitle{Beispiel}
+		Übertragen von 
+		${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ 
+		als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $.
+		
+		
+			Versende $ (p(1),p(2),\dots,p(7))$
+			\visible<2->{ = (\textcolor{green}{8},} 
+			\only<2>{\textcolor{green}{15},}
+			\only<3>{\textcolor{red}{50},}
+			\only<2>{\textcolor{green}{26},}
+			\only<3>{\textcolor{red}{37},}
+			\visible<2->{\textcolor{green}{41}, \textcolor{green}{60}, 
+			\textcolor{green}{83}, \textcolor{green}{110})}	
+			\only<2>{\includegraphics[scale = 1.2]{images/polynom1.pdf}}
+			\only<3>{\includegraphics[scale = 1.2]{images/polynom2.pdf}}
+			\visible<3>{
+			\newline
+			\textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+	\frametitle{Parameter}
+	\begin{center}
+		\begin{tabular}{ c c c } 
+			\hline
+			Nutzlas & Fehler & Versenden \\
+			\hline 
+			3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 
+			4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\
+\visible<1->{3}& 
+\visible<1->{3}& 
+\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ 
+			&&\\
+\visible<1->{$k$} & 
+\visible<1->{$t$} & 
+\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ 
+			\hline
+			&&\\
+			&&\\
+			\multicolumn{3}{l} {
+				\visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!}
+			}
+		\end{tabular}
+	\end{center}	
+	\end{frame}
+
+%-------------------------------------------------------------------------------	
+
+\section{Diskrete Fourier Transformation}
+	\begin{frame}
+		\frametitle{Idee}
+		\begin{itemize}
+			\item Fourier-transformieren
+			\item Übertragung
+			\item Rücktransformieren
+		\end{itemize}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\begin{figure}
+			\only<1>{
+			\includegraphics[width=0.9\linewidth]{images/fig1.pdf}
+			}
+			\only<2>{
+			\includegraphics[width=0.9\linewidth]{images/fig2.pdf}
+			}
+			\only<3>{
+			\includegraphics[width=0.9\linewidth]{images/fig3.pdf}
+			}
+			\only<4>{
+			\includegraphics[width=0.9\linewidth]{images/fig4.pdf}
+			}
+			\only<5>{
+			\includegraphics[width=0.9\linewidth]{images/fig5.pdf}
+			}
+			\only<6>{
+			\includegraphics[width=0.9\linewidth]{images/fig6.pdf}
+			}
+			\only<7>{
+			\includegraphics[width=0.9\linewidth]{images/fig7.pdf}
+			}
+	\end{figure}
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+	\frametitle{Diskrete Fourier Transformation}
+	\begin{itemize}
+	\item Diskrete Fourier-Transformation gegeben durch:
+	\visible<1->{
+	\[
+	\label{ft_discrete}
+	\hat{c}_{k} 
+	= \frac{1}{N} \sum_{n=0}^{N-1}
+	{f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn}
+	\]}
+	\visible<2->{
+	\item Ersetzte
+	\[
+	w = e^{-\frac{2\pi j}{N} k}
+	\]}
+	\visible<3->{
+	\item Wenn $N$ konstant:
+	\[
+	\hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
+	\]}
+	\end{itemize}
+	\end{frame}	
+
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+	\frametitle{Diskrete Fourier Transformation}
+	\[
+	\begin{pmatrix}
+		\hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n
+	\end{pmatrix}
+	= \frac{1}{N}
+	\begin{pmatrix}
+		w^0 & w^0   & w^0  & \dots  &w^0     \\
+		w^0 & w^1   &w^2   & \dots  &w^{N-1}     \\ 
+		w^0 & w^2   &w^4   & \dots  &w^{2(N-1)}  \\ 
+		\vdots   & \vdots     &\vdots     &\ddots  &\vdots       \\
+		w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots  &w^{(N-1)(N-1)}  \\ 
+	\end{pmatrix}
+	\begin{pmatrix}
+		\textcolor{blue}{f_0}  	\\
+		\textcolor{blue}{f_1}	\\
+		\textcolor{blue}{f_2}	\\	 
+ 		\vdots  \\
+	 	0  \\ 
+	\end{pmatrix}
+	\]
+	\end{frame}
+%-------------------------------------------------------------------------------	
+
+	\begin{frame}
+		\frametitle{Probleme und Fragen}
+		
+			Wie wird der Fehler lokalisiert?
+		\visible<2>{
+			\newline
+			Indem in einem endlichen Körper gerechnet wird.
+		}
+	\end{frame}	
+
+%-------------------------------------------------------------------------------	
+
+
+\section{Reed-Solomon in Endlichen Körpern}	
+
+	\begin{frame}
+		\frametitle{Reed-Solomon in Endlichen Körpern}
+		
+		\begin{itemize}
+			\onslide<1->{\item Warum endliche Körper?}
+			
+			\onslide<2->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler}
+			
+			\onslide<3->{\qquad digitale Fehlerkorrektur}
+			
+			%\onslide<4->{\qquad bessere Laufzeit}
+			
+			\vspace{10pt}
+			
+			\onslide<4->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil}
+			
+			\vspace{10pt}
+			
+			\onslide<5->{\item aus Fehlerkorrekturteil die Fehlerstellen finden}
+			
+			\onslide<6->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom}
+			
+%			\vspace{10pt}
+			
+%			\onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet}
+			
+%			Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet
+						
+		\end{itemize}
+		
+%		TODO
+		
+%		erklärung und einführung der endlichen körper, was wollen wir erreichen?
+		
+%		wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann
+		
+%		sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Definition eines Beispiels}
+		
+		\begin{itemize}
+			
+			\onslide<1->{\item endlicher Körper $q = 11$}
+			
+			\onslide<2->{ist eine Primzahl}
+
+			\onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$}
+			
+			\vspace{10pt}
+			
+			\onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen}
+			
+			\onslide<5->{$n = q - 1 = 10$ Zahlen}
+			
+			\vspace{10pt}
+			
+			\onslide<6->{\item Max.~Fehler $t = 2$}
+			
+			\onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können}
+			
+			\vspace{10pt}
+			
+			\onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen}
+			
+			\onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen}
+			
+			\onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$}
+			
+			\onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$}
+			
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Codierung eines Beispiels}
+	\begin{frame}
+		\frametitle{Codierung}
+		
+		\begin{itemize}
+			\onslide<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation}
+			
+			\vspace{10pt}
+			
+			\onslide<2->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$}
+			
+			\vspace{10pt}
+			
+			\onslide<3->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken}
+			
+			\onslide<4->{$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$}
+			
+			\vspace{10pt}
+			
+			\onslide<5->{\item Wir wählen $a = 8$}
+			
+			\onslide<6->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$}
+			
+			\onslide<7->{$8$ ist eine primitive Einheitswurzel}
+			
+			\vspace{10pt}
+			
+			\onslide<8->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$}
+			
+			\onslide<9->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben}
+			
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Codierung}
+		
+		\begin{itemize}
+			\onslide<1->{\item Übertragungsvektor $v$}
+			
+			\onslide<2->{\item $v = A \cdot m$}
+			
+		\end{itemize}
+		
+		\[
+		\onslide<3->{
+		v = \begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+			8^0&	8^3&	8^6&	8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+			8^0&	8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\
+		\end{pmatrix}
+		}
+		\]
+		
+		\begin{itemize}
+			\onslide<4->{\item $v = [5,3,6,5,2,10,2,7,10,4]$}
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Decodierung ohne Fehler}
+	\begin{frame}
+		\frametitle{Decodierung ohne Fehler}
+		
+		\begin{itemize}
+			\onslide<1->{\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$}
+			
+			\vspace{10pt}
+			
+			\onslide<2->{\item Wir suchen die Inverse der Matrix $A$}
+			
+			\vspace{10pt}
+			
+		\end{itemize}
+		
+		\begin{columns}[t]
+		\begin{column}{0.55\textwidth}		
+		\onslide<3->{ Inverse der Fouriertransformation}
+		\vspace{10pt}
+		\onslide<4->{
+		\[
+		F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt
+		\]
+		}
+		\vspace{10pt}
+		\onslide<5->{
+		\[
+		\mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega
+		\]
+		}
+		\end{column}
+		\begin{column}{0.45\textwidth}		
+		\onslide<6->{Inverse von $a$}
+		
+		\vspace{10pt}
+		
+		\onslide<7->{
+		\[
+		8^{1} \Rightarrow 8^{-1}
+		\]
+		}
+	
+		\onslide<8->{Inverse finden wir über den Eulkidischen Algorithmus}
+		\vspace{10pt}
+		\end{column}
+		\end{columns}		
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Der Euklidische Algorithmus}
+	
+	\begin{columns}[t]
+	\begin{column}{0.50\textwidth}	
+	
+	Recap aus der Vorlesung:
+	
+	Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$
+	
+	\begin{tabular}{rcl}
+		$a  b$ &$\equiv$& $1 \mod p$\\
+		$a  b$ &$=$& $1 + n  p$\\
+		$a  b - n  p$ &$=$& $1$\\
+		&&\\
+		$\operatorname{ggT}(a,p)$&$=$& $1$\\
+		$sa + tp$&$=$& $1$\\
+		$b$&$=$&$s$\\
+		$n$&$=$&$-t$
+	\end{tabular}
+	
+	\end{column}
+	\begin{column}{0.50\textwidth}	
+	
+	\begin{center}
+	\onslide<1->{
+	\begin{tabular}{| c | c c | c | r r |}
+		\hline
+		$k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\
+		\hline 
+		& & & & $1$& $0$\\
+		$0$& $8$& $11$& $0$& $0$& $1$\\
+		$1$& $11$& $8$& $1$& $1$& $0$\\
+		$2$& $8$& $3$& $2$& $-1$& $1$\\
+		$3$& $3$& $2$& $1$& $3$& $-2$\\
+		$4$& $2$& $1$& $2$& \textcolor<2->{blue}{$-4$}& \textcolor<2->{red}{$3$}\\
+		$5$& $1$& $0$& & $11$& $-8$\\
+		\hline
+	\end{tabular}	
+	}
+
+	\vspace{10pt}
+	
+	\begin{tabular}{rcl}
+		\onslide<3->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\
+		\onslide<4->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\
+		\onslide<5->{$8^{-1}$ &$=$& $7$}
+		
+	\end{tabular}
+	
+	\end{center}
+	
+	\end{column}
+	\end{columns}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Decodierung mit Inverser Matrix}	
+		
+		\begin{itemize}
+			\onslide<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$}
+			
+			\onslide<2->{\item $m = 1/10 \cdot A^{-1} \cdot v$}
+			
+			\onslide<3->{\item $m = 10 \cdot A^{-1} \cdot v$}
+			
+		\end{itemize}
+		\onslide<4->{
+		\[
+		m = 10 \cdot \begin{pmatrix}
+			7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0\\
+			7^0&	7^1&	7^2&	7^3&	7^4&	7^5&	7^6&	7^7&    7^8&	7^9\\
+			7^0&	7^2&	7^4&	7^6&	7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\
+			7^0&	7^3&	7^6&	7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\
+			7^0&	7^4&	7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\
+			7^0&	7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\
+			7^0&	7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\
+			7^0&	7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\
+			7^0&	7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\
+			7^0&	7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\
+		\end{pmatrix}
+		\]
+		}
+		
+		\begin{itemize}
+			\onslide<5->{\item $m = [0,0,0,0,4,7,2,5,8,1]$}
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+\section{Decodierung mit Fehler}
+	\begin{frame}
+		\frametitle{Decodierung mit Fehler - Ansatz}	
+		
+		\begin{itemize}
+			\onslide<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$}
+			
+			\onslide<2->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$}
+			
+			\onslide<3->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$}
+			
+		\end{itemize}
+		
+		\onslide<4->{Wie finden wir die Fehler?}
+		
+		\begin{itemize}
+			\onslide<5->{\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$}
+			
+			\onslide<6->{\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$}
+			
+			%\only<7->{\item $e(X) = r(X) - m(X)$}
+			
+			\onslide<7->{\item $e(X) = r(X) - m(X)$}
+			
+		\end{itemize}	
+
+		\begin{center}
+			\onslide<8->{
+		\begin{tabular}{c c c c c c c c c c c}
+			\hline
+			$i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\
+			\hline
+			$r(a^{i})$& \onslide<9->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\
+			$m(a^{i})$& \onslide<10->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\
+			$e(a^{i})$& \onslide<11->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\
+			\hline
+		\end{tabular}	
+			}
+		\end{center}
+	
+		
+		\begin{itemize}
+			\onslide<12->{\item Alle Stellen, die nicht Null sind, sind Fehler}
+		\end{itemize}
+		
+
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Nullstellen des Fehlerpolynoms finden}
+		
+		\begin{itemize}
+			\onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$}
+			
+			\vspace{10pt}
+			
+			\onslide<2->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$}
+			
+			\vspace{10pt}
+			
+			\onslide<3->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$}
+			
+			\vspace{10pt}
+			
+			\onslide<4->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$}
+			
+			\vspace{10pt}
+			
+			\onslide<5->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat}
+			
+			\vspace{10pt}
+			
+			\onslide<6->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$}
+				
+		\end{itemize}
+			
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Nullstellen des Fehlerpolynoms finden}
+		
+		\begin{itemize}
+			
+			\onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$}
+			
+			\vspace{10pt}
+			
+			\onslide<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$}
+			
+			\vspace{10pt}
+			
+			\onslide<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$}
+			
+			\vspace{10pt}
+			
+			\onslide<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$}
+			
+			\vspace{10pt}
+			
+			\onslide<1->{\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können}
+			
+			\vspace{10pt}
+			
+			\onslide<2->{$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $
+			
+			\qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$}
+			
+			\onslide<3->{$= d(X) \cdot e(X)$}
+			
+			\vspace{10pt}
+			
+			\onslide<4->{\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$}
+			
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Kennen wir $e(X)$?}
+		
+		\begin{itemize}
+			
+			\onslide<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite}
+			
+			\vspace{10pt}
+			
+			\onslide<2->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?}
+			
+			\vspace{10pt}
+			
+			\onslide<3->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$
+			
+			In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat}
+			
+			\vspace{10pt}
+			
+			\onslide<4->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$}
+			
+			\vspace{10pt}
+			
+			\onslide<5->{\item $f(X) = X^{10} - 1 = X^{10} + 10$}
+			
+			\vspace{10pt}
+			
+			\onslide<6->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen}
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Der Euklidische Algorithmus (nochmal)}
+		
+		\onslide<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$}
+		\onslide<2->{
+		\[
+		\arraycolsep=1.4pt
+		\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+			X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\
+			X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& &  & & & &   & & & & & &   & &  & & \\ \cline{1-9}
+			&& 3X^9&+& 8X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\
+			&& 3X^9&+& 2X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\ \cline{3-9}
+			& &    & &6X^8&+&0X^7&+&p(X)& &   & & & & & &   & &  & & \\
+		\end{array}
+		\]
+		}
+		\onslide<3->{
+		\[
+		\arraycolsep=1.4pt
+		\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+			5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\
+			5X^9&+& 0X^8&+& p(X)& & & & & &  & & & &   & & & & & &   & &  & & \\ \cline{1-5}
+			&& 7X^8&+& p(X)& & & & & &   & & & & & &   & &  & & \\
+		\end{array}
+		\]
+		}
+		\vspace{10pt}
+		
+		\onslide<4->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$}
+		
+		\vspace{10pt}
+		
+		\onslide<5->{	$\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen	}
+		
+	\end{frame}
+		
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Der Erweiterte Euklidische Algorithmus}
+		
+		\begin{center}
+			
+		\begin{tabular}{| c | c | c c |}
+			\hline
+			$k$ &  $q_i$ & $e_i$ & $f_i$\\
+			\hline 
+			& & $0$& $1$\\
+			$0$& $9X + 5$& $1$& $0$\\
+			$1$& $10X + 3$& $9X+5$& $1$\\
+			$2$& & \textcolor<2->{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\
+			\hline
+		\end{tabular}	
+			
+		\end{center}
+		
+		\vspace{10pt}
+		
+		\begin{tabular}{ll}
+			\onslide<3->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\}
+			\onslide<4->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\}
+			\onslide<5->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$}
+		\end{tabular}
+		
+		\vspace{10pt}
+		
+		\onslide<6->{
+		\begin{center}
+			$a^i = 5 \qquad \Rightarrow \qquad i = 3$
+			
+			$a^i = 6 \qquad \Rightarrow \qquad i = 8$
+		\end{center}
+		}
+
+	\onslide<7->{$d(X) = (X-a^3)(X-a^8)$}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Nachricht Rekonstruieren}	
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\begin{itemize}
+			
+			\onslide<1->{\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$}
+			
+			\onslide<2->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$}
+			
+		\end{itemize}
+		\onslide<3->{
+		\[
+		\textcolor{gray}{
+		\begin{pmatrix}
+			 a^0 \\ a^1 \\ a^2 \\ \textcolor<4->{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor<4->{red}{a^8} \\ a^9 \\
+		\end{pmatrix}}
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ \textcolor<4->{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor<4->{red}{1} \\ 4 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+			\textcolor<4->{red}{8^0}&	\textcolor<4->{red}{8^3}&	\textcolor<4->{red}{8^6}&	\textcolor<4->{red}{8^9}& \textcolor<4->{red}{8^{12}}& \textcolor<4->{red}{8^{15}}& \textcolor<4->{red}{8^{18}}& \textcolor<4->{red}{8^{21}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{27}}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+			\textcolor<4->{red}{8^0}&	\textcolor<4->{red}{8^8}& \textcolor<4->{red}{8^{16}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{32}}& \textcolor<4->{red}{8^{40}}& \textcolor<4->{red}{8^{48}}& \textcolor<4->{red}{8^{56}}& \textcolor<4->{red}{8^{64}}& \textcolor<4->{red}{8^{72}}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\
+		\end{pmatrix}
+		\]
+		}
+		
+		\begin{itemize}
+			\onslide<5->{\item Fehlerstellen entfernen}
+		\end{itemize}
+		
+	\end{frame}	
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    \textcolor<4->{green}{8^0}&    \textcolor<4->{green}{8^0}&    \textcolor<4->{green}{8^0}&    \textcolor<4->{green}{8^0}\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	\textcolor<4->{green}{8^6}&	 \textcolor<4->{green}{8^7}&     \textcolor<4->{green}{8^8}&	  \textcolor<4->{green}{8^9}\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& \textcolor<4->{green}{8^{12}}& \textcolor<4->{green}{8^{14}}& \textcolor<4->{green}{8^{16}}& \textcolor<4->{green}{8^{18}}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<4->{green}{8^{24}}& \textcolor<4->{green}{8^{28}}& \textcolor<4->{green}{8^{32}}& \textcolor<4->{green}{8^{36}}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<4->{green}{8^{30}}& \textcolor<4->{green}{8^{35}}& \textcolor<4->{green}{8^{40}}& \textcolor<4->{green}{8^{45}}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<4->{green}{8^{36}}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{48}}& \textcolor<4->{green}{8^{54}}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{49}}& \textcolor<4->{green}{8^{56}}& \textcolor<4->{green}{8^{63}}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<4->{green}{8^{54}}& \textcolor<4->{green}{8^{63}}& \textcolor<4->{green}{8^{72}}& \textcolor<4->{green}{8^{81}}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\onslide<3->{\item Nullstellen entfernen}
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<3->{red}{7} \\ \textcolor<3->{red}{4} \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+			\textcolor<3->{red}{8^0}&	\textcolor<3->{red}{8^7}& \textcolor<3->{red}{8^{14}}& \textcolor<3->{red}{8^{21}}& \textcolor<3->{red}{8^{28}}& \textcolor<3->{red}{8^{35}}\\
+			\textcolor<3->{red}{8^0}&	\textcolor<3->{red}{8^9}& \textcolor<3->{red}{8^{18}}& \textcolor<3->{red}{8^{27}}& \textcolor<3->{red}{8^{36}}& \textcolor<3->{red}{8^{45}}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\vspace{5pt}
+		
+		\begin{itemize}
+			\onslide<2->{\item Matrix in eine Quadratische Form bringen}
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\vspace{5pt}
+		
+		\begin{itemize}
+			\onslide<2->{\item Matrix Invertieren}
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			1&  1& 1&  1& 1&  1\\
+			1&  8& 9&  6& 4& 10\\
+			1&  9& 4&  3& 5&  1\\
+			1&  4& 5&  9& 3&  1\\
+			1& 10& 1& 10& 1& 10\\
+			1&  3& 9&  5& 4&  1\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{center}
+			\onslide<2->{$\Downarrow$}
+		\end{center}
+		\[
+		\onslide<3->{
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			 6&  4&  4&  6& 2&  1\\
+			 2&  7& 10&  3& 4&  7\\
+			 1&  8&  9&  8& 3&  4\\
+			 3&  6&  6&  4& 5&  9\\
+			10& 10&  9&  8& 1&  6\\
+			 1&  9&  6&  4& 7&  6\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		}
+		\]
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			6&  4&  4&  6& 2&  1\\
+			2&  7& 10&  3& 4&  7\\
+			1&  8&  9&  8& 3&  4\\
+			3&  6&  6&  4& 5&  9\\
+			10& 10&  9&  8& 1&  6\\
+			1&  9&  6&  4& 7&  6\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\onslide<2->{\item $m = [4,7,2,5,8,1]$}
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+
+\end{document}
diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc
new file mode 100644
index 0000000..095b5e6
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/RS.toc
@@ -0,0 +1,9 @@
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+\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2}
+\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3}
+\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4}
+\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5}
+\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6}
+\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7}
+\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8}
diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.aux b/buch/papers/reedsolomon/RS presentation/RS_handout.aux
new file mode 100644
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new file mode 100644
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diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.out b/buch/papers/reedsolomon/RS presentation/RS_handout.out
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+\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2
+\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3
+\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4
+\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5
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+\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7
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diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz
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+\documentclass[11pt,aspectratio=169]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[ngerman]{babel}
+\usepackage{tikz}
+\usetheme{Hannover}
+
+\begin{document}
+	\author{Joshua Bär und Michael Steiner}
+	\title{Reed-Solomon-Code}
+	\subtitle{}
+	\logo{}
+	\institute{OST Ostschweizer Fachhochschule}
+	\date{26.04.2021}
+	\subject{Mathematisches Seminar}
+	%\setbeamercovered{transparent}
+	\setbeamercovered{invisible}
+	\setbeamertemplate{navigation symbols}{}
+	\begin{frame}[plain]
+		\maketitle
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Einführung}
+	\begin{frame}
+		\frametitle{Reed-Solomon-Code:}
+		\begin{itemize}
+		\item Für Übertragung von Daten
+		\item Ermöglicht Korrektur von Übertragungsfehler
+		\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.
+		\end{itemize}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Polynom Ansatz}
+	\begin{frame}
+		\begin{itemize}
+			\item $2, 1, 5$ versenden und auf 2 Fehler absichern
+		\end{itemize}
+		\frametitle{Beispiel}
+		Übertragen von 
+		${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ 
+		als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $.
+		\newline
+		Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, 
+		\textcolor{red}{50}, \textcolor{red}{37},
+		\textcolor{green}{41}, \textcolor{green}{60}, 
+		\textcolor{green}{83}, \textcolor{green}{110})$
+		\includegraphics[scale = 1.2]{images/polynom2.pdf}
+		\newline
+		\textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+	\frametitle{Parameter}
+	\begin{center}
+		\begin{tabular}{ c c c } 
+			\hline
+			Nutzlas & Fehler & Versenden \\
+			\hline 
+			3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 
+			4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\
+			3& 3& 9 Werte eines Polynoms vom Grad 2 \\ 
+			&&\\
+			$k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ 
+			\hline
+			&&\\
+			&&\\
+			\multicolumn{3}{l} {
+			Ausserdem können bis zu $2t$ Fehler erkannt werden!
+			}
+		\end{tabular}
+	\end{center}	
+	\end{frame}
+
+%-------------------------------------------------------------------------------	
+
+\section{Diskrete Fourier Transformation}
+	\begin{frame}
+		\frametitle{Idee}
+		\begin{itemize}
+			\item Fourier-transformieren
+			\item Übertragung
+			\item Rücktransformieren
+		\end{itemize}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\begin{figure}
+			\only<1>{
+			\includegraphics[width=0.9\linewidth]{images/fig1.pdf}
+			}
+			\only<2>{
+			\includegraphics[width=0.9\linewidth]{images/fig2.pdf}
+			}
+			\only<3>{
+			\includegraphics[width=0.9\linewidth]{images/fig3.pdf}
+			}
+			\only<4>{
+			\includegraphics[width=0.9\linewidth]{images/fig4.pdf}
+			}
+			\only<5>{
+			\includegraphics[width=0.9\linewidth]{images/fig5.pdf}
+			}
+			\only<6>{
+			\includegraphics[width=0.9\linewidth]{images/fig6.pdf}
+			}
+			\only<7>{
+			\includegraphics[width=0.9\linewidth]{images/fig7.pdf}
+			}
+	\end{figure}
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+	\frametitle{Diskrete Fourier Transformation}
+	\begin{itemize}
+	\item Diskrete Fourier-Transformation gegeben durch:
+	
+	\[
+	\label{ft_discrete}
+	\hat{c}_{k} 
+	= \frac{1}{N} \sum_{n=0}^{N-1}
+	{f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn}
+	\]
+
+	\item Ersetzte
+	\[
+	w = e^{-\frac{2\pi j}{N} k}
+	\]
+
+	\item Wenn $N$ konstant:
+	\[
+	\hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
+	\]
+	\end{itemize}
+	\end{frame}	
+
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+	\frametitle{Diskrete Fourier Transformation}
+	\[
+	\begin{pmatrix}
+		\hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n
+	\end{pmatrix}
+	= \frac{1}{N}
+	\begin{pmatrix}
+		w^0 & w^0   & w^0  & \dots  &w^0     \\
+		w^0 & w^1   &w^2   & \dots  &w^{N-1}     \\ 
+		w^0 & w^2   &w^4   & \dots  &w^{2(N-1)}  \\ 
+		\vdots   & \vdots     &\vdots     &\ddots  &\vdots       \\
+		w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots  &w^{(N-1)(N-1)}  \\ 
+	\end{pmatrix}
+	\begin{pmatrix}
+		\textcolor{blue}{f_0}  	\\
+		\textcolor{blue}{f_1}	\\
+		\textcolor{blue}{f_2}	\\	 
+ 		\vdots  \\
+	 	0  \\ 
+	\end{pmatrix}
+	\]
+	\end{frame}
+%-------------------------------------------------------------------------------	
+
+	\begin{frame}
+		\frametitle{Probleme und Fragen}
+		
+			Wie wird der Fehler lokalisiert?
+			\newline
+			Indem in einem endlichen Körper gerechnet wird.
+		
+	\end{frame}	
+
+%-------------------------------------------------------------------------------	
+
+
+\section{Reed-Solomon in Endlichen Körpern}	
+
+	\begin{frame}
+		\frametitle{Reed-Solomon in Endlichen Körpern}
+		
+		\begin{itemize}
+			\item Warum endliche Körper?
+			
+			\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler
+			
+			\qquad digitale Fehlerkorrektur
+			
+			%\onslide<4->{\qquad bessere Laufzeit}
+			
+			\vspace{10pt}
+			
+			\item Nachricht = Nutzdaten + Fehlerkorrekturteil
+			
+			\vspace{10pt}
+			
+			\item aus Fehlerkorrekturteil die Fehlerstellen finden
+			
+			\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom
+			
+%			\vspace{10pt}
+			
+%			\onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet}
+			
+%			Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet
+						
+		\end{itemize}
+		
+%		TODO
+		
+%		erklärung und einführung der endlichen körper, was wollen wir erreichen?
+		
+%		wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann
+		
+%		sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Definition eines Beispiels}
+		
+		\begin{itemize}
+			
+			\item endlicher Körper $q = 11$
+			
+			ist eine Primzahl
+
+			beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$
+			
+			\vspace{10pt}
+			
+			\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen
+			
+			$n = q - 1 = 10$ Zahlen
+			
+			\vspace{10pt}
+			
+			\item Max.~Fehler $t = 2$
+			
+			maximale Anzahl von Fehler, die wir noch korrigieren können
+			
+			\vspace{10pt}
+			
+			\item Nutzlast $k = n -2t = 6$ Zahlen
+			
+			Fehlerkorrkturstellen $2t = 4$ Zahlen
+			
+			Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$
+			
+			als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$
+			
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Codierung eines Beispiels}
+	\begin{frame}
+		\frametitle{Codierung}
+		
+		\begin{itemize}
+			\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation
+			
+			\vspace{10pt}
+			
+			\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$
+			
+			\vspace{10pt}
+			
+			\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken
+			
+			$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$
+			
+			\vspace{10pt}
+			
+			\item Wir wählen $a = 8$
+			
+			$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$
+			
+			$8$ ist eine primitive Einheitswurzel
+			
+			\vspace{10pt}
+			
+			\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$
+			
+			$\Rightarrow$ \qquad können wir auch als Matrix schreiben
+			
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Codierung}
+		
+		\begin{itemize}
+			\item Übertragungsvektor $v$
+			
+			\item $v = A \cdot m$
+			
+		\end{itemize}
+		
+		\[
+		v = \begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+			8^0&	8^3&	8^6&	8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+			8^0&	8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\item $v = [5,3,6,5,2,10,2,7,10,4]$
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Decodierung ohne Fehler}
+	\begin{frame}
+		\frametitle{Decodierung ohne Fehler}
+		
+		\begin{itemize}
+			\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$
+			
+			\vspace{10pt}
+			
+			\item Wir suchen die Inverse der Matrix $A$
+			
+			\vspace{10pt}
+			
+		\end{itemize}
+		
+		\begin{columns}[t]
+		\begin{column}{0.55\textwidth}		
+		Inverse der Fouriertransformation
+		\vspace{10pt}
+
+		\[
+		F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt
+		\]
+
+		\vspace{10pt}
+
+		\[
+		\mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega
+		\]
+
+		\end{column}
+		\begin{column}{0.45\textwidth}		
+		Inverse von $a$
+		
+		\vspace{10pt}
+		
+		\[
+		8^{1} \Rightarrow 8^{-1}
+		\]
+	
+		Inverse finden wir über den Eulkidischen Algorithmus
+		\vspace{10pt}
+		\end{column}
+		\end{columns}		
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Der Euklidische Algorithmus}
+	
+	\begin{columns}[t]
+	\begin{column}{0.50\textwidth}	
+	
+	Recap aus der Vorlesung:
+	
+	Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$
+	
+	\begin{tabular}{rcl}
+		$a  b$ &$\equiv$& $1 \mod p$\\
+		$a  b$ &$=$& $1 + n  p$\\
+		$a  b - n  p$ &$=$& $1$\\
+		&&\\
+		$\operatorname{ggT}(a,p)$&$=$& $1$\\
+		$sa + tp$&$=$& $1$\\
+		$b$&$=$&$s$\\
+		$n$&$=$&$-t$
+	\end{tabular}
+	
+	\end{column}
+	\begin{column}{0.50\textwidth}	
+	
+	\begin{center}
+
+	\begin{tabular}{| c | c c | c | r r |}
+		\hline
+		$k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\
+		\hline 
+		& & & & $1$& $0$\\
+		$0$& $8$& $11$& $0$& $0$& $1$\\
+		$1$& $11$& $8$& $1$& $1$& $0$\\
+		$2$& $8$& $3$& $2$& $-1$& $1$\\
+		$3$& $3$& $2$& $1$& $3$& $-2$\\
+		$4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\
+		$5$& $1$& $0$& & $11$& $-8$\\
+		\hline
+	\end{tabular}	
+	
+
+	\vspace{10pt}
+	
+	\begin{tabular}{rcl}
+		$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\
+		$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\
+		$8^{-1}$ &$=$& $7$
+		
+	\end{tabular}
+	
+	\end{center}
+	
+	\end{column}
+	\end{columns}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Decodierung mit Inverser Matrix}	
+		
+		\begin{itemize}
+			\item $v = [5,3,6,5,2,10,2,7,10,4]$
+			
+			\item $m = 1/10 \cdot A^{-1} \cdot v$
+			
+			\item $m = 10 \cdot A^{-1} \cdot v$
+			
+		\end{itemize}
+
+		\[
+		m = 10 \cdot \begin{pmatrix}
+			7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0\\
+			7^0&	7^1&	7^2&	7^3&	7^4&	7^5&	7^6&	7^7&    7^8&	7^9\\
+			7^0&	7^2&	7^4&	7^6&	7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\
+			7^0&	7^3&	7^6&	7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\
+			7^0&	7^4&	7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\
+			7^0&	7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\
+			7^0&	7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\
+			7^0&	7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\
+			7^0&	7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\
+			7^0&	7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\item $m = [0,0,0,0,4,7,2,5,8,1]$
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------		
+\section{Decodierung mit Fehler}
+	\begin{frame}
+		\frametitle{Decodierung mit Fehler - Ansatz}	
+		
+		\begin{itemize}
+			\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$
+			
+			\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$
+			
+			\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$
+			
+		\end{itemize}
+		
+		Wie finden wir die Fehler?
+		
+		\begin{itemize}
+			\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$
+			
+			\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$
+			
+			\item $e(X) = r(X) - m(X)$
+			
+		\end{itemize}	
+
+		\begin{center}
+
+		\begin{tabular}{c c c c c c c c c c c}
+			\hline
+			$i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\
+			\hline
+			$r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\
+			$m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\
+			$e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\
+			\hline
+		\end{tabular}	
+
+		\end{center}
+		
+		\begin{itemize}
+			\item Alle Stellen, die nicht Null sind, sind Fehler
+		\end{itemize}
+		
+
+	\end{frame}
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Nullstellen des Fehlerpolynoms finden}
+		
+		\begin{itemize}
+			\item Satz von Fermat: $f(X) = X^{q-1}-1=0$
+			
+			\vspace{10pt}
+			
+			\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$
+			
+			\vspace{10pt}
+			
+			\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$
+			
+			\vspace{10pt}
+			
+			\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$
+			
+			\vspace{10pt}
+			
+			\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat
+			
+			\vspace{10pt}
+			
+			$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$
+				
+		\end{itemize}
+			
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Nullstellen des Fehlerpolynoms finden}
+		
+		\begin{itemize}
+			
+			\item Satz von Fermat: $f(X) = X^{q-1}-1=0$
+			
+			\vspace{10pt}
+			
+			\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$
+			
+			\vspace{10pt}
+			
+			\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$
+			
+			\vspace{10pt}
+			
+			\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$
+			
+			\qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$
+			
+			\vspace{10pt}
+			
+			\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können
+			
+			\vspace{10pt}
+			
+			$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $
+			
+			\qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$
+			
+			$= d(X) \cdot e(X)$
+			
+			\vspace{10pt}
+			
+			\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$
+			
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Kennen wir $e(X)$?}
+		
+		\begin{itemize}
+			
+			\item $e(X)$ ist unbekannt auf der Empfängerseite
+			
+			\vspace{10pt}
+			
+			\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?
+			
+			\vspace{10pt}
+			
+			\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$
+			
+			In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat
+			
+			\vspace{10pt}
+			
+			\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$
+			
+			\vspace{10pt}
+			
+			\item $f(X) = X^{10} - 1 = X^{10} + 10$
+			
+			\vspace{10pt}
+			
+			\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Der Euklidische Algorithmus (nochmal)}
+		
+		$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$
+		
+		\[
+		\arraycolsep=1.4pt
+		\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+			X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\
+			X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& &  & & & &   & & & & & &   & &  & & \\ \cline{1-9}
+			&& 3X^9&+& 8X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\
+			&& 3X^9&+& 2X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\ \cline{3-9}
+			& &    & &6X^8&+&0X^7&+&p(X)& &   & & & & & &   & &  & & \\
+		\end{array}
+		\]
+
+		\[
+		\arraycolsep=1.4pt
+		\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+			5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\
+			5X^9&+& 0X^8&+& p(X)& & & & & &  & & & &   & & & & & &   & &  & & \\ \cline{1-5}
+			&& 7X^8&+& p(X)& & & & & &   & & & & & &   & &  & & \\
+		\end{array}
+		\]
+		
+		\vspace{10pt}
+		
+		$\operatorname{ggT}(f(X),e(X)) = 6X^8$
+		
+		\vspace{10pt}
+		
+		$\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen
+		
+	\end{frame}
+		
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Der Erweiterte Euklidische Algorithmus}
+		
+		\begin{center}
+			
+		\begin{tabular}{| c | c | c c |}
+			\hline
+			$k$ &  $q_i$ & $e_i$ & $f_i$\\
+			\hline 
+			& & $0$& $1$\\
+			$0$& $9X + 5$& $1$& $0$\\
+			$1$& $10X + 3$& $9X+5$& $1$\\
+			$2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\
+			\hline
+		\end{tabular}	
+			
+		\end{center}
+		
+		\vspace{10pt}
+		
+		\begin{tabular}{ll}
+			Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\
+			Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\
+			Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$
+		\end{tabular}
+		
+		\vspace{10pt}
+		
+		\begin{center}
+			$a^i = 5 \qquad \Rightarrow \qquad i = 3$
+			
+			$a^i = 6 \qquad \Rightarrow \qquad i = 8$
+		\end{center}
+		
+
+	$d(X) = (X-a^3)(X-a^8)$
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Nachricht Rekonstruieren}	
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\begin{itemize}
+			
+			\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$
+			
+			\item $d(X) = (X-\textcolor{red}{a^3})(X-\textcolor{red}{a^8})$
+			
+		\end{itemize}
+	
+		\[
+		\textcolor{gray}{
+		\begin{pmatrix}
+			 a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\
+		\end{pmatrix}}
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+			\textcolor{red}{8^0}&	\textcolor{red}{8^3}&	\textcolor{red}{8^6}&	\textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+			\textcolor{red}{8^0}&	\textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\item Fehlerstellen entfernen
+		\end{itemize}
+		
+	\end{frame}	
+%-------------------------------------------------------------------------------		
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    \textcolor{green}{8^0}&    \textcolor{green}{8^0}&    \textcolor{green}{8^0}&    \textcolor{green}{8^0}\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	\textcolor{green}{8^6}&	   \textcolor{green}{8^7}&    \textcolor{green}{8^8}&    \textcolor{green}{8^9}\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\
+			8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\
+			8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\item Nullstellen entfernen
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+			\textcolor{red}{8^0}&	\textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\
+			\textcolor{red}{8^0}&	\textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\vspace{5pt}
+		
+		\begin{itemize}
+			\item Matrix in eine Quadratische Form bringen
+		\end{itemize}
+	
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+			8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+			8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+			8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+			8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+			8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\vspace{5pt}
+		
+		\begin{itemize}
+			\item Matrix Invertieren
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			1&  1& 1&  1& 1&  1\\
+			1&  8& 9&  6& 4& 10\\
+			1&  9& 4&  3& 5&  1\\
+			1&  4& 5&  9& 3&  1\\
+			1& 10& 1& 10& 1& 10\\
+			1&  3& 9&  5& 4&  1\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{center}
+			$\Downarrow$
+		\end{center}
+		\[
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			 6&  4&  4&  6& 2&  1\\
+			 2&  7& 10&  3& 4&  7\\
+			 1&  8&  9&  8& 3&  4\\
+			 3&  6&  6&  4& 5&  9\\
+			10& 10&  9&  8& 1&  6\\
+			 1&  9&  6&  4& 7&  6\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		\]
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+	\begin{frame}
+		\frametitle{Rekonstruktion der Nachricht}
+		
+		\[
+		\begin{pmatrix}
+			m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			6&  4&  4&  6& 2&  1\\
+			2&  7& 10&  3& 4&  7\\
+			1&  8&  9&  8& 3&  4\\
+			3&  6&  6&  4& 5&  9\\
+			10& 10&  9&  8& 1&  6\\
+			1&  9&  6&  4& 7&  6\\
+		\end{pmatrix}
+		\cdot
+		\begin{pmatrix}
+			5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+		\end{pmatrix}
+		\]
+		
+		\begin{itemize}
+			\item $m = [4,7,2,5,8,1]$
+		\end{itemize}
+		
+	\end{frame}
+%-------------------------------------------------------------------------------
+
+\end{document}
diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.toc b/buch/papers/reedsolomon/RS presentation/RS_handout.toc
new file mode 100644
index 0000000..ce1bdc2
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/RS_handout.toc
@@ -0,0 +1,9 @@
+\babel@toc {ngerman}{}
+\beamer@sectionintoc {1}{Einführung}{2}{0}{1}
+\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2}
+\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3}
+\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4}
+\beamer@sectionintoc {5}{Codierung eines Beispiels}{18}{0}{5}
+\beamer@sectionintoc {6}{Decodierung ohne Fehler}{20}{0}{6}
+\beamer@sectionintoc {7}{Decodierung mit Fehler}{23}{0}{7}
+\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{29}{0}{8}
diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.pdf b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf
new file mode 100644
index 0000000..abde60c
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf
diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.svg b/buch/papers/reedsolomon/RS presentation/images/fig1.svg
new file mode 100644
index 0000000..8682b56
--- /dev/null
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@@ -0,0 +1,180 @@
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+    ><text x="-7.5" xml:space="preserve" y="14" style="stroke:none;"
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+      >0</text
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
+      >0.4</text
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
+      >0.6</text
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
+      >0.8</text
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+    ><text x="-8" xml:space="preserve" y="5.5" style="stroke:none;"
+      >1</text
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
+      >1.2</text
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
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+    ><text x="-19" xml:space="preserve" y="5.5" style="stroke:none;"
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diff --git a/buch/papers/reedsolomon/RS presentation/images/fig4.pdf b/buch/papers/reedsolomon/RS presentation/images/fig4.pdf
new file mode 100644
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diff --git a/buch/papers/reedsolomon/RS presentation/images/fig4.svg b/buch/papers/reedsolomon/RS presentation/images/fig4.svg
new file mode 100644
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diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.pdf b/buch/papers/reedsolomon/RS presentation/images/polynom1.pdf
new file mode 100644
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diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz b/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz
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diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1.tex
new file mode 100644
index 0000000..db83daa
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+% polynome1
+%-------------------
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\newcommand{\teiler}{40}
+\begin{document}
+	%Übertragen von den Zahlen 
+	%\textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} 
+	%als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline
+	%Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, 
+	%	\textcolor{green}{15}, \textcolor{green}{26},
+	%	\textcolor{green}{ 41}, \textcolor{green}{60}, 
+	%	\textcolor{green}{83}, \textcolor{green}{110})$
+
+
+\begin{tikzpicture}[>=latex,thick]
+
+\draw[color=blue, line width=1.4pt] 
+plot[domain=0:8, samples=100]
+	({\x},{(2*\x^2+1*\x+5)/\teiler});
+\draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}];
+\draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}];
+\def\punkt#1{
+	\fill[color=green] #1 circle[radius=0.08];
+	\draw #1 circle[radius=0.07];
+}
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+\punkt{(7,110/\teiler)}
+%\draw[color=gray,line width=1pt,dashed] 
+%plot[domain=0.5:7, samples=100]
+%({\x},{(0.1958*\x^2-1.2875*\x+3.0417)});
+%\def\erpunkt#1{
+%	\fill[color=red] #1 circle[radius=0.08];
+%	\draw #1 circle[radius=0.07];
+%}
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+%\erpunkt{(3,0.9414)}
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+
+
+
+\end{tikzpicture}
+\end{document}
+
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diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf
new file mode 100644
index 0000000..05f4ba0
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf
diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz b/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz
new file mode 100644
index 0000000..24859aa
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz
diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.tex b/buch/papers/reedsolomon/RS presentation/images/polynom2.tex
new file mode 100644
index 0000000..aa792ce
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/images/polynom2.tex
@@ -0,0 +1,57 @@
+% polynome2
+%-------------------
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\newcommand{\teiler}{40}
+\begin{document}
+	%Übertragen von den Zahlen 
+	%\textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} 
+	%als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline
+	%Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, 
+	%	\textcolor{green}{15}, \textcolor{green}{26},
+	%	\textcolor{green}{ 41}, \textcolor{green}{60}, 
+	%	\textcolor{green}{83}, \textcolor{green}{110})$
+	
+	
+	\begin{tikzpicture}[>=latex,thick]
+		
+		\draw[color=blue, line width=1.4pt] 
+		plot[domain=0:8, samples=100]
+		({\x},{(2*\x^2+1*\x+5)/\teiler});
+		\draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}];
+		\draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}];
+		\def\punkt#1{
+			\fill[color=green] #1 circle[radius=0.08];
+			\draw #1 circle[radius=0.07];
+		}
+		\punkt{(1,8/\teiler)}
+		%\punkt{(2,15/\teiler)}
+		%\punkt{(3,26/\teiler)}
+		\punkt{(4,41/\teiler)}
+		\punkt{(5,60/\teiler)}
+		\punkt{(6,83/\teiler)}
+		\punkt{(7,110/\teiler)}
+		\draw[color=gray,line width=1pt,dashed] 
+		plot[domain=0.5:7, samples=100]
+		({\x},{(0.1958*\x^2-1.2875*\x+3.0417)});
+		\def\erpunkt#1{
+			\fill[color=red] #1 circle[radius=0.08];
+			\draw #1 circle[radius=0.07];
+		}
+		\erpunkt{(2,50/\teiler)}
+		\erpunkt{(3,0.9414)}
+
+		
+		\draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}];
+		\draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}];
+		
+		
+		
+		
+	\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex
new file mode 100644
index 0000000..0339d9c
--- /dev/null
+++ b/buch/papers/reedsolomon/codebsp.tex
@@ -0,0 +1,197 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Codierung eines Beispiels
+\label{reedsolomon:section:codebsp}}
+\rhead{Codierung eines Beispiels}
+
+Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir die einzelnen Probleme und ihre Lösungen anhand eines Beispiels betrachten.
+Da wir in endlichen Körpern rechnen, werden wir zuerst solch einen Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6 (verweis auf eine Definition im Buch ohne label)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen.
+Wir legen für unser Beispiel den endlichen Körper $\mathbb{F}_{q}$ mit $q = 11$ fest.
+Zur Hilfestellung können dazu die beiden Tabellen \ref{reedsolomon:subsection:adtab} und
+\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten die Resultate der arithmetischen Operationen im Körper $\mathbb{F}_{11}$, die durchgeführt werden können.
+Aus der Definition der endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen und somit $11 = 0$ gelten muss.
+
+% OLD TEXT
+%Alle folgenden Berechnungen wurden mit den beiden Restetabellen \ref{reedsolomon:subsection:adtab} und \ref{reedsolomon:subsection:mptab} durchgeführt.
+%Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen.
+
+% die beiden Restetabellen von F_11
+%\input{papers/reedsolomon/restetabelle1}
+%\input{papers/reedsolomon/restetabelle2}
+
+Die Menge uns zur Verfügung stehender Zahlen legt auch fest, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann.
+Der Nachrichtenblock im Beispiel besteht aus
+\[
+n = q - 1 = 10 \text{ Zahlen},
+\]
+wobei die null weggelassen wird. Wenn wir versuchen würden, mit der null zu codieren, so stellen wir fest, dass wir wieder null an der gleichen Stelle erhalten und somit wäre die Codierung nicht eindeutig.
+
+% Notes
+%Da bei allen Codes, die codiert werden wird an der gleichen Stelle eine Nullstelle auftreten.
+
+% Old Text
+%Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel
+%\[
+%n = q - 1 = 10 \text{ Zahlen}.
+%\]
+
+Im nächsten Schritt bestimmen wir, wie viele Fehler $t$ maximal während der Übertragung auftreten dürfen, damit wir sie noch korrigieren können.
+Unser Beispielcode sollte in der Lage sein
+\[
+t = 2
+\]
+Fehlerstellen korrigieren zu können.
+
+Die Grösse des Nutzdatenteils hängt von der Grösse des Nachrichtenblocks sowie der Anzahl der Fehlerkorrekturstellen ab. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig.
+Bei maximal 2 Fehler können wir noch
+\[
+k = n - 2t = 6\text{ Zahlen}
+\]
+übertragen. 
+
+Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist, aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann.
+
+Wir legen nun für das Beispiel die Nachricht
+\[
+m = [0,0,0,0,4,7,2,5,8,1]
+\]
+fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Stellen für die Fehlerkorrektur zuständig sind.
+Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht fehlerfrei übertragen worden.
+
+Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachricht auch noch als Polynom
+\[
+m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1
+\] 
+dar.
+
+% Old Text
+%Die Nachricht können wir auch als Polynom 
+%\[
+%m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1
+%\] 
+%darstellen.
+
+\subsection{Der Ansatz der diskreten Fouriertransformation
+	\label{reedsolomon:subsection:diskFT}}
+
+In einem vorherigen Abschnitt \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert.
+Wir wählen deshalb eine Zahl $a$, die die gleichen Aufgaben haben soll wie $e^{\frac{j}{2 \pi}}$ in der diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ ist. Dazu soll die Potenz von $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um
+\[
+\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}
+\]
+in
+\[
+\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}.
+\]
+umzuschreiben.
+% Old Text
+%Wir suchen also eine Zahl $a$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann.
+%Dazu schreiben wir
+%\[
+%\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}
+%\]
+%um in 
+%\[
+%\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}.
+%\]
+%
+%Wenn wir alle möglichen Werte für $a$ einsetzen, also
+%\begin{align}
+%a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\
+%a = 1 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\} \\
+%a = 2 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\} \\
+%a = 3 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\} \\
+%a = 4 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\} \\
+%a = 5 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\} \\
+%a = 6 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\} \\
+%a = 7 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\} \\
+%a = 8 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\} \\
+%a = 9 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\} \\
+%a = 10 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}
+%\end{align}
+
+\subsubsection{Die primitiven Einheitswurzeln
+	\label{reedsolomon:subsection:primsqrt}}
+
+Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen
+\begin{center}
+\begin{tabular}{c c c c c c c}
+$a = 1$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 2$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 3$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 4$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 5$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 6$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 7$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 8$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 9$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+$a = 10$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\
+\end{tabular}
+\end{center}
+%\begin{center}
+%\begin{tabular}{c r c l}
+%%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\
+%$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\
+%$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\
+%$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\
+%$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\
+%$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\
+%$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\
+%$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\
+%$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\
+%$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\
+%$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$	
+%\end{tabular}
+%\end{center}
+so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em primitive Einheitswurzel \em genannt. 
+Wenden wir diese Vorgehensweise auch für andere endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzel auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a = 8$.
+
+\subsubsection{Bildung einer Transformationsmatrix
+	\label{reedsolomon:subsection:transMat}}
+
+Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben, ergibt sich daraus 
+%
+%Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen.
+%
+\begin{center}
+	\begin{tabular}{c}
+		$m(8^0) = 4 \cdot 1^5 + 7 \cdot 1^4 + 2 \cdot 1^3 + 5 \cdot 1^2 + 8 \cdot 1^1 + 1 = 5$ \\
+		$m(8^1) = 4 \cdot 8^5 + 7 \cdot 8^4 + 2 \cdot 8^3 + 5 \cdot 8^2 + 8 \cdot 8^1 + 1 = 3$ \\
+		\vdots \\
+		$m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$
+	\end{tabular}
+\end{center}
+als unser Übertragungsvektor. 
+
+\subsection{Allgemeine Codierung
+	\label{reedsolomon:subsection:algCod}}
+Um das Ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen, die unsere Transformationsmatrix $A$ bildet.
+
+Für die allgemeine Codierung benötigen wir die Nachricht $m$, die codiert werden soll, sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung:
+\[
+v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+	8^0&	8^3&	8^6&	8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+	8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+	8^0&	8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\
+	8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\
+\end{pmatrix}
+.
+\]
+Für unseren Übertragungsvektor resultiert
+\[
+v = [5,3,6,5,2,10,2,7,10,4],
+\]
+den wir jetzt über einen beliebigen Nachrichtenkanal versenden können.
diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex
new file mode 100644
index 0000000..a46d7da
--- /dev/null
+++ b/buch/papers/reedsolomon/decmitfehler.tex
@@ -0,0 +1,319 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Decodierung: Ansatz mit Fehlerkorrektur
+\label{reedsolomon:section:decmitfehler}}
+\rhead{Decodierung mit Fehler}
+Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Übertragungsvektor fehlerlos versendet und empfangen wurde.
+In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen.
+Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt.
+In diesem Abschnitt betrachten wir somit die Idee der Fehlerkorrektur und wie wir diese auf unser Beispiel anwenden können.
+
+Der Übertragungskanal im Beispiel weisst jetzt den Fehlervektor 
+\[
+u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0]
+\]
+auf.
+Senden wir jetzt unser Übertragungsvektor $v$ durch diesen Kanal addiert sich der Fehlervektor $u$ auf unsere Übertragung und wir erhalten 
+\begin{center}
+	
+	\begin{tabular}{c | c r }
+		$v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\
+		$u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\
+		\hline
+		$w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\
+	\end{tabular}
+	
+	% alternative design
+	%\begin{tabular}{c | c cccccccccccc }
+	%	$v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\
+	%	$u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\
+	%	\hline
+	%	$w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\
+	%\end{tabular}
+	
+\end{center}
+als neuen, fehlerbehafteten Übertragungsvektor $w$ auf der Empfängerseite.
+% Old Text
+%In diesem Abschnitt gehen wir genauer darauf ein, wie der Reed-Solomon-Code eine solche Feherkorrektur vornimt. 
+%
+%In diesem Abschnitt betrachten wir das Problem, dass während der Übertragung des Übertragungsvektors von unserem Beispiel 
+%
+%
+%Zu diesem Zweck wurden Fehler korrigierende Codes entwickelt.
+%
+%Dieser Optimalfall kann jedoch mit keinem Übertragungskanal garantiert werden
+%
+%
+%Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist.
+%Wir legen daher den Fehlervektor
+%\[
+%u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0]
+%\]
+%fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit
+%
+%\begin{center}
+%
+%\begin{tabular}{c | c r }
+%	$v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\
+%	$u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\
+%	\hline
+%	$w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\
+%\end{tabular}
+%
+%% alternative design
+%%\begin{tabular}{c | c cccccccccccc }
+%%	$v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\
+%%	$u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\
+%%	\hline
+%%	$w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\
+%%\end{tabular}
+%
+%\end{center}
+%als Übertragungsvektor auf der Empfängerseite erhalten. 
+Als Empfänger wissen wir jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden.
+Wir stellen jedoch recht schnell fest, dass am decodierten Nachrichtenblock
+\[
+r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7]
+\]
+etwas nicht in Ordnung ist, denn die vorderen vier Fehlerkorrekturstellen haben nicht mehr den Wert null.
+Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass der Übertragungsvektor fehlerhaft empfangen wurde.
+% Old Text
+%Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir
+%\[
+%r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7].
+%\]
+Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche Stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind.
+
+\subsection{Das Fehlerstellenpolynom $d(X)$
+	\label{reedsolomon:subsection:fehlerpolynom}}
+Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die fehlerhaften von den korrekten Stellen im Übertragungsvektor unterscheiden zu können. 
+In einem ersten Versuch berechnen wir die Differenz $d$ des empfangenen und dem gesendeten Übertragungsvektor mit
+%Alle Stellen in $d$, die nicht null sind sind demnach fehler. 
+%
+%In einem ersten Versuch könnten wir $d$ berechnen mit
+\begin{center}
+\begin{tabular}{r c l}
+	$m(X)$ & $=$ & $4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \\
+	$r(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ \\
+	$d(X)$ & $=$ & $r(X) - m(X)$
+\end{tabular}
+\end{center}
+und nennen $d(X)$ als unseres Fehlerstellenpolynom. Dieses Polynom soll uns sagen, welche Stellen korrekt und welche fehlerhaft sind. 
+
+Durch das verwenden von $m(X)$ stossen wir auf weitere Probleme, da wir den Nachrichtenvektor auf der Empfängerseite nicht kennen (unser Ziel ist es ja genau diesen zu finden). Dieses Problem betrachten wir im Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} genauer. Um die Überlegungen in den folgenden Abschnitten besser zu verstehen sei $m(X)$ bekannt auf der Empfängerseite.
+
+%Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen.
+
+Setzen wir jetzt unsere Einheitswurzel aus dem Beispiel ein so erhalten wir
+% Old Text
+%\begin{align}
+%	m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\
+%	r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\
+%	e(X) & = r(X) - m(X).
+%\end{align}
+%Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir
+\begin{center}
+\begin{tabular}{c c c c c c c c c c c}
+	\hline
+	$i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\
+	\hline
+	$r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\
+	$m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\
+	$d(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\
+	\hline
+\end{tabular}
+\end{center}
+und damit die Information, dass allen Stellen, die nicht Null sind, Fehler enthalten. 
+Aus der Tabelle lesen wir, das in unserem Beispiel die Fehler an der Stelle drei und acht zu finden sind.
+
+Für das einfache Bestimmen von Hand mag dies ja noch ausreichen, jedoch können wir mit diesen Stellen nicht das Lokatorpolynom bestimmen, denn dafür bräuchten wir alle Nullstellen, an denen es Fehler gegeben hat (also sozusagen genau das umgekehrte). Um dies zu erreichen wenden wir eine andere Herangehensweise und nehmen uns den Satz von Fermat sowie den kleinsten gemeinsamen Teiler zur Hilfe.
+
+\subsection{Mit dem grössten gemeinsamen Teiler auf Nullstellenjagd
+\label{reedsolomon:subsection:ggT}}
+
+Zuerst betrachten wir den Satz von Fermat, dessen Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass
+\[
+f(X) = X^{q-1} -1 = 0
+\] 
+gilt für jedes $X$. Setzen wir das $q$ von unserem Beispiel ein
+\[
+f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\}
+\]
+und stellen dies als Faktorisierung dar. So ergibt sich die Darstellung 
+\[
+f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9).
+\]
+Zur Überprüfung können wir unsere Einheitswurzel in $a$ einsetzen und werden sehen, dass wir für $f(X) = 0$ erhalten werden.
+
+Wir können jetzt auch $d(X)$ nach der gleichen Überlegung darstellen als 
+\[
+d(X) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \cdot p(x),
+\]
+wobei diese Darstellung nicht mehr alle Nullstellen umfasst wie es noch in $f(X)$ der Fall war. 
+Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ zusammen.
+Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $d(X)$ berechnen, so erhalten wir mit 
+\[
+\operatorname{ggT}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9)
+\]
+eine Liste von Nullstellen, an denen es keine Fehler gegeben hat.
+Dies scheint zuerst nicht sehr hilfreich zu sein, da wir für das Lokatorpolynom ja eine Liste der Nullstellen suchen, an denen es Fehler gegeben hat. Aus diesem Grund berechnen wir im nächsten Schritt das kleinste gemeinsame Vielfache von $f(X)$ und $d(X)$. 
+
+%Wir werden auch feststellen, das unsere Bemühungen bisher nicht umsonst waren.  
+
+\subsection{Mit dem kgV fehlerhafte Nullstellen finden
+	\label{reedsolomon:subsection:kgV}}
+
+Das kgV hat nämlich die Eigenschaft sämtliche Nullstellen zu finden, also nicht nur die fehlerhaften sondern auch die korrekten, was in 
+\[
+\operatorname{kgV}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X).
+\]
+ersichtlich ist.
+Aus dem vorherigen Abschnitt wissen wir auch, dass $d(X)$ alle korrekten Nullstellen beinhaltet. Teilen wir das kgV jetzt auf in 
+\[
+\operatorname{kgV}(f(X),d(X)) = d(X) \cdot l(X)
+\]
+sollten wir für $l(X)$ eine Liste mit allen fehlerhaften Nullstellen erhalten.
+Somit ist 
+\[
+l(X) = (X-a^3)(X-a^8)
+\]
+unser gesuchtes Lokatorpolynom. 
+Es scheint so als müssten wir nur noch an den besagten Stellen den Übertragungsvektor korrigieren und wir währen fertig mit der Fehlerkorrektur.
+Jedoch haben wir noch ein grundlegendes Problem, dass zu Beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben, auf der Empfängerseite aber nicht kennen.
+
+\subsection{Der problematische Nachrichtenvektor $m(X)$
+	\label{reedsolomon:subsection:nachrichtenvektor}}
+
+In Abschnitt \ref{reedsolomon:section:decmitfehler} haben wir
+\[
+d(X) = r(X) - m(X)
+\]
+in Abhängigkeit von $m(X)$ berechnet. 
+Jedoch haben wir ausser acht gelassen, dass $m(X)$ auf der Empfängerseite nicht existiert und somit gänzlich unbekannt ist.
+Es scheint so als würde dieser Lösungsansatz, den wir bisher verfolgt haben, nicht funktioniert.
+Wir könnten uns höchstens noch fragen, ob wir tatsächlich nichts über den Nachrichtenvektor im Beispiel wissen. Wenn wir noch einmal den Vektor betrachten als
+\[
+m = [0,0,0,0,4,7,2,5,8,1]
+\]
+fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten $2t$ (im Beispiel vier) stellen.
+Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten $k$ stellen (im Beispiel sechs) für uns unbekannt, dargestellt als
+\[
+m = [0,0,0,0,?,?,?,?,?,?].
+\]
+Nach der Definition des Reed-Solomon-Codes soll an genau diesen vier Stellen auch die Information befinden, wo die Fehlerstellen liegen. Daher reicht es auch aus
+% darum werden die stellen auch als fehlerkorrekturstellen bezeichnet
+\[
+d(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)
+\]
+so zu berechnen, dass wir die wichtigen vier Stellen kennen, der Rest des Polynoms jedoch im unbekannten Restpolynom $p(X)$ enthalten ist. 
+
+\subsection{Die Berechnung der Fehlerstellen
+	\label{reedsolomon:subsection:nachrichtenvektor}}
+
+Um die Fehlerstellen zu berechnen wenden wir die gleiche Vorgehensweise wie zuvor an, also zuerst den ggT, danach berechnen wir das kgV um am Ende das Lokatorpolynom zu erhalten.
+
+\subsubsection{Schritt 1: ggT}
+
+Wir berechnen den ggT von $f(X)$ und $d(X)$ mit
+\begin{center}
+\begin{tabular}{r c l}
+	$f(X)$ & $=$ & $X^{10} - 1 = X^{10} + 10$ \\
+	$d(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$
+\end{tabular}
+\end{center} 
+%
+%
+%
+%Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus
+%\[
+%e(X) = r(X) - m(X),
+%\]
+%wobei $m(X)$ auf der Empfängerseite unbekannt ist.
+%Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt.
+%Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen.
+%Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen.
+%\[
+%m = [0,0,0,0,?,?,?,?,?,?]
+%\]
+%An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert.
+%
+%Wir können $e(X)$ also bestimmen als
+%\[
+%e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)
+%\]
+%wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und
+%\[
+%f(X) = X^{10} - 1 = X^{10} + 10
+%\]
+%ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen.
+%Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so
+%
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+	X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\
+	X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& &  & & & &   & & & & & &   & &  & & \\ \cline{1-9}
+	&& 3X^9&+& 8X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\
+	&& 3X^9&+& 2X^8&+& 9X^7&+& p(X)& &   & & & & & &   & &  & & \\ \cline{3-9}
+	& &    & &6X^8&+&0X^7&+&p(X)& &   & & & & & &   & &  & & \\
+\end{array}
+\]
+
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr}
+	5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\
+	5X^9&+& 0X^8&+& p(X)& & & & & &  & & & &   & & & & & &   & &  & & \\ \cline{1-5}
+	&& 7X^8&+& p(X)& & & & & &   & & & & & &   & &  & & \\
+\end{array}
+\]
+und erhalten
+\[
+\operatorname{ggT}(f(X),e(X)) = 6X^8.
+\]
+
+\subsubsection{Schritt 2: kgV}
+
+Mit dem Resultat das wir vom ggT erhalten haben können wir jetzt das kgV berechnen. Dazu können wir jetzt den erweiterten Euklidischen Algorithmus verwenden, den wir in Abschnitt \ref{buch:subsection:daskgv} kennengelernt haben.
+%
+%Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. 
+%
+%Aus diesem erweiterten Euklidischen Algorithmus erhalten wir 
+\begin{center}
+	
+	\begin{tabular}{| c | c | c c |}
+		\hline
+		$k$ &  $q_i$ & $e_i$ & $f_i$\\
+		\hline 
+		& & $0$& $1$\\
+		$0$& $9X + 5$& $1$& $0$\\
+		$1$& $10X + 3$& $9X+5$& $1$\\
+		$2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\
+		\hline
+	\end{tabular}	
+	
+\end{center}
+Daraus erhalten wir die Faktoren
+\[
+l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6).
+\]
+\subsubsection{Schritt 3: Fehlerstellen bestimmen}
+Unser gesuchtes Lokatorpolynom hat also die Form
+\[
+l(X) = (X-a^i)(X-a^j).
+\]
+Also brauchen wir nur noch $i$ und $j$ zu berechnen und wir haben unsere gesuchten Fehlerstellen.
+Diese bekommen wir recht einfach mit
+\begin{center}
+	$a^i = 5 \qquad \Rightarrow \qquad i = 3$
+	
+	$a^j = 6 \qquad \Rightarrow \qquad j = 8$.
+\end{center}
+Schlussendlich erhalten wir
+\[
+d(X) = (X-a^3)(X-a^8)
+\]
+als unser Lokatorpolynom mit den fehlerhaften Stellen.
diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex
new file mode 100644
index 0000000..0470db0
--- /dev/null
+++ b/buch/papers/reedsolomon/decohnefehler.tex
@@ -0,0 +1,207 @@
+%
+% teil3.tex -- Beispiel-File für Teil 3
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Decodierung: Ansatz ohne Fehler
+\label{reedsolomon:section:decohnefehler}}
+\rhead{Decodierung ohne Fehler}
+
+In diesem Abschnitt betrachten wie die Überlegung, wie wir auf der Empfängerseite die Nachricht aus dem empfangenen Übertragungsvektor erhalten. Nach einer einfachen Überlegung müssen wir den Übertragungsvektor decodieren, was auf den ersten Blick nicht allzu kompliziert sein sollte, solange wir davon ausgehen können, dass es während der Übertragung keine Fehler gegeben hat. Wir betrachten deshalb den Übertragungskanal als fehlerfrei.
+
+Der Übertragungsvektor empfangen wir also als
+\[
+v = [5,3,6,5,2,10,2,7,10,4].
+\]
+% Old Text
+%Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei.
+%Wir erhalten also unseren Übertragungsvektor
+%\[
+%v = [5,3,6,5,2,10,2,7,10,4].
+%\]
+Nach einem banalen Ansatz ist die Decodierung die Inverse der Codierung. Dank der Matrixschreibweise lässt sich dies relativ einfach umsetzen.
+% Old Text
+%Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können.
+%Ein banaler Ansatz ist das Invertieren der Glechung
+\[
+v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v
+\]
+Nur stellt sich jetzt die Frage, wie wir die Inverse von $A$ berechnen.
+Dazu können wir wiederum den Ansatz der Fouriertransformation uns zur Hilfe nehmen,
+jedoch betrachten wir jetzt deren Inverse.
+Definiert ist sie als
+\[
+F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega.
+\]
+Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:sfaktor} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. 
+\[
+8^1 \qquad \rightarrow \qquad 8^{-1}
+\]
+Mit einem solchen Problem haben wir uns bereits in Abschnitt \ref{buch:section:euklid} befasst und so den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können.
+
+% Old Text
+%Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können.
+
+\subsection{Inverse der primitiven Einheitswurzel
+\label{reedsolomon:subsection:invEinh}}
+
+Die Funktionsweise des euklidischen Algorithmus ist im Abschnitt \ref{buch:section:euklid} ausführlich beschrieben.
+Für unsere Anwendung wählen wir die Parameter $a = 8$ und $b = 11$ ($\mathbb{F}_{11}$).
+Daraus erhalten wir 
+
+\begin{center}
+
+\begin{tabular}{| c | c c | c | r r |}
+	\hline
+	$k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\
+	\hline 
+	& & & & $1$& $0$\\
+	$0$& $8$& $11$& $0$& $0$& $1$\\
+	$1$& $11$& $8$& $1$& $1$& $0$\\
+	$2$& $8$& $3$& $2$& $-1$& $1$\\
+	$3$& $3$& $2$& $1$& $3$& $-2$\\
+	$4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\
+	$5$& $1$& $0$& & $11$& $-8$\\
+	\hline
+\end{tabular}
+
+\end{center}
+\begin{center}
+
+\begin{tabular}{rcl}
+	$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\
+	$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\
+	$8^{-1}$ &$=$& $7$
+	
+\end{tabular}
+
+\end{center}
+als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir, indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen:
+\[
+\begin{pmatrix}
+	8^0 & 8^0 & 8^0 & 8^0 & \dots & 8^0 \\
+	8^0 & 8^{-1} & 8^{-2} & 8^{-3} & \dots & 8^{-9} \\
+	8^0 & 8^{-2} & 8^{-4} & 8^{-6} & \dots & 8^{-18} \\
+	8^0 & 8^{-3} & 8^{-6} & 8^{-9} & \dots & 8^{-27} \\
+ 	\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
+	8^0 & 8^{-9} & 8^{-18} & 8^{-27} & \dots & 8^{-81} \\
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\begin{pmatrix}
+	7^0 & 7^0 & 7^0 & 7^0 & \dots & 7^0 \\
+	7^0 & 7^{1} & 7^{2} & 7^{3} & \dots & 7^{9} \\
+	7^0 & 7^{2} & 7^{4} & 7^{6} & \dots & 7^{18} \\
+	7^0 & 7^{3} & 7^{6} & 7^{9} & \dots & 7^{27} \\
+	\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
+	7^0 & 7^{9} & 7^{18} & 7^{27} & \dots & 7^{81} \\
+\end{pmatrix}
+\] 
+
+\subsection{Der Faktor $s$
+	\label{reedsolomon:subsection:sfaktor}}
+Die diskrete Fouriertransformation benötigt für die Inverse einen Vorfaktor von $\frac{1}{2\pi}$.
+Primitiv nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls einen benötigen.
+Nur stellt sich jetzt die Frage, wie wir diesen Vorfaktor in unserem Fall ermitteln können.
+Dafür betrachten wir eine Regel aus der Linearen Algebra, nämlich dass
+
+\[
+A \cdot A^{-1} = E
+\]
+entsprechen muss.
+Ist dies nicht der Fall, so benötigt $A^{-1}$ eben genau diesen Korrekturfaktor und ändert die Gleichung so zu
+\begin{equation}
+	A \cdot s \cdot A^{-1} = E.
+	\label{reedsolomon:equation:sfaktor}
+\end{equation}
+%\[
+%A \cdot s \cdot A^{-1} = E.
+%\]
+Somit sollte es für uns ein leichtes Spiel sein, $s$ für unser Beispiel zu ermitteln:
+\[
+\begin{pmatrix}
+	8^0 & 8^0 & 8^0 & \dots & 8^0 \\
+	8^0 & 8^1 & 8^2 & \dots & 8^9 \\
+	8^0 & 8^2 & 8^4 & \dots & 8^{18} \\
+	\vdots & \vdots & \vdots & \ddots & \vdots \\
+	8^0 & 8^9 & 8^{18} & \dots & 8^{81} \\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	7^0 & 7^0 & 7^0 & \dots & 7^0 \\
+	7^0 & 7^{1} & 7^{2} & \dots & 7^{9} \\
+	7^0 & 7^{2} & 7^{4} & \dots & 7^{18} \\
+	\vdots & \vdots & \vdots & \ddots & \vdots \\
+	7^0 & 7^{9} & 7^{18} & \dots & 7^{81} \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	10 & 0 & 0 & \dots & 0 \\
+	0 & 10 & 0 & \dots & 0 \\
+	0 & 0 & 10 & \dots & 0 \\
+	\vdots & \vdots & \vdots  & \ddots & \vdots \\
+	0 & 0 & 0 & \dots & 10 \\
+\end{pmatrix}
+\]
+Aus der letzten Matrix folgt, dass wir
+\[
+s = \dfrac{1}{10}
+\]
+als unseren Vorfaktor setzen müssen um die Gleichung \ref{reedsolomon:equation:sfaktor} zu erfüllen. Da wir in $\mathbb{F}_{11}$ nur mit ganzen Zahlen arbeiten schreiben wir $\frac{1}{10}$ in $10^{-1}$ um und bestimmen diese Inverse erneut mit dem euklidischen Algorithmus und erhalten für $10^{-1} = 10$ als unseren Vorfaktor in $\mathbb{F}_{11}$.
+%
+%erfüllt wird. Wir schreiben den Bruch um in $\frac{1}{10} = 10^{-1}$ und wenden darauf erneut den euklidischen Algorithmus an und erhalten somit den Vorfaktor $10^{-1} = 10 = s$ in $\mathbb{F}_{11}$.
+%
+%Um $s$ eindeutig zu bestimmen müssen wir $\frac{1}{10}$ nur noch in den Bereich von $\mathbb{F}_{11}$ verschieben. Wie sich herausstellt können wir das recht einfach bewerkstelligen, da $\frac{1}{10} = 10^{-1}$ entspricht. Daraus können wir $s$ mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt.
+%
+%Da $s$ jetzt ein Bruch ist brauchen wir ihn nur noch in $\mathbb{F}_{11}$ zu schieben. Praktischerweise können wir $\frac{1}{10} = 10^{-1}$ darstellen 
+%
+%Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt.
+%
+%Daher nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls ein solcher Vorfaktor benötigen. Dieser Faktor hat seinen Ursprung in der Gleichung
+%\[
+%A \cdot A^{-1} = E.
+%\]
+%Sollte diese Gleichung nicht aufgehen, so muss die Inverse mit 
+\subsection{Allgemeine Decodierung
+	\label{reedsolomon:subsection:algdec}}
+
+Wir haben jetzt alles für eine erfolgreiche Rücktransformation vom empfangenen Nachrichtenvektor beisammen. Die allgemeine Gleichung für die Rücktransformation lautet
+\[
+m = s \cdot A^{-1} \cdot v.
+\]
+Setzen wir nun die Werte ein in
+%
+%Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor
+%\[
+%m = \textcolor{red}{s} \cdot A^{-1} \cdot v
+%\]
+%den wir noch bestimmen müssen. 
+%Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt
+%\[
+%s = \frac{1}{10}.
+%\]
+%Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit
+\[
+m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix}
+	7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0&    7^0\\
+	7^0&	7^1&	7^2&	7^3&	7^4&	7^5&	7^6&	7^7&    7^8&	7^9\\
+	7^0&	7^2&	7^4&	7^6&	7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\
+	7^0&	7^3&	7^6&	7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\
+	7^0&	7^4&	7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\
+	7^0&	7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\
+	7^0&	7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\
+	7^0&	7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\
+	7^0&	7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\
+	7^0&	7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\
+\end{pmatrix}
+\]
+und wir erhalten
+\[
+m = [0,0,0,0,4,7,2,5,8,1]
+\]
+als unsere Nachricht zurück.
\ No newline at end of file
diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex
new file mode 100644
index 0000000..19e5dd4
--- /dev/null
+++ b/buch/papers/reedsolomon/endlichekoerper.tex
@@ -0,0 +1,23 @@
+%
+% teil1.tex -- Beispiel-File für das Paper
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Reed-Solomon in Endlichen Körpern
+\label{reedsolomon:section:endlichekoerper}}
+\rhead{Reed-Solomon in endlichen Körpern}
+\[
+\textcolor{red}{\text{TODO: (warten auf den 1. Teil)}}
+\]
+Das Rechnen in endlichen Körpern bietet einige Vorteile:
+
+\begin{itemize}
+	\item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten.
+	
+	\item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. 
+	
+\end{itemize}
+
+Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen.
+
+Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. 
diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex
new file mode 100644
index 0000000..b006f21
--- /dev/null
+++ b/buch/papers/reedsolomon/hilfstabellen.tex
@@ -0,0 +1,19 @@
+%
+% hilfstabellen.tex
+% Autor: Michael Steiner
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Hilfstabellen für $\mathbb{F}_{11}$
+	\label{reedsolomon:section:hilfstabellen}}
+\rhead{Hilfstabellen}
+
+Um das rechnen  zu erleichtern findet man in diesem Abschnitt die Resultate, die bei der Addition und der Multiplikation in $\mathbb{F}_{11}$ resultieren.
+
+\subsection{Additionstabelle
+	\label{reedsolomon:subsection:adtab}}	
+\input{papers/reedsolomon/restetabelle1.tex}
+
+\subsection{Multiplikationstabelle
+	\label{reedsolomon:subsection:mptab}}
+\input{papers/reedsolomon/restetabelle2.tex}
\ No newline at end of file
diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex
index 8219b63..4e2fd60 100644
--- a/buch/papers/reedsolomon/main.tex
+++ b/buch/papers/reedsolomon/main.tex
@@ -1,10 +1,10 @@
 %
 % main.tex -- Paper zum Thema <reedsolomon>
 %
-% (c) 2020 Hochschule Rapperswil
+% (c) 2021 Joshua Bär und Michael Steiner, Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:reedsolomon}}
-\lhead{Thema}
+\chapter{Reed-Solomon-Code\label{chapter:reedsolomon}}
+\lhead{Reed-Solomon-Code}
 \begin{refsection}
 \chapterauthor{Joshua Bär und Michael Steiner}
 
@@ -27,10 +27,25 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
 Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
 \end{itemize}
 
+% Joshua
 \input{papers/reedsolomon/teil0.tex}
 \input{papers/reedsolomon/teil1.tex}
 \input{papers/reedsolomon/teil2.tex}
 \input{papers/reedsolomon/teil3.tex}
 
+% Michael
+\input{papers/reedsolomon/endlichekoerper}
+\input{papers/reedsolomon/codebsp}
+\input{papers/reedsolomon/decohnefehler}
+\input{papers/reedsolomon/decmitfehler}
+\input{papers/reedsolomon/rekonstruktion}
+\input{papers/reedsolomon/zusammenfassung}
+%\input{papers/reedsolomon/anwendungen}		-> geplant
+\input{papers/reedsolomon/hilfstabellen}
+
+\nocite{reedsolomon:weitz}
+\nocite{reedsolomon:informationkommunikation}
+%\nocite{reedsolomon:mendezmueller}
+
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib
index 38613bd..731bd35 100644
--- a/buch/papers/reedsolomon/references.bib
+++ b/buch/papers/reedsolomon/references.bib
@@ -4,32 +4,22 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{reedsolomon:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
+@online{reedsolomon:weitz,
+	title = {Fehlerkorrektur mit Reed-Solomon-Codes},
+	url = {https://youtu.be/uOLW43OIZJ0},
+	date = {2021-06-10},
+	year = {2021},
+	month = {6},
+	day = {10}
 }
 
-@book{reedsolomon:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
-}
-
-@article{reedsolomon:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@book{reedsolomon:informationkommunikation,
+	title = {Information und Kommunikation},
+	author = {Markus Hufschmid},
+	publisher = {Teubner},
+	year = {2007},
+	isbn = {978-3-8351-0122-7},
+	inseries = {},
+	volume = {1}
 }
 
diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex
new file mode 100644
index 0000000..04e748c
--- /dev/null
+++ b/buch/papers/reedsolomon/rekonstruktion.tex
@@ -0,0 +1,188 @@
+%
+% rekonstruktion.tex
+% Autor: Michael Steiner
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Nachricht Rekonstruieren
+\label{reedsolomon:section:rekonstruktion}}
+\rhead{Rekonstruktion der Nachricht}
+Im letzten Abschnitt haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können.
+Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren.
+Das Lokatorpolynom
+\[
+l(X) = (X - a^3)(X-a^8)
+\]
+markiert dabei diese fehlerhaften Stellen im Übertragungsvektor
+\[
+w = [5,3,6,8,2,10,2,7,1,4].
+\]
+Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor ursprünglich codiert haben.
+Unser Ziel ist es wie auch schon im Abschnitt \ref{reedsolomon:section:decohnefehler} eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. 
+Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Abschnitt \ref{reedsolomon:section:decohnefehler} angewendet haben.
+
+Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen:
+\[
+\textcolor{gray}{
+	\begin{pmatrix}
+		a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\
+\end{pmatrix}}
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&	8^9\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+	\textcolor{red}{8^0}&	\textcolor{red}{8^3}&	\textcolor{red}{8^6}&	\textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+	8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+	\textcolor{red}{8^0}&	\textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\
+	8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\
+\end{pmatrix}
+.
+\]
+Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher keinen weiterer Nutzen. 
+Aus diesem Grund werden diese Stellen aus dem Vektor entfernt, was wir hier ohne Probleme machen können, da dieser Code ja über Fehlerkorrekturstellen verfügt, deren Aufgabe es ist, eine bestimmte Anzahl an Fehler kompensieren zu können.
+Die dazugehörigen Zeilen in der Matrix werden ebenfalls entfernt, da die Matrix gleich viele Zeilen wie im Übertragungsvektor aufweisen muss, damit man ihn decodieren kann.
+
+Daraus resultiert
+\[
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	8^6&	8^7&    8^8&    8^9\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\
+	8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\
+	8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\
+\end{pmatrix}
+.
+\]
+Die Matrix ist jedoch nicht mehr quadratisch, was eine Rekonstruktion durch Inversion ausschliesst. 
+Um die quadratische Form wieder herzustellen müssen wir zwei Spalten aus der Matrix entfernen.
+Wir kennen aber das Resultat aus den letzten vier Spalten, da wir wissen, das die Nachricht aus Nutzdatenteil und Fehlerkorrekturteil besteht, wobei der letzteres bekanntlich aus lauter Nullstellen besteht.
+\[
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0&    \textcolor{darkgreen}{8^0}&    \textcolor{darkgreen}{8^0}&    \textcolor{darkgreen}{8^0}&    \textcolor{darkgreen}{8^0}\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5&	\textcolor{darkgreen}{8^6}&	   \textcolor{darkgreen}{8^7}&    \textcolor{darkgreen}{8^8}&    \textcolor{darkgreen}{8^9}\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}& \textcolor{darkgreen}{8^{12}}& \textcolor{darkgreen}{8^{14}}& \textcolor{darkgreen}{8^{16}}& \textcolor{darkgreen}{8^{18}}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{darkgreen}{8^{24}}& \textcolor{darkgreen}{8^{28}}& \textcolor{darkgreen}{8^{32}}& \textcolor{darkgreen}{8^{36}}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{darkgreen}{8^{30}}& \textcolor{darkgreen}{8^{35}}& \textcolor{darkgreen}{8^{40}}& \textcolor{darkgreen}{8^{45}}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{darkgreen}{8^{36}}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{48}}& \textcolor{darkgreen}{8^{54}}\\
+	8^0&	8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{49}}& \textcolor{darkgreen}{8^{56}}& \textcolor{darkgreen}{8^{63}}\\
+	8^0&	8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{darkgreen}{8^{54}}& \textcolor{darkgreen}{8^{63}}& \textcolor{darkgreen}{8^{72}}& \textcolor{darkgreen}{8^{81}}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{darkgreen}{m_6} \\ \textcolor{darkgreen}{m_7} \\ \textcolor{darkgreen}{m_8} \\ \textcolor{darkgreen}{m_9} \\
+\end{pmatrix}
+\]
+Wir nehmen die entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem
+\[
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+	\textcolor{red}{8^0}&	\textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\
+	\textcolor{red}{8^0}&	\textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+\end{pmatrix}
+.
+\]
+Die roten Zeilen können wir aufgrund der Überbestimmtheit ebenfalls entfernen und erhalten so die gesuchte quadratische Matrix
+\[
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	8^0&    8^0&    8^0&    8^0&    8^0&    8^0\\
+	8^0&	8^1&	8^2&	8^3&	8^4&	8^5\\
+	8^0&	8^2&	8^4&	8^6&	8^8& 8^{10}\\
+	8^0&	8^4&	8^8& 8^{12}& 8^{16}& 8^{20}\\
+	8^0&	8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\
+	8^0&	8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+\end{pmatrix}
+.
+\]
+Nun können wir den Gauss-Algorithmus anwenden um die Matrix zu Invertieren.
+\[
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	1&  1& 1&  1& 1&  1\\
+	1&  8& 9&  6& 4& 10\\
+	1&  9& 4&  3& 5&  1\\
+	1&  4& 5&  9& 3&  1\\
+	1& 10& 1& 10& 1& 10\\
+	1&  3& 9&  5& 4&  1\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\begin{pmatrix}
+	m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\
+\end{pmatrix}
+=
+\begin{pmatrix}
+	6&  4&  4&  6& 2&  1\\
+	2&  7& 10&  3& 4&  7\\
+	1&  8&  9&  8& 3&  4\\
+	3&  6&  6&  4& 5&  9\\
+	10& 10&  9&  8& 1&  6\\
+	1&  9&  6&  4& 7&  6\\
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+	5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\
+\end{pmatrix}
+\]
+Multiplizieren wir nun aus, erhalten wir unseren Nutzdatenteil
+\[
+m = [4,7,2,5,8,1]
+\]
+zurück, den wir ursprünglich versendet haben.
+
+Wir möchten noch anmerken, dass es mehrere Wege für die Rekonstruktion des Nutzdatenteils gibt, diese aber alle auf dem Lokatorpolynom basieren. 
+
diff --git a/buch/papers/reedsolomon/restetabelle1.tex b/buch/papers/reedsolomon/restetabelle1.tex
new file mode 100644
index 0000000..3969ef2
--- /dev/null
+++ b/buch/papers/reedsolomon/restetabelle1.tex
@@ -0,0 +1,176 @@
+% created by Michael Steiner
+%
+% Restetabelle von F_11: Addition
+
+% alternatives design
+%\begin{figure}
+%\begin{center}
+%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|}
+%\hline
+%+&0&1&2&3&4&5&6&7&8&9&10\\
+%\hline
+%0&0&1&2&3&4&5&6&7&8&9&10\\
+%1&1&2&3&4&5&6&7&8&9&10&0\\
+%2&2&3&4&5&6&7&8&9&10&0&1\\
+%3&3&4&5&6&7&8&9&10&0&1&2\\
+%4&4&5&6&7&8&9&10&0&1&2&3\\
+%5&5&6&7&8&9&10&0&1&2&3&4\\
+%6&6&7&8&9&10&0&1&2&3&4&5\\
+%7&7&8&9&10&0&1&2&3&4&5&6\\
+%8&8&9&10&0&1&2&3&4&5&6&7\\
+%9&9&10&0&1&2&3&4&5&6&7&8\\
+%10&10&0&1&2&3&4&5&6&7&8&9\\
+%\hline
+%\end{tabular}
+%\end{center}
+%\end{figure}
+
+\begin{center}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.45]
+\fill[color=gray!40] (0,0) rectangle (18,-1.5);
+\fill[color=gray!40] (0,0) rectangle (1.5,-18);	
+\draw[step = 1.5, gray,very thin] (0,0) grid (18,-18);
+\draw[very thick] (0,0) rectangle (18,-18);
+\draw[very thick] (0,-1.5) -- (18,-1.5);
+\draw[very thick] (1.5,0) -- (1.5,-18);
+\node at (0.75,-0.75) {$+$};
+\foreach \x in {0,...,10}
+	\node at (2.25+\x*1.5,-0.75) {$\x$};
+\foreach \y in {0,...,10}
+	\node at (0.75,-2.25+\y*-1.5) {$\y$};
+% Row 0
+\node at ( 2.25,-2.25) {$0$};
+\node at ( 3.75,-2.25) {$1$};
+\node at ( 5.25,-2.25) {$2$};
+\node at ( 6.75,-2.25) {$3$};
+\node at ( 8.25,-2.25) {$4$};
+\node at ( 9.75,-2.25) {$5$};
+\node at (11.25,-2.25) {$6$};
+\node at (12.75,-2.25) {$7$};
+\node at (14.25,-2.25) {$8$};
+\node at (15.75,-2.25) {$9$};
+\node at (17.25,-2.25) {$10$};
+% Row 1
+\node at ( 2.25,-3.75) {$1$};
+\node at ( 3.75,-3.75) {$2$};
+\node at ( 5.25,-3.75) {$3$};
+\node at ( 6.75,-3.75) {$4$};
+\node at ( 8.25,-3.75) {$5$};
+\node at ( 9.75,-3.75) {$6$};
+\node at (11.25,-3.75) {$7$};
+\node at (12.75,-3.75) {$8$};
+\node at (14.25,-3.75) {$9$};
+\node at (15.75,-3.75) {$10$};
+\node at (17.25,-3.75) {$0$};
+% Row 2
+\node at ( 2.25,-5.25) {$2$};
+\node at ( 3.75,-5.25) {$3$};
+\node at ( 5.25,-5.25) {$4$};
+\node at ( 6.75,-5.25) {$5$};
+\node at ( 8.25,-5.25) {$6$};
+\node at ( 9.75,-5.25) {$7$};
+\node at (11.25,-5.25) {$8$};
+\node at (12.75,-5.25) {$9$};
+\node at (14.25,-5.25) {$10$};
+\node at (15.75,-5.25) {$0$};
+\node at (17.25,-5.25) {$1$};
+% Row 3
+\node at ( 2.25,-6.75) {$3$};
+\node at ( 3.75,-6.75) {$4$};
+\node at ( 5.25,-6.75) {$5$};
+\node at ( 6.75,-6.75) {$6$};
+\node at ( 8.25,-6.75) {$7$};
+\node at ( 9.75,-6.75) {$8$};
+\node at (11.25,-6.75) {$9$};
+\node at (12.75,-6.75) {$10$};
+\node at (14.25,-6.75) {$0$};
+\node at (15.75,-6.75) {$1$};
+\node at (17.25,-6.75) {$2$};
+% Row 4
+\node at ( 2.25,-8.25) {$4$};
+\node at ( 3.75,-8.25) {$5$};
+\node at ( 5.25,-8.25) {$6$};
+\node at ( 6.75,-8.25) {$7$};
+\node at ( 8.25,-8.25) {$8$};
+\node at ( 9.75,-8.25) {$9$};
+\node at (11.25,-8.25) {$10$};
+\node at (12.75,-8.25) {$0$};
+\node at (14.25,-8.25) {$1$};
+\node at (15.75,-8.25) {$2$};
+\node at (17.25,-8.25) {$3$};
+% Row 5
+\node at ( 2.25,-9.75) {$5$};
+\node at ( 3.75,-9.75) {$6$};
+\node at ( 5.25,-9.75) {$7$};
+\node at ( 6.75,-9.75) {$8$};
+\node at ( 8.25,-9.75) {$9$};
+\node at ( 9.75,-9.75) {$10$};
+\node at (11.25,-9.75) {$0$};
+\node at (12.75,-9.75) {$1$};
+\node at (14.25,-9.75) {$2$};
+\node at (15.75,-9.75) {$3$};
+\node at (17.25,-9.75) {$4$};
+% Row 6
+\node at ( 2.25,-11.25) {$6$};
+\node at ( 3.75,-11.25) {$7$};
+\node at ( 5.25,-11.25) {$8$};
+\node at ( 6.75,-11.25) {$9$};
+\node at ( 8.25,-11.25) {$10$};
+\node at ( 9.75,-11.25) {$0$};
+\node at (11.25,-11.25) {$1$};
+\node at (12.75,-11.25) {$2$};
+\node at (14.25,-11.25) {$3$};
+\node at (15.75,-11.25) {$4$};
+\node at (17.25,-11.25) {$5$};
+% Row 7
+\node at ( 2.25,-12.75) {$7$};
+\node at ( 3.75,-12.75) {$8$};
+\node at ( 5.25,-12.75) {$9$};
+\node at ( 6.75,-12.75) {$10$};
+\node at ( 8.25,-12.75) {$0$};
+\node at ( 9.75,-12.75) {$1$};
+\node at (11.25,-12.75) {$2$};
+\node at (12.75,-12.75) {$3$};
+\node at (14.25,-12.75) {$4$};
+\node at (15.75,-12.75) {$5$};
+\node at (17.25,-12.75) {$6$};
+% Row 8
+\node at ( 2.25,-14.25) {$8$};
+\node at ( 3.75,-14.25) {$9$};
+\node at ( 5.25,-14.25) {$10$};
+\node at ( 6.75,-14.25) {$0$};
+\node at ( 8.25,-14.25) {$1$};
+\node at ( 9.75,-14.25) {$2$};
+\node at (11.25,-14.25) {$3$};
+\node at (12.75,-14.25) {$4$};
+\node at (14.25,-14.25) {$5$};
+\node at (15.75,-14.25) {$6$};
+\node at (17.25,-14.25) {$7$};
+% Row 9
+\node at ( 2.25,-15.75) {$9$};
+\node at ( 3.75,-15.75) {$10$};
+\node at ( 5.25,-15.75) {$0$};
+\node at ( 6.75,-15.75) {$1$};
+\node at ( 8.25,-15.75) {$2$};
+\node at ( 9.75,-15.75) {$3$};
+\node at (11.25,-15.75) {$4$};
+\node at (12.75,-15.75) {$5$};
+\node at (14.25,-15.75) {$6$};
+\node at (15.75,-15.75) {$7$};
+\node at (17.25,-15.75) {$8$};
+% Row 10
+\node at ( 2.25,-17.25) {$10$};
+\node at ( 3.75,-17.25) {$0$};
+\node at ( 5.25,-17.25) {$1$};
+\node at ( 6.75,-17.25) {$2$};
+\node at ( 8.25,-17.25) {$3$};
+\node at ( 9.75,-17.25) {$4$};
+\node at (11.25,-17.25) {$5$};
+\node at (12.75,-17.25) {$6$};
+\node at (14.25,-17.25) {$7$};
+\node at (15.75,-17.25) {$8$};
+\node at (17.25,-17.25) {$9$};
+\end{tikzpicture}
+
+\end{center}
diff --git a/buch/papers/reedsolomon/restetabelle2.tex b/buch/papers/reedsolomon/restetabelle2.tex
new file mode 100644
index 0000000..1a9815c
--- /dev/null
+++ b/buch/papers/reedsolomon/restetabelle2.tex
@@ -0,0 +1,176 @@
+% created by Michael Steiner
+%
+% Restetabelle von F_11: Multiplikation
+
+% alternatives design
+%\begin{figure}
+%\begin{center}
+%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|}
+%\hline
+%\cdot&0&1&2&3&4&5&6&7&8&9&10\\
+%\hline
+%0&0&0&0&0&0&0&0&0&0&0&0\\
+%1&0&1&2&3&4&5&6&7&8&9&10\\
+%2&0&2&4&6&8&10&1&3&5&7&9\\
+%3&0&3&6&9&1&4&7&10&2&5&8\\
+%4&0&4&8&1&5&9&2&6&10&3&7\\
+%5&0&5&10&4&9&3&8&2&7&1&6\\
+%6&0&6&1&7&2&8&3&9&4&10&5\\
+%7&0&7&3&10&6&2&9&5&1&8&4\\
+%8&0&8&5&2&10&7&4&1&9&6&3\\
+%9&0&9&7&5&3&1&10&8&6&4&2\\
+%10&0&10&9&8&7&6&5&4&3&2&1\\
+%\hline
+%\end{tabular}
+%\end{center}
+%\end{figure}
+
+\begin{center}
+	
+	\begin{tikzpicture}[>=latex,thick,scale=0.45]
+		\fill[color=gray!40] (0,0) rectangle (18,-1.5);
+		\fill[color=gray!40] (0,0) rectangle (1.5,-18);	
+		\draw[step = 1.5, gray,very thin] (0,0) grid (18,-18);
+		\draw[very thick] (0,0) rectangle (18,-18);
+		\draw[very thick] (0,-1.5) -- (18,-1.5);
+		\draw[very thick] (1.5,0) -- (1.5,-18);
+		\node at (0.75,-0.75) {$\cdot$};
+		\foreach \x in {0,...,10}
+		\node at (2.25+\x*1.5,-0.75) {$\x$};
+		\foreach \y in {0,...,10}
+		\node at (0.75,-2.25+\y*-1.5) {$\y$};
+		% Row 0
+		\node at ( 2.25,-2.25) {$0$};
+		\node at ( 3.75,-2.25) {$0$};
+		\node at ( 5.25,-2.25) {$0$};
+		\node at ( 6.75,-2.25) {$0$};
+		\node at ( 8.25,-2.25) {$0$};
+		\node at ( 9.75,-2.25) {$0$};
+		\node at (11.25,-2.25) {$0$};
+		\node at (12.75,-2.25) {$0$};
+		\node at (14.25,-2.25) {$0$};
+		\node at (15.75,-2.25) {$0$};
+		\node at (17.25,-2.25) {$0$};
+		% Row 1
+		\node at ( 2.25,-3.75) {$0$};
+		\node at ( 3.75,-3.75) {$1$};
+		\node at ( 5.25,-3.75) {$2$};
+		\node at ( 6.75,-3.75) {$3$};
+		\node at ( 8.25,-3.75) {$4$};
+		\node at ( 9.75,-3.75) {$5$};
+		\node at (11.25,-3.75) {$6$};
+		\node at (12.75,-3.75) {$7$};
+		\node at (14.25,-3.75) {$8$};
+		\node at (15.75,-3.75) {$9$};
+		\node at (17.25,-3.75) {$10$};
+		% Row 2
+		\node at ( 2.25,-5.25) {$0$};
+		\node at ( 3.75,-5.25) {$2$};
+		\node at ( 5.25,-5.25) {$4$};
+		\node at ( 6.75,-5.25) {$6$};
+		\node at ( 8.25,-5.25) {$8$};
+		\node at ( 9.75,-5.25) {$10$};
+		\node at (11.25,-5.25) {$1$};
+		\node at (12.75,-5.25) {$3$};
+		\node at (14.25,-5.25) {$5$};
+		\node at (15.75,-5.25) {$7$};
+		\node at (17.25,-5.25) {$9$};
+		% Row 3
+		\node at ( 2.25,-6.75) {$0$};
+		\node at ( 3.75,-6.75) {$3$};
+		\node at ( 5.25,-6.75) {$6$};
+		\node at ( 6.75,-6.75) {$9$};
+		\node at ( 8.25,-6.75) {$1$};
+		\node at ( 9.75,-6.75) {$4$};
+		\node at (11.25,-6.75) {$7$};
+		\node at (12.75,-6.75) {$10$};
+		\node at (14.25,-6.75) {$2$};
+		\node at (15.75,-6.75) {$5$};
+		\node at (17.25,-6.75) {$8$};
+		% Row 4
+		\node at ( 2.25,-8.25) {$0$};
+		\node at ( 3.75,-8.25) {$4$};
+		\node at ( 5.25,-8.25) {$8$};
+		\node at ( 6.75,-8.25) {$1$};
+		\node at ( 8.25,-8.25) {$5$};
+		\node at ( 9.75,-8.25) {$9$};
+		\node at (11.25,-8.25) {$2$};
+		\node at (12.75,-8.25) {$6$};
+		\node at (14.25,-8.25) {$10$};
+		\node at (15.75,-8.25) {$3$};
+		\node at (17.25,-8.25) {$7$};
+		% Row 5
+		\node at ( 2.25,-9.75) {$0$};
+		\node at ( 3.75,-9.75) {$5$};
+		\node at ( 5.25,-9.75) {$10$};
+		\node at ( 6.75,-9.75) {$4$};
+		\node at ( 8.25,-9.75) {$9$};
+		\node at ( 9.75,-9.75) {$3$};
+		\node at (11.25,-9.75) {$8$};
+		\node at (12.75,-9.75) {$2$};
+		\node at (14.25,-9.75) {$7$};
+		\node at (15.75,-9.75) {$1$};
+		\node at (17.25,-9.75) {$6$};
+		% Row 6
+		\node at ( 2.25,-11.25) {$0$};
+		\node at ( 3.75,-11.25) {$6$};
+		\node at ( 5.25,-11.25) {$1$};
+		\node at ( 6.75,-11.25) {$7$};
+		\node at ( 8.25,-11.25) {$2$};
+		\node at ( 9.75,-11.25) {$8$};
+		\node at (11.25,-11.25) {$3$};
+		\node at (12.75,-11.25) {$9$};
+		\node at (14.25,-11.25) {$4$};
+		\node at (15.75,-11.25) {$10$};
+		\node at (17.25,-11.25) {$5$};
+		% Row 7
+		\node at ( 2.25,-12.75) {$0$};
+		\node at ( 3.75,-12.75) {$7$};
+		\node at ( 5.25,-12.75) {$3$};
+		\node at ( 6.75,-12.75) {$10$};
+		\node at ( 8.25,-12.75) {$6$};
+		\node at ( 9.75,-12.75) {$2$};
+		\node at (11.25,-12.75) {$9$};
+		\node at (12.75,-12.75) {$5$};
+		\node at (14.25,-12.75) {$1$};
+		\node at (15.75,-12.75) {$8$};
+		\node at (17.25,-12.75) {$4$};
+		% Row 8
+		\node at ( 2.25,-14.25) {$0$};
+		\node at ( 3.75,-14.25) {$8$};
+		\node at ( 5.25,-14.25) {$5$};
+		\node at ( 6.75,-14.25) {$2$};
+		\node at ( 8.25,-14.25) {$10$};
+		\node at ( 9.75,-14.25) {$7$};
+		\node at (11.25,-14.25) {$4$};
+		\node at (12.75,-14.25) {$1$};
+		\node at (14.25,-14.25) {$9$};
+		\node at (15.75,-14.25) {$6$};
+		\node at (17.25,-14.25) {$3$};
+		% Row 9
+		\node at ( 2.25,-15.75) {$0$};
+		\node at ( 3.75,-15.75) {$9$};
+		\node at ( 5.25,-15.75) {$7$};
+		\node at ( 6.75,-15.75) {$5$};
+		\node at ( 8.25,-15.75) {$3$};
+		\node at ( 9.75,-15.75) {$1$};
+		\node at (11.25,-15.75) {$10$};
+		\node at (12.75,-15.75) {$8$};
+		\node at (14.25,-15.75) {$6$};
+		\node at (15.75,-15.75) {$4$};
+		\node at (17.25,-15.75) {$2$};
+		% Row 10
+		\node at ( 2.25,-17.25) {$0$};
+		\node at ( 3.75,-17.25) {$10$};
+		\node at ( 5.25,-17.25) {$9$};
+		\node at ( 6.75,-17.25) {$8$};
+		\node at ( 8.25,-17.25) {$7$};
+		\node at ( 9.75,-17.25) {$6$};
+		\node at (11.25,-17.25) {$5$};
+		\node at (12.75,-17.25) {$4$};
+		\node at (14.25,-17.25) {$3$};
+		\node at (15.75,-17.25) {$2$};
+		\node at (17.25,-17.25) {$1$};
+	\end{tikzpicture}
+	
+\end{center}
\ No newline at end of file
diff --git a/buch/papers/reedsolomon/zusammenfassung.tex b/buch/papers/reedsolomon/zusammenfassung.tex
new file mode 100644
index 0000000..568356f
--- /dev/null
+++ b/buch/papers/reedsolomon/zusammenfassung.tex
@@ -0,0 +1,15 @@
+\section{Zusammenfassung 
+	\label{reedsolomon:section:zf}}
+\rhead{Zusammenfassung}
+Dieser Abschnitt beinhaltet eine Übersicht über die Funktionsweise eines Reed-Solomon-Codes für beliebige endliche Körper.
+
+TODO:
+
+\subsubsection{Schritt 1: primitives Element}
+
+\subsubsection{Schritt 2: Codierung}
+
+\subsubsection{Schritt 3: Decodierung ohne Fehler}
+
+\subsubsection{Schritt 4: Decodierung mit Fehler}
+
diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex
new file mode 100644
index 0000000..b1588ff
--- /dev/null
+++ b/buch/papers/spannung/Einleitung.tex
@@ -0,0 +1,89 @@
+\section{Einleitung\label{spannung:section:Einleitung}}
+\rhead{Einleitung}
+Das Hook'sche Gesetz beschreibt die Beziehung von Spannung und Dehnung von linear-elastischen Materialien im Eindimensionalen.
+In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben.
+Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen.
+In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand.
+Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus.
+Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen.
+Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden.
+Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch.
+
+Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen.
+Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen.
+In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein,
+wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden, z.~B.~\cite{spannung:Grundlagen-der-Geotechnik}.
+
+\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}}
+\rhead{Spannungsausbreitung}
+Die Geotechnik ist eine Ingenieurdisziplin, bei welcher man Erdbau und den Erdbau tangierende Bauwerke dimensioniert.
+Sie beinhaltet aber auch die statische Beurteilung von Boden und Fels.
+
+Belastet man den Boden mit einer Spannung
+\[
+\sigma
+=
+\frac{F}{A}
+,
+\]
+so wird diese in den Boden geleitet und von diesem kompensiert.
+Im Boden entstehen unterschiedlich hohe Zusatzspannungen.
+Diese Zusatzspannung breitet sich räumlich im Boden aus.
+Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bild4} breitet sich die Zusatzspannung zwiebelartig aus.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png}
+	\caption{Ausbreitung der Zusatzspannung im Boden infolge einfacher Flächenlast}
+	\label{fig:Bild4}
+\end{figure}
+
+Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{spannung:Bild5}).
+Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden.
+Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$.
+Je nach Modell werden noch andere Parameter berücksichtigt.
+Das können beispielsweise jenste Bodenkennwerte oder auch der Wassergehalt sein.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png}
+	\caption{Funktionen der Spannung und Dehnung im Zusammenhang mit der Tiefe}
+	\label{fig:Bild5}
+\end{figure}
+
+Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung des Bodens einher, welche eine Setzung bedeutet.
+Im einfachsten Fall kann modellhaft mit
+\[
+\varepsilon
+=
+\frac{\sigma}{E}
+\]
+die Setzung an einem Punkt an der Bodenoberfläche mit
+\[
+s
+=
+\int_{0}^{\infty}\varepsilon\enspace dt
+\]
+berechnet werden mit:
+\begin{align*}
+	\varepsilon &= \text{Dehnung [$-$]}                                     \\
+	     \sigma &= \text{Spannung [\si{\kilo\pascal}]}                      \\
+	          E &= \text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]}\\
+	          t &= \text{Tiefe [\si{\meter}]}                               \\
+	          s &= \text{Setzung, Absenkung [m].}
+\end{align*}
+Diese Zusammenhänge sind wie erwähnt unter anderem im  Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben.
+In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen.
+Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{spannung:Bild3}).
+Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen.
+Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$.
+Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden.
+Diese beruht auf Annahmen nach Hooke auf einem linear-elastischen Boden.
+Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png}
+	\caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat}
+	\label{fig:Bild3}
+\end{figure}
diff --git a/buch/papers/spannung/Grafiken/Bild1.png b/buch/papers/spannung/Grafiken/Bild1.png
new file mode 100644
index 0000000..32b627e
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/Bild1.png
diff --git a/buch/papers/spannung/Grafiken/Bild2.png b/buch/papers/spannung/Grafiken/Bild2.png
new file mode 100644
index 0000000..d1321a4
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/Bild2.png
diff --git a/buch/papers/spannung/Grafiken/Bild3.png b/buch/papers/spannung/Grafiken/Bild3.png
new file mode 100644
index 0000000..8ca72a1
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/Bild3.png
diff --git a/buch/papers/spannung/Grafiken/Bild4.png b/buch/papers/spannung/Grafiken/Bild4.png
new file mode 100644
index 0000000..526ee7b
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/Bild4.png
diff --git a/buch/papers/spannung/Grafiken/Bild5.png b/buch/papers/spannung/Grafiken/Bild5.png
new file mode 100644
index 0000000..6ee004d
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/Bild5.png
diff --git a/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png
new file mode 100644
index 0000000..31505bd
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png
diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png
new file mode 100644
index 0000000..2c359e6
--- /dev/null
+++ b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png
diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex
index 585a423..bbdf730 100644
--- a/buch/papers/spannung/main.tex
+++ b/buch/papers/spannung/main.tex
@@ -4,33 +4,18 @@
 % (c) 2020 Hochschule Rapperswil
 %
 \chapter{Thema\label{chapter:spannung}}
-\lhead{Thema}
+\lhead{Dreiachsiger Spannungszustand}
 \begin{refsection}
 \chapterauthor{Adrian Schuler und Thomas Reichlin}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
+% TODO Text
 
+\input{papers/spannung/Einleitung.tex}
 \input{papers/spannung/teil0.tex}
 \input{papers/spannung/teil1.tex}
 \input{papers/spannung/teil2.tex}
 \input{papers/spannung/teil3.tex}
+\input{papers/spannung/teil4.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/spannung/references.bib b/buch/papers/spannung/references.bib
index ed5703c..02f8d09 100644
--- a/buch/papers/spannung/references.bib
+++ b/buch/papers/spannung/references.bib
@@ -4,27 +4,46 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{spannung:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
+@online{spannung:Tensor,
+	title = {Tensor},
+	url = {https://de.wikipedia.org/wiki/Tensor},
+	date = {2021-05-29},
+	year = {2021},
+	month = {5},
 	day = {6}
 }
 
-@book{spannung:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
+@online{spannung:Voigtsche-Notation,
+	title = {Voigtsche Notation},
+	url = {https://de.wikipedia.org/wiki/Voigtsche_Notation},
+	date = {2021-05-29},
+	year = {2021},
+	month = {5},
+	day = {6}
+}
+
+@book{spannung:Grundlagen-der-Geotechnik,
+	title = {Grundlagen der Geotechnik},
+	author = {Hans-Henning Schmidt and Roland F. Buchmaier and Carola Vogt-Breyer},
+	publisher = {Springer Fachmedien Wiesbaden GmbH},
+	year = {2017},
+	isbn = {978-3-658-14930-7},
+	inseries = {Geotechnik nach Eurocode},
+	volume = {5}
+}
+
+@book{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik,
+	title = {Stoffgesetze und numerische Modellierung in der Geotechnik},
+	author = {Carlo Rabaiotti and Alessio Höttges},
+	publisher = {Hochschule Rapperswil},
+	year = {2021},
+	isbn = {},
+	inseries = {},
+	volume = {}
 }
 
 @article{spannung:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
+        author = { Tabea Méndez and Andreas Müller },
         title = { Noncommutative harmonic analysis and image registration },
         journal = { Appl. Comput. Harmon. Anal.},
         year = 2019,
diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex
index cf47a18..7647252 100644
--- a/buch/papers/spannung/teil0.tex
+++ b/buch/papers/spannung/teil0.tex
@@ -1,22 +1,82 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{spannung:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{spannung:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
+\section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}}
+\rhead{Der Spannungszustand}
+Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{spannung:Bild2}).
+Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material.
+Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt.
+Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist.
 
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
+\begin{figure}
+	\centering
+	\includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png}
+	\caption{Infinitesimales Bodenelement mit den 9 Spannungen}
+	\label{fig:Bild2}
+\end{figure}
 
+Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten im Betrag die selben Spannungen aufweisen,
+sodass der Elementarwürfel im Gleichgewicht ist.
+Wäre dieses Gleichgewicht nicht vorhanden, käme es zu Verschiebungen und Drehungen.
+Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$.
+Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt.
+So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind.
+Normalspannungen wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche.
+Alle Beträge dieser neun Spannungen am Elementarwürfel bilden den Spannungszustand.
+Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden.
+Daher gibt es auch den entsprechenden Dehnungszustand.
 
+
+\section{Spannungszustand\label{spannung:section:Spannungsustand}}
+\rhead{Spannungszustand}
+
+Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}).
+Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand.
+Nach Hooke gilt:
+\[
+F
+\sim
+\Delta l
+.
+\]
+Teilt man beide Seiten durch die Konstanten $A$ und $l_0$, erhält man
+\[
+\frac{F}{A}
+=
+\sigma
+\sim
+\varepsilon
+=
+\frac{\Delta l}{l_0}
+\]
+und somit
+\[
+\sigma
+\sim
+\varepsilon
+,
+\]
+mit
+\begin{align*}
+	l_0 &= \text{Länge zu Beginn [\si{\meter}]} \\
+	  A &= \text{Fläche [\si{\meter\squared}].}
+\end{align*}
+Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen.
+Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$.
+\begin{figure}
+	\centering
+	\includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png}
+	\caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe}
+	\label{fig:Bild1}
+\end{figure}
+Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich der eindimensionale Fall mit
+\[
+\sigma
+=
+E\cdot\varepsilon
+\]
+beschreiben.
+Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch
+\[
+E
+=
+\frac{\Delta\sigma}{\Delta\varepsilon}
+\]
+ausgedrückt werden.
\ No newline at end of file
diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex
index 95e6f0a..74516c1 100644
--- a/buch/papers/spannung/teil1.tex
+++ b/buch/papers/spannung/teil1.tex
@@ -1,55 +1,24 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{spannung:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{spannung:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{spannung:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{spannung:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{spannung:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
+\section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}}
+\rhead{Skalare, Vektoren, Matrizen und Tensoren}
+Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden.
+Allerdings sind noch höhere Stufen dieser Objekte beinhaltet.
+Ein Skalar, ein Vektor oder eine Matrix ist daher auch ein Tensor.
+Ein Skalar ist ein Tensor 0. Stufe.
+Mit einem Vektor können mehrere Skalare auf einmal beschrieben werden.
+Ein Vektor hat daher die Stufe 1 und ist höherstufig als ein Skalar.
+Mit einer Matrix können wiederum mehrere Vektoren auf einmal beschrieben werden.
+Eine Matrix hat daher die Stufe 2 und ist noch höherstufig als ein Vektor.
+Versteht man diese Stufen, so versteht man den Sinn des Begriffs Tensor.
 
+Jede Stufe von Tensoren verlangt andere Rechenregeln.
+So zeigt sich auch der Nachteil von Tensoren mit Stufen höher als 2.
+Man ist also bestrebt höherstufige Tensoren mit Skalaren, Vektoren oder Matrizen zu beschreiben.
 
+Der Begriff Tensor wurde 1840 von Rowan Hamilton in die Mathematik eingeführt.
+James Clerk Maxwell hat bereits mit Tensoren operiert, ohne den Begriff Tensor gekannt zu haben.
+Erst Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert.
+Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschrieben.
+Auch Albert Einstein hat solche Tensoren eingesetzt,
+um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können.
+\cite{spannung:Tensor}
+\cite{spannung:Voigtsche-Notation}
diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex
index 37d3242..6326eab 100644
--- a/buch/papers/spannung/teil2.tex
+++ b/buch/papers/spannung/teil2.tex
@@ -1,40 +1,494 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{spannung:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{spannung:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
+\section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}}
+\rhead{Dreiachsiger Spannungszustand}
+Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen.
+\begin{figure}
+	\centering
+	\includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png}
+	\caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen}
+	\label{fig:infinitesimalerWuerfel}
+\end{figure}
+Ein Tensor 0. Stufe, sprich ein Skalar, kann lediglich den 1D Spannungszustand beschreiben.
+Um den 3D Spannungszustandes als ein mathematisches Objekt darstellen zu können, wird ein Tensor 2. Stufe, sprich eine Matrix, eingesetzt.
+Die Spannungen sind durch die zwei Indizes
+\[
+i, j\in\left\{1, 2, 3\right\}
+\]
+definiert.
+Daher ergeben sich die neun Spannungen.
+Die nachfolgenden Zusammenhänge sind in \cite{spannung:Voigtsche-Notation} beschrieben.
+Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als $3\times3$ Matrix mit
+\[
+\overline{\sigma}
+=
+\sigma_{ij}
+=
+\begin{pmatrix}
+	\sigma_{11} & \sigma_{12} & \sigma_{13} \\ 
+	\sigma_{21} & \sigma_{22} & \sigma_{23} \\
+	\sigma_{31} & \sigma_{32} & \sigma_{33}
+\end{pmatrix}
+\]
+dargestellt werden und beschreibt somit den gesamten Spannungszustand.
+Die Dehnungen wirken in die gleichen Richtungen wie die korrespondierenden Spannungen und sind durch die zwei Indizes
+\[
+k, l\in\left\{1, 2, 3\right\}
+\]
+definiert.
+Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\times3$ Matrix mit
+\[
+\overline{\varepsilon}
+=
+\varepsilon_{kl}
+=
+\begin{pmatrix}
+	\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ 
+	\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
+	\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
+\end{pmatrix}
+\]
+dargestellt werden und beschreibt den gesamten Dehnungszustand.
 
+Der Spannungs- und Dehnungstensor 2. Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist.
+Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt.
+So ergibt sich der Spannungsvektor
 
+\[
+\overline{\sigma}
+=
+\sigma_{ij}
+=
+\begin{pmatrix}
+	\sigma_{11} & \sigma_{12} & \sigma_{13} \\ 
+	\sigma_{21} & \sigma_{22} & \sigma_{23} \\
+	\sigma_{31} & \sigma_{32} & \sigma_{33}
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\vec{\sigma}
+=
+\begin{pmatrix}
+	\sigma_{11}\\
+	\sigma_{12}\\
+	\sigma_{13}\\
+	\sigma_{21}\\
+	\sigma_{22}\\
+	\sigma_{23}\\
+	\sigma_{31}\\
+	\sigma_{32}\\
+	\sigma_{33}
+\end{pmatrix}
+\]
+und Dehnungsvektor
+\[
+\overline{\varepsilon}
+=
+\varepsilon_{kl}
+=
+\begin{pmatrix}
+	\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ 
+	\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
+	\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\vec{\varepsilon}
+=
+\begin{pmatrix}
+	\varepsilon_{11} \\
+	\varepsilon_{12} \\
+	\varepsilon_{13} \\
+	\varepsilon_{21} \\
+	\varepsilon_{22} \\
+	\varepsilon_{23} \\
+	\varepsilon_{31} \\
+	\varepsilon_{32} \\
+	\varepsilon_{33}
+\end{pmatrix}
+.
+\]
+Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2. Stufe ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4. Stufe benötigt.
+Dieser ist im 1D Spannungszustand ein Tensor 0. Stufe und somit ein Skalar, der Elastizitätsmodul $E$.
+
+Dieser Elastizitätstensor 4. Stufe kann als Tensor 2. Stufe, sprich als Matrix, dargestellt werden.
+So wird die Spannungsgleichung stark vereinfacht, da nun eine Matrix auf einen Vektor operiert.
+Dieser Tensor muss für eine Spannung jeden Einfluss aus allen 9 Dehnungen mit Konstanten erfassen.
+Dies bedeutet um eine von 9 Spannungen berechnen zu können müssen alle 9 Dehnung mit unterschiedlichen Faktoren summiert werden.
+Es ergeben sich $9^2$ Einträge, welches mit den 4 Indizes
+\[
+i, j, k, l\in\left\{1, 2, 3\right\}
+,
+\]
+die zueinander verknüpft werden müssen, zu begründen ist.
+Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein
+\[
+\overline{\overline{C}}
+=
+C_{ijkl}
+=
+\begin{pmatrix}
+C_{1111} & C_{1112} & C_{1113} & C_{1121} & C_{1122} & C_{1123} & C_{1131} & C_{1132} & C_{1133} \\
+C_{1211} & C_{1212} & C_{1213} & C_{1221} & C_{1222} & C_{1223} & C_{1231} & C_{1232} & C_{1233} \\
+C_{1311} & C_{1312} & C_{1313} & C_{1321} & C_{1322} & C_{1323} & C_{1331} & C_{1332} & C_{1333} \\
+C_{2111} & C_{2112} & C_{2113} & C_{2121} & C_{2122} & C_{2123} & C_{2131} & C_{2132} & C_{2133} \\
+C_{2211} & C_{2212} & C_{2213} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\
+C_{2311} & C_{2312} & C_{2313} & C_{2321} & C_{2322} & C_{2323} & C_{2331} & C_{2332} & C_{2333} \\
+C_{3111} & C_{3112} & C_{3113} & C_{3121} & C_{3122} & C_{3123} & C_{3131} & C_{3132} & C_{3133} \\
+C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{3232} & C_{3233} \\
+C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333}
+\end{pmatrix}
+\]
+geschrieben werden kann.
+Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein.
+Folglich gilt:
+\[
+\overline{\overline{C}}
+=
+\overline{\overline{C}}~^{T}
+.
+\]
+Die allgemeine Spannungsgleichung lautet nun:
+\[
+\vec\sigma
+=
+\overline{\overline{C}}\cdot\vec{\varepsilon}
+.
+\]
+
+Als Indexnotation
+\[
+\sigma_{ij}
+=
+\sum_{k=1}^3
+\sum_{l=1}^3
+C_{ijkl}\cdot\varepsilon_{kl}
+\]
+kann dies ebenfalls geschrieben werden.
+
+Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert.
+Da dieser Modul durch die eindimensionale Betrachtung definiert ist,
+muss für die dreidimensionale Betrachtung eine weitere Kennzahl eingeführt werden.
+Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche durch
+\[
+\nu
+=
+\frac{\varepsilon_q}{\varepsilon}
+=
+\frac{\Delta b}{b_0}
+\]
+und
+\begin{align*}
+	\varepsilon     &= \text{Längsdehnung [$-$]} \\
+	\varepsilon_q   &= \text{Querdehnung [$-$]}
+\end{align*}
+definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich
+\[
+\begin{pmatrix}
+	\sigma_{11}\\
+	\sigma_{12}\\
+	\sigma_{13}\\
+	\sigma_{21}\\
+	\sigma_{22}\\
+	\sigma_{23}\\
+	\sigma_{31}\\
+	\sigma_{32}\\
+	\sigma_{33}
+\end{pmatrix}
+=
+\frac{E}{(1+\nu)(1-2\nu)}
+\begin{pmatrix}
+	1-2\nu &          0 &          0 &          0 &    \nu &          0 &          0 &          0 & \nu   \\
+	     0 &\frac{1}{4} &          0 &\frac{1}{4} &      0 &          0 &          0 &          0 & 0     \\
+	     0 &          0 &\frac{1}{4} &          0 &      0 &          0 &\frac{1}{4} &          0 & 0     \\
+	     0 &\frac{1}{4} &          0 &\frac{1}{4} &      0 &          0 &          0 &          0 & 0     \\
+	   \nu &          0 &          0 &          0 & 1-2\nu &          0 &          0 &          0 & \nu   \\
+  	     0 &          0 &          0 &          0 &      0 &\frac{1}{4} &          0 &\frac{1}{4} & 0     \\
+	     0 &          0 &\frac{1}{4} &          0 &      0 &          0 &\frac{1}{4} &          0 & 0     \\
+	     0 &          0 &          0 &          0 &      0 &\frac{1}{4} &          0 &\frac{1}{4} & 0     \\
+	   \nu &          0 &          0 &          0 &    \nu &          0 &          0 &          0 & 1-2\nu
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{11} \\
+	\varepsilon_{12} \\
+	\varepsilon_{13} \\
+	\varepsilon_{21} \\
+	\varepsilon_{22} \\
+	\varepsilon_{23} \\
+	\varepsilon_{31} \\
+	\varepsilon_{32} \\
+	\varepsilon_{33}
+\end{pmatrix}
+.
+\]
+Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als
+\[
+\sigma_{22}
+=
+\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{11}+\frac{E}{(1+\nu)}\cdot\varepsilon_{22}+\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{33}
+\]
+berechnen.
+
+Man betrachte nun die Eigenschaften des Elastizitätstensors.
+Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab.
+Es ergeben sich keine Blöcke mit einheitlichen Einträgen.
+
+Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind.
+Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten.
+Diese Symmetrie setzt daher voraus, dass
+\[
+\sigma_{12}
+=
+\sigma_{21}
+,
+\qquad
+\sigma_{13}
+=
+\sigma_{31}
+,
+\qquad
+\sigma_{23}
+=
+\sigma_{32}
+\]
+und folglich auch
+\[
+\varepsilon_{12}
+=
+\varepsilon_{21}
+,
+\qquad
+\varepsilon_{13}
+=
+\varepsilon_{31}
+,
+\qquad
+\varepsilon_{23}
+=
+\varepsilon_{32}
+\]
+gilt.
+Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche-Notation} ausgenutzt, um die Gleichung vereinfachen zu können.
+Durch diese Symmetrie gilt
+\[
+\overline{\sigma}
+=
+\begin{pmatrix}
+	\sigma_{11} & \sigma_{12} & \sigma_{13} \\ 
+	\sigma_{21} & \sigma_{22} & \sigma_{23} \\
+	\sigma_{31} & \sigma_{32} & \sigma_{33}
+\end{pmatrix}
+=
+\begin{pmatrix}
+	\sigma_{11} & \sigma_{12} & \sigma_{13} \\ 
+  	            & \sigma_{22} & \sigma_{23} \\
+	 \text{sym} &             & \sigma_{33} 
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\vec{\sigma}
+=
+\begin{pmatrix}
+    \sigma_{11}\\
+	\sigma_{22}\\
+	\sigma_{33}\\
+	\sigma_{23}\\
+	\sigma_{13}\\
+	\sigma_{12}
+\end{pmatrix}
+\]
+und entsprechend
+\[
+\overline{\varepsilon}
+=
+\begin{pmatrix}
+	\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ 
+	\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
+	\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
+\end{pmatrix}
+=
+\begin{pmatrix}
+	\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ 
+	                 & \varepsilon_{22} & \varepsilon_{23} \\
+	\text{sym}       &                  & \varepsilon_{33}
+\end{pmatrix}
+\qquad
+\Rightarrow
+\qquad
+\vec{\varepsilon}
+=
+\begin{pmatrix}
+	\varepsilon_{11} \\
+	\varepsilon_{22} \\
+	\varepsilon_{33} \\
+	\varepsilon_{23} \\
+	\varepsilon_{13} \\
+	\varepsilon_{12}
+\end{pmatrix}
+.
+\]
+
+Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je sechs Einträgen darstellen.
+Der Elastizitätstensor kann entsprechend auf eine $6\times6$ Matrix reduziert werden.
+Es lässt sich nun eine reduzierte allgemeine Spannungsgleichung mit
+\[
+\vec{\sigma}
+=
+\overline{\overline{C}}\cdot\vec{\varepsilon}
+\]
+beziehungsweise
+\[
+\begin{pmatrix}
+	\sigma_{11} \\
+	\sigma_{22} \\
+	\sigma_{33} \\
+	\sigma_{23} \\
+	\sigma_{13} \\
+	\sigma_{12}
+\end{pmatrix}
+=
+\begin{pmatrix}
+	C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\
+	C_{2211} & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\
+	C_{3311} & C_{3322} & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\
+	C_{2311} & C_{2322} & C_{2333} & C_{2323} & C_{2313} & C_{2312} \\
+	C_{1311} & C_{1322} & C_{1333} & C_{1323} & C_{1313} & C_{1312} \\
+	C_{1211} & C_{1222} & C_{1233} & C_{1223} & C_{1213} & C_{1212}
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{11} \\
+	\varepsilon_{22} \\
+	\varepsilon_{33} \\
+	\varepsilon_{23} \\
+	\varepsilon_{13} \\
+	\varepsilon_{12}
+\end{pmatrix}
+\]
+beschreiben.
+Die Spannung $\sigma_{11}$ beispielsweise erhält man, wenn man die sechs Produkte aus den Konstanten $C$ und Dehnungen $\varepsilon$ summiert.
+Die Symmetrieeigenschaft des Elastizitätstensors bleibt auch hier erhalten.
+Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit
+
+\[
+\begin{pmatrix}
+	\sigma_{11} \\
+	\sigma_{22} \\
+	\sigma_{33} \\
+	\sigma_{23} \\
+	\sigma_{13} \\
+	\sigma_{12}
+\end{pmatrix}
+=
+\begin{pmatrix}
+	  C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\
+	           & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\
+	           &          & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ 
+	           &          &          & C_{2323} & C_{2313} & C_{2312} \\ 
+               &          &          &          & C_{1313} & C_{1312} \\
+    \text{sym} &          &          &          &          & C_{1212} 
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{11} \\
+	\varepsilon_{22} \\
+	\varepsilon_{33} \\
+	\varepsilon_{23} \\
+	\varepsilon_{13} \\
+	\varepsilon_{12}
+\end{pmatrix}
+\]
+beschreiben.
+Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert.
+Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist:
+\begin{equation}
+\begin{pmatrix}
+	\sigma_{11}\\
+	\sigma_{22}\\
+	\sigma_{33}\\
+	\sigma_{23}\\
+	\sigma_{13}\\
+	\sigma_{12}
+\end{pmatrix}
+=
+\frac{E}{(1+\nu)(1-2\nu)}
+\begin{pmatrix}
+	1- 2\nu & \nu     & \nu     & 0           & 0           & 0\\
+	    \nu & 1- 2\nu & \nu     & 0           & 0           & 0\\
+        \nu & \nu     & 1- 2\nu & 0           & 0           & 0\\
+          0 & 0       & 0       & \frac{1}{2} & 0           & 0\\
+          0 & 0       & 0       & 0           & \frac{1}{2} & 0\\
+          0 & 0       & 0       & 0           & 0           & \frac{1}{2}
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{11}\\
+	\varepsilon_{22}\\
+	\varepsilon_{33}\\
+	\varepsilon_{23}\\
+	\varepsilon_{13}\\
+	\varepsilon_{12}
+\end{pmatrix}
+.
+\label{spannung:Spannungsgleichung}
+\end{equation}
+
+Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also,
+dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben.
+Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben.
+Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die  $3\times3$ Blöcke.
+
+Man betrachte als Beispiel die Berechnung von $\sigma_{33}$.
+Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ haben.
+Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss.
+Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss.
+
+Von  $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können.
+Dadurch erhält man die Dehnungsgleichung:
+
+\[
+\vec{\varepsilon}
+=
+\overline{\overline{C}}\mathstrut^{-1}\cdot \vec{\sigma}
+\]
+
+\[
+\begin{pmatrix}
+	\varepsilon_{11}\\
+	\varepsilon_{22}\\
+	\varepsilon_{33}\\
+	\varepsilon_{23}\\
+	\varepsilon_{13}\\
+	\varepsilon_{12}
+\end{pmatrix}
+=
+\frac{1}{E}
+\begin{pmatrix}
+	   1 & -\nu & -\nu & 0      & 0      & 0     \\
+	-\nu &    1 & -\nu & 0      & 0      & 0     \\
+	-\nu & -\nu &    1 & 0      & 0      & 0     \\
+ 	   0 &    0 &    0 & 2+2\nu & 0      & 0     \\
+	   0 &    0 &    0 &      0 & 2+2\nu & 0     \\
+	   0 &    0 &    0 &      0 & 0      & 2+2\nu
+\end{pmatrix}
+\begin{pmatrix}
+	\sigma_{11}\\
+	\sigma_{22}\\
+	\sigma_{33}\\
+	\sigma_{23}\\
+	\sigma_{13}\\
+	\sigma_{12}
+\end{pmatrix}
+.
+\]
+Die zwei $3\times3$ Blöcke links unten und rechts oben sind folglich noch vorhanden.
+Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man die Dehnung
+\[
+\varepsilon_{22}
+=
+\frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33}
+=
+\frac{1}{E}\cdot(\sigma_{22}-\nu\cdot\sigma_{11}-\nu\cdot\sigma_{33})
+.
+\]
+Diese hängt wieder am meisten von $\sigma_{22}$ ab.
+Ist die Querdehnung $\nu$ grösser, so wird die Dehnung $\varepsilon_{22}$ reduziert.
+Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich $\nu=0.5$.
diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex
index ce7d50f..3e456c3 100644
--- a/buch/papers/spannung/teil3.tex
+++ b/buch/papers/spannung/teil3.tex
@@ -1,40 +1,108 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{spannung:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
+\section{Die geotechnischen Invarianten\label{spannung:section:Die geotechnischen Invarianten}}
+\rhead{Die geotechnischen Invarianten}
+In vielen Fällen in der Geotechnik und auch in Versuchen hat man gleichmässige Belastungen über eine grössere Fläche.
+Durch eine solche Belastung auf den Boden, entstehen gleichermassen Spannungen in Richtung $2$ und $3$,
+wenn man von einem isotropen Bodenmaterial ausgeht.
+Folglich gilt:
 
-\subsection{De finibus bonorum et malorum
-\label{spannung:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
+\[
+\sigma_{22}
+=
+\sigma_{33}
+.
+\]
+Dadurch wird der Spannungszustand vereinfacht.
+Diesen vereinfachten Spannungszustand kann man mit den zwei geotechnischen Invarianten abbilden.
+Die erste Invariante ist die volumetrische Spannung
+\begin{equation}
+p
+=
+\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}
+\label{spannung:Invariante_p}
+,
+\end{equation}
+welche als arithmetisches Mittel aller Normalspannungen im infinitesimalen Würfel definiert ist.
+Die zweite Invariante ist die deviatorische Spannung
+\begin{equation}
+q
+=
+\sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}}
+\label{spannung:Invariante_q}
+.
+\end{equation}
+Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt.
+Die hydrostatische Spannung $p$ kann gemäss Gleichung \eqref{spannung:Invariante_p} als
+\[
+p
+=
+\frac{\sigma_{11}+2\sigma_{33}}{3}
+\]
+vereinfacht werden.
+Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als
+\[
+q
+=
+\sigma_{11}-\sigma_{33}
+\]
+vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten.
 
+Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} berechnet werden.
+Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren.
+Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten.
+So ergibt sich
+\[
+\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p}
+=
+\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}}
+\]
+und
+\[
+\overbrace{\sigma_{11}-\sigma_{33}}^{q}
+=
+\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}}
+.
+\]
+Die Faktoren mit den Dehnungskomponenten können so mit
+\[
+\varepsilon_{v}
+=
+(\varepsilon_{11} - 2\varepsilon_{33})
+\qquad
+\text{und}
+\qquad
+\varepsilon_{s}
+=
+\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})
+\]
+eingeführt werden, mit
+\begin{align*}
+	\varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\
+	\varepsilon_{s} &= \text{Deviatorische Dehnung [-].}
+\end{align*}
+Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden.
+Die deviatorische Dehnung $\varepsilon_{s}$  kann mit einer Verzerrung verglichen werden.
 
+Diese zwei Gleichungen kann man durch die Matrixschreibweise
+\begin{equation}
+\begin{pmatrix}
+	q\\
+	p
+\end{pmatrix}
+=
+\begin{pmatrix}
+	\frac{3E}{2(1+\nu)} &                   0 \\
+	                  0 & \frac{E}{3(1-2\nu)}
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{s}\\
+	\varepsilon_{v}
+\end{pmatrix}
+\label{spannung:Matrixschreibweise}
+\end{equation}
+vereinfachen.
+Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor.
+Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden.
+
+Mit dieser Formel \eqref{spannung:Matrixschreibweise} lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen.
+Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch.
+Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben.
diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex
new file mode 100644
index 0000000..2f2e4ce
--- /dev/null
+++ b/buch/papers/spannung/teil4.tex
@@ -0,0 +1,79 @@
+\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}}
+\rhead{Oedometer-Versuch}
+Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden.
+Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen.
+Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$.
+Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken.
+Die seitliche Verschiebung des Materials wird durch einen Stahlring verhindert.
+Die Probe wird sich so stetig verdichten.
+Das Volumen nimmt ab und die Dehnung nimmt immer mehr zu.
+Unter diesen Bedingungen wird der oedometrische Elastizitätsmodul mit steigender Dehnung zunehmen.
+
+Da im Boden das umgebende Material ähnlich eine seitliche Verformung verhindert,
+bildet dieser oedometrische Elastizitätsmodul die Realität besser ab, als der gewöhnliche Elastizitätsmodul.
+Durch dieses Verhindern des seitlichen Ausbrechens ist
+\[
+\varepsilon_{22}
+=
+\varepsilon_{33}
+=
+0
+\]
+aber auch
+\[
+\sigma_{22}
+=
+\sigma_{33}
+\neq 0
+.
+\]
+Die Spannung $\sigma_{11}$ wird durch die aufgebrachte Kraft mit
+\[
+\sigma_{11}
+=
+\frac{F}{A}
+\]
+und die Dehnung $\varepsilon_{11}$ jeweils mit den entsprechenden Setzungen berechnet.
+Diese Randbedingungen können in die vereinfachte Gleichung \eqref{spannung:Matrixschreibweise} eingesetzt werden.
+Diese lautet nun:
+\[
+\begin{pmatrix}
+	\sigma_{11}-\sigma_{33} \\
+	\sigma_{11}+2\sigma_{33}
+\end{pmatrix}
+=
+\begin{pmatrix}
+	\frac{E_{OED}}{(1+\nu)} &                             0 \\
+                          0 & \frac{E_{OED}}{3(1-2\nu)}
+\end{pmatrix}
+\begin{pmatrix}
+	\varepsilon_{11}\\
+	\varepsilon_{11}
+\end{pmatrix}
+.
+\]
+Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen
+\[
+\sigma_{11}-\sigma_{33}
+=
+\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11}
+\]
+und
+\[
+\sigma_{11}+2\sigma_{33}
+=
+\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11}
+\]
+berechnen.
+Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen.
+Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden.
+Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{spannung:DiagrammOedometer-Versuch}).
+Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark.
+Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png}
+	\caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material}
+	\label{fig:DiagrammOedometer-Versuch}
+\end{figure}
\ No newline at end of file
diff --git a/buch/papers/verkehr/Makefile.inc b/buch/papers/verkehr/Makefile.inc
index 7bd8de1..876d0df 100644
--- a/buch/papers/verkehr/Makefile.inc
+++ b/buch/papers/verkehr/Makefile.inc
@@ -3,12 +3,10 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-dependencies-verkehr =						\
+dependencies-verkehr =							\
 	papers/verkehr/packages.tex					\
-	papers/verkehr/main.tex					\
-	papers/verkehr/references.bib				\
-	papers/verkehr/teil0.tex					\
-	papers/verkehr/teil1.tex					\
-	papers/verkehr/teil2.tex					\
-	papers/verkehr/teil3.tex
+	papers/verkehr/main.tex						\
+	papers/verkehr/section1.tex					\
+	papers/verkehr/section2.tex					\
+	papers/verkehr/references.bib
 
diff --git a/buch/papers/verkehr/figures/chart_Vr1.png b/buch/papers/verkehr/figures/chart_Vr1.png
new file mode 100644
index 0000000..14d4eca
--- /dev/null
+++ b/buch/papers/verkehr/figures/chart_Vr1.png
diff --git a/buch/papers/verkehr/figures/chart_Vr2.png b/buch/papers/verkehr/figures/chart_Vr2.png
new file mode 100644
index 0000000..2d68681
--- /dev/null
+++ b/buch/papers/verkehr/figures/chart_Vr2.png
diff --git a/buch/papers/verkehr/figures/chart_pathDiff.png b/buch/papers/verkehr/figures/chart_pathDiff.png
new file mode 100644
index 0000000..02bded7
--- /dev/null
+++ b/buch/papers/verkehr/figures/chart_pathDiff.png
diff --git a/buch/papers/verkehr/figures/dist_display6.png b/buch/papers/verkehr/figures/dist_display6.png
new file mode 100644
index 0000000..3056f43
--- /dev/null
+++ b/buch/papers/verkehr/figures/dist_display6.png
diff --git a/buch/papers/verkehr/figures/network_aStar.png b/buch/papers/verkehr/figures/network_aStar.png
new file mode 100644
index 0000000..5a681bd
--- /dev/null
+++ b/buch/papers/verkehr/figures/network_aStar.png
diff --git a/buch/papers/verkehr/figures/network_dij.png b/buch/papers/verkehr/figures/network_dij.png
new file mode 100644
index 0000000..d9348d7
--- /dev/null
+++ b/buch/papers/verkehr/figures/network_dij.png
diff --git a/buch/papers/verkehr/main.tex b/buch/papers/verkehr/main.tex
index 332ee7e..6348993 100644
--- a/buch/papers/verkehr/main.tex
+++ b/buch/papers/verkehr/main.tex
@@ -4,33 +4,13 @@
 % (c) 2020 Hochschule Rapperswil
 %
 \chapter{Thema\label{chapter:verkehr}}
-\lhead{Thema}
+\lhead{Verkehrsfluss und Verkehrsnetze}
 \begin{refsection}
-\chapterauthor{Hans Muster}
+\chapterauthor{Pascal Andreas Schmid und Robine Luchsinger}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
-
-\input{papers/verkehr/teil0.tex}
-\input{papers/verkehr/teil1.tex}
-\input{papers/verkehr/teil2.tex}
-\input{papers/verkehr/teil3.tex}
+\input{papers/verkehr/section1.tex}
+\input{papers/verkehr/section2.tex}
+\input{papers/verkehr/section3.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex
new file mode 100644
index 0000000..6a5dc28
--- /dev/null
+++ b/buch/papers/verkehr/section1.tex
@@ -0,0 +1,70 @@
+\section{Einführung}
+\label{section:verkehr/einfuehrung}
+
+\subsection{Verkehrsnetze}
+Das Verkehrsnetz besteht aus allen Anlagen, auf oder unter der Erdoberfläche, auf denen eine räumliche Fortbewegung von Personen oder auch Gütern stattfindet. Verkehrsnetze sind ein Bestandteil der Verkehrsinfrastruktur, die auf topografischen Karten festgehalten werden. Sie umfassen den Schienenverkehr, alle Strassen und Wege, wie auch Flugplätze und alle dazugehörigen Bauwerke. 
+Aus verkehrsgeografischer Sicht besteht das Verkehrsnetz aus Kanten, Knotenpunkten und dem Hinterland. Die Knotenpunkte werden auch hier durch die Kanten verbunden, die den Verkehrsstrom aufnehmen, wobei das Hinterland durch einzelne Knoten versorgt wird. Die Aufteilung in Kanten und Knotenpunkte ermöglicht eine Vereinfachung komplexer Verkehrsnetze, damit sie mittels der Graphentheorie untersucht werden können.
+Grundsätzlich können kurze Wege zwischen den Knotenpunkten das Ziel beim
+Aufbau eines Verkehrsnetzes sein. Es kann aber auch versucht werden, die Bau- und Unterhaltskosten des Verkehrsnetzes in einem gewissen Rahmen zu halten. Aus diesen Vorgaben ergibt sich dann, je nach dem was gewünscht wird, eine grob- oder feinmaschige Struktur des Netzes. 
+Ziel ist aber ein möglichst wirtschaftliches und optimales Verkehrsnetz.
+
+\subsection{Suchalgorithmen}
+
+\subsubsection{Dijkstra-Algorithmus}
+Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Infomratikprofessor Edsger Dijkstra. Den Algorithmus hat er im Jahr 1959 erfunden.
+Der Algorithmus von Dijkstra ist ein Greedy-Algorithmus (gieriger Algorithmus), der schrittweise einen Folgezustand auswählt, damit beim Zeitpunkt der Wahl der grösste Gewinn bzw. das beste Ergebnis erzielt werden kann. 
+Trotz der Schnelligkeit der Greedy-Algorithmen, können viele Probleme nicht optimal gelöst werden.
+Vereinfacht wird beim Dijkstra-Algorithmus, ausgehend von einem Startknoten so lange dem kürzesten Pfad gefolgt, bis der Zielknoten erreicht wird. Dabei muss für jeden besuchten Knoten die Kostenfunktion als auch der Pfad dahin (vorheriger Knoten) gespeichert werden.
+Dadurch wird hingegen garantiert, dass, wenn der Zielknoten erreicht wird, auch der kürzeste Pfad gefunden wurde.
+Grundlegende Voraussetzung für den Dijkstra-Algorithmus ist die strikte Positivität der Kantengewichte. Andernfalls würde ein wiederholtes Ablaufen einer Kante mit negativem Gewicht zu einer stetigen Reduktion der Kostenfunktion führen, was zu einer unendlichen Schlaufe führen würde.
+
+\subsubsection{A*-Algorithmus}
+Suchalgorithmen werden nach einfachen (uninformierte) und heuristischen (informierten) Algorithmen unterschieden. Während einfache Algorithmen den Suchraum intuitiv durchsuchen, beziehen heuristische Algorithmen Wissen über den Suchraum mit ein.
+Der A*-Algorithmus geht auf seine Erfinder Peter Hart, Nils Nilsson und Bertram Raphael zurück, die den Algorithmus erstmals im Jahr 1968 beschrieben.
+Der A*-Algorithmus ist ein heuristischer Suchalgorithmus, der den kürzesten Pfad zwischen zwei Knoten in einem Graphen mit positiven Kantengewichten berechnet.
+Im Gegensatz zu einfachen Suchalgorithmen, wird beim A*-Algorithmus eine Schätzfunktion, die sogenannte Heuristik, verwendet. Dies ermöglicht ein zielgerichtetes Suchen und gleichzeitig wird die Laufzeit verringert. 
+Ausserdem findet der A*-Algorithmus immer eine optimale Lösung, sofern eine vorhanden ist.
+Der A*-Algorithmus wird als Verallgemeinerung gehandhabt und gilt als Erweiterung des Dijkstra-Algorithmus. 
+=======
+
+\subsubsection{Floyd-Warshall-Algorithmus}
+Der Floyd-Warshall-Algorithmus wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt.
+Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die kürzesten , beziehungsweise die optimalsten Wege zwischen allen Paaren von Knoten berechnet, sofern der Graph keinen negativen Kreis (Zyklus) aufweist.
+Ein Kreis in einem Graphen ist ein Weg, bei dem Start- und Endpunkt den gleichen Knoten aufweisen. Dieser wird negativ, wenn die Summe der gewichteten Kanten kleiner als Null wird.
+
+\subsubsection{Euklidische Heuristik}
+Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar.
+Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant.
+
+\subsection{PageRank-Algorithmus}
+Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc..
+Beim PageRank-Algorithmus handelt es sich um den Algorithmus von Google, aus dem die Google-Matrix abgeleitet wird.
+Die Google-Matrix ist eine immens grosse Matrix mit Millionen Zeilen und Spalten, die für die schnelle und vor allem exakte Bestimmung der PageRanks (Gewichtung) eine grosse Bedeutung hat.
+Der PageRank-Algorithmus analysiert und gewichtet beispielsweise die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur. 
+Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\
+Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche gilt.
+
+%THEORIE...
+Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen.
+
+\begin{equation}
+A_{i,j}=\left\{ \begin{matrix}
+1 & \text{Kante von $j$ nach $i$} \\ 0 & \text{keine Kante von $j$ nach $i$}  
+\end{matrix}
+ \right.
+\label{verkehr:Adja}
+\end{equation}
+
+
+Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1...n\right\}\end{equation}
+Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet.
+Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt.
+\begin{equation} P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \end{equation}
+Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt:
+\begin{equation} \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1...n\right\} \end{equation}
+Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet.
+Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das "erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt.
+\begin{equation} \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\end{equation}
+somit
+\begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation}
+Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ und stellt das abschliessende Ranking dar.
diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex
new file mode 100644
index 0000000..638d9dd
--- /dev/null
+++ b/buch/papers/verkehr/section2.tex
@@ -0,0 +1,55 @@
+\section{Versuchsreihe}
+\label{section:verkehr/versuchsreihe}
+
+Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt.
+Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert.
+Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt.
+Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen.
+
+Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind.
+
+\subsection{Einfluss der Knotenzahl auf die Rechenzeit}
+\label{verkehr:Knotenzahl}
+
+\begin{figure}
+\centering
+\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr1.png}
+
+\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.}
+\label{verkehr:Vr1}
+\end{figure}
+
+In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt.
+Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist.
+Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt.
+
+\begin{figure}
+\centering
+\includegraphics[width=12cm]{papers/verkehr/figures/chart_pathDiff.png}
+
+\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.}
+\label{verkehr:pathDifference}
+\end{figure}
+
+
+\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit}
+
+\begin{figure}
+\centering
+\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\
+\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.}
+\label{verkehr:Vr2}
+\end{figure}
+
+Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\
+Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen  \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären.
+
+\begin{figure}
+\centering
+\includegraphics[width=6cm]{papers/verkehr/figures/network_dij.png}\qquad
+\includegraphics[width=6cm]{papers/verkehr/figures/network_aStar.png}
+\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.}
+\label{verkehr:Comparison}
+\end{figure}
+
+In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde.
diff --git a/buch/papers/verkehr/section3.tex b/buch/papers/verkehr/section3.tex
new file mode 100644
index 0000000..99a0d92
--- /dev/null
+++ b/buch/papers/verkehr/section3.tex
@@ -0,0 +1,8 @@
+\section{Ausblick}
+\subsection{Optimierungsprobleme bei Graphen}
+Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\
+Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten.  
+
+\subsection{Wahl der Heuristik}
+Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\
+In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden.
diff --git a/buch/papers/verkehr/teil0.tex b/buch/papers/verkehr/teil0.tex
deleted file mode 100644
index 5031841..0000000
--- a/buch/papers/verkehr/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{verkehr:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{verkehr:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/verkehr/teil1.tex b/buch/papers/verkehr/teil1.tex
deleted file mode 100644
index 855aef8..0000000
--- a/buch/papers/verkehr/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{verkehr:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{verkehr:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{verkehr:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{verkehr:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{verkehr:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/verkehr/teil2.tex b/buch/papers/verkehr/teil2.tex
deleted file mode 100644
index 5170ded..0000000
--- a/buch/papers/verkehr/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{verkehr:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{verkehr:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/verkehr/teil3.tex b/buch/papers/verkehr/teil3.tex
deleted file mode 100644
index 8f79154..0000000
--- a/buch/papers/verkehr/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{verkehr:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{verkehr:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/test1.tex b/buch/test1.tex
deleted file mode 100644
index 8345230..0000000
--- a/buch/test1.tex
+++ /dev/null
@@ -1,93 +0,0 @@
-%
-% test1.tex -- Test 1
-%
-% (c) 2012 Prof. Dr. Andreas Mueller, HSR
-%
-%\documentclass[a4paper,12pt]{book}
-\documentclass[a4paper,12pt]{article}
-\usepackage{geometry}
-\geometry{papersize={210mm,297mm},total={165mm,260mm}}
-\usepackage{ngerman}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{times}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsfonts}
-\usepackage{amsthm}
-\usepackage{graphicx}
-\usepackage{fancyhdr}
-\usepackage{textcomp}
-\usepackage[all]{xy}
-\usepackage{txfonts}
-\usepackage{alltt}
-\usepackage{verbatim}
-\usepackage{paralist}
-\usepackage{makeidx}
-\usepackage{array}
-\usepackage{hyperref}
-\usepackage{caption}
-\usepackage{subcaption}
-\usepackage{standalone}
-\usepackage{environ}
-\usepackage{tikz}
-\input{../common/linsys.tex}
-\newcounter{beispiel}
-\newenvironment{beispiele}{
-\bgroup\smallskip\parindent0pt\bf Beispiele\egroup
-
-\begin{list}{\arabic{beispiel}.}
-  {\usecounter{beispiel}
-  \setlength{\labelsep}{5mm}
-  \setlength{\rightmargin}{0pt}
-}}{\end{list}}
-\newcounter{uebungsaufgabe}
-% environment fuer uebungsaufgaben
-\newenvironment{uebungsaufgaben}{
-\begin{list}{\arabic{uebungsaufgabe}.}
-  {\usecounter{uebungsaufgabe}
-  \setlength{\labelwidth}{2cm}
-  \setlength{\leftmargin}{0pt}
-  \setlength{\labelsep}{5mm}
-  \setlength{\rightmargin}{0pt}
-  \setlength{\itemindent}{0pt}
-}}{\end{list}\vfill\pagebreak}
-\newenvironment{teilaufgaben}{
-\begin{enumerate}
-\renewcommand{\labelenumi}{\alph{enumi})}
-}{\end{enumerate}}
-% Loesung
-\NewEnviron{loesung}{%
-\begin{proof}[L"osung]%
-\renewcommand{\qedsymbol}{$\bigcirc$}
-\BODY
-\end{proof}}
-\NewEnviron{bewertung}{\relax}
-\NewEnviron{diskussion}{
-\BODY
-}
-\RenewEnviron{loesung}{\relax}
-\RenewEnviron{diskussion}{\relax}
-\newenvironment{hinweis}{%
-\renewcommand{\qedsymbol}{}
-\begin{proof}[Hinweis]}{\end{proof}}
-
-\begin{document}
-{\parindent0pt\hbox to\hsize{%
-Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
-\vspace{0.5cm}
-
-\section*{Kurztest 1}
-
-\begin{uebungsaufgaben}
-
-\item
-\input chapters/30-endlichekoerper/uebungsaufgaben/3003.tex
-\item
-\input chapters/30-endlichekoerper/uebungsaufgaben/3004.tex
-\item
-\input chapters/30-endlichekoerper/uebungsaufgaben/3005.tex
-
-\end{uebungsaufgaben}
-
-\end{document}
diff --git a/buch/test2.tex b/buch/test2.tex
deleted file mode 100644
index ea842ce..0000000
--- a/buch/test2.tex
+++ /dev/null
@@ -1,91 +0,0 @@
-%
-% test2.tex -- Test 2
-%
-% (c) 2012 Prof. Dr. Andreas Mueller, HSR
-%
-%\documentclass[a4paper,12pt]{book}
-\documentclass[a4paper,12pt]{article}
-\usepackage{geometry}
-\geometry{papersize={210mm,297mm},total={165mm,260mm}}
-\usepackage{ngerman}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{times}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{amsfonts}
-\usepackage{amsthm}
-\usepackage{graphicx}
-\usepackage{fancyhdr}
-\usepackage{textcomp}
-\usepackage[all]{xy}
-\usepackage{txfonts}
-\usepackage{alltt}
-\usepackage{verbatim}
-\usepackage{paralist}
-\usepackage{makeidx}
-\usepackage{array}
-\usepackage{hyperref}
-\usepackage{caption}
-\usepackage{subcaption}
-\usepackage{standalone}
-\usepackage{environ}
-\usepackage{tikz}
-\input{../common/linsys.tex}
-\newcounter{beispiel}
-\newenvironment{beispiele}{
-\bgroup\smallskip\parindent0pt\bf Beispiele\egroup
-
-\begin{list}{\arabic{beispiel}.}
-  {\usecounter{beispiel}
-  \setlength{\labelsep}{5mm}
-  \setlength{\rightmargin}{0pt}
-}}{\end{list}}
-\newcounter{uebungsaufgabe}
-% environment fuer uebungsaufgaben
-\newenvironment{uebungsaufgaben}{
-\begin{list}{\arabic{uebungsaufgabe}.}
-  {\usecounter{uebungsaufgabe}
-  \setlength{\labelwidth}{2cm}
-  \setlength{\leftmargin}{0pt}
-  \setlength{\labelsep}{5mm}
-  \setlength{\rightmargin}{0pt}
-  \setlength{\itemindent}{0pt}
-}}{\end{list}\vfill\pagebreak}
-\newenvironment{teilaufgaben}{
-\begin{enumerate}
-\renewcommand{\labelenumi}{\alph{enumi})}
-}{\end{enumerate}}
-% Loesung
-\NewEnviron{loesung}{%
-\begin{proof}[L"osung]%
-\renewcommand{\qedsymbol}{$\bigcirc$}
-\BODY
-\end{proof}}
-\NewEnviron{bewertung}{\relax}
-\NewEnviron{diskussion}{
-\BODY
-}
-\RenewEnviron{loesung}{\relax}
-\RenewEnviron{diskussion}{\relax}
-\newenvironment{hinweis}{%
-\renewcommand{\qedsymbol}{}
-\begin{proof}[Hinweis]}{\end{proof}}
-
-\begin{document}
-{\parindent0pt\hbox to\hsize{%
-Name: \hbox to7cm{\dotfill} Vorname: \dotfill}}
-\vspace{0.5cm}
-
-\section*{Kurztest 2}
-
-\begin{uebungsaufgaben}
-
-\item
-\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex
-\item
-\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex
-
-\end{uebungsaufgaben}
-
-\end{document}
diff --git a/vorlesungen/10_mseliealgebra/slides.tex b/vorlesungen/10_mseliealgebra/slides.tex
index 0fceaff..936139a 100644
--- a/vorlesungen/10_mseliealgebra/slides.tex
+++ b/vorlesungen/10_mseliealgebra/slides.tex
@@ -9,13 +9,27 @@
 \folie{7/einparameter.tex}
 \folie{7/ableitung.tex}
 \folie{7/liealgebra.tex}
+% XXX Beispiele von Lie-Algebren
+\folie{7/liealgbeispiel.tex}
+% XXX Vektorprodukt als Lie-Algebra
+\folie{7/vektorlie.tex}
 \folie{7/kommutator.tex}
+% XXX kommutator rechnerisch
+\folie{7/bch.tex}
 
 \section{Exponentialabbildung}
 \folie{7/dg.tex}
+\folie{7/logarithmus.tex}
+% Interpolation
+\folie{7/interpolation.tex}
+
+\section{Integration}
+\folie{7/haar.tex}
+\ifthenelse{\boolean{presentation}}{
+\folie{7/mannigfaltigkeit.tex}
+}{}
+\folie{7/integration.tex}
 
 % LOG Reihe
-% Interpolation
 % Mittelung auf einer Lie-Gruppe
-% Vektorprodukt als Lie-Gruppe
 
diff --git a/vorlesungen/11_msegraphen/Makefile b/vorlesungen/11_msegraphen/Makefile
new file mode 100644
index 0000000..36f1877
--- /dev/null
+++ b/vorlesungen/11_msegraphen/Makefile
@@ -0,0 +1,33 @@
+#
+# Makefile -- graphen
+#
+# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+all:	graphen-handout.pdf MathSemMSE-11-graphen.pdf
+
+include ../slides/Makefile.inc
+
+SOURCES = common.tex slides.tex $(slides)
+
+MathSemMSE-11-graphen.pdf:	MathSemMSE-11-graphen.tex $(SOURCES)
+	pdflatex MathSemMSE-11-graphen.tex
+
+graphen-handout.pdf:	graphen-handout.tex $(SOURCES)
+	pdflatex graphen-handout.tex
+
+thumbnail:	thumbnail.jpg # fix1.jpg
+
+thumbnail.pdf:	MathSemMSE-11-graphen.pdf
+	pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-11-graphen.pdf 1
+thumbnail.jpg:	thumbnail.pdf
+	convert -density 300 thumbnail.pdf \
+                -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
+
+fix1.pdf:	MathSemMSE-11-graphen.pdf
+	pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-11-graphen.pdf 1
+fix1.jpg:	fix1.pdf
+	convert -density 300 fix1.pdf \
+                -resize 1920x1080 -units PixelsPerInch fix1.jpg
+
diff --git a/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex
new file mode 100644
index 0000000..6d4ff2d
--- /dev/null
+++ b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex
@@ -0,0 +1,14 @@
+%
+% MathSem-11-msegraphen.tex -- Präsentation 
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\begin{frame}
+\titlepage
+\end{frame}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/11_msegraphen/common.tex b/vorlesungen/11_msegraphen/common.tex
new file mode 100644
index 0000000..67cf5e5
--- /dev/null
+++ b/vorlesungen/11_msegraphen/common.tex
@@ -0,0 +1,16 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\input{../common/common.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[Graphen]{Spektrale Graphentheorie}
+\author[A.~Müller]{Prof. Dr. Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
diff --git a/vorlesungen/11_msegraphen/graphen-handout.tex b/vorlesungen/11_msegraphen/graphen-handout.tex
new file mode 100644
index 0000000..58d0aff
--- /dev/null
+++ b/vorlesungen/11_msegraphen/graphen-handout.tex
@@ -0,0 +1,11 @@
+%
+% msegraphen-handout.tex -- Handout XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/11_msegraphen/slides.tex b/vorlesungen/11_msegraphen/slides.tex
new file mode 100644
index 0000000..b3d1519
--- /dev/null
+++ b/vorlesungen/11_msegraphen/slides.tex
@@ -0,0 +1,35 @@
+%
+% slides.tex -- XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\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/chrind.tex}
+\folie{8/chrindprop.tex}
+\folie{8/chroma1.tex}
+\folie{8/amax.tex}
+\folie{8/subgraph.tex}
+\folie{8/chrwilf.tex}
+\folie{8/weitere.tex}
diff --git a/vorlesungen/12_msewkeitsmatrizen/Makefile b/vorlesungen/12_msewkeitsmatrizen/Makefile
new file mode 100644
index 0000000..ec420cd
--- /dev/null
+++ b/vorlesungen/12_msewkeitsmatrizen/Makefile
@@ -0,0 +1,42 @@
+#
+# Makefile -- wkeitsmatrizen
+#
+# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+all:	wkeitsmatrizen-handout.pdf MathSemMSE-12-wkeitsmatrizen.pdf
+
+include ../slides/Makefile.inc
+
+SOURCES = common.tex slides.tex $(slides)
+
+MathSemMSE-12-wkeitsmatrizen.pdf:	MathSemMSE-12-wkeitsmatrizen.tex $(SOURCES)
+	pdflatex MathSemMSE-12-wkeitsmatrizen.tex
+
+wkeitsmatrizen-handout.pdf:	wkeitsmatrizen-handout.tex $(SOURCES)
+	pdflatex wkeitsmatrizen-handout.tex
+
+thumbnail:	thumbnail.jpg # fix1.jpg
+
+thumbnail.pdf:	MathSemMSE-12-wkeitsmatrizen.pdf
+	pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-12-wkeitsmatrizen.pdf 1
+thumbnail.jpg:	thumbnail.pdf
+	convert -density 300 thumbnail.pdf \
+                -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
+
+fix1.pdf:	MathSemMSE-12-wkeitsmatrizen.pdf
+	pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-12-wkeitsmatrizen.pdf 1
+fix1.jpg:	fix1.pdf
+	convert -density 300 fix1.pdf \
+                -resize 1920x1080 -units PixelsPerInch fix1.jpg
+
+parts:	part1.pdf part2.pdf
+
+part1.pdf:	MathSemMSE-12-wkeitsmatrizen.pdf
+	pdfjam --outfile part1.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-12-wkeitsmatrizen.pdf 1-160
+
+part2.pdf:	MathSemMSE-12-wkeitsmatrizen.pdf
+	pdfjam --outfile part2.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-12-wkeitsmatrizen.pdf 161-211
diff --git a/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex b/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex
new file mode 100644
index 0000000..e526731
--- /dev/null
+++ b/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex
@@ -0,0 +1,14 @@
+%
+% MathSem-12-msewkeitsmatrizen.tex -- Präsentation 
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\begin{frame}
+\titlepage
+\end{frame}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/12_msewkeitsmatrizen/common.tex b/vorlesungen/12_msewkeitsmatrizen/common.tex
new file mode 100644
index 0000000..2de59e8
--- /dev/null
+++ b/vorlesungen/12_msewkeitsmatrizen/common.tex
@@ -0,0 +1,16 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\input{../common/common.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[W'keitsmatrizen]{Wahrscheinlichkeitsmatrizen}
+\author[A.~Müller]{Prof. Dr. Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
diff --git a/vorlesungen/12_msewkeitsmatrizen/slides.tex b/vorlesungen/12_msewkeitsmatrizen/slides.tex
new file mode 100644
index 0000000..e0f8e9c
--- /dev/null
+++ b/vorlesungen/12_msewkeitsmatrizen/slides.tex
@@ -0,0 +1,35 @@
+%
+% slides.tex -- XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Google}
+\folie{9/google.tex}
+
+
+
+\section{Markov-Ketten}
+\folie{9/markov.tex}
+\folie{9/stationaer.tex}
+\folie{9/irreduzibel.tex}
+
+\section{Perron-Frobenius}
+\folie{9/pf/positiv.tex}
+\folie{9/pf/primitiv.tex}
+\folie{9/pf/trennung.tex}
+\folie{9/pf/vergleich.tex}
+\folie{9/pf/vergleich3d.tex}
+\folie{9/pf/dreieck.tex}
+\folie{9/pf/folgerungen.tex}
+\folie{9/pf.tex}
+\folie{9/potenz.tex}
+
+\section{Parrondo}
+\folie{9/parrondo/uebersicht.tex}
+\folie{9/parrondo/erwartung.tex}
+\folie{9/parrondo/spiela.tex}
+\folie{9/parrondo/spielb.tex}
+\folie{9/parrondo/spielbmod.tex}
+\folie{9/parrondo/kombiniert.tex}
+\folie{9/parrondo/deformation.tex}
+
diff --git a/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex b/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex
new file mode 100644
index 0000000..ae577c3
--- /dev/null
+++ b/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex
@@ -0,0 +1,11 @@
+%
+% msewkeitsmatrizen-handout.tex -- Handout XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/13_msegraphwavelets/Makefile b/vorlesungen/13_msegraphwavelets/Makefile
new file mode 100644
index 0000000..6dba66c
--- /dev/null
+++ b/vorlesungen/13_msegraphwavelets/Makefile
@@ -0,0 +1,33 @@
+#
+# Makefile -- graphwavelets
+#
+# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+all:	graphwavelets-handout.pdf MathSemMSE-13-graphwavelets.pdf
+
+include ../slides/Makefile.inc
+
+SOURCES = common.tex slides.tex $(slides)
+
+MathSemMSE-13-graphwavelets.pdf:	MathSemMSE-13-graphwavelets.tex $(SOURCES)
+	pdflatex MathSemMSE-13-graphwavelets.tex
+
+graphwavelets-handout.pdf:	graphwavelets-handout.tex $(SOURCES)
+	pdflatex graphwavelets-handout.tex
+
+thumbnail:	thumbnail.jpg # fix1.jpg
+
+thumbnail.pdf:	MathSemMSE-13-graphwavelets.pdf
+	pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-13-graphwavelets.pdf 1
+thumbnail.jpg:	thumbnail.pdf
+	convert -density 300 thumbnail.pdf \
+                -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
+
+fix1.pdf:	MathSemMSE-13-graphwavelets.pdf
+	pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-13-graphwavelets.pdf 1
+fix1.jpg:	fix1.pdf
+	convert -density 300 fix1.pdf \
+                -resize 1920x1080 -units PixelsPerInch fix1.jpg
+
diff --git a/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex b/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex
new file mode 100644
index 0000000..112d952
--- /dev/null
+++ b/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex
@@ -0,0 +1,14 @@
+%
+% MathSem-13-msegraphwavelets.tex -- Präsentation 
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\begin{frame}
+\titlepage
+\end{frame}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/13_msegraphwavelets/common.tex b/vorlesungen/13_msegraphwavelets/common.tex
new file mode 100644
index 0000000..b9799f0
--- /dev/null
+++ b/vorlesungen/13_msegraphwavelets/common.tex
@@ -0,0 +1,16 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\input{../common/common.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[SGWT]{Wavelets auf Graphen}
+\author[A.~Müller]{Prof. Dr. Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
diff --git a/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex b/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex
new file mode 100644
index 0000000..98789e5
--- /dev/null
+++ b/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex
@@ -0,0 +1,11 @@
+%
+% msegraphwavelets-handout.tex -- Handout XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/13_msegraphwavelets/slides.tex b/vorlesungen/13_msegraphwavelets/slides.tex
new file mode 100644
index 0000000..3fd38f2
--- /dev/null
+++ b/vorlesungen/13_msegraphwavelets/slides.tex
@@ -0,0 +1,44 @@
+%
+% slides.tex -- XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+% Funktionen auf einem Graphen
+\folie{8/wavelets/funktionen.tex}
+
+% Laplace-Basis auf dem Graphen
+\folie{8/wavelets/laplacebasis.tex}
+
+% Fourier-Transformation auf einem Graphen
+\folie{8/wavelets/fourier.tex}
+
+% Lokalisierung in Standardbasis und Fourier-Basis
+\folie{8/wavelets/lokalisierungsvergleich.tex}
+
+% Lokalisierung im Frequenzraum
+\folie{8/wavelets/frequenzlokalisierung.tex}
+
+% Dilatation im Frequenzraum
+\folie{8/wavelets/dilatation.tex}
+
+% Dilatation in Matrixform
+\folie{8/wavelets/matrixdilatation.tex}
+
+% Funktionen g und h
+\folie{8/wavelets/gundh.tex}
+\ifthenelse{\boolean{presentation}}{
+\folie{8/wavelets/dilbei.tex}
+}{}
+
+% Wavelet Frame
+\folie{8/wavelets/frame.tex}
+
+% Framekonstante
+\folie{8/wavelets/framekonstanten.tex}
+
+% Kugel-Beispiel
+\ifthenelse{\boolean{presentation}}{
+\folie{8/wavelets/beispiel.tex}
+}{}
+
diff --git a/vorlesungen/14_msehilbertraum/Makefile b/vorlesungen/14_msehilbertraum/Makefile
new file mode 100644
index 0000000..e5de69c
--- /dev/null
+++ b/vorlesungen/14_msehilbertraum/Makefile
@@ -0,0 +1,33 @@
+#
+# Makefile -- hilbertraum
+#
+# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+all:	hilbertraum-handout.pdf MathSemMSE-14-hilbertraum.pdf
+
+include ../slides/Makefile.inc
+
+SOURCES = common.tex slides.tex $(slides)
+
+MathSemMSE-14-hilbertraum.pdf:	MathSemMSE-14-hilbertraum.tex $(SOURCES)
+	pdflatex MathSemMSE-14-hilbertraum.tex
+
+hilbertraum-handout.pdf:	hilbertraum-handout.tex $(SOURCES)
+	pdflatex hilbertraum-handout.tex
+
+thumbnail:	thumbnail.jpg # fix1.jpg
+
+thumbnail.pdf:	MathSemMSE-14-hilbertraum.pdf
+	pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-14-hilbertraum.pdf 1
+thumbnail.jpg:	thumbnail.pdf
+	convert -density 300 thumbnail.pdf \
+                -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
+
+fix1.pdf:	MathSemMSE-14-hilbertraum.pdf
+	pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
+		MathSemMSE-14-hilbertraum.pdf 1
+fix1.jpg:	fix1.pdf
+	convert -density 300 fix1.pdf \
+                -resize 1920x1080 -units PixelsPerInch fix1.jpg
+
diff --git a/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex b/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex
new file mode 100644
index 0000000..b06500c
--- /dev/null
+++ b/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex
@@ -0,0 +1,14 @@
+%
+% MathSem-14-msehilbertraum.tex -- Präsentation 
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{true}
+\begin{document}
+\begin{frame}
+\titlepage
+\end{frame}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/14_msehilbertraum/common.tex b/vorlesungen/14_msehilbertraum/common.tex
new file mode 100644
index 0000000..a9089bf
--- /dev/null
+++ b/vorlesungen/14_msehilbertraum/common.tex
@@ -0,0 +1,16 @@
+%
+% common.tex -- gemeinsame definition
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\input{../common/packages.tex}
+\input{../common/common.tex}
+\mode<beamer>{%
+\usetheme[hideothersubsections,hidetitle]{Hannover}
+}
+\beamertemplatenavigationsymbolsempty
+\title[Hilbertraum]{Hilbertraum}
+\author[A.~Müller]{Prof.~Dr.~Andreas Müller}
+\date[]{}
+\newboolean{presentation}
+
diff --git a/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex b/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex
new file mode 100644
index 0000000..3dc7abf
--- /dev/null
+++ b/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex
@@ -0,0 +1,11 @@
+%
+% msehilbertraum-handout.tex -- Handout XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\documentclass[handout,aspectratio=169]{beamer}
+\input{common.tex}
+\setboolean{presentation}{false}
+\begin{document}
+\input{slides.tex}
+\end{document}
diff --git a/vorlesungen/14_msehilbertraum/slides.tex b/vorlesungen/14_msehilbertraum/slides.tex
new file mode 100644
index 0000000..19925db
--- /dev/null
+++ b/vorlesungen/14_msehilbertraum/slides.tex
@@ -0,0 +1,31 @@
+%
+% slides.tex -- XXX
+%
+% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\section{Hilbertraum}
+\folie{2/hilbertraum/definition.tex}
+\folie{2/hilbertraum/l2beispiel.tex}
+\folie{2/hilbertraum/basis.tex}
+\folie{2/hilbertraum/plancherel.tex}
+
+\section{Beispiele}
+\folie{2/hilbertraum/l2.tex}
+
+\section{Riesz-Darstellungssatz}
+\folie{2/hilbertraum/riesz.tex}
+\folie{2/hilbertraum/rieszbeispiel.tex}
+
+\section{$A^*$}
+\folie{2/hilbertraum/adjungiert.tex}
+\folie{2/hilbertraum/spektral.tex}
+
+\section{PDE und Hilbertraum}
+\folie{2/hilbertraum/sturm.tex}
+\folie{2/hilbertraum/laplace.tex}
+\folie{2/hilbertraum/qm.tex}
+\folie{2/hilbertraum/energie.tex}
+\folie{2/hilbertraum/sobolev.tex}
+
+
diff --git a/vorlesungen/99_vortraege/Makefile b/vorlesungen/99_vortraege/Makefile
index 8a5751c..69ec665 100644
--- a/vorlesungen/99_vortraege/Makefile
+++ b/vorlesungen/99_vortraege/Makefile
@@ -15,7 +15,8 @@ MathSem-99-vortraege.pdf:	MathSem-99-vortraege.tex $(SOURCES)
 vortraege-handout.pdf:	vortraege-handout.tex $(SOURCES)
 	pdflatex vortraege-handout.tex
 
-thumbnail:	thumbnail1.jpg thumbnail2.jpg thumbnail3.jpg thumbnail4.jpg 
+thumbnail:	thumbnail1.jpg thumbnail2.jpg thumbnail3.jpg thumbnail4.jpg \
+		thumbnail5.jpg
 
 thumbnail1.pdf:	MathSem-99-vortraege.pdf
 	pdfjam --outfile thumbnail1.pdf --papersize '{16cm,9cm}' \
@@ -45,6 +46,13 @@ thumbnail4.jpg:	thumbnail4.pdf
 	convert -density 300 thumbnail4.pdf \
                 -resize 1920x1080 -units PixelsPerInch thumbnail4.jpg
 
+thumbnail5.pdf:	MathSem-99-vortraege.pdf
+	pdfjam --outfile thumbnail5.pdf --papersize '{16cm,9cm}' \
+		MathSem-99-vortraege.pdf 5
+thumbnail5.jpg:	thumbnail5.pdf
+	convert -density 300 thumbnail5.pdf \
+                -resize 1920x1080 -units PixelsPerInch thumbnail5.jpg
+
 fix1.pdf:	MathSem-99-vortraege.pdf
 	pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
 		MathSem-99-vortraege.pdf 1
diff --git a/vorlesungen/99_vortraege/MathSem-99-vortraege.tex b/vorlesungen/99_vortraege/MathSem-99-vortraege.tex
index c962e90..ddcfcf8 100644
--- a/vorlesungen/99_vortraege/MathSem-99-vortraege.tex
+++ b/vorlesungen/99_vortraege/MathSem-99-vortraege.tex
@@ -61,5 +61,18 @@ Fabio Viecelli, Lukas Zogg: Erdbebenmessung
 \end{center}
 \end{frame}
 
+\title[Vorträge]{31. Mai: Vorträge}
+\begin{frame}
+\titlepage
+\vspace{-2cm}
+\begin{center}
+Marc Kühne: Munkres-Algorithmus
+\phantom{blubb} \\
+\phantom{blubb} \\
+\phantom{blubb} \\
+Michael Schmid: Schnelle Matrixmultiplikation
+\end{center}
+\end{frame}
+
 \input{slides.tex}
 \end{document}
diff --git a/vorlesungen/punktgruppen/.gitignore b/vorlesungen/punktgruppen/.gitignore
new file mode 100644
index 0000000..3633a3d
--- /dev/null
+++ b/vorlesungen/punktgruppen/.gitignore
@@ -0,0 +1,20 @@
+# directories
+__pycache__
+media/Tex
+
+media/images/crystal
+media/images/freezeframes
+
+media/videos
+media/audio
+
+media/Punktgruppen
+media/Punktgruppen.mp4
+
+build
+
+# files
+script.log
+slides.log
+slides.vrb
+missfont.log
diff --git a/vorlesungen/punktgruppen/Makefile b/vorlesungen/punktgruppen/Makefile
new file mode 100644
index 0000000..302e976
--- /dev/null
+++ b/vorlesungen/punktgruppen/Makefile
@@ -0,0 +1,18 @@
+TEX=xelatex
+TEXARGS=--output-directory=build --halt-on-error --shell-escape
+
+all: slides.pdf script.pdf media
+
+.PHONY: clean
+clean:
+	@rm -rfv build
+
+%.pdf: %.tex
+	mkdir -p build
+	$(TEX) $(TEXARGS) $<
+	$(TEX) $(TEXARGS) $<
+	cp build/$@ .
+
+media:
+	poetry install
+	poetry run manim -ql crystals.py
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
new file mode 100644
index 0000000..4a9836a
--- /dev/null
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -0,0 +1,611 @@
+from manim import *
+
+import math as m
+import numpy as np
+import itertools as it
+
+# configure style
+config.background_color = '#202020'
+config.tex_template.add_to_preamble(
+    r"\usepackage[p,osf]{scholax}"
+    r"\usepackage{amsmath}"
+    r"\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}"
+)
+
+# scenes
+class Geometric2DSymmetries(Scene):
+    def construct(self):
+        self.wait(5)
+
+        self.intro()
+        self.cyclic()
+        self.dihedral()
+        self.circle()
+
+    def intro(self):
+        # create square
+        square = Square()
+        square.set_fill(PINK, opacity=.5)
+        self.play(SpinInFromNothing(square))
+        self.wait()
+
+        # the action of doing nothing
+        action = MathTex(r"\mathbb{1}")
+        self.play(Write(action))
+        self.play(ApplyMethod(square.scale, 1.2))
+        self.play(ApplyMethod(square.scale, 1/1.2))
+        self.play(FadeOut(action))
+        self.wait()
+
+        # show some reflections
+        axis = DashedLine(2 * LEFT, 2 * RIGHT)
+        sigma = MathTex(r"\sigma")
+        sigma.next_to(axis, RIGHT)
+
+        self.play(Create(axis))
+        self.play(Write(sigma))
+        self.play(ApplyMethod(square.flip, RIGHT))
+
+        for d in [UP + RIGHT, UP]:
+            self.play(
+                Rotate(axis, PI/4),
+                Rotate(sigma, PI/4, about_point=ORIGIN))
+
+            self.play(Rotate(sigma, -PI/4), run_time=.5)
+            self.play(ApplyMethod(square.flip, d))
+
+        self.play(FadeOutAndShift(sigma), Uncreate(axis))
+
+        # show some rotations
+        dot = Dot(UP + RIGHT)
+        figure = VGroup(square, dot)
+
+        rot = MathTex(r"r")
+        self.play(Write(rot), Create(dot))
+
+        last = rot
+        for newrot in map(MathTex, [r"r", r"r^2", r"r^3"]):
+            self.play(
+                ReplacementTransform(last, newrot),
+                Rotate(figure, PI/2, about_point=ORIGIN))
+            self.wait(.5)
+            last = newrot
+
+        self.play(Uncreate(dot), FadeOut(square), FadeOut(last))
+
+
+    def cyclic(self):
+        # create symmetric figure
+        figure = VGroup()
+        prev = [1.5, 0, 0]
+        for i in range(1,6):
+            pos = [
+                1.5*m.cos(2 * PI/5 * i),
+                1.5*m.sin(2 * PI/5 * i),
+                0
+            ]
+
+            if prev:
+                line = Line(prev, pos)
+                figure.add(line)
+
+            dot = Dot(pos, radius=.1)
+            if i == 5:
+                dot.set_fill(RED)
+
+            prev = pos
+            figure.add(dot)
+
+        group = MathTex(r"G = \langle r \rangle")
+        self.play(Write(group), run_time = 2)
+        self.wait(3)
+
+        self.play(ApplyMethod(group.to_edge, UP))
+
+        actions = map(MathTex, [
+            r"\mathbb{1}", r"r", r"r^2",
+            r"r^3", r"r^4", r"\mathbb{1}", r"r"])
+
+        action = next(actions, MathTex(r"r"))
+
+        self.play(Create(figure))
+        self.play(Write(action))
+        self.wait()
+
+        for i in range(5):
+            newaction = next(actions, MathTex(r"r"))
+            self.play(
+                ReplacementTransform(action, newaction),
+                Rotate(figure, 2*PI/5, about_point=ORIGIN))
+            action = newaction
+
+        self.wait()
+        newaction = next(actions, MathTex(r"r"))
+        self.play(
+            ReplacementTransform(action, newaction),
+            Rotate(figure, 2*PI/5, about_point=ORIGIN))
+        action = newaction
+        self.wait(2)
+
+        self.play(Uncreate(figure), FadeOut(action))
+
+        whole_group = MathTex(
+            r"G = \langle r \rangle" 
+            r"= \left\{\mathbb{1}, r, r^2, r^3, r^4 \right\}")
+
+        self.play(ApplyMethod(group.move_to, ORIGIN))
+        self.play(ReplacementTransform(group, whole_group))
+        self.wait(5)
+
+        cyclic = MathTex(
+            r"C_n = \langle r \rangle" 
+            r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}")
+
+        cyclic_title = Tex(r"Zyklische Gruppe")
+        cyclic_title.next_to(cyclic, UP * 2)
+
+        cyclic.scale(1.2)
+        cyclic_title.scale(1.2)
+
+        self.play(ReplacementTransform(whole_group, cyclic))
+        self.play(FadeInFrom(cyclic_title, UP))
+
+        self.wait(5)
+        self.play(FadeOut(cyclic), FadeOut(cyclic_title))
+
+    def dihedral(self):
+        # create square
+        square = Square()
+        square.set_fill(PINK, opacity=.5)
+
+        # generator equation
+        group = MathTex(
+            r"G = \langle \sigma, r \,|\,",
+            r"\sigma^2 = \mathbb{1},",
+            r"r^4 = \mathbb{1},",
+            r"(\sigma r)^2 = \mathbb{1} \rangle")
+
+        self.play(Write(group), run_time = 2)
+        self.wait(5)
+
+        self.play(ApplyMethod(group.to_edge, UP))
+        self.play(FadeIn(square))
+        self.wait()
+
+        # flips
+        axis = DashedLine(2 * LEFT, 2 * RIGHT)
+        sigma = MathTex(r"\sigma^2 = \mathbb{1}")
+        sigma.next_to(axis, RIGHT)
+        self.play(Create(axis), Write(sigma))
+        self.play(ApplyMethod(square.flip, RIGHT))
+        self.play(ApplyMethod(square.flip, RIGHT))
+        self.play(Uncreate(axis), FadeOut(sigma))
+
+        # rotations
+        dot = Dot(UP + RIGHT)
+        rot = MathTex(r"r^4 = \mathbb{1}")
+        rot.next_to(square, DOWN * 3)
+
+        figure = VGroup(dot, square)
+
+        self.play(Write(rot), Create(dot))
+        for i in range(4):
+            self.play(Rotate(figure, PI/2))
+        self.play(FadeOut(rot), Uncreate(dot))
+
+        # rotation and flip
+        action = MathTex(r"(\sigma r)^2 = \mathbb{1}")
+        action.next_to(square, DOWN * 5)
+
+        dot = Dot(UP + RIGHT)
+        axis = DashedLine(2 * LEFT, 2 * RIGHT)
+        self.play(Create(dot), Create(axis), Write(action))
+
+        figure = VGroup(dot, square)
+
+        for i in range(2):
+            self.play(Rotate(figure, PI/2))
+            self.play(ApplyMethod(figure.flip, RIGHT))
+            self.wait()
+
+        self.play(Uncreate(dot), Uncreate(axis), FadeOut(action))
+        self.play(FadeOut(square))
+
+        # equation for the whole
+        whole_group = MathTex(
+            r"G &= \langle \sigma, r \,|\,"
+            r"\sigma^2 = r^4 = (\sigma r)^2 = \mathbb{1} \rangle \\"
+            r"&= \left\{"
+            r"\mathbb{1}, r, r^2, r^3, \sigma, \sigma r, \sigma r^2, \sigma r^3"
+            r"\right\}")
+
+        self.play(ApplyMethod(group.move_to, ORIGIN))
+        self.play(ReplacementTransform(group, whole_group))
+        self.wait(2)
+
+        dihedral = MathTex(
+            r"D_n &= \langle \sigma, r \,|\,"
+            r"\sigma^2 = r^n = (\sigma r)^2 = \mathbb{1} \rangle \\"
+            r"&= \left\{"
+            r"\mathbb{1}, r, r^2, \dots, \sigma, \sigma r, \sigma r^2, \dots"
+            r"\right\}")
+
+        dihedral_title = Tex(r"Diedergruppe: Symmetrien eines \(n\)-gons")
+        dihedral_title.next_to(dihedral, UP * 2)
+
+        dihedral.scale(1.2)
+        dihedral_title.scale(1.2)
+
+        self.play(ReplacementTransform(whole_group, dihedral))
+        self.play(FadeInFrom(dihedral_title, UP))
+
+        self.wait(5)
+        self.play(FadeOut(dihedral), FadeOut(dihedral_title))
+
+    def circle(self):
+        circle = Circle(radius=2)
+        dot = Dot()
+        dot.move_to(2 * RIGHT)
+
+        figure = VGroup(circle, dot)
+        group_name = MathTex(r"C_\infty")
+
+        # create circle
+        self.play(Create(circle))
+        self.play(Create(dot))
+
+        # move it around
+        self.play(Rotate(figure, PI/3))
+        self.play(Rotate(figure, PI/6))
+        self.play(Rotate(figure, -PI/3))
+
+        # show name
+        self.play(Rotate(figure, PI/4), Write(group_name))
+        self.wait()
+        self.play(Uncreate(figure))
+
+        nsphere = MathTex(r"C_\infty \cong S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}")
+        nsphere_title = Tex(r"Kreisgruppe")
+        nsphere_title.next_to(nsphere, 2 * UP)
+
+        nsphere.scale(1.2)
+        nsphere_title.scale(1.2)
+
+        self.play(ReplacementTransform(group_name, nsphere))
+        self.play(FadeInFrom(nsphere_title, UP))
+
+        self.wait(5)
+        self.play(FadeOut(nsphere_title), FadeOut(nsphere))
+        self.wait(2)
+
+
+class Geometric3DSymmetries(ThreeDScene):
+    def construct(self):
+        self.improper_rotation()
+        self.tetrahedron()
+
+    def improper_rotation(self):
+        # changes the source of the light and camera
+        self.renderer.camera.light_source.move_to(3*IN) 
+        self.set_camera_orientation(phi=0, theta=0)
+
+        # initial square
+        square = Square()
+        square.set_fill(PINK, opacity=.5)
+
+        self.play(SpinInFromNothing(square))
+        self.wait(2)
+
+        for i in range(4):
+            self.play(Rotate(square, PI/2))
+            self.wait(.5)
+
+        self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES)
+
+        # create rotation axis
+        axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5])
+
+        axis_name = MathTex(r"r \in C_4")
+        # move to yz plane
+        axis_name.rotate(PI/2, axis = RIGHT)
+        axis_name.next_to(axis, OUT)
+
+        self.play(Create(axis))
+        self.play(Write(axis_name))
+        self.wait()
+
+        # create sphere from slices
+        cyclic_slices = []
+        for i in range(4):
+            colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E]
+            cyclic_slices.append(ParametricSurface(
+                    lambda u, v: np.array([
+                        np.sqrt(2) * np.cos(u) * np.cos(v),
+                        np.sqrt(2) * np.cos(u) * np.sin(v),
+                        np.sqrt(2) * np.sin(u)
+                    ]),
+                    v_min=PI/4 + PI/2 * i,
+                    v_max=PI/4 + PI/2 * (i + 1),
+                    u_min=-PI/2, u_max=PI/2,
+                    checkerboard_colors=colors, resolution=(10,5)))
+
+        self.play(FadeOut(square), *map(Create, cyclic_slices))
+
+        cyclic_sphere = VGroup(*cyclic_slices)
+        for i in range(4):
+            self.play(Rotate(cyclic_sphere, PI/2))
+            self.wait()
+
+        new_axis_name = MathTex(r"r \in D_4")
+        # move to yz plane
+        new_axis_name.rotate(PI/2, axis = RIGHT)
+        new_axis_name.next_to(axis, OUT)
+        self.play(ReplacementTransform(axis_name, new_axis_name))
+
+        # reflection plane
+        self.play(FadeOut(cyclic_sphere), FadeIn(square))
+        plane = ParametricSurface(
+                lambda u, v: np.array([u, 0, v]),
+                u_min = -2, u_max = 2,
+                v_min = -2, v_max = 2,
+                fill_opacity=.3, resolution=(1,1))
+
+        plane_name = MathTex(r"\sigma \in D_4")
+        # move to yz plane
+        plane_name.rotate(PI/2, axis = RIGHT)
+        plane_name.next_to(plane, OUT + RIGHT)
+
+        self.play(Create(plane))
+        self.play(Write(plane_name))
+        self.wait()
+
+        self.move_camera(phi = 25 * DEGREES, theta = -75 * DEGREES)
+        self.wait()
+
+        condition = MathTex(r"(\sigma r)^2 = \mathbb{1}")
+        condition.next_to(square, DOWN);
+
+        self.play(Write(condition))
+        self.play(Rotate(square, PI/2))
+        self.play(Rotate(square, PI, RIGHT))
+
+        self.play(Rotate(square, PI/2))
+        self.play(Rotate(square, PI, RIGHT))
+        self.play(FadeOut(condition))
+
+        self.move_camera(phi = 75 * DEGREES, theta = -80 * DEGREES)
+
+        # create sphere from slices
+        dihedral_slices = []
+        for i in range(4):
+            for j in range(2):
+                colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E]
+                dihedral_slices.append(ParametricSurface(
+                        lambda u, v: np.array([
+                            np.sqrt(2) * np.cos(u) * np.cos(v),
+                            np.sqrt(2) * np.cos(u) * np.sin(v),
+                            np.sqrt(2) * np.sin(u)
+                        ]),
+                        v_min=PI/2 * j + PI/4 + PI/2 * i,
+                        v_max=PI/2 * j + PI/4 + PI/2 * (i + 1),
+                        u_min=-PI/2 if j == 0 else 0,
+                        u_max=0 if j == 0 else PI/2,
+                        checkerboard_colors=colors, resolution=(10,5)))
+
+        dihedral_sphere = VGroup(*dihedral_slices)
+
+        self.play(FadeOut(square), Create(dihedral_sphere))
+
+        for i in range(2):
+            self.play(Rotate(dihedral_sphere, PI/2))
+            self.play(Rotate(dihedral_sphere, PI, RIGHT))
+            self.wait()
+
+        self.wait(2)
+        self.play(*map(FadeOut, [dihedral_sphere, plane, plane_name, new_axis_name]), FadeIn(square))
+        self.wait(3)
+        self.play(*map(FadeOut, [square, axis]))
+        self.wait(3)
+
+    def tetrahedron(self):
+        tet = Tetrahedron(edge_length=2)
+        self.play(FadeIn(tet))
+        
+        self.move_camera(phi = 75 * DEGREES, theta = -100 * DEGREES)
+        self.begin_ambient_camera_rotation(rate=.1)
+
+        axes = []
+        for coord in tet.vertex_coords:
+            axes.append((-2 * coord, 2 * coord))
+
+        lines = [
+            Line3D(start=s, end=e) for s, e in axes
+        ]
+
+        self.play(*map(Create, lines))
+        self.wait()
+
+        for axis in axes:
+            self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+            self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+
+        self.wait(5)
+        self.stop_ambient_camera_rotation()
+        self.wait()
+        self.play(*map(Uncreate, lines))
+        self.play(FadeOut(tet))
+        self.wait(5)
+
+
+class AlgebraicSymmetries(Scene):
+    def construct(self):
+        self.wait(5)
+        self.cyclic()
+        # self.matrices()
+
+    def cyclic(self):
+        # show the i product
+        product = MathTex(
+            r"1", r"\cdot i &= i \\",
+            r"i \cdot i &= -1 \\",
+            r"-1 \cdot i &= -i \\",
+            r"-i \cdot i &= 1")
+        product.scale(1.5)
+
+        for part in product:
+            self.play(Write(part))
+            self.wait()
+
+        self.play(ApplyMethod(product.scale, 1/1.5))
+
+        # gather in group
+        group = MathTex(r"G = \left\{ 1, i, -1, -i \right\}")
+        self.play(ReplacementTransform(product, group))
+        self.wait(2)
+
+        # show C4
+        grouppow =  MathTex(
+            r"G &= \left\{ 1, i, i^2, i^3 \right\} \\",
+            r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}")
+        self.play(ReplacementTransform(group, grouppow[0]))
+        self.wait(2)
+
+        self.play(Write(grouppow[1]))
+        self.wait(4)
+        
+        self.play(ApplyMethod(grouppow.to_edge, UP))
+
+        # define morphisms
+        morphism = MathTex(r"\phi: C_4 \to G \\")
+        morphism.shift(UP)
+        self.play(Write(morphism))
+        self.wait()
+
+        # show an example
+        mappings = MathTex(
+            r"\phi(\mathbb{1}) &= 1 \\",
+            r"\phi(r) &= i \\",
+            r"\phi(r^2) &= i^2 \\",
+            r"\phi(r^3) &= i^3 \\")
+        mappings.next_to(morphism, 2 * DOWN)
+
+        self.play(Write(mappings))
+        self.wait(3)
+        self.play(FadeOutAndShift(mappings, DOWN))
+
+        # more general definition
+        homomorphism = MathTex(
+            r"\phi(r\circ \mathbb{1}) &= \phi(r)\cdot\phi(\mathbb{1}) \\",
+            r"&= i\cdot 1")
+        homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT)
+        for part in homomorphism:
+            self.play(Write(part))
+            self.wait()
+
+        hom_bracegrp = VGroup(morphism, homomorphism)
+
+        self.play(
+            ApplyMethod(grouppow.shift, 3 * LEFT),
+            ApplyMethod(hom_bracegrp.shift, 3 * LEFT))
+
+        hom_brace = Brace(hom_bracegrp, direction=RIGHT)
+        hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT)
+        hom_text_short = MathTex(r"\mathrm{Hom}(C_4, G)").next_to(hom_brace.get_tip(), RIGHT)
+
+        self.play(Create(hom_brace))
+        self.play(Write(hom_text))
+        self.wait()
+        self.play(ReplacementTransform(hom_text, hom_text_short))
+        self.wait()
+
+        # add the isomorphism part
+        isomorphism = Tex(r"\(\phi\) ist bijektiv")
+        isomorphism.next_to(homomorphism, DOWN).align_to(homomorphism, LEFT)
+        self.play(Write(isomorphism))
+
+        iso_bracegrp = VGroup(hom_bracegrp, isomorphism)
+
+        iso_brace = Brace(iso_bracegrp, RIGHT)
+        iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT)
+        iso_text_short = MathTex("C_4 \cong G").next_to(iso_brace.get_tip(), RIGHT)
+
+        self.play(
+            ReplacementTransform(hom_brace, iso_brace),
+            ReplacementTransform(hom_text_short, iso_text))
+        self.wait()
+
+        self.play(ReplacementTransform(iso_text, iso_text_short))
+        self.wait()
+
+        # create a group for the whole
+        morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short)
+
+        self.play(
+            ApplyMethod(grouppow.to_edge, LEFT),
+            ApplyMethod(morphgrp.to_edge, LEFT))
+
+        # draw a complex plane
+        plane = ComplexPlane(x_range = [-2.5, 2.5])
+        coordinates = plane.get_coordinate_labels(1, -1, 1j, -1j)
+
+        roots = list(map(lambda p: Dot(p, fill_color=PINK), (
+            [1, 0, 0], [0, 1, 0], [-1, 0, 0], [0, -1, 0]
+        )))
+
+        arrow = CurvedArrow(
+            1.5 * np.array([m.cos(10 * DEGREES), m.sin(10 * DEGREES), 0]),
+            1.5 * np.array([m.cos(80 * DEGREES), m.sin(80 * DEGREES), 0]))
+        arrowtext = MathTex("\cdot i")
+        arrowtext.move_to(2 / m.sqrt(2) * (UP + RIGHT))
+
+        square = Square().rotate(PI/4).scale(1/m.sqrt(2))
+        square.set_fill(PINK).set_opacity(.4)
+
+        figuregrp = VGroup(plane, square, arrow, arrowtext, *coordinates, *roots)
+        figuregrp.to_edge(RIGHT)
+
+        self.play(Create(plane))
+        self.play(
+            *map(Create, roots),
+            *map(Write, coordinates))
+        self.wait()
+        self.play(FadeIn(square), Create(arrow), Write(arrowtext))
+
+        for _ in range(4):
+            self.play(Rotate(square, PI/2))
+            self.wait(.5)
+
+        self.play(
+            *map(FadeOut, (square, arrow, arrowtext)),
+            *map(FadeOut, coordinates),
+            *map(FadeOut, roots))
+        self.play(Uncreate(plane))
+        self.play(
+            FadeOutAndShift(grouppow, RIGHT),
+            FadeOutAndShift(morphgrp, RIGHT))
+
+        modulo = MathTex(
+            r"\phi: C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\"
+            r"\phi(\mathbb{1} \circ r^2) &= 0 + 2 \pmod 4").scale(1.5)
+        self.play(Write(modulo))
+        self.wait(2)
+
+        self.play(FadeOut(modulo))
+        self.wait(3)
+
+    def matrices(self):
+        question = MathTex(
+            r"D_n &\cong \,? \\"
+            r"S_n &\cong \,? \\"
+            r"A_n &\cong \,?").scale(1.5)
+
+        answer = MathTex(
+            r"D_n &\cong \,?\\"
+            r"S_4 &\cong \mathrm{Aut}(Q_8) \\"
+            r"A_5 &\cong \mathrm{PSL}_2 (5)").scale(1.5)
+
+        self.play(Write(question))
+        self.wait()
+        self.play(ReplacementTransform(question, answer))
+
+        self.wait(3)
diff --git a/vorlesungen/punktgruppen/media/images/nosignal.jpg b/vorlesungen/punktgruppen/media/images/nosignal.jpg
new file mode 100644
index 0000000..2beeb8b
--- /dev/null
+++ b/vorlesungen/punktgruppen/media/images/nosignal.jpg
diff --git a/vorlesungen/punktgruppen/poetry.lock b/vorlesungen/punktgruppen/poetry.lock
new file mode 100644
index 0000000..069d270
--- /dev/null
+++ b/vorlesungen/punktgruppen/poetry.lock
@@ -0,0 +1,743 @@
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+name = "certifi"
+version = "2020.12.5"
+description = "Python package for providing Mozilla's CA Bundle."
+category = "main"
+optional = false
+python-versions = "*"
+
+[[package]]
+name = "chardet"
+version = "4.0.0"
+description = "Universal encoding detector for Python 2 and 3"
+category = "main"
+optional = false
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
+
+[[package]]
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+description = "Composable command line interface toolkit"
+category = "main"
+optional = false
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
+
+[[package]]
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+version = "1.2.2"
+description = "Extends click.Group to invoke a command without explicit subcommand name"
+category = "main"
+optional = false
+python-versions = "*"
+
+[package.dependencies]
+click = "*"
+
+[[package]]
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+version = "0.7.1"
+description = "Option groups and subcommand help sections for pallets/click"
+category = "main"
+optional = false
+python-versions = ">=3.6"
+
+[package.dependencies]
+click = ">=7.0,<9.0"
+
+[[package]]
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+description = "Cross-platform colored terminal text."
+category = "main"
+optional = false
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+
+[[package]]
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+description = "converts and manipulates various color representation (HSL, RVB, web, X11, ...)"
+category = "main"
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+python-versions = "*"
+
+[package.extras]
+test = ["nose"]
+
+[[package]]
+name = "commonmark"
+version = "0.9.1"
+description = "Python parser for the CommonMark Markdown spec"
+category = "main"
+optional = false
+python-versions = "*"
+
+[package.extras]
+test = ["flake8 (==3.7.8)", "hypothesis (==3.55.3)"]
+
+[[package]]
+name = "decorator"
+version = "4.4.2"
+description = "Decorators for Humans"
+category = "main"
+optional = false
+python-versions = ">=2.6, !=3.0.*, !=3.1.*"
+
+[[package]]
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+description = "Portable OpenGL Context"
+category = "main"
+optional = false
+python-versions = "*"
+
+[[package]]
+name = "idna"
+version = "2.10"
+description = "Internationalized Domain Names in Applications (IDNA)"
+category = "main"
+optional = false
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
+
+[[package]]
+name = "importlib-metadata"
+version = "4.0.1"
+description = "Read metadata from Python packages"
+category = "main"
+optional = false
+python-versions = ">=3.6"
+
+[package.dependencies]
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+
+[package.extras]
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+testing = ["pytest (>=4.6)", "pytest-checkdocs (>=2.4)", "pytest-flake8", "pytest-cov", "pytest-enabler (>=1.0.1)", "packaging", "pep517", "pyfakefs", "flufl.flake8", "pytest-black (>=0.3.7)", "pytest-mypy", "importlib-resources (>=1.3)"]
+
+[[package]]
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+category = "main"
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+
+[package.dependencies]
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+
+[[package]]
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+description = "Bindings for Pango for using with Manim."
+category = "main"
+optional = false
+python-versions = ">=3.6"
+
+[[package]]
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+description = "Python bindings for the mapbox earcut C++ polygon triangulation library."
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+
+[package.dependencies]
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+
+[[package]]
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+description = "ModernGL: High performance rendering for Python 3"
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+optional = false
+python-versions = "*"
+
+[package.dependencies]
+glcontext = ">=2,<3"
+
+[[package]]
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+[package.dependencies]
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+[package.extras]
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+
+[[package]]
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+description = "Multiple dispatch"
+category = "main"
+optional = false
+python-versions = "*"
+
+[package.dependencies]
+six = "*"
+
+[[package]]
+name = "networkx"
+version = "2.5.1"
+description = "Python package for creating and manipulating graphs and networks"
+category = "main"
+optional = false
+python-versions = ">=3.6"
+
+[package.dependencies]
+decorator = ">=4.3,<5"
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+[package.extras]
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+description = "NumPy is the fundamental package for array computing with Python."
+category = "main"
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+
+[[package]]
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+[[package]]
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+[[package]]
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+[[package]]
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+description = "Pygments is a syntax highlighting package written in Python."
+category = "main"
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+
+[[package]]
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diff --git a/vorlesungen/punktgruppen/pyproject.toml b/vorlesungen/punktgruppen/pyproject.toml
new file mode 100644
index 0000000..527eb57
--- /dev/null
+++ b/vorlesungen/punktgruppen/pyproject.toml
@@ -0,0 +1,15 @@
+[tool.poetry]
+name = "presentation"
+version = "0.1.0"
+description = ""
+authors = ["Nao Pross <np@0hm.ch>"]
+
+[tool.poetry.dependencies]
+python = "^3.7"
+manim = "^0.6.0"
+
+[tool.poetry.dev-dependencies]
+
+[build-system]
+requires = ["poetry-core>=1.0.0"]
+build-backend = "poetry.core.masonry.api"
diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf
new file mode 100644
index 0000000..91993fb
--- /dev/null
+++ b/vorlesungen/punktgruppen/script.pdf
diff --git a/vorlesungen/punktgruppen/script.tex b/vorlesungen/punktgruppen/script.tex
new file mode 100644
index 0000000..bc50e21
--- /dev/null
+++ b/vorlesungen/punktgruppen/script.tex
@@ -0,0 +1,214 @@
+\documentclass[a4paper]{article}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+
+\usepackage[cm]{manuscript}
+\usepackage{xcolor}
+
+\newcommand{\scene}[1]{\par\noindent[ #1 ]\par}
+\newenvironment{totranslate}{\color{blue!70!black}}{}
+
+\begin{document}
+\section{Das sind wir}
+\scene{Camera}
+
+\section{Ablauf}
+Zuerst werden wir Symmetrien in 2 Dimensionen anschauen, dann \"uberlegen wir
+kurz was es heisst f\"ur eine Symmetrie ``algebraisch'' zu sein. Von da aus
+kommt die dritte Dimension hinzu, die man besser mit Matrizen verstehen kann.
+Mit der aufgebauten Theorie werden wir versuchen Kristalle zu klassifizieren.
+Und zum Schluss kommen wir zu Anwendungen, welche f\"ur Ingenieure von
+Interesse sind.
+
+\section{intro}
+\scene{Spontan}
+
+\section{2D Geometrie}
+\scene{Intro}
+Wir fangen mit den 2 dimensionalen Symmetrien an, da man sie sich am
+einfachsten vorstellen kann. Eine Symmetrie eines Objektes beschreibt eine
+Aktion, welche nachdem sie auf das Objekt wirkt, das Objekt wieder gleich
+aussehen l\"asst.
+
+\scene{Viereck}
+Die einfachste Aktion, ist das Viereck zu nehmen, und wieder hinzulegen.
+Eine andere Aktion k\"onnte sein, das Objekt um eine Achse zu spiegeln,
+oder eine Rotation um 90 Grad.
+
+\scene{Zyklische Gruppe}
+Fokussieren wir uns auf die einfachste Klassen von Symmetrien: diejenigen die
+von einer reinen Drehung generiert werden. Wir sammeln diese in einer Gruppe
+\(G\), und notieren das sie von eine Rotation \(r\) generiert worden sind, mit
+diesen spitzen Klammern.
+
+Nehmen wir als Beispiel dieses Pentagon. Wenn wir \(r\) 5-mal anwenden, ist es
+dasselbe als wenn wir nichts gemacht h\"atten. Wenn wir es noch ein 6. mal
+drehen, entspricht dies dasselbe wie \(r\) nur 1 mal zu nutzen.
+
+\scene{Notation}
+So, die Gruppe setzt sich zusammen aus dem neutralen Element, und den Potenzen
+1 bis 4 von \(r\). Oder im allgemein Gruppen mit dieser Struktur, in welcher die
+Aktion \(n-1\) mal angewendet werden kann, heissen ``Zyklische Gruppe''.
+
+\scene{Diedergruppe}
+Nehmen wir nun auch noch die Spiegeloperation \(\sigma\) dazu. Weil wir jetzt 2
+Operationen haben, m\"ussen wir auch im Generator schreiben wie sie
+zusammenh\"angen.  Schauen wir dann uns genauer diesen Ausdr\"uck an.  Zweimal
+Spielegeln ist \"aquivalent zum neutralen Element, sowie 4 mal um 90 Grad
+drehen und 2 Drehspiegelungen, welche man auch Inversion nennt.
+
+\scene{Notation}
+Daraus k\"onnen wir wieder die ganze Gruppe erzeugen, die im allgemeinen den
+Symmetrien eines \(n\)-gons entsprechen.
+
+\scene{Kreisgruppe}
+Bis jetzt hatten wir nur diskrete Symmetrien, was nicht zwingend der Fall sein
+muss. Ein Ring kann man kontinuierlich drehen, und sieht dabei immer gleich
+aus.
+
+Diese Symmetrie ist auch als Kreisgruppe bekannt, die man sch\"on mit dem
+komplexen Einheitskreis definieren kann.
+
+\section{Algebra}
+\scene{Produkt mit \(i\)}
+\"Uberlegen wir uns eine spezielle algebraische Operation: Multiplikation mit
+der imagin\"aren Einheit. \(1\) mal \(i\) ist gleich \(i\). Wieder mal \(i\)
+ist \(-1\), dann \(-i\) und schliesslich kommen wir z\"uruck auf \(1\).  Diese
+fassen wir in eine Gruppe \(G\) zusammen. Oder sch\"oner geschrieben:. Sieht das
+bekannt aus?
+
+\scene{Morphismen}
+Das Gefühl, dass es sich um dasselbe handelt, kann wie folgt formalisiert
+werden.  Sei \(\phi\) eine Funktion von \(C_4\) zu \(G\) und ordnen wir zu
+jeder Symmetrieoperation ein Element aus \(G\). Wenn man die Zuordnung richtig
+definiert, dann sieht man die folgende Eigenschaft: Eine Operation nach eine
+andere zu nutzen, und dann die Funktion des Resultats zu nehmen, ist gleich wie
+die Funktion der einzelnen Operazionen zu nehmen und die Resultate zu
+multiplizieren. Dieses Ergebnis ist so bemerkenswert, dass es in der Mathematik
+einen Namen bekommen hat: Homorphismus, von griechisch "homos" dasselbe und
+"morphe" Form.  Manchmal auch so geschrieben. Ausserdem, wenn \(\phi\) eins zu
+eins ist, heisst es \emph{Iso}morphismus: "iso" gleiche Form. Was man
+typischerweise mit diesem Symbol schreibt.
+
+\scene{Animation}
+Sie haben wahrscheinlich schon gesehen, worauf das hinausläuft.  Dass die
+zyklische Gruppe \(C_4\) und \(G\) isomorph sind ist nicht nur Fachjargon der
+mathematik, sondern sie haben wirklich die selbe Struktur.
+
+\scene{Modulo}
+Das Beispiel mit der komplexen Einheit, war wahrscheinlich nicht so
+\"uberraschend. Aber was merkw\"urdig ist, ist das Beziehungen zwischen
+Symmetrien und Algebra auch in Bereichen gefunden werden, welche auf den ersten
+Blick, nicht geomerisch erscheinen. Ein R\"atsel für die Neugierigen: die Summe
+in der Modulo-Arithmetik.  Als Hinweis: Um die Geometrie zu finden denken Sie
+an einer Uhr.
+
+\section{3D Geometrie}
+2 Dimensionen sind einfacher zu zeichnen, aber leider leben wir im 3
+dimensionalen Raum.
+
+\scene{Zyklische Gruppe}
+Wenn wir unser bekanntes Viereck mit seiner zyklischer Symmetrie in 3
+Dimensionen betrachten, k\"onnen wir seine Drehachse sehen.
+
+\scene{Diedergruppe}
+Um auch noch die andere Symmetrie des Rechteckes zu sehen, ben\"otigen wir eine
+Spiegelachse \(\sigma\), die hier eine Spiegelebene ist.
+
+\scene{Transition}
+Um die Punktsymmetrien zu klassifizieren orientiert man sich an einer Achse, um
+welche sich die meisten Symmetrien drehen. Das geht aber nicht immer, wie beim
+Tetraeder.
+
+\scene{Tetraedergruppe}
+Diese Geometrie hat 4 gleichwertige Symmetrieachsen, die eben eine
+Symmetriegruppe aufbauen, welche kreativer weise Tetraedergruppe genannt wird.
+Vielleicht fallen Ihnnen weitere Polygone ein mit dieser Eigenschaft, bevor wir
+zum n\"achsten Thema weitergehen.
+
+\section{Matrizen}
+\scene{Titelseite}
+Nun gehen wir kurz auf den Thema unseres Seminars ein: Matrizen.  Das man mit
+Matrizen Dinge darstellen kann, ist keine Neuigkeit mehr, nach einem
+Semester MatheSeminar.  Also überrascht es wohl auch keinen, das man alle
+punktsymmetrischen Operationen auch mit Matrizen Formulieren kann.
+
+\scene{Matrizen}
+
+Sei dann \(G\) unsere Symmetrie Gruppe, die unsere abstrakte Drehungen und
+Spiegelungen enth\"ahlt. Die Matrix Darstellung dieser Gruppe, ist eine
+Funktion gross \(\Phi\), von \(G\) zur orthogonalen Gruppe \(O(3)\), die zu
+jeder Symmetrie Operation klein \(g\) eine Matrix gross \(\Phi_g\) zuordnet.
+
+Zur Erinnerung, die Orthogonale Gruppe ist definiert als die Matrizen, deren
+transponierte auch die inverse ist. Da diese Volumen und Distanzen erhalten,
+natuerlich nur bis zu einer Vorzeichenumkehrung, macht es Sinn, dass diese
+Punksymmetrien genau beschreiben.
+
+Nehmen wir die folgende Operationen als Beispiele. Die Matrix der trivialen
+Operation, dass heisst nichts zu machen, ist die Einheitsmatrix. Eine
+Spiegelung ist dasselbe aber mit einem Minus, und Drehungen sind uns schon
+dank Herrn M\"uller bekannt.
+
+\section{Kristalle}
+\scene{Spontan}
+
+\section{Piezo}
+\scene{Spontan}
+
+\section{Licht}
+Als Finale, haben wir ein schwieriges Problem aus der Physik. Das Ziel dieser
+Folie ist nicht jedes Zeichen zu versehen, sondern zu zeigen wie man von hier
+weiter gehen kann. Wir mochten sehen wie sich Licht in einem Kristall verhaltet.
+Genauer, wir m\"ochten die Amplitude einer
+elektromagnetischer Welle in einem Kristall beschreiben.
+
+Das Beispiel richtet sich mehr an Elektrotechnik Studenten, aber die Theorie
+ist die gleiche bei mechanischen Wellen in Materialien mit einer
+Spannungstensor wie dem, den wir letzte Woche gesehen haben.
+% Ganz grob gesagt, ersetzt man E durch Xi und epsilon durch das Sigma.
+
+Um eine Welle zu beschreiben, verwenden wir die Helmholtz-Gleichung, die einige
+von uns bereits in anderen Kursen gel\"ost haben.  Schwierig wird aber dieses
+Problem, wenn der Term vor der Zeitableitung ein Tensor ist (f\"ur uns eine Matrix).
+
+Zur Vereinfachung werden wir eine ebene Welle verwenden. Setzt man dieses E in
+die Helmholtz-Gleichung ein, erhält man folgendes zurück: ein Eigenwertproblem.
+
+Physikalisch bedeutet dies, dass die Welle in diesem Material ihre Amplitude in
+Abhängigkeit von der Ausbreitungsrichtung ändert.  Und die Eigenwerte sagen
+aus, wie stark die Amplitude der Welle in jeder Richtung skaliert wird.
+
+Ich sagte, in jede Richtung skaliert, aber welche Richtungen genau?
+Physikalisch hängt das von der kristallinen Struktur des Materials ab, aber
+mathematisch können wir sagen: in Richtung der Eigenvektoren!  Aber diesen
+Eigenraum zu finden, in dem die Eigenvektoren wohnen, ist beliebig schwierig. 
+
+Hier kommt unsere Gruppentheorie zu Hilfe. Wir können die Symmetrien unseres
+Kristalls zur Hilfe nehmen. Zu jeder dieser Symmetrien lässt sich bekanntlich eine
+einfache Matrix finden, deren Eigenraum ebenfalls relativ leicht zu finden ist.
+Zum Beispiel ist der Eigenraum der Rotation \(r\), die Rotationsachse, für die
+Reflexion \(\sigma\) eine Ebene, und so weiter.
+
+Nun ist die Frage, ob man diese Eingenraume der Symmetrienoperationen
+kombinieren kann um den Eigenraum des physikalisches Problems zu finden.
+
+Aber leider ist meine Zeit abgelaufen in der Recherche, also müssen Sie mir 2
+Dingen einfach glauben, erstens dass es einen Weg gibt, und zweitens dass eher
+nicht so schlimm ist, wenn man die Notation einmal gelernt hat.
+
+Nachdem wir an, wir haben den Eigenraum U gefunden, dann können wir einen
+(Eigen)Vektor E daraus nehmen und in ihm direkt lambda ablesen. Das sagt uns,
+wie die Amplitude der Welle, in diese Richtung gedämpft wurde.
+
+Diese Methode ist nicht spezifisch für dieses Problem, im Gegenteil, ich habe
+gesehen, dass sie in vielen Bereichen eingesetzt wird, wie z.B.:
+Kristallographie, Festkörperphysik, Molekülschwingungen in der Quantenchemie
+und numerische Simulationen von Membranen.
+
+\section{Outro}
+\scene{Camera}
+
+\end{document}
+% vim:et ts=2 sw=2:
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
new file mode 100644
index 0000000..e1769f8
--- /dev/null
+++ b/vorlesungen/punktgruppen/slides.pdf
diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex
new file mode 100644
index 0000000..cd3d7d7
--- /dev/null
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -0,0 +1,895 @@
+\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer}
+
+% language
+\usepackage{polyglossia}
+\setmainlanguage{german}
+
+% pretty drawings
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\usetikzlibrary{external}
+\tikzexternalize[
+  mode = graphics if exists,
+  figure list = true,
+  prefix=build/
+]
+
+% Theme
+\beamertemplatenavigationsymbolsempty
+
+% set look
+\usetheme{default} 
+\usecolortheme{fly}
+\usefonttheme{serif}
+
+%% Set font
+\usepackage[p,osf]{scholax}
+\usepackage{amsmath}
+\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}
+
+% set colors
+\definecolor{background}{HTML}{202020}
+
+\setbeamercolor{normal text}{fg=white, bg=background}
+\setbeamercolor{structure}{fg=white}
+
+\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35}
+
+\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg}
+\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75}
+\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50}
+\setbeamercolor*{palette quaternary}{fg=white,bg=background}
+
+\setbeamercolor*{block title}{parent=structure}
+\setbeamercolor*{block body}{fg=background, bg=}
+
+\setbeamercolor*{framesubtitle}{fg=white}
+
+\setbeamertemplate{section page}
+{
+  \begin{center}
+    \Huge
+    \insertsection
+  \end{center}
+}
+\AtBeginSection{\frame{\sectionpage}}
+
+% Macros
+\newcommand{\ten}[1]{#1}
+
+% Metadata
+\title{\LARGE \scshape Punktgruppen und Kristalle}
+\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz}
+\institute{Hochschule f\"ur Technik OST, Rapperswil}
+\date{10. Mai 2021}
+
+% Slides
+\begin{document}
+\frame{
+  \titlepage
+  \vfill
+  \begin{center}
+    \small \color{gray}
+    Slides: \texttt{s.0hm.ch/ctBsD}
+  \end{center}
+}
+\frame{\tableofcontents}
+
+\frame{
+  \begin{itemize}
+    \item Was heisst \emph{Symmetrie} in der Mathematik? \pause
+    \item Wie kann ein Kristall modelliert werden? \pause
+    \item Aus der Physik: Licht, Piezoelektrizit\"at \pause
+  \end{itemize}
+  \begin{center}
+    \begin{tikzpicture}
+      \begin{scope}[
+          node distance = 0cm
+        ]
+        \node[
+          rectangle, fill = gray!40!background,
+          minimum width = 3cm, minimum height = 2cm,
+        ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+        \node[
+          draw, rectangle, thick, white, fill = red!50,
+          minimum width = 3cm, minimum height = 1mm,
+          above = of body
+        ] (pos) {};
+
+        \node[
+          draw, rectangle, thick, white, fill = blue!50,
+          minimum width = 3cm, minimum height = 1mm,
+          below = of body
+        ] (neg) {};
+
+        \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+        \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+        \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n);
+      \end{scope}
+      \begin{scope}[
+          node distance = 0cm,
+          xshift = 7cm
+        ]
+        \node[
+          rectangle, fill = gray!40!background,
+          minimum width = 3cm, minimum height = 1.5cm,
+        ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+        \node[
+          draw, rectangle, thick, white, fill = red!50,
+          minimum width = 3cm, minimum height = 1mm,
+          above = of body
+        ] (pos) {};
+
+        \node[
+          draw, rectangle, thick, white, fill = blue!50,
+          minimum width = 3cm, minimum height = 1mm,
+          below = of body
+        ] (neg) {};
+
+        \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1);
+        \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1);
+
+        \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+        \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+        \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n);
+      \end{scope}
+    \end{tikzpicture}
+  \end{center}
+}
+
+\section{2D Symmetrien}
+%% Made in video
+{
+  \usebackgroundtemplate{
+    \includegraphics[height=\paperheight]{media/images/nosignal}}
+  \frame{}
+}
+
+\section{Algebraische Symmetrien}
+%% Made in video
+\frame{
+  \begin{columns}[T]
+    \begin{column}{.5\textwidth}
+      Produkt mit \(i\)
+      \begin{align*}
+        1 \cdot i &= i \\
+        i \cdot i &= -1 \\
+        -1 \cdot i &= -i \\
+        -i \cdot i &= 1
+      \end{align*}
+      \pause
+      %
+      Gruppe
+      \begin{align*}
+        G &= \left\{
+          1, i, -1, -i
+        \right\} \\
+        &= \left\{
+          1, i, i^2, i^3
+        \right\} \\
+        C_4 &= \left\{
+          \mathbb{1}, r, r^2, r^3
+        \right\}
+      \end{align*}
+      \pause
+    \end{column}
+    \begin{column}{.5\textwidth}
+      Darstellung \(\phi : C_4 \to G\)
+      \begin{align*}
+        \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\
+        \phi(r) &= i & \phi(r^3) &= i^3
+      \end{align*}
+      \pause
+      %
+      Homomorphismus
+      \begin{align*}
+        \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\
+        &= i \cdot 1
+      \end{align*}
+      \pause
+      %
+      \(\phi\) ist bijektiv \(\implies C_4 \cong G\)
+			\pause
+			%
+			\begin{align*}
+				\psi : C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\
+				\psi(\mathbb{1}\circ r^2) &= 0 + 2 \pmod{4}
+			\end{align*}
+    \end{column}
+  \end{columns}
+}
+
+\section{3D Symmetrien}
+%% Made in video
+{
+  \usebackgroundtemplate{
+    \includegraphics[height=\paperheight]{media/images/nosignal}}
+  \frame{}
+}
+
+\section{Matrizen}
+\frame{
+  \begin{columns}[T]
+    \begin{column}{.5\textwidth}
+      Symmetriegruppe
+      \[
+        G = \left\{\mathbb{1}, r, \sigma, \dots \right\}
+      \]
+      \pause
+      Matrixdarstellung
+      \begin{align*}
+        \Phi : G &\to O(3) \\
+        g &\mapsto \Phi_g
+      \end{align*}
+      \pause
+      Orthogonale Gruppe
+      \[
+        O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\}
+      \]
+    \end{column}
+    \pause
+    \begin{column}{.5\textwidth}
+      \begin{align*}
+        \Phi_\mathbb{1} &= \begin{pmatrix}
+          1 & 0 & 0 \\
+          0 & 1 & 0 \\
+          0 & 0 & 1
+        \end{pmatrix} = I \\[1em]
+        \Phi_\sigma &= \begin{pmatrix}
+          1 & 0 & 0 \\
+          0 & -1 & 0 \\
+          0 & 0 & 1
+        \end{pmatrix} \\[1em]
+        \Phi_r &= \begin{pmatrix}
+          \cos \alpha & -\sin \alpha & 0 \\
+          \sin \alpha & \cos \alpha & 0 \\
+          0 & 0 & 1
+        \end{pmatrix}
+      \end{align*}
+    \end{column}
+  \end{columns}
+}
+
+\section{Kristalle}
+\begin{frame}[fragile]{}
+  \begin{columns}
+    \onslide<1->{
+      \begin{column}{.5\textwidth}
+        \begin{center}
+            \begin{tikzpicture}[
+                dot/.style = {
+                  draw, circle, thick, white, fill = gray!40!background,
+                  minimum size = 2mm,
+                  inner sep = 0pt,
+                  outer sep = 1mm,
+                },
+              ]
+
+              \begin{scope}
+                \clip (-2,-2) rectangle (3,4);
+                \foreach \y in {-7,-6,...,7} {
+                  \foreach \x in {-7,-6,...,7} {
+                    \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {};
+                  }
+                }
+              \end{scope}
+              \draw[white, thick] (-2, -2) rectangle (3,4);
+
+              \draw[red!80!background, thick, ->] 
+                (N00) to node[midway, below] {\(\vec{a}_1\)} (N10);
+              \draw[cyan!80!background, thick, ->]
+                (N00) to node[midway, left] {\(\vec{a}_2\)} (N01);
+            \end{tikzpicture}
+        \end{center}
+      \end{column}
+    }
+    \begin{column}{.5\textwidth}
+      \onslide<2->{
+        Kristallgitter:
+        \(n_i \in \mathbb{Z}\),
+      }
+      \onslide<3->{
+        \(\vec{a}_i \in \mathbb{R}^3\)
+      }
+      \onslide<2->{
+        \[
+          \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 \onslide<3->{+ n_3 \vec{a}_3}
+        \]
+      }
+      \vspace{1cm}
+
+      \onslide<4->{
+        Invariant unter Translation
+        \[
+          Q_i(\vec{r}) = \vec{r} + \vec{a}_i
+        \]
+      }
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{}
+  \begin{columns}[T]
+    \begin{column}{.5\textwidth}
+      \onslide<1->{
+        Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien?
+      }
+      \begin{center}
+        \begin{tikzpicture}[
+            dot/.style = {
+              draw, circle, thick, white, fill = gray!40!background,
+              minimum size = 2mm,
+              inner sep = 0pt,
+              outer sep = 1mm,
+            },
+          ]
+
+          \onslide<2->{
+            \node[dot] (A1) at (0,0) {};
+            \node[below left] at (A1) {\(A\)};
+          }
+
+          \onslide<3->{
+            \node[dot] (A2) at (2.5,0) {};
+            \node[below right] at (A2) {\(A'\)};
+
+            \draw[red!80!background, thick, ->] 
+              (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
+          }
+
+          \onslide<4->{
+            \node[dot] (B1) at (120:2.5) {};
+            \node[above left] at (B1) {\(B\)};
+
+            \draw[green!70!background, thick, ->]
+              (A1) ++(.5,0) arc (0:120:.5)
+              node[midway, above, xshift=1mm] {\(C_n\)};
+            \draw[red!80!background, dashed, thick, ->] (A1) to (B1);
+          }
+
+          \onslide<5->{
+            \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
+            \node[above right] at (B2) {\(B'\)};
+
+            \draw[green!70!background, thick, dashed, ->]
+              (A2) ++(-.5,0) arc (180:60:.5);
+            \draw[red!80!background, dashed, thick, ->] (A2) to (B2);
+          }
+
+          \onslide<6->{
+            \draw[yellow!80!background, thick, ->]
+              (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
+          }
+
+          \onslide<10->{
+		    \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {};
+			\draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {};
+            \node[above left, xshift=-2mm] at (Xl) {\(x\)};
+            \node[above right, xshift= 2mm] at (Xr) {\(x\)};
+          }
+        \end{tikzpicture}
+      \end{center}
+    \end{column}
+    \begin{column}{.5\textwidth}
+      \onslide<7->{
+        Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\)
+      }
+      \begin{align*}
+        \onslide<9->{q' = n q \onslide<10->{&= q + 2x \\}}
+        \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\}
+        \onslide<12->{n &= 1 - 2\cos\alpha}
+      \end{align*}
+      \onslide<13->{
+        Somit muss
+        \begin{align*}
+          \alpha &= \cos^{-1}\left(\frac{1-n}{2}\right) \\[1em]
+          \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} \\
+          n &\in \left\{ 1, 2, 3, 4, 6 \right\}
+        \end{align*}
+      }
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}[fragile]{M\"ogliche Kristallstrukturen}
+  \begin{center}
+    \begin{tikzpicture}[]
+      \node[circle, dashed, draw = gray, 
+        thick, fill = background,
+        minimum size = 4cm] {};
+      \node[gray] at (.9,-1.2) {674};
+
+      \node[circle, draw = white, thick,
+        fill = orange!40!background,
+        xshift = -3mm, yshift = 2mm,
+        minimum size = 2.75cm,
+        outer sep = 1mm] (A) {};
+      \node[white, yshift = 2mm] at (A) {230};
+      \node[white, font=\large, above right = of A] (Al) {Raumgruppe};
+      \draw[white, thick, ->] (Al.west) to[out=180, in=60] (A);
+
+      \node[circle, draw = white, thick, 
+        fill = red!20!background,
+        xshift = -5mm, yshift = -5mm,
+        minimum size = 1cm,
+        outer sep = 1mm] (B) {32};
+      \node[white, font=\large, below left = of B, xshift=-4mm] (Bl) {Kristallklassen};
+      \draw[white, thick, ->] (Bl.east) to[out = 0, in = 180] (B);
+    \end{tikzpicture}
+  \end{center}
+\end{frame}
+
+{
+  \usebackgroundtemplate[fragile]{
+    \begin{tikzpicture}[
+        overlay,
+        xshift = .45\paperwidth,
+        yshift = .47\paperheight,
+        classcirc/.style = {
+          draw = gray, thick, circle,
+          minimum size = 12mm,
+          inner sep = 0pt, outer sep = 0pt,
+        },
+        classlabel/.style = {
+          below right = 5mm
+        },
+        round/.style = {
+          draw = yellow, thick, circle,
+          minimum size = 1mm,
+          inner sep = 0pt, outer sep = 0pt,
+        },
+        cross/.style = {
+          cross out, draw = magenta, thick,
+          minimum size = 1mm, 
+          inner sep = 0pt, outer sep = 0pt
+        },
+      ]
+      \matrix [row sep = 3mm, column sep = 0mm] {
+        \node[classcirc] (C1)  {} node[classlabel] {\(C_{1}\)};  & 
+        \node[classcirc] (C2)  {} node[classlabel] {\(C_{2}\)};  &
+        \node[classcirc] (C3)  {} node[classlabel] {\(C_{3}\)};  &
+        \node[classcirc] (Ci)  {} node[classlabel] {\(C_{i}\)};  &
+
+        \node[classcirc] (Cs)  {} node[classlabel] {\(C_{s}\)};  &
+        \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; &
+        \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; &
+        \node[classcirc] (D2)  {} node[classlabel] {\(D_{2}\)};  \\
+
+        \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; &
+        \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & 
+        \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & 
+        \node[classcirc] (D3)  {} node[classlabel] {\(D_{3}\)};  &
+
+        \node[classcirc] (C4)  {} node[classlabel] {\(C_{4}\)};  &
+        \node[classcirc] (C6)  {} node[classlabel] {\(C_{6}\)};  &
+        \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; &
+        \node[classcirc] (S4)  {} node[classlabel] {\(S_{4}\)};  \\
+
+        \node[classcirc] (S3)  {} node[classlabel] {\(S_{3}\)};  &
+        \node[classcirc, dashed] (T)   {} node[classlabel] {\(T_{}\)};   &
+        \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; &
+        \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; &
+
+        \node[classcirc, dashed] (Th)  {} node[classlabel] {\(T_{h}\)};  &
+        \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; &
+        \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; &
+        \node[classcirc, dashed] (Td)  {} node[classlabel] {\(T_{d}\)};  \\
+        
+        \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; &
+        \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; &
+        \node[classcirc, dashed] (O)   {} node[classlabel] {\(O_{}\)};   &
+        \node[classcirc] (D4)  {} node[classlabel] {\(D_{4}\)};  &
+
+        \node[classcirc] (D6)  {} node[classlabel] {\(D_{6}\)};  &
+        \node[classcirc, dashed] (Oh)  {} node[classlabel] {\(O_{h}\)};  &
+        \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; &
+        \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\
+      };
+
+
+      \node[cross] at ($(C1)+(4mm,0)$) {};
+
+      
+      \node[cross] at ($(C2)+(4mm,0)$) {};
+      \node[cross] at ($(C2)-(4mm,0)$) {};
+
+
+      \node[cross] at ($(C3)+(  0:4mm)$) {};
+      \node[cross] at ($(C3)+(120:4mm)$) {};
+      \node[cross] at ($(C3)+(240:4mm)$) {};
+
+
+      \node[cross] at ($(Ci)+(4mm,0)$) {};
+      \node[round] at ($(Ci)-(4mm,0)$) {};
+
+
+      \node[cross] at ($(Cs)+(4mm,0)$) {};
+      \node[round] at ($(Cs)+(4mm,0)$) {};
+
+
+      \node[cross] at ($(C3i)+(  0:4mm)$) {};
+      \node[cross] at ($(C3i)+(120:4mm)$) {};
+      \node[cross] at ($(C3i)+(240:4mm)$) {};
+      \node[round] at ($(C3i)+( 60:4mm)$) {};
+      \node[round] at ($(C3i)+(180:4mm)$) {};
+      \node[round] at ($(C3i)+(300:4mm)$) {};
+
+
+      \node[cross] at ($(C2h)+(4mm,0)$) {};
+      \node[cross] at ($(C2h)-(4mm,0)$) {};
+      \node[round] at ($(C2h)+(4mm,0)$) {};
+      \node[round] at ($(C2h)-(4mm,0)$) {};
+
+
+      \node[cross] at ($(D2)+( 20:4mm)$) {};
+      \node[cross] at ($(D2)+(200:4mm)$) {};
+      \node[round] at ($(D2)+(160:4mm)$) {};
+      \node[round] at ($(D2)+(340:4mm)$) {};
+
+
+      \foreach \x in {0, 120, 240} {
+        \node[cross] at ($(D3d)+({\x+15}:4mm)$) {};
+        \node[cross] at ($(D3d)+({\x-15}:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 180} {
+        \node[cross] at ($(C2v)+({\x+15}:4mm)$) {};
+        \node[cross] at ($(C2v)+({\x-15}:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 180} {
+        \node[cross] at ($(D2h)+({\x+15}:4mm)$) {};
+        \node[cross] at ($(D2h)+({\x-15}:4mm)$) {};
+        \node[round] at ($(D2h)+({\x+15}:4mm)$) {};
+        \node[round] at ($(D2h)+({\x-15}:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 120, 240} {
+        \node[cross] at ($(D3)+({\x+15}:4mm)$) {};
+        \node[round] at ($(D3)+({\x-15}:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 90, 180, 270} {
+        \node[cross] at ($(C4)+(\x:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 60, 120, 180, 240, 300} {
+        \node[cross] at ($(C6)+(\x:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 120, 240} {
+        \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {};
+        \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {};
+        \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {};
+        \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {};
+      }
+
+
+      \node[cross] at ($(S4)+(4mm,0)$) {};
+      \node[cross] at ($(S4)-(4mm,0)$) {};
+      \node[round] at ($(S4)+(0,4mm)$) {};
+      \node[round] at ($(S4)-(0,4mm)$) {};
+
+
+      \foreach \x in {0, 120, 240} {
+        \node[cross] at ($(S3)+(\x:4mm)$) {};
+        \node[round] at ($(S3)+(\x:4mm)$) {};
+      }
+
+
+      %% TODO: T
+
+
+      \foreach \x in {0, 90, 180, 270} {
+        \node[cross] at ($(C4h)+(\x:4mm)$) {};
+        \node[round] at ($(C4h)+(\x:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 60, 120, 180, 240, 300} {
+        \node[cross] at ($(C6h)+(\x:4mm)$) {};
+        \node[round] at ($(C6h)+(\x:4mm)$) {};
+      }
+
+
+      %% TODO: Th
+
+
+      \foreach \x in {0, 90, 180, 270} {
+        \node[cross] at ($(C4v)+(\x+15:4mm)$) {};
+        \node[cross] at ($(C4v)+(\x-15:4mm)$) {};
+      }
+
+
+
+      \foreach \x in {0, 60, 120, 180, 240, 300} {
+        \node[cross] at ($(C6v)+(\x+10:4mm)$) {};
+        \node[cross] at ($(C6v)+(\x-10:4mm)$) {};
+      }
+
+
+      %% TODO: Td
+
+
+      \foreach \x in {0, 180} {
+        \node[cross] at ($(D2d)+({\x+15}:4mm)$) {};
+        \node[round] at ($(D2d)+({\x-15}:4mm)$) {};
+
+        \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {};
+        \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 120, 240} {
+        \node[cross] at ($(D3h)+({\x+15}:4mm)$) {};
+        \node[cross] at ($(D3h)+({\x-15}:4mm)$) {};
+        \node[round] at ($(D3h)+({\x+15}:4mm)$) {};
+        \node[round] at ($(D3h)+({\x-15}:4mm)$) {};
+      }
+
+
+      %% TODO: O
+
+
+      \foreach \x in {0, 90, 180, 270} {
+        \node[cross] at ($(D4)+({\x+15}:4mm)$) {};
+        \node[round] at ($(D4)+({\x-15}:4mm)$) {};
+      }
+
+      \foreach \x in {0, 60, 120, 180, 240, 300} {
+        \node[cross] at ($(D6)+({\x+10}:4mm)$) {};
+        \node[round] at ($(D6)+({\x-10}:4mm)$) {};
+      }
+
+
+      % TODO Oh
+
+
+      \foreach \x in {0, 90, 180, 270} {
+        \node[cross] at ($(D4h)+(\x+15:4mm)$) {};
+        \node[cross] at ($(D4h)+(\x-15:4mm)$) {};
+        \node[round] at ($(D4h)+(\x+15:4mm)$) {};
+        \node[round] at ($(D4h)+(\x-15:4mm)$) {};
+      }
+
+
+      \foreach \x in {0, 60, 120, 180, 240, 300} {
+        \node[cross] at ($(D6h)+({\x+10}:4mm)$) {};
+        \node[cross] at ($(D6h)+({\x-10}:4mm)$) {};
+        \node[round] at ($(D6h)+({\x+10}:4mm)$) {};
+        \node[round] at ($(D6h)+({\x-10}:4mm)$) {};
+      }
+    \end{tikzpicture}
+  }
+  \begin{frame}[fragile]{}
+  \end{frame}
+}
+
+\section{Anwendungen}
+\begin{frame}[fragile]{}
+  \centering
+  \begin{tikzpicture}[
+      box/.style = {
+        rectangle, thick, draw = white, fill = darkgray!50!background,
+        minimum height = 1cm, outer sep = 2mm,
+      },
+    ]
+
+    \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] {
+      & \node (A) {32 Kristallklassen}; \\
+      \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\
+      & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\
+    };
+
+    \draw[thick, ->] (A.west) to[out=180, in=90] (B.north);
+    \draw[thick, ->] (A.south) to (C);
+    \draw[thick, ->] (C.south) to (D.north);
+    \draw[thick, ->] (C.east) to[out=0, in=90] (E.north);
+  \end{tikzpicture}
+\end{frame}
+
+\begin{frame}[fragile]{}
+  \begin{tikzpicture}[
+      overlay, xshift = 1.5cm, yshift = 1.5cm,
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = white, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum};
+    \pause
+
+    \begin{scope}
+      \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
+        \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\
+        \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+        \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\
+      };
+      \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+    \end{scope}
+    \pause
+
+    \begin{scope}[xshift=11cm]
+      \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
+        \node[charge, \t] (C\i) at (\x:1.5cm) {};
+      }
+
+      \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {};
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+    \end{scope}
+    \pause
+
+	%% 
+    \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+
+	%% hex with vertical pressure
+    \begin{scope}[xshift=11cm, yshift=-4.5cm]
+      \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+
+      \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+      \foreach \d in {C1, C2} {
+        \draw[orange, very thick, <-] (\d) to ++(0,.7);
+      }
+
+      \foreach \d in {C4, C5} {
+        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+      }
+
+      \node[white] (E) {\(\vec{E}_p\)};
+      \begin{scope}[node distance = .5mm]
+        \node[red!50, right = of E] {\(+\)};
+        \node[blue!50, left = of E] {\(-\)};
+      \end{scope}
+      % \draw[gray, thick, dotted] (E) to ++(0,2);
+      % \draw[gray, thick, dotted] (E) to ++(0,-2);
+    \end{scope}
+	\pause
+
+	%% square with vertical pressure 
+    \begin{scope}[yshift=-4.5cm]
+      \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
+        \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\
+        \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+        \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\
+      };
+
+      \foreach \d in {NW, N, NE} {
+        \draw[orange, very thick, <-] (\d) to ++(0,.7);
+      }
+
+      \foreach \d in {SW, S, SE} {
+        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+      }
+
+      \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+    \end{scope}
+    \pause
+
+	%% hex with horizontal pressure
+    \begin{scope}[xshift=5.5cm, yshift=-4.5cm]
+      \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+
+      \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+      \draw[orange, very thick, <-] (C6) to ++(.7,0);
+      \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+
+      \node[white] (E) {\(\vec{E}_p\)};
+      \begin{scope}[node distance = .5mm]
+        \node[blue!50, right = of E] {\(-\)};
+        \node[red!50, left = of E] {\(+\)};
+      \end{scope}
+      % \draw[gray, thick, dotted] (E) to ++(0,2);
+      % \draw[gray, thick, dotted] (E) to ++(0,-2);
+    \end{scope}
+    \pause
+	  
+
+  \end{tikzpicture}
+\end{frame}
+
+\frame{
+  \frametitle{Licht in Kristallen}
+  \begin{columns}[T]
+    \begin{column}{.45\textwidth}
+      \onslide<2->{
+        Helmholtz Wellengleichung
+        \[
+          \nabla^2 \vec{E} = \ten{\varepsilon}\mu
+          \frac{\partial^2}{\partial t^2} \vec{E}
+        \]
+      }
+      \onslide<3->{
+        Ebene Welle
+        \[
+          \vec{E} = \vec{E}_0 \exp\left[i
+            \left(\vec{k}\cdot\vec{r} - \omega t \right)\right]
+        \]
+      }
+      \onslide<4->{
+        Anisotropisch Dielektrikum
+        \[
+          (\ten{K}\ten{\varepsilon})\vec{E} 
+            = \frac{k^2}{\mu \omega^2} \vec{E}
+          \implies
+          \Phi \vec{E} = \lambda \vec{E}
+        \]
+      }
+    \end{column}
+    \begin{column}{.55\textwidth}
+      \onslide<5->{
+        Eingenraum
+        \begin{align*}
+          U_\lambda &= \left\{ v : \Phi v = \lambda v \right\}
+          = \mathrm{null}\left(\Phi - \lambda I\right)
+        \end{align*}
+      }\onslide<6->{
+        Symmetriegruppe und Darstellung
+        \begin{align*}
+          G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\
+          &\Phi : G \to O(n)
+        \end{align*}
+      }\onslide<7->{
+        Kann man \(U_\lambda\) von \(G\) herauslesen?
+        \only<7>{
+          \[
+            U_\lambda \stackrel{?}{=} f\left(\bigoplus_{g \in G} \Phi_g\right)
+          \]
+        }\only<8>{
+          \begin{align*}
+            \mathrm{Tr}\left[\Phi_r(g)\right] 
+              &= \sum_i n_i \mathrm{Tr}\left[\Psi_i(g)\right] \\
+            |G| &= \sum_i\mathrm{Tr}\left[\Psi_i(\mathbb{1})\right]
+          \end{align*}
+        }
+      }
+    \end{column}
+  \end{columns}
+}
+
+% \begin{frame}[fragile]
+%   \centering
+%   \tdplotsetmaincoords{70}{110}
+%   \begin{tikzpicture}[scale=2, tdplot_main_coords]
+%     \node[draw=white, thick, minimum size = 3cm, circle] {};
+%     % \foreach \x in {0, 120, 240} {
+%     % }
+%   \end{tikzpicture}
+% \end{frame}
+
+
+\end{document}
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc
index c857fec..cbd4dfe 100644
--- a/vorlesungen/slides/2/Makefile.inc
+++ b/vorlesungen/slides/2/Makefile.inc
@@ -17,5 +17,19 @@ chapter2 =								\
 	../slides/2/frobeniusanwendung.tex				\
 	../slides/2/quotient.tex					\
 	../slides/2/quotientv.tex					\
+	../slides/2/hilbertraum/definition.tex				\
+	../slides/2/hilbertraum/l2beispiel.tex				\
+	../slides/2/hilbertraum/basis.tex				\
+	../slides/2/hilbertraum/plancherel.tex				\
+	../slides/2/hilbertraum/l2.tex					\
+	../slides/2/hilbertraum/riesz.tex				\
+	../slides/2/hilbertraum/rieszbeispiel.tex			\
+	../slides/2/hilbertraum/adjungiert.tex				\
+	../slides/2/hilbertraum/spektral.tex				\
+	../slides/2/hilbertraum/sturm.tex				\
+	../slides/2/hilbertraum/laplace.tex				\
+	../slides/2/hilbertraum/qm.tex					\
+	../slides/2/hilbertraum/energie.tex				\
+	../slides/2/hilbertraum/sobolev.tex				\
 	../slides/2/chapter.tex
 
diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex
index 49e656a..d3714c3 100644
--- a/vorlesungen/slides/2/chapter.tex
+++ b/vorlesungen/slides/2/chapter.tex
@@ -15,3 +15,17 @@
 \folie{2/frobeniusanwendung.tex}
 \folie{2/quotient.tex}
 \folie{2/quotientv.tex}
+\folie{2/hilbertraum/definition.tex}
+\folie{2/hilbertraum/l2beispiel.tex}
+\folie{2/hilbertraum/basis.tex}
+\folie{2/hilbertraum/plancherel.tex}
+\folie{2/hilbertraum/l2.tex}
+\folie{2/hilbertraum/riesz.tex}
+\folie{2/hilbertraum/rieszbeispiel.tex}
+\folie{2/hilbertraum/adjungiert.tex}
+\folie{2/hilbertraum/spektral.tex}
+\folie{2/hilbertraum/sturm.tex}
+\folie{2/hilbertraum/laplace.tex}
+\folie{2/hilbertraum/qm.tex}
+\folie{2/hilbertraum/energie.tex}
+\folie{2/hilbertraum/sobolev.tex}
diff --git a/vorlesungen/slides/2/hilbertraum/adjungiert.tex b/vorlesungen/slides/2/hilbertraum/adjungiert.tex
new file mode 100644
index 0000000..da41576
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/adjungiert.tex
@@ -0,0 +1,83 @@
+%
+% adjungiert.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Adjungierter Operator}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2->
+$A\colon H\to L$ lineare Abbildung zwischen Hilberträumen, $y\in L$
+\item<3->
+\[
+H\to\mathbb{C}
+:
+x\mapsto \langle y, Ax\rangle_L
+\]
+ist eine lineare Abbildung $H\to\mathbb{C}$
+\item<4->
+Nach dem Darstellungssatz gibt es $v\in H$ mit
+\[
+\langle y,Ax\rangle_L = \langle v,x\rangle_H
+\quad 
+\forall x\in H
+\]
+\end{itemize}
+\uncover<5->{%
+Die Abbildung 
+\[
+L\to H
+:
+y\mapsto v =: A^*y
+\]
+heisst {\em adjungierte Abbildung}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Endlichdimensional (Matrizen)}
+\[
+A^* = \overline{A}^t
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<7->{%
+\begin{block}{Selbstabbildungen}
+Für Operatoren $A\colon H\to H$ ist $A^*\colon H\to H$
+\[
+\langle x,Ay\rangle
+=
+\langle A^*x, y\rangle
+\quad
+\forall x,y\in H
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<9->{%
+\begin{block}{Selbstadjungierte Operatoren}
+\[
+A=A^*
+\uncover<10->{\;\Leftrightarrow\;
+\langle x,Ay \rangle
+=
+\langle A^*x,y \rangle}
+\uncover<11->{=
+\langle Ax,y \rangle}
+\]
+\uncover<12->{Matrizen:
+\begin{itemize}
+\item<13-> hermitesch
+\item<14-> für reelle Hilberträume: symmetrisch
+\end{itemize}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/basis.tex b/vorlesungen/slides/2/hilbertraum/basis.tex
new file mode 100644
index 0000000..022fa07
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/basis.tex
@@ -0,0 +1,65 @@
+%
+% basis.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Hilbert-Basis}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Menge $\mathcal{B}=\{b_k|k>0\}$ ist eine Hilbertbasis, wenn
+\begin{itemize}
+\item<2-> $\mathcal{B}$ ist orthonormiert: $\langle b_k,b_l\rangle=\delta_{kl}$
+\item<3-> Der Unterraum $\langle b_k|k>0\rangle\subset H$ ist
+dicht:
+Jeder Vektor von $H$ kann beliebig genau durch Linearkombinationen von $b_k$
+approximiert werden.
+\end{itemize}
+\uncover<4->{%
+Ein Hilbertraum mit einer Hilbertbasis heisst {\em separabel}}
+\end{block}
+\uncover<5->{%
+\begin{block}{Endlichdimensional}
+Der Algorithmus bricht nach endlich vielen Schritten ab.
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Konstruktion}
+Iterativ: $\mathcal{B}_0=\emptyset$
+\begin{enumerate}
+\item<7-> $V_k = \langle \mathcal{B}_k \rangle$
+\item<8-> Wenn $V_k\ne H$, wähle einen Vektor
+\begin{align*}
+x\in V_k^{\perp}
+&=
+\{
+x\in H\;|\; x\perp V_k
+\}
+\\
+&=
+\{x\in H\;|\;
+x\perp y\;\forall y\in V_k
+\}
+\end{align*}
+\item<9-> $b_{k+1} = x/\|x\|$
+\[
+\mathcal{B}_{k+1} = \mathcal{B}_k\cup \{b_{k+1}\}
+\]
+\end{enumerate}
+\uncover<10->{%
+Wenn $H$ separabel ist, dann ist
+\[
+\mathcal{B} = \bigcup_{k} \mathcal{B}_k
+\]
+eine Hilbertbasis für $H$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/definition.tex b/vorlesungen/slides/2/hilbertraum/definition.tex
new file mode 100644
index 0000000..d101637
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/definition.tex
@@ -0,0 +1,63 @@
+%
+% definition.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Hilbertraum --- Definition}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\mathbb{C}$-Hilbertraum $H$}
+\begin{enumerate}
+\item<2-> $\mathbb{C}$-Vektorraum, muss nicht endlichdimensional sein
+\item<3-> Sesquilineares Skalarprodukt
+\[
+\langle \cdot,\cdot\rangle
+\colon H \to \mathbb{C}: (x,y) \mapsto \langle x,y\rangle
+\]
+Dazugehörige Norm:
+\[
+\|x\| = \sqrt{\langle x,x\rangle}
+\]
+\item<4-> Vollständigkeit: jede Cauchy-Folge konvergiert
+\end{enumerate}
+\uncover<5->{%
+Ohne Vollständigkeit: {\em Prähilbertraum}}
+\end{block}
+\uncover<6->{%
+\begin{block}{$\mathbb{R}$-Hilbertraum}
+Vollständiger $\mathbb{R}$-Vektorraum mit bilinearem Skalarprodukt
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Vollständigkeit}
+\begin{itemize}
+\item<8-> $(x_n)_{n\in\mathbb{N}}$ ist eine Cauchy-Folge:
+Für alle $\varepsilon>0$ gibt es $N>0$ derart, dass
+\[
+\| x_n-x_m\| < \varepsilon\quad\forall n,m>N
+\]
+\item<9-> Grenzwert existiert: $\exists x\in H$ derart, dass es für alle
+$\varepsilon >0$ ein $N>0$ gibt derart, dass
+\[
+\|x_n-x\|<\varepsilon\quad\forall n>N
+\]
+\end{itemize}
+\end{block}}
+\uncover<10->{%
+\begin{block}{Cauchy-Schwarz-Ungleichung}
+\[
+|\langle x,y\rangle|
+\le  \|x\| \cdot \|y\|
+\]
+Gleichheit für linear abhängige $x$ und $y$
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/energie.tex b/vorlesungen/slides/2/hilbertraum/energie.tex
new file mode 100644
index 0000000..202a7c5
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/energie.tex
@@ -0,0 +1,67 @@
+%
+% energie.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Energie --- Zeitentwicklung --- Schrödinger}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.30\textwidth}
+\uncover<2->{%
+\begin{block}{Totale Energie}
+Hamilton-Funktion
+\begin{align*}
+H
+&=
+\frac12mv^2 + V(x)
+\\
+&=
+\frac{p^2}{2m} + V(x)
+\end{align*}
+\end{block}}
+\uncover<3->{%
+\begin{block}{Quantisierungsregel}
+\begin{align*}
+\text{Variable}&\to \text{Operator}
+\\
+x_k & \to x_k
+\\
+p_k & \to \frac{\hbar}{i} \frac{\partial}{\partial x_k}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.66\textwidth}
+\uncover<4->{%
+\begin{block}{Energie-Operator}
+\[
+H
+=
+-\frac{\hbar^2}{2m}\Delta + V(x)
+\]
+\end{block}}
+\uncover<5->{%
+\begin{block}{Eigenwertgleichung}
+\[
+-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t) = E\psi(x,t)
+\]
+Zeitunabhängige Schrödingergleichung
+\end{block}}
+\uncover<6->{%
+\begin{block}{Zeitabhängigkeit = Schrödingergleichung}
+\[
+-\frac{\hbar}{i}
+\frac{\partial}{\partial t}
+\psi(x,t)
+=
+-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t)
+\]
+\uncover<7->{Eigenwertgleichung durch Separation von $t$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/l2.tex b/vorlesungen/slides/2/hilbertraum/l2.tex
new file mode 100644
index 0000000..bd744ab
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/l2.tex
@@ -0,0 +1,61 @@
+%
+% l2.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$L^2$-Hilbertraum}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2->
+Vektorraum: Funktionen
+\[
+f\colon [a,b] \to \mathbb{C}
+\]
+\item<3->
+Sesquilineares Skalarprodukt
+\[
+\langle f,g\rangle
+=
+\int_a^b \overline{f(x)}\, g(x) \,dx
+\]
+\item<4->
+Norm:
+\[
+\|f\|^2 = \int_a^b |f(x)|^2\,dx
+\]
+\item<5->
+Vollständigkeit?
+\uncover<6->{$\rightarrow$
+Lebesgue Konvergenz-Satz}
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Vollständigkeit}
+\begin{itemize}
+\item
+Funktioniert nicht für Riemann-Integral
+\item<8->
+Erweiterung des Integrals auf das sogenannte Lebesgue-Integral (nach
+Henri Lebesgue)
+\item<9->
+Abzählbare Mengen spielen keine Rolle $\rightarrow$ Nullmengen
+\item<10->
+Funktionen $\rightarrow$ Klassen von Funktionen, die sich auf einer Nullmenge
+unterscheiden
+\item<11->
+Konvergenz-Satz von Lebesgue $\rightarrow$ es funktioniert
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/l2beispiel.tex b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex
new file mode 100644
index 0000000..3ae44af
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex
@@ -0,0 +1,82 @@
+%
+% l2beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beispiele: $\mathbb{R},\mathbb{R}^2,\dots,\mathbb{R}^n,\dots,l^2$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2-> Quadratsummierbare Folgen von komplexen Zahlen
+\[
+l^2
+=
+\biggl\{
+(x_k)_{k\in\mathbb{N}}\,\bigg|\, \sum_{k=0}^\infty |x_k|^2 < \infty
+\biggr\}
+\]
+\item<3-> Skalarprodukt:
+\begin{align*}
+\langle x,y\rangle
+&=
+\sum_{k=0}^\infty \overline{x}_ky_k,
+&
+\uncover<4->{\|x\|^2 = \sum_{k=0}^\infty |x_k|^2}
+\end{align*}
+\item<5-> Vollständigkeit,
+Konvergenz: Cauchy-Schwarz-Ungleichung
+\[
+\biggl|
+\sum_{k=0}^\infty \overline{x}_ky_k
+\biggr|
+\le
+\sum_{k=0}^\infty |x_k|^2
+\sum_{l=0}^\infty |y_l|^2
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Standardbasisvektoren}
+\begin{align*}
+e_i
+&=
+(0,\dots,0,\underset{\underset{\textstyle i}{\textstyle\uparrow}}{1},0,\dots)
+\\
+\uncover<7->{(e_i)_k &= \delta_{ik}}
+\end{align*}
+\uncover<8->{sind orthonormiert:
+\begin{align*}
+\langle e_i,e_j\rangle
+&=
+\sum_k \overline{\delta}_{ik}\delta_{jk}
+\uncover<9->{=
+\delta_{ij}}
+\end{align*}}
+\end{block}}
+\vspace{-16pt}
+\uncover<10->{%
+\begin{block}{Analyse}
+$x_k$ kann mit Skalarprodukten gefunden werden:
+\begin{align*}
+\hat{x}_i
+=
+\langle e_i,x\rangle
+&\uncover<11->{=
+\sum_{k=0}^\infty \overline{\delta}_{ik}  x_k}
+\uncover<12->{=
+x_i}
+\end{align*}
+\uncover<13->{(Fourier-Koeffizienten)}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/laplace.tex b/vorlesungen/slides/2/hilbertraum/laplace.tex
new file mode 100644
index 0000000..8f6b196
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/laplace.tex
@@ -0,0 +1,66 @@
+%
+% laplace.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Höhere Dimension}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Problem}
+Gegeben: $\Omega\subset\mathbb{R}^n$ ein Gebiet
+\\
+Gesucht: Lösungen von $\Delta u=0$ mit $u_{|\partial\Omega}=0$
+\end{block}
+\uncover<2->{%
+\begin{block}{Funktionen}
+Hilbertraum $H$ der Funktionen $f:\overline{\Omega}\to\mathbb{C}$
+mit $f_{|\partial\Omega}=0$
+\end{block}}
+\uncover<3->{%
+\begin{block}{Skalarprodukt}
+\[
+\langle f,g\rangle
+=
+\int_{\Omega} \overline{f}(x) g(x)\,d\mu(x)
+\]
+\end{block}}
+\uncover<4->{%
+\begin{block}{Laplace-Operator}
+\[
+\Delta \psi = \operatorname{div}\operatorname{grad}\psi
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<5->{%
+\begin{block}{Selbstadjungiert}
+\begin{align*}
+\langle f,\Delta g\rangle
+&\uncover<6->{=
+\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)}
+\\
+&\uncover<7->{=
+\int_{\partial\Omega}
+\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)}
+\\
+&\uncover<7->{\qquad
+-
+\int_{\Omega}
+\operatorname{grad}\overline{f}(x)\cdot \operatorname{grad}g(x)
+\,d\mu(x)}
+\\
+&\uncover<8->{=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)}
+\\
+&\uncover<9->{=
+\langle \Delta f,g\rangle}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex
new file mode 100644
index 0000000..73dd46b
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex
@@ -0,0 +1,102 @@
+%
+% plancherel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Plancherel-Gleichung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Hilbertraum mit Hilbert-Basis}
+$H$ Hilbertraum mit Hilbert-Basis
+$\mathcal{B}=\{b_k\;|\; k>0\}$, $x\in H$
+\end{block}
+\uncover<2->{%
+\begin{block}{Analyse: Fourier-Koeffizienten}
+\begin{align*}
+a_k = \hat{x}_k &=\langle b_k, x\rangle
+\\
+\uncover<3->{\hat{x}&=\mathcal{F}x}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Synthese: Fourier-Reihe}
+\begin{align*}
+\tilde{x}
+&=
+\sum_k a_k b_k
+\uncover<5->{=
+\sum_k \langle x,b_k\rangle b_k}
+\end{align*}
+\end{block}}
+\vspace{-6pt}
+\uncover<6->{%
+\begin{block}{Analyse von $\tilde{x}$}
+\begin{align*}
+\langle b_l,\tilde{x}\rangle
+&=
+\biggl\langle
+b_l,\sum_{k}\langle b_k,x\rangle b_k
+\biggr\rangle
+\uncover<7->{=
+\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle}
+\uncover<8->{=
+\sum_k \langle b_k,x\rangle\delta_{kl}}
+\uncover<9->{=
+\langle b_l,x\rangle}
+\uncover<10->{=
+\hat{x}_l}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Plancherel-Gleichung}
+\begin{align*}
+\|\tilde{x}\|^2
+&=
+\langle \tilde{x},\tilde{x}\rangle
+=
+\biggl\langle
+\sum_k \hat{x}_kb_k,
+\sum_l \hat{x}_lb_l
+\biggr\rangle
+\\
+&\uncover<12->{=
+\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle}
+\uncover<13->{=
+\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl}}
+\\
+\uncover<14->{
+\|\tilde{x}\|^2
+&=
+\sum_k |\hat{x}_k|^2}
+\uncover<15->{=
+\|\hat{x}\|_{l^2}^2}
+\uncover<16->{=
+\|\mathcal{F}x\|_{l^2}^2}
+\end{align*}
+\end{block}}
+\vspace{-12pt}
+\uncover<17->{%
+\begin{block}{Isometrie}
+\begin{align*}
+\mathcal{F}
+\colon
+H \to l^2
+\colon
+x\mapsto \hat{x}
+\end{align*}
+\uncover<18->{Alle separablen Hilberträume sind isometrisch zu $l^2$ via
+%Fourier-Transformation
+$\mathcal{F}$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/qm.tex b/vorlesungen/slides/2/hilbertraum/qm.tex
new file mode 100644
index 0000000..a108121
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/qm.tex
@@ -0,0 +1,90 @@
+%
+% qm.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Anwendung: Quantenmechanik}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zustände (Wellenfunktion)}
+$L^2$-Funktionen auf $\mathbb{R}^3$
+\[
+\psi\colon\mathbb{R}^3\to\mathbb{C}
+\]
+\end{block}
+\vspace{-6pt}
+\uncover<2->{%
+\begin{block}{Wahrscheinlichkeitsinterpretation}
+\[
+|\psi(x)|^2 = \left\{
+\begin{minipage}{4.6cm}\raggedright
+Wahrscheinlichkeitsdichte für Position $x$ des Teilchens
+\end{minipage}\right.
+\]
+\end{block}}
+\vspace{-6pt}
+\uncover<3->{%
+\begin{block}{Skalarprodukt}
+\[
+\langle\psi,\psi\rangle
+=
+\int_{\mathbb{R}^3} |\psi(x)|^2\,dx = 1
+\]
+\end{block}}
+\vspace{-6pt}
+\uncover<4->{%
+\begin{block}{Messgrösse $A$}
+Selbstadjungierter Operator $A$
+\\
+\uncover<5->{$\rightarrow$
+Hilbertbasis $|i\rangle$ von EV von $A$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Überlagerung}
+\begin{align*}
+|\psi\rangle
+&=
+\sum_i
+w_i|i\rangle
+\\
+\uncover<7->{\langle \psi|\psi\rangle
+&=
+\sum_i |w_i|^2 \qquad\text{(Plancherel)}}
+\end{align*}
+\uncover<8->{%
+$|w_i|^2=|\langle \psi|i\rangle|^2$ Wahrscheinlichkeit für Zustand $|i\rangle$
+}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Erwartungswert}
+\begin{align*}
+E(A)
+&\uncover<10->{=
+\sum_i |w_i|^2 \alpha_i}
+\uncover<11->{=
+\sum_i \overline{w}_i\alpha_i w_i }
+\hspace{5cm}
+\\
+&\only<12>{=
+\sum_{i,j} \overline{w}_j\alpha_i w_i \langle j|i\rangle}
+\uncover<13->{=
+\sum_{i} \overline{w}_j\langle j| \sum_i \alpha_i w_i |i\rangle}
+\\
+&\uncover<14->{=
+\sum_{i,j} \overline{w}_j w_i \langle j|
+A|i\rangle}
+\uncover<15->{=
+\langle \psi| A |\psi\rangle}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/riesz.tex b/vorlesungen/slides/2/hilbertraum/riesz.tex
new file mode 100644
index 0000000..437fb3c
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/riesz.tex
@@ -0,0 +1,76 @@
+%
+% riesz.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Darstellungssatz von Riesz}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Dualraum}
+$V$ ein Vektorraum, $V^*$ der Raum aller Linearformen
+\[
+f\colon V\to \mathbb{C}
+\]
+\end{block}
+\uncover<3->{%
+\begin{block}{Beispiel: $l^\infty$}
+$l^\infty=\text{beschränkte Folgen in $\mathbb{C}$}$,
+Linearformen:
+\begin{align*}
+\uncover<4->{
+f(x)
+&=
+\sum_{i=0}^\infty f_ix_i}
+\\
+\uncover<5->{
+\|f\|
+&=
+\sup_{\|x\|_{\infty}\le 1}
+|f(x)|}
+\uncover<6->{=
+\sum_{k\in\mathbb{N}} |f_k|}
+\\
+\uncover<7->{
+\Rightarrow
+l^{\infty*}
+&=
+l^1}
+\uncover<9->{\qquad(\ne l^2)}
+\\
+\uncover<8->{
+&=\{\text{summierbare Folgen in $\mathbb{C}$}\}
+}
+\end{align*}
+
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Beispiel: $\mathbb{C}^n$}
+${\mathbb{C}^n}^* = \mathbb{C}^n$
+\end{block}}
+\uncover<10->{%
+\begin{theorem}[Riesz]
+Zu einer stetigen Linearform $f\colon H\to\mathbb{C}$ gibt es $v\in H$ mit
+\[
+f(x) = \langle v,x\rangle
+\quad\forall x\in H
+\]
+und $\|f\| = \|v\|$
+\end{theorem}}
+\uncover<11->{%
+\begin{block}{Dualraum von $H$}
+$H^*=H$
+\end{block}}%
+\uncover<12->{%
+Der Hilbertraum ist die ``intuitiv richtige, unendlichdimensionale''
+Verallgemeinerung von $\mathbb{C}^n$}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex
new file mode 100644
index 0000000..de9383f
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex
@@ -0,0 +1,107 @@
+%
+% rieszbeispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Linearform auf $L^2$-Funktionen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Linearform auf $\mathbb{C}^n$}
+\begin{align*}
+{\color{blue}x}&=\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix},
+&
+f({\color{blue}x})
+&= 
+\begin{pmatrix}f_1&f_2&\dots&f_n\end{pmatrix} {\color{blue}x}
+\\
+\uncover<2->{
+{\color{red}v}&=
+\rlap{$
+\begin{pmatrix}
+\overline{f}_1&\overline{f}_2&\dots&\overline{f}_n
+\end{pmatrix}^t
+\uncover<3->{\;\Rightarrow\;
+f({\color{blue}x})=\langle {\color{red}v},{\color{blue}x}\rangle}
+$}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Linearform auf $L^2([a,b])$}
+\begin{align*}
+{\color{red}x}&\in L^2([a,b])
+\\
+\uncover<5->{
+f&\colon L^2([a,b]) \to \mathbb{C}
+: {\color{red}x} \mapsto f({\color{red}x})}
+\intertext{\uncover<6->{Riesz-Darstellungssatz: $\exists {\color{blue}v}\in L^2([a,b])$}}
+\uncover<7->{f({\color{red}x})
+&=
+\int_a^b {\color{blue}\overline{v}(t)}{\color{red}x(t)}\,dt}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}[xshift=-3.5cm]
+\def\s{0.058}
+\foreach \n in {0,...,5}{
+\uncover<3->{
+	\draw[color=red,line width=3pt]
+		({\n+\s},{1/(\n+0.5)}) -- ({\n+\s},0);
+	\node[color=red] at ({\n},{-0.2+1/(\n+0.5)})
+		[above right] {$v_\n\mathstrut$};
+}
+	\draw[color=blue,line width=3pt]
+		({\n-\s},{0.4+0.55*sin(200*\n)+0.25*\n}) -- ({\n-\s},0);
+	\node[color=blue] at ({\n},{-0.2+0.4+0.55*sin(200*\n)+0.25*\n})
+		[above left] {$x_\n\mathstrut$};
+}
+\draw[->] (-0.6,0) -- (6,0) coordinate[label={$n$}];
+\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}];
+\foreach \n in {0,...,5}{
+	\fill (\n,0) circle[radius=0.08];
+	\node at (\n,0) [below] {$\n$\strut};
+}
+\node at (5.6,0) [below] {$\cdots$\strut};
+\end{scope}
+\uncover<4->{
+\begin{scope}[xshift=3.5cm]
+\uncover<7->{
+\fill[color=red!40,opacity=0.5] 
+	plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)})
+	--
+	(5,0) -- (0,0) -- cycle;
+}
+\fill[color=blue!40,opacity=0.5]
+	plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x})
+	-- (5,0) -- (0,0) -- cycle;
+\uncover<7->{
+\draw[color=red,line width=1.4pt]
+	plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)});
+\node[color=red] at (0,2) [right] {$x(t)$};
+}
+
+\draw[color=blue,line width=1.4pt]
+	plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x});
+\node[color=blue] at (4.5,2) [right]{$v(t)$};
+
+\draw[->] (-0.6,0) -- (6.0,0) coordinate[label={$t$}];
+\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}];
+\draw (0.0,-0.1) -- (0.0,0.1);
+\node at (0.0,0) [below] {$a$\strut};
+\draw (5.0,-0.1) -- (5.0,0.1);
+\node at (5.0,0) [below] {$b$\strut};
+\end{scope}
+}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/sobolev.tex b/vorlesungen/slides/2/hilbertraum/sobolev.tex
new file mode 100644
index 0000000..828d34d
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/sobolev.tex
@@ -0,0 +1,51 @@
+%
+% sobolev.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Sobolev-Raum}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Vektorrraum $W$}
+Funktionen $f\colon \Omega\to\mathbb{C}$
+\begin{itemize}
+\item<2->
+$f\in L^2(\Omega)$
+\item<3->
+$\nabla f\in L^2(\Omega)$
+\item<4->
+homogene Randbedingungen:
+$f_{|\partial \Omega}=0$
+\end{itemize}
+\end{block}
+\uncover<5->{%
+\begin{block}{Skalarprodukt}
+\begin{align*}
+\langle f,g\rangle_W
+&\uncover<6->{=
+\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)}
+\\
+&\uncover<7->{\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)}
+\\
+&\uncover<8->{=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<9->{%
+\begin{block}{Vollständigkeit}
+\dots
+\end{block}}
+\uncover<10->{%
+\begin{block}{Anwendung}
+``Ein Hilbertraum für jedes partielle Differentialgleichungsproblem''
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/spektral.tex b/vorlesungen/slides/2/hilbertraum/spektral.tex
new file mode 100644
index 0000000..b561b69
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/spektral.tex
@@ -0,0 +1,91 @@
+%
+% spektral.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektraltheorie für selbstadjungierte Operatoren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Voraussetzungen}
+\begin{itemize}
+\item
+Hilbertraum $H$
+\item
+$A\colon H\to H$ linear
+\end{itemize}
+\end{block}
+\uncover<2->{%
+\begin{block}{Eigenwerte}
+$x\in H$ ein EV von $A$ zum EW $\lambda\ne 0$
+\begin{align*}
+\uncover<3->{\langle x,x\rangle
+&=
+\frac1{\lambda}
+\langle x,\lambda x\rangle}
+\uncover<3->{=
+\frac1{\lambda}
+\langle x,Ax\rangle}
+\\
+&\uncover<4->{=
+\frac1{\lambda}
+\langle Ax,x\rangle}
+\uncover<5->{=
+\frac{\overline{\lambda}}{\lambda}
+\langle x,x\rangle}
+\\
+\uncover<6->{\frac{\overline{\lambda}}{\lambda}&=1
+\quad\Rightarrow\quad
+\overline{\lambda} = \lambda}
+\uncover<7->{\quad\Rightarrow\quad
+\lambda\in\mathbb{R}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Orthogonalität}
+$u,v$ EV zu EW $\mu,\lambda\in \mathbb{R}\setminus\{0\}$, $\overline{\mu}=\mu\ne\lambda$
+\begin{align*}
+\uncover<9->{
+\langle u,v\rangle
+&=
+\frac{1}{\mu}
+\langle \mu u,v\rangle}
+\uncover<10->{=
+\frac{1}{\mu}
+\langle Au,v\rangle}
+\\
+&\uncover<11->{=
+\frac{1}{\mu}
+\langle u,Av\rangle}
+\uncover<12->{=
+\frac{1}{\mu}
+\langle u,\lambda v\rangle}
+\uncover<13->{=
+\frac{\lambda}{\mu}
+\langle u,v\rangle}
+\\
+\uncover<14->{\Rightarrow
+\;
+0
+&=
+\underbrace{\biggl(\frac{\lambda}{\mu}-1\biggr)}_{\displaystyle \ne 0}
+\langle u,v\rangle}
+\uncover<15->{\;\Rightarrow\;
+\langle u,v\rangle = 0}
+\end{align*}
+\uncover<16->{EV zu verschiedenen EW sind orthogonal}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<17->{%
+\begin{block}{Spektralsatz}
+Es gibt eine Hilbertbasis von $H$ aus Eigenvektoren von $A$
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex
new file mode 100644
index 0000000..a6865ab
--- /dev/null
+++ b/vorlesungen/slides/2/hilbertraum/sturm.tex
@@ -0,0 +1,58 @@
+%
+% sturm.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Sturm-Liouville-Problem}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Wellengleichung}
+Saite mit variabler Massedichte führt auf die DGL
+\[
+-y''(t) + q(t) y(t) = \lambda y(t),
+\quad
+q(t) > 0
+\]
+mit Randbedingungen $y(0)=y(1)=0$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Sturm-Liouville-Operator}
+\[
+A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p
+\]
+auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten
+\[
+f(0)=f(1)=0
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Selbstadjungiert}
+\begin{align*}
+\langle f,Ag \rangle
+&\uncover<4->{=
+\langle f,-D^2 g\rangle + \langle f,qg\rangle
+=
+-
+\int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt
++\langle f,qg\rangle}
+\\
+&\uncover<5->{=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0}
++\int_0^1 \overline{f}'(t)g'(t)\,dt
++\langle f,qg\rangle}
+\uncover<6->{=-\int_0^1 \overline{f}''(t)g(t)\,dt
++\langle qf,g\rangle}
+\\
+&\uncover<7->{=\langle Af,g\rangle}
+\end{align*}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 7512612..ffd5091 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -16,13 +16,20 @@ chapter5 =								\
 	../slides/7/einparameter.tex					\
 	../slides/7/ableitung.tex					\
 	../slides/7/liealgebra.tex					\
+	../slides/7/liealgbeispiel.tex					\
+	../slides/7/vektorlie.tex					\
 	../slides/7/kommutator.tex					\
+	../slides/7/bch.tex						\
 	../slides/7/dg.tex						\
+	../slides/7/interpolation.tex					\
+	../slides/7/exponentialreihe.tex				\
+	../slides/7/logarithmus.tex					\
 	../slides/7/zusammenhang.tex					\
 	../slides/7/quaternionen.tex					\
 	../slides/7/qdreh.tex						\
 	../slides/7/ueberlagerung.tex					\
 	../slides/7/hopf.tex						\
 	../slides/7/haar.tex						\
+	../slides/7/integration.tex					\
 	../slides/7/chapter.tex
 
diff --git a/vorlesungen/slides/7/bch.tex b/vorlesungen/slides/7/bch.tex
new file mode 100644
index 0000000..0148dc4
--- /dev/null
+++ b/vorlesungen/slides/7/bch.tex
@@ -0,0 +1,76 @@
+%
+% bch.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Baker-Campbell-Hausdorff-Formel}
+$g(t),h(t)\in G
+\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit }
+g(t)=\exp At, h(t)=\exp Bt}$
+\uncover<3->{%
+\begin{align*}
+g(t)
+&=
+I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots,
+&
+h(t)
+&=
+I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots
+\end{align*}}
+\uncover<5->{%
+\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{-1}h(t)^{-1}$}
+\begin{align*}
+\uncover<6->{c(t)
+&=
+\biggl(
+  {\color<7,9-11,13-15,19-21>{red}I}
+  + {\color<8,16-19>{red}A}t
+  + \frac{{\color<12>{red}A^2}t^2}{2!}
+  + \dots
+\biggr)
+\biggl(
+  {\color<7,8,10-12,14-15,17-18,21>{red}I}
+  + {\color<9,16,19-20>{red}B}t
+  + \frac{{\color<13>{red}B^2}t^2}{2!}
+  + \dots
+\biggr)
+\exp(-{\color<10,14,17,19,21>{red}A}t)
+\exp(-{\color<11,15,18,20-21>{red}B}t)
+}
+\\
+&\uncover<7->{={\color<7>{red}I}}
+\uncover<8->{+t(
+   \uncover<8->{  {\color<8>{red}A}}
+   \uncover<9->{+ {\color<9>{red}B}}
+   \uncover<10->{- {\color<10>{red}A}}
+   \uncover<11->{- {\color<11>{red}B}}
+)}
+\uncover<12->{+\frac{t^2}{2!}(
+  \uncover<12->{  {\color<12>{red}A^2}}
+  \uncover<13->{+ {\color<13>{red}B^2}}
+  \uncover<14->{+ {\color<14>{red}A^2}}
+  \uncover<15->{+ {\color<15>{red}B^2}}
+)}
+\\
+&\phantom{\mathstrut=I}
+\uncover<12->{+t^2(
+     \uncover<16->{  {\color<16>{red}AB}}
+     \uncover<17->{- {\color<17>{red}A^2}}
+     \uncover<18->{- {\color<18>{red}AB}}
+     \uncover<19->{- {\color<19>{red}BA}}
+     \uncover<20->{- {\color<20>{red}B^2}}
+     \uncover<21->{+ {\color<21>{red}AB}}
+)}
+\uncover<22->{+t^3(\dots)+\dots}
+\\
+&\uncover<23->{=
+I + \frac{t^2}{2}[A,B] + o(t^3)
+}
+\end{align*}}
+\end{block}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
index 1c78ccc..3736e0f 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -15,11 +15,18 @@
 \folie{7/einparameter.tex}
 \folie{7/ableitung.tex}
 \folie{7/liealgebra.tex}
+\folie{7/liealgbeispiel.tex}
+\folie{7/vektorlie.tex}
 \folie{7/kommutator.tex}
+\folie{7/bch.tex}
 \folie{7/dg.tex}
+\folie{7/interpolation.tex}
+\folie{7/exponentialreihe.tex}
+\folie{7/logarithmus.tex}
 \folie{7/zusammenhang.tex}
 \folie{7/quaternionen.tex}
 \folie{7/qdreh.tex}
 \folie{7/ueberlagerung.tex}
 \folie{7/hopf.tex}
 \folie{7/haar.tex}
+\folie{7/integration.tex}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex
index 4447bac..f9528a4 100644
--- a/vorlesungen/slides/7/dg.tex
+++ b/vorlesungen/slides/7/dg.tex
@@ -45,7 +45,7 @@ Ableitung von $\gamma(t)$ an der Stelle $t$:
 \vspace{-10pt}
 \uncover<7->{%
 \begin{block}{Differentialgleichung}
-\vspace{-10pt}
+%\vspace{-10pt}
 \[
 \dot{\gamma}(t) = \gamma(t) A
 \quad
@@ -66,7 +66,7 @@ Exponentialfunktion
 \vspace{-5pt}
 \uncover<9->{%
 \begin{block}{Kontrolle: Tangentialvektor berechnen}
-\vspace{-10pt}
+%\vspace{-10pt}
 \begin{align*}
 \frac{d}{dt}e^{At}
 &\uncover<10->{=
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
index 5171085..a32affd 100644
--- a/vorlesungen/slides/7/einparameter.tex
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -41,7 +41,7 @@ D_{x,t+s}
 \begin{column}{0.48\textwidth}
 \uncover<5->{%
 \begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 \begin{pmatrix}
 1&s\\
@@ -61,7 +61,7 @@ D_{x,t+s}
 \vspace{-12pt}
 \uncover<6->{%
 \begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 \begin{pmatrix}
 e^s&0\\0&e^{-s}
@@ -78,7 +78,7 @@ e^{t+s}&0\\0&e^{-(t+s)}
 \vspace{-12pt}
 \uncover<7->{%
 \begin{block}{Gemischt}
-\vspace{-12pt}
+%\vspace{-12pt}
 \begin{gather*}
 A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
 \\
diff --git a/vorlesungen/slides/7/exponentialreihe.tex b/vorlesungen/slides/7/exponentialreihe.tex
new file mode 100644
index 0000000..b1aeda6
--- /dev/null
+++ b/vorlesungen/slides/7/exponentialreihe.tex
@@ -0,0 +1,24 @@
+%
+% exponentialreihe.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Exponentialreihe}
+\begin{align*}
+h(s) &= \exp(tA_0 + sB) = \sum_{k=0}^\infty \frac{1}{k!} (tA_0 + sB)^k
+\\
+&=
+I + (tA_0 + sB) + \frac{1}{2!}(t^2A_0^2 + ts(A_0B + BA_0) + s^2B^2)
++ \frac{1}{3!}(t^3A_0^3 + t^2s(A_0^2B + A_0BA_0 + BA_0^2) + \dots)
++ \dots
+\\
+\frac{dg(s)}{ds}
+&=
+B + \frac1{2!}t(A_0B+BA_0) + \frac{1}{3!}t^2(A_0^2B+A_0BA_0+BA_0^2) + \dots
+\end{align*}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
index cc67c8a..6f99bc3 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 # 
-all:	rodriguez.jpg 
+all:	rodriguez.jpg  test.png
 
 rodriguez.png:	rodriguez.pov
 	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
@@ -16,4 +16,14 @@ commutator:	commutator.ini commutator.pov common.inc
 jpg:
 	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
 
+dreibein/timestamp:	interpolation.m
+	octave interpolation.m
+	touch dreibein/timestamp
 
+test.png:	test.pov drehung.inc dreibein/d025.inc dreibein/timestamp
+	povray +A0.1 -W1080 -H1080 -Otest.png test.pov
+
+dreibein/d025.inc:	dreibein/timestamp
+
+animation:
+	povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini
diff --git a/vorlesungen/slides/7/images/drehung.inc b/vorlesungen/slides/7/images/drehung.inc
new file mode 100644
index 0000000..c9b4bb7
--- /dev/null
+++ b/vorlesungen/slides/7/images/drehung.inc
@@ -0,0 +1,142 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.23;
+#declare O = <0, 0, 0>;
+#declare at = 0.02;
+
+camera {
+	location <8.5, 2, 6.5>
+	look_at <0, 0, 0>
+	right x * imagescale
+	up y * imagescale
+}
+
+//light_source {
+//	<-14, 20, -50> color White
+//	area_light <1,0,0> <0,0,1>, 10, 10
+//	adaptive 1
+//	jitter
+//}
+
+light_source {
+	<41, 20, 10> 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, 1.0 * 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
+#declare r = 1.0;
+
+arrow(< -r-0.2,    0.0,    0  >, <  r+0.2,    0.0,    0.0 >, at, Gray)
+arrow(<    0.0,    0.0, -r-0.2>, <    0.0,    0.0,  r+0.2 >, at, Gray)
+arrow(<    0.0, -r-0.2,    0  >, <    0.0,  r+0.2,    0.0 >, at, Gray)
+
+#declare farbeX = rgb<1.0,0.2,0.6>;
+#declare farbeY = rgb<0.0,0.8,0.4>;
+#declare farbeZ = rgb<0.4,0.6,1.0>;
+
+#declare farbex = rgb<1.0,0.0,0.0>;
+#declare farbey = rgb<0.0,0.6,0.0>;
+#declare farbez = rgb<0.0,0.0,1.0>;
+
+#macro quadrant(X, Y, Z)
+	intersection {
+		sphere { O, 0.5 }
+		plane { -X, 0 }
+		plane { -Y, 0 }
+		plane { -Z, 0 }
+		pigment {
+			color rgb<1.0,0.6,0.2>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+	arrow(O, X, 1.1*at, farbex)
+	arrow(O, Y, 1.1*at, farbey)
+	arrow(O, Z, 1.1*at, farbez)
+#end
+
+#macro drehung(X, Y, Z)
+//	intersection {
+//		sphere { O, 0.5 }
+//		plane { -X, 0 }
+//		plane { -Y, 0 }
+//		plane { -Z, 0 }
+//		pigment {
+//			color Gray
+//		}
+//		finish {
+//			specular 0.95
+//			metallic
+//		}
+//	}
+	arrow(O, 1.1*X, 0.9*at, farbeX)
+	arrow(O, 1.1*Y, 0.9*at, farbeY)
+	arrow(O, 1.1*Z, 0.9*at, farbeZ)
+#end
+
+#macro achse(H)
+	cylinder { H, -H, at
+		pigment {
+			color rgb<0.6,0.4,0.2>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+	cylinder { 0.003 * H, -0.003 * H, 1
+		pigment {
+			color rgbt<0.6,0.4,0.2,0.5>
+		}
+		finish {
+			specular 0.95
+			metallic
+		}
+	}
+#end
diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini
new file mode 100644
index 0000000..f07c079
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.ini
@@ -0,0 +1,8 @@
+Input_File_Name=interpolation.pov
+Initial_Frame=0
+Final_Frame=50
+Initial_Clock=0
+Final_Clock=50
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m
new file mode 100644
index 0000000..31554e8
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.m
@@ -0,0 +1,54 @@
+#
+# interpolation.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 50;
+global A;
+global B;
+
+A = (pi / 2) * [
+  0, 0,  0;
+  0, 0, -1;
+  0, 1,  0
+];
+g0 = expm(A)
+
+B = (pi / 2) * [
+  0, 0, 1;
+  0, 0, 0;
+ -1, 0, 0
+];
+g1 = expm(B)
+
+function retval = g(t)
+	global A;
+	global B;
+	retval = expm((1-t)*A+t*B);
+endfunction
+
+function dreibein(fn, M, funktion)
+	fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n",
+		funktion,
+		M(1,1), M(3,1), M(2,1),
+		M(1,2), M(3,2), M(2,2),
+		M(1,3), M(3,3), M(2,3));
+endfunction
+
+G = g1 * inverse(g0);
+[V, lambda] = eig(G);
+H = real(V(:,3));
+
+D = logm(g1*inverse(g0));
+
+for i = (0:N)
+	filename = sprintf("dreibein/d%03d.inc", i);
+	fn = fopen(filename, "w");
+	t = i/N;
+	dreibein(fn, g(t), "quadrant");
+	dreibein(fn, expm(t*D)*g0, "drehung");
+	fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1));
+	fclose(fn);
+endfor
+
diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov
new file mode 100644
index 0000000..71e0257
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.pov
@@ -0,0 +1,10 @@
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+
+#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc");
+#include filename
+
diff --git a/vorlesungen/slides/7/images/test.pov b/vorlesungen/slides/7/images/test.pov
new file mode 100644
index 0000000..5707be1
--- /dev/null
+++ b/vorlesungen/slides/7/images/test.pov
@@ -0,0 +1,7 @@
+//
+// test.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+#include "dreibein/d025.inc"
diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex
new file mode 100644
index 0000000..525e6de
--- /dev/null
+++ b/vorlesungen/slides/7/integration.tex
@@ -0,0 +1,66 @@
+%
+% integration.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Invariante Integration}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Koordinatenwechsel}
+Die Koordinatentransformation
+$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$
+hat die Ableitungsmatrix
+\[
+t_{ij}
+=
+\frac{\partial y_i}{\partial x_j}
+\]
+\uncover<2->{%
+$n$-faches Integral
+\begin{gather*}
+\int\dots\int
+h(f(x))
+\det
+\biggl(
+\frac{\partial y_i}{\partial x_j}
+\biggr)
+\,dx_1\,\dots dx_n
+\\
+=
+\int\dots\int
+h(y)
+\,dy_1\,\dots dy_n
+\end{gather*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{auf einer Lie-Gruppe}
+Koordinatenwechsel sind Multiplikationen mit einer
+Matrix $g\in G$
+\end{block}}
+\uncover<4->{%
+\begin{block}{Volumenelement in $I$}
+Man muss nur das Volumenelement in $I$ in einem beliebigen
+Koordinatensystem definieren:
+\[
+dV = dy_1\,\dots\,dy_n
+\]
+\end{block}}
+\uncover<5->{%
+\begin{block}{Volumenelement in $g$}
+\[
+\text{``\strut}g\cdot dV\text{\strut''}
+=
+\det(g) \, dy_1\,\dots\,dy_n
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex
new file mode 100644
index 0000000..249ee26
--- /dev/null
+++ b/vorlesungen/slides/7/interpolation.tex
@@ -0,0 +1,112 @@
+%
+% interpolation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Interpolation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen
+$g_0\in\operatorname{SO}(3)$
+und
+$g_1\in\operatorname{SO}(3)$:
+\[
+g_0=g(0)
+\quad\wedge\quad
+g_1=g(1)
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Lösung}
+$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$
+\begin{align*}
+\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)}
+\\
+\uncover<5->{A(t)^t
+&=(1-t)A_0^t + tA_1^t}
+\\
+&\uncover<6->{=
+-(1-t)A_0 - t A_1}
+\uncover<7->{=
+-A(t)}
+\\
+\uncover<9->{\Rightarrow
+g(t) &= \exp A(t) \in \operatorname{SO}(3)}
+\\
+&\uncover<10->{\ne 
+\exp (\log(g_1g_0^{-1})t) g_0}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Animation}
+\centering
+\ifthenelse{\boolean{presentation}}{
+\bild{12}{i00}
+\bild{13}{i01}
+\bild{14}{i02}
+\bild{15}{i03}
+\bild{16}{i04}
+\bild{17}{i05}
+\bild{18}{i06}
+\bild{19}{i07}
+\bild{20}{i08}
+\bild{21}{i09}
+\bild{22}{i10}
+\bild{23}{i11}
+\bild{24}{i12}
+\bild{25}{i13}
+\bild{26}{i14}
+\bild{27}{i15}
+\bild{28}{i16}
+\bild{29}{i17}
+\bild{30}{i18}
+\bild{31}{i19}
+\bild{32}{i20}
+\bild{33}{i21}
+\bild{34}{i22}
+\bild{35}{i23}
+\bild{36}{i24}
+\bild{37}{i25}
+\bild{38}{i26}
+\bild{39}{i27}
+\bild{40}{i28}
+\bild{41}{i29}
+\bild{42}{i30}
+\bild{43}{i31}
+\bild{44}{i32}
+\bild{45}{i33}
+\bild{46}{i34}
+\bild{47}{i35}
+\bild{48}{i36}
+\bild{49}{i37}
+\bild{50}{i38}
+\bild{51}{i39}
+\bild{52}{i40}
+\bild{53}{i41}
+\bild{54}{i42}
+\bild{55}{i43}
+\bild{56}{i44}
+\bild{57}{i45}
+\bild{58}{i46}
+\bild{59}{i47}
+\bild{60}{i48}
+\bild{61}{i49}
+\bild{62}{i50}
+}{
+\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png}
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/liealgbeispiel.tex b/vorlesungen/slides/7/liealgbeispiel.tex
new file mode 100644
index 0000000..a17de40
--- /dev/null
+++ b/vorlesungen/slides/7/liealgbeispiel.tex
@@ -0,0 +1,78 @@
+%
+% liealgbeispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra Beispiele}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{sl}_2(\mathbb{R})$}
+Spurlose Matrizen:
+\[
+\operatorname{sl}_2(\mathbb{R})
+=
+\{A\in M_n(\mathbb{R})\;|\; \operatorname{Spur}A=0\}
+\]
+\end{block}
+\begin{block}{Lie-Algebra?}
+Nachrechnen: $[A,B]\in \operatorname{sl}_2(\mathbb{R})$:
+\begin{align*}
+\operatorname{Spur}([A,B])
+&=
+\operatorname{Spur}(AB-BA)
+\\
+&=
+\operatorname{Spur}(AB)-\operatorname{Spur}(BA)
+\\
+&=
+\operatorname{Spur}(AB)-\operatorname{Spur}(AB)
+\\
+&=0
+\end{align*}
+$\Rightarrow$ $\operatorname{sl}_2(\mathbb{R})$ ist eine Lie-Algebra
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{so}(n)$}
+Antisymmetrische Matrizen:
+\[
+\operatorname{so}(n)
+=
+\{A\in M_n(\mathbb{R})
+\;|\;
+A=-A^t
+\}
+\]
+\end{block}
+\begin{block}{Lie-Algebra?}
+Nachrechnen: $A,B\in \operatorname{so}(n)$
+\begin{align*}
+[A,B]^t
+&=
+(AB-BA)^t
+\\
+&=
+B^tA^t - A^tB^t
+\\
+&=
+(-B)(-A)-(-A)(-B)
+\\
+&=
+BA-AB
+=
+-(AB-BA)
+\\
+&=
+-[A,B]
+\end{align*}
+$\Rightarrow$ $\operatorname{so}(n)$ ist eine Lie-Algebra
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex
new file mode 100644
index 0000000..58065d7
--- /dev/null
+++ b/vorlesungen/slides/7/logarithmus.tex
@@ -0,0 +1,82 @@
+%
+% logarithmus.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Logarithmus}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Taylor-Reihe}
+\begin{align*}
+\frac{d}{dx}\log(1+x)
+&= \frac{1}{1+x}
+\\
+\uncover<2->{
+\Rightarrow\quad
+\log (1+x)
+&=
+\int_0^x \frac{1}{1+t}\,dt}
+\end{align*}
+\begin{align*}
+\uncover<3->{\frac{1}{1+t}
+&=
+1-t+t^2-t^3+\dots}
+\\
+\uncover<4->{\log(1+x)
+&=\int_0^x
+1-t+t^2-t^3+\dots
+\,dt}
+\\
+&\only<5>{=
+x-\frac{x^2}{2}  + \frac{x^3}{3} - \frac{x^4}4 + \dots}
+\uncover<6->{=
+\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}}
+\\
+\uncover<7->{\log (I+A)
+&=
+\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Konvergenzradius}
+Polstelle bei $x=-1$
+\(
+\varrho =1
+\)
+\end{block}}
+\vspace{-5pt}
+\begin{block}{\uncover<9->{Alternative: Spektraltheorie}}
+\uncover<9->{
+Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$
+definiert:}
+\vspace{-15pt}
+\uncover<8->{
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+	\fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1);
+}
+\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}];
+\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}];
+\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1];
+\draw[color=blue] (1,0) circle[radius=1];
+\uncover<9->{
+	\draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0);
+}
+\fill (1,0) circle[radius=0.08];
+\node at (2.3,1.9) {$\mathbb{C}$};
+\node at (1,0) [below] {$1$};
+\end{tikzpicture}
+\end{center}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/vektorlie.tex b/vorlesungen/slides/7/vektorlie.tex
new file mode 100644
index 0000000..621a832
--- /dev/null
+++ b/vorlesungen/slides/7/vektorlie.tex
@@ -0,0 +1,206 @@
+%
+% viktorlie.tex -- slide template
+%
+% (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{Vektorprodukt als Lie-Algebra}
+%\vspace{-10pt}
+\centering
+\begin{tikzpicture}[>=latex,thick]
+\arraycolsep=2.4pt
+\def\Ax{0}
+\def\Ux{4.1}
+\def\Kx{7.2}
+\def\Rx{13.1}
+
+\def\Lx{2.2}
+\def\Ly{0}
+\def\Lz{-2.2}
+
+\fill[color=red!20] (\Ax,{\Lx-1.55}) rectangle ({\Ux-0.1},{\Lx+0.55});
+\fill[color=red!20] (\Ux,{\Lx-1.55}) rectangle ({\Kx-0.1},{\Lx+0.55});
+\fill[color=red!20] (\Kx,{\Lx-1.55}) rectangle ({\Rx},{\Lx+0.55});
+
+\fill[color=darkgreen!20] (\Ax,{\Ly-1.55}) rectangle ({\Ux-0.1},{\Ly+0.55});
+\fill[color=darkgreen!20] (\Ux,{\Ly-1.55}) rectangle ({\Kx-0.1},{\Ly+0.55});
+\fill[color=darkgreen!20] (\Kx,{\Ly-1.55}) rectangle ({\Rx},{\Ly+0.55});
+
+\fill[color=blue!20] (\Ax,{\Lz-1.55}) rectangle ({\Ux-0.1},{\Lz+0.55});
+\fill[color=blue!20] (\Ux,{\Lz-1.55}) rectangle ({\Kx-0.1},{\Lz+0.55});
+\fill[color=blue!20] (\Kx,{\Lz-1.55}) rectangle ({\Rx},{\Lz+0.55});
+
+\coordinate (A) at (\Ax,3.2);
+\coordinate (Ax) at (\Ax,\Lx);
+\coordinate (Ay) at (\Ax,\Ly);
+\coordinate (Az) at (\Ax,\Lz);
+
+\node at (A) [right]
+	{\usebeamercolor[fg]{title}Drehmatrix, $\operatorname{SO}(n)$\strut};
+
+\node at (Ax) [right] {$\displaystyle\tiny
+D_{x,\alpha}=\begin{pmatrix}
+1&0&0\\
+0&\cos\alpha&-\sin\alpha\\
+0&\sin\alpha&\cos\alpha
+\end{pmatrix}$};
+
+\node at (Ay) [right] {$\displaystyle\tiny
+D_{y,\alpha}=\begin{pmatrix}
+\cos\alpha&0&\sin\alpha\\
+0&1&0\\
+-\sin\alpha&0&\cos\alpha
+\end{pmatrix}$};
+
+\node at (Az) [right] {$\displaystyle\tiny
+D_{z,\alpha}=\begin{pmatrix}
+\cos\alpha&-\sin\alpha&0\\
+\sin\alpha&\cos\alpha&0\\
+0&0&1
+\end{pmatrix}$};
+
+\coordinate (U) at (\Ux,3.2);
+\coordinate (Ux) at (\Ux,\Lx);
+\coordinate (Uy) at (\Ux,\Ly);
+\coordinate (Uz) at (\Ux,\Lz);
+\coordinate (Ex) at (\Ux,{\Lx-1});
+\coordinate (Ey) at (\Ux,{\Ly-1});
+\coordinate (Ez) at (\Ux,{\Lz-1});
+
+\uncover<2->{
+\node at (U) [right]
+	{\usebeamercolor[fg]{title}Ableitung, $\operatorname{so}(n)$\strut};
+
+\node at (Ux) [right] {$\displaystyle\tiny
+U_x=\begin{pmatrix*}[r]
+0&0&0\\
+0&0&-1\\
+0&1&0
+\end{pmatrix*}
+$};
+
+\node at (Uy) [right] {$\displaystyle\tiny
+U_y=\begin{pmatrix*}[r]
+0&0&1\\
+0&0&0\\
+-1&0&0
+\end{pmatrix*}
+$};
+
+\node at (Uz) [right] {$\displaystyle\tiny
+U_z=\begin{pmatrix*}[r]
+0&-1&0\\
+1&0&0\\
+0&0&0
+\end{pmatrix*}
+$};
+}
+
+\uncover<9->{
+\node at (Ex) [right] {$\displaystyle
+\, e_x = \tiny\begin{pmatrix}1\\0\\0\end{pmatrix}
+$};
+
+\node at (Ey) [right] {$\displaystyle
+\, e_y = \tiny\begin{pmatrix}0\\1\\0\end{pmatrix}
+$};
+
+\node at (Ez) [right] {$\displaystyle
+\, e_z = \tiny\begin{pmatrix}0\\0\\1\end{pmatrix}
+$};
+}
+
+\coordinate (K) at (\Kx,3.2);
+\coordinate (Kx) at (\Kx,\Lx);
+\coordinate (Ky) at (\Kx,\Ly);
+\coordinate (Kz) at (\Kx,\Lz);
+\coordinate (Vx) at (\Kx,{\Lx-1});
+\coordinate (Vy) at (\Kx,{\Ly-1});
+\coordinate (Vz) at (\Kx,{\Lz-1});
+
+\uncover<3->{
+\node at (K) [right]
+	{\usebeamercolor[fg]{title}Kommutator\strut};
+
+\node at (Kx) [right] {$\displaystyle
+\begin{aligned}
+[U_y,U_z] &\uncover<4->{= 
+{\tiny
+\begin{pmatrix}
+0&0&0\\
+0&0&0\\
+0&1&0
+\end{pmatrix}}
+\uncover<5->{\mathstrut-
+\tiny
+\begin{pmatrix}
+0&0&0\\
+0&0&1\\
+0&0&0
+\end{pmatrix}}}
+\uncover<6->{=U_x}
+\end{aligned}
+$};
+}
+
+\uncover<7->{
+\node at (Ky) [right] {$\displaystyle
+\begin{aligned}
+[U_z,U_x] &= 
+{\tiny
+\begin{pmatrix}
+0&0&1\\
+0&0&0\\
+0&0&0
+\end{pmatrix}
+-
+\begin{pmatrix}
+0&0&0\\
+0&0&0\\
+1&0&0
+\end{pmatrix}}
+=U_y
+\end{aligned}
+$};
+}
+
+\uncover<8->{
+\node at (Kz) [right] {$\displaystyle
+\begin{aligned}
+[U_x,U_y] &= 
+{\tiny
+\begin{pmatrix}
+0&0&0\\
+1&0&0\\
+0&0&0
+\end{pmatrix}
+-
+\begin{pmatrix}
+0&1&0\\
+0&0&0\\
+0&0&0
+\end{pmatrix}}
+=U_z
+\end{aligned}
+$};
+}
+
+\uncover<10->{
+\node at (Vx) [right] {$\displaystyle \phantom{]}e_y\times e_z = e_x$};
+}
+
+\uncover<11->{
+\node at (Vy) [right] {$\displaystyle \phantom{]}e_z\times e_x = e_y$};
+}
+
+\uncover<12->{
+\node at (Vz) [right] {$\displaystyle \phantom{]}e_x\times e_y = e_z$};
+}
+
+\end{tikzpicture}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
index d46dc7f..6ac5665 100644
--- a/vorlesungen/slides/8/Makefile.inc
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -28,5 +28,25 @@ chapter8 =								\
 	../slides/8/tokyo/bahn0.tex					\
 	../slides/8/tokyo/bahn1.tex					\
 	../slides/8/tokyo/bahn2.tex					\
+	../slides/8/chrind.tex						\
+	../slides/8/chrindprop.tex					\
+	../slides/8/chroma1.tex						\
+	../slides/8/amax.tex						\
+	../slides/8/subgraph.tex					\
+	../slides/8/chrwilf.tex						\
+	../slides/8/weitere.tex						\
+	../slides/8/wavelets/funktionen.tex				\
+	../slides/8/wavelets/laplacebasis.tex				\
+	../slides/8/wavelets/vektoren.tex				\
+	../slides/8/wavelets/fourier.tex				\
+	../slides/8/wavelets/lokalisierungsvergleich.tex		\
+	../slides/8/wavelets/frequenzlokalisierung.tex			\
+	../slides/8/wavelets/dilatation.tex				\
+	../slides/8/wavelets/matrixdilatation.tex			\
+	../slides/8/wavelets/gundh.tex					\
+	../slides/8/wavelets/dilbei.tex					\
+	../slides/8/wavelets/frame.tex					\
+	../slides/8/wavelets/framekonstanten.tex			\
+	../slides/8/wavelets/beispiel.tex				\
 	../slides/8/chapter.tex
 
diff --git a/vorlesungen/slides/8/amax.tex b/vorlesungen/slides/8/amax.tex
new file mode 100644
index 0000000..951400a
--- /dev/null
+++ b/vorlesungen/slides/8/amax.tex
@@ -0,0 +1,86 @@
+%
+% amax.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\alpha_{\text{max}}$ und $d$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Definition}
+$\alpha_{\text{max}}$ ist der grösste Eigenwert der Adjazenzmatrix
+\end{block}
+\uncover<2->{
+\begin{block}{Fakten}
+\begin{itemize}
+\item<3->
+Der Eigenwert $\alpha_{\text{max}}$ ist einfach
+\item<4->
+Es gibt einen positiven Eigenvektor $f$ zum Eigenwert $\alpha_{\text{max}}$
+\item<5->
+$f$ maximiert
+\[
+\frac{\langle Af,f\rangle}{\langle f,f\rangle}
+=
+\alpha_{\text{max}}
+\]
+\end{itemize}
+Herkunft: Perron-Frobenius-Theorie positiver Matrizen (nächste Woche)
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<6->{%
+\begin{block}{Mittlerer Grad}
+\[
+\overline{d}
+=
+\frac1{n} \sum_{v} \operatorname{deg}(v)
+\le
+\alpha_{\text{max}}
+\le
+d
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item Konstante Funktion $1$ anstelle von $f$:
+\[
+\frac{\langle A1,1\rangle}{\langle 1,1\rangle}
+\uncover<8->{=
+\frac{\sum_v \operatorname{deg}(v)}{n}}
+\uncover<9->{=
+\overline{d}}
+\uncover<10->{\le
+\alpha_{\text{max}}}
+\]
+\item<11-> Komponenten von $Af$ summieren:
+\begin{align*}
+\uncover<12->{
+\alpha_{\text{max}}
+f(v) &= (Af)(v)}\uncover<13->{ = \sum_{u\sim v} f(u)}
+\\
+\uncover<14->{\alpha_{\text{max}}
+\sum_{v}f(v)
+&=
+\sum_v
+\operatorname{deg}(v) f(v)}
+\\
+&\uncover<15->{\le
+d\sum_v f(v)}
+\;
+\uncover<16->{\Rightarrow
+\;
+\alpha_{\text{max}} \le d}
+\end{align*}
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
index 6a0b13f..69b7231 100644
--- a/vorlesungen/slides/8/chapter.tex
+++ b/vorlesungen/slides/8/chapter.tex
@@ -30,3 +30,24 @@
 \folie{8/tokyo/bahn1.tex}
 \folie{8/tokyo/bahn2.tex}
 
+\folie{8/chrind.tex}
+\folie{8/chrindprop.tex}
+\folie{8/chroma1.tex}
+\folie{8/amax.tex}
+\folie{8/subgraph.tex}
+\folie{8/chrwilf.tex}
+\folie{8/weitere.tex}
+
+\folie{8/wavelets/funktionen.tex}
+\folie{8/wavelets/laplacebasis.tex}
+\folie{8/wavelets/fourier.tex}
+\folie{8/wavelets/lokalisierungsvergleich.tex}
+\folie{8/wavelets/frequenzlokalisierung.tex}
+\folie{8/wavelets/dilatation.tex}
+\folie{8/wavelets/matrixdilatation.tex}
+\folie{8/wavelets/gundh.tex}
+\folie{8/wavelets/frame.tex}
+\folie{8/wavelets/dilbei.tex}
+\folie{8/wavelets/framekonstanten.tex}
+\folie{8/wavelets/beispiel.tex}
+
diff --git a/vorlesungen/slides/8/chrind.tex b/vorlesungen/slides/8/chrind.tex
new file mode 100644
index 0000000..bd406ab
--- /dev/null
+++ b/vorlesungen/slides/8/chrind.tex
@@ -0,0 +1,231 @@
+%
+% chrind.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Chromatische Zahl und Unabhängigkeitszahl}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Chromatische Zahl}
+$\operatorname{chr}(G)=\mathstrut$
+minimale Anzahl Farben, die zum Einfärben eines Graphen $G$ nötig sind derart,
+dass benachbarte Knoten verschiedene Farben haben.
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\only<1>{
+        \fill[color=white] (PA) circle[radius=\r];
+        \fill[color=white] (PB) circle[radius=\r];
+        \fill[color=white] (PC) circle[radius=\r];
+        \fill[color=white] (PD) circle[radius=\r];
+        \fill[color=white] (PE) circle[radius=\r];
+        \fill[color=white] (QA) circle[radius=\r];
+        \fill[color=white] (QB) circle[radius=\r];
+        \fill[color=white] (QC) circle[radius=\r];
+        \fill[color=white] (QD) circle[radius=\r];
+        \fill[color=white] (QE) circle[radius=\r];
+}
+
+\only<2->{
+        \fill[color=blau] (PA) circle[radius=\r];
+        \fill[color=rot] (PB) circle[radius=\r];
+        \fill[color=rot] (PC) circle[radius=\r];
+        \fill[color=gruen] (PD) circle[radius=\r];
+        \fill[color=gruen] (PE) circle[radius=\r];
+
+        \fill[color=rot] (QA) circle[radius=\r];
+        \fill[color=blau] (QB) circle[radius=\r];
+        \fill[color=gruen] (QC) circle[radius=\r];
+        \fill[color=rot] (QD) circle[radius=\r];
+        \fill[color=blau] (QE) circle[radius=\r];
+}
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius=\r];
+\draw (PC) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius=\r];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{chr} G = 3$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Unabhängigkeitszahl}
+$\operatorname{ind}(G)=\mathstrut$
+maximale Anzahl nicht benachbarter Knoten
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+\definecolor{gelb}{rgb}{0,0,1}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\foreach \n in {1,...,7}{
+	\only<\n>{\node[color=white] at (1,2.9) {$\n$};}
+}
+
+\fill[color=white] (PA) circle[radius=\r];
+\fill[color=white] (PB) circle[radius=\r];
+\fill[color=white] (PC) circle[radius=\r];
+\fill[color=white] (PD) circle[radius=\r];
+\fill[color=white] (PE) circle[radius=\r];
+\fill[color=white] (QA) circle[radius=\r];
+\fill[color=white] (QB) circle[radius=\r];
+\fill[color=white] (QC) circle[radius=\r];
+\fill[color=white] (QD) circle[radius=\r];
+\fill[color=white] (QE) circle[radius=\r];
+
+\only<4->{
+        \fill[color=rot] (QA) circle[radius={1.5*\r}];
+        \fill[color=rot!40] (QB) circle[radius=\r];
+        \fill[color=rot!40] (QE) circle[radius=\r];
+        \fill[color=rot!40] (PA) circle[radius=\r];
+}
+
+\only<5->{
+        \fill[color=blau] (PB) circle[radius={1.5*\r}];
+        \fill[color=blau!40] (PD) circle[radius=\r];
+        \fill[color=blau!40] (PE) circle[radius=\r];
+        \fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r];
+}
+
+\only<6->{
+        \fill[color=gruen] (PC) circle[radius={1.5*\r}];
+        \fill[color=gruen!40] (QC) circle[radius=\r];
+        \fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r];
+        \fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r];
+}
+
+\only<7->{
+        \fill[color=gelb] (QD) circle[radius={1.5*\r}];
+        \fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r];
+        \fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r];
+        \fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r];
+}
+
+\only<-3|handout:0>{
+	\draw (QA) circle[radius=\r];
+}
+\only<4->{
+	\draw (QA) circle[radius={1.5*\r}];
+}
+
+\only<-4|handout:0>{
+	\draw (PB) circle[radius=\r];
+}
+\only<5->{
+	\draw (PB) circle[radius={1.5*\r}];
+}
+
+\only<-5|handout:0>{
+	\draw (PC) circle[radius=\r];
+}
+\only<6->{
+	\draw (PC) circle[radius={1.5*\r}];
+}
+
+\only<-6|handout:0>{
+	\draw (QD) circle[radius=\r];
+}
+\only<7->{
+	\draw (QD) circle[radius={1.5*\r}];
+}
+
+\draw (PA) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\only<4|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 1$};
+}
+\only<5|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 2$};
+}
+\only<6|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 3$};
+}
+\only<7->{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 4$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chrindprop.tex b/vorlesungen/slides/8/chrindprop.tex
new file mode 100644
index 0000000..094588c
--- /dev/null
+++ b/vorlesungen/slides/8/chrindprop.tex
@@ -0,0 +1,62 @@
+%
+% chrindprop.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zusammenhang zwischen $\operatorname{chr}G$ und $\operatorname{ind}G$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Proposition}
+Ist $G$ ein Graph mit $n$ Knoten, dann gilt
+\[
+\operatorname{chr}G
+\cdot
+\operatorname{ind}G
+\ge n
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Beispiel}
+Peterson-Graph $K$ hat $n=10$ Knoten:
+\[
+\operatorname{chr}(K)
+\cdot
+\operatorname{ind}(K)
+=
+3\cdot 4
+\ge 
+10
+=
+n
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<3->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<4-> eine minimale Färbung hat $\operatorname{chr}(G)$ Farben
+\item<5-> Sie teilt die Knoten in $\operatorname{chr}(G)$
+gleichfarbige Mengen auf
+\item<6-> Jede einfarbige Menge von Knoten ist unabhängig, d.~h.~sie 
+besteht aus Knoten, die nicht miteinander verbunden sind.
+\item<7-> Jede einfarbige Menge enthält höchstens $\operatorname{ind}(G)$
+\item<8-> Die Gesamtzahl der Knoten ist
+\[
+n\uncover<9->{=\sum_{\text{Farbe}}\underbrace{|V_{\text{Farbe}}|}_{\le \operatorname{ind}(G)}}
+\uncover<10->{\le
+\operatorname{chr}(G)
+\cdot
+\operatorname{ind}(G)}
+\]
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chroma1.tex b/vorlesungen/slides/8/chroma1.tex
new file mode 100644
index 0000000..6a55704
--- /dev/null
+++ b/vorlesungen/slides/8/chroma1.tex
@@ -0,0 +1,56 @@
+%
+% chroma1.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Schranke für $\operatorname{chr}(G)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Proposition}
+Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt
+\[
+\operatorname{chr}(G) \le d + 1
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item<3->
+Peterson-Graph $G$: maximaler Grad ist $d=3$, aber
+\[
+\operatorname{chr}(G)
+=
+3
+< d+1=4
+\]
+\item<4->
+Voller Graph $V$: maximaler Grad ist $d=n-1$,
+\[
+\operatorname{chr}(V) = n = d+1
+\]
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\begin{proof}[Beweis]
+Mit vollständiger Induktion, d.~h.~Annahme: Graphen mit $<n$ Knoten und
+maximalem Grad $d$ lassen sich mit höchstens $d+1$ Farben färben.
+\begin{itemize}
+\item<5-> $X$ ein Graph mit $n$ Knoten
+\item<6-> entferne den Knoten $v\in X$, $X'=X\setminus\{v\}$
+\item<7-> $X'$ lässt sich mit höchstens $d+1$ Farben einfärben
+\item<8-> $v$ hat höchstens $d$ Nachbarn, die höchsten $d$ verschiedene
+Farben haben
+\item<9-> Es bleibt eine Farbe für $v$
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chrwilf.tex b/vorlesungen/slides/8/chrwilf.tex
new file mode 100644
index 0000000..7edb10e
--- /dev/null
+++ b/vorlesungen/slides/8/chrwilf.tex
@@ -0,0 +1,115 @@
+%
+% chrwilf.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\def\kante#1#2{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2);
+}
+\def\knoten#1#2{
+	\uncover<8->{
+		\fill[color=#2!30] (#1) circle[radius=0.2];
+		\draw[color=#2] (#1) circle[radius=0.2];
+	}
+	\only<-7>{
+		\draw (#1) circle[radius=0.2];
+	}
+}
+\def\R{1.5}
+\definecolor{rot}{rgb}{1,0,0}
+\definecolor{gruen}{rgb}{0,0.6,0}
+\definecolor{blau}{rgb}{0,0,1}
+\begin{frame}[t]
+\frametitle{Schranke für die chromatische Zahl}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Wilf)}
+$\uncover<2->{\operatorname{chr}(X) \le 1+}\alpha_{\text{max}} \le\uncover<2->{ 1 + }d$
+\end{block}
+\uncover<3->{%
+\begin{block}{Beispiel}
+\begin{align*}
+\uncover<4->{d&= 4}
+&&\uncover<5->{\Rightarrow& \operatorname{chr}(G) &\le 5}\\
+\uncover<6->{\alpha_{\text{max}} &= 
+2.9565}
+&&\uncover<7->{\Rightarrow& \operatorname{chr}(G) &\le 3}\\
+\uncover<4->{\overline{d} &= \frac{24}{9}=\rlap{$2.6666$}}
+\end{align*}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (A) at (0:\R);
+\coordinate (B) at (40:\R);
+\coordinate (C) at (80:\R);
+\coordinate (D) at (120:\R);
+\coordinate (E) at (160:\R);
+\coordinate (F) at (200:\R);
+\coordinate (G) at (240:\R);
+\coordinate (H) at (280:\R);
+\coordinate (I) at (320:\R);
+
+\knoten{A}{rot}
+\knoten{B}{blau}
+\knoten{C}{gruen}
+\knoten{D}{blau}
+\knoten{E}{rot}
+\knoten{F}{blau}
+\knoten{G}{rot}
+\knoten{H}{gruen}
+\knoten{I}{blau}
+
+\kante{A}{B}
+\kante{B}{C}
+\kante{C}{D}
+\kante{D}{E}
+\kante{E}{F}
+\kante{F}{G}
+\kante{G}{H}
+\kante{H}{I}
+\kante{I}{A}
+
+\kante{A}{C}
+\kante{A}{D}
+\kante{D}{G}
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<9->{%
+\begin{proof}[Beweis]
+Induktion nach der Grösse $n$ des Graphen.
+\begin{itemize}
+\item<10->
+Entferne $v\in X$ mit minimalem Grad: $X'=X\setminus \{v\}$
+\item<11->
+Induktionsannahme:
+\[
+\operatorname{chr}(X')
+\le
+1+
+\alpha_{\text{max}}' 
+\]
+\item<12->
+$X'$ kann mit höhcstens $1+\alpha_{\text{max}}'\le 1+\alpha_{\text{max}}$
+Farben eingefärbt werden.
+\item<13->
+Wegen
+\(
+\deg(v) \le \overline{d} \le \alpha_{\text{max}}
+\)
+hat $v$ höchstens $\alpha_{\text{max}}$ Nachbarn, um $v$ zu färben,
+braucht man also höchstens $1+\alpha_{\text{max}}$ Farben.
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
index 952c85b..10f88cd 100644
--- a/vorlesungen/slides/8/inzidenz.tex
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -5,6 +5,8 @@
 %
 \bgroup
 \definecolor{darkgreen}{rgb}{0,0.6,0}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
 \begin{frame}[t]
 \frametitle{Inzidenz- und Adjazenzmatrix}
 \vspace{-20pt}
@@ -67,7 +69,7 @@
 \vspace{-10pt}
 \uncover<5->{%
 \begin{block}{Definition}
-\vspace{-15pt}
+%\vspace{-15pt}
 \begin{align*}
 B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\
 A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$}
diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex
index 5f2f51a..43e5330 100644
--- a/vorlesungen/slides/8/inzidenzd.tex
+++ b/vorlesungen/slides/8/inzidenzd.tex
@@ -5,6 +5,8 @@
 %
 \bgroup
 \definecolor{darkgreen}{rgb}{0,0.6,0}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
 \begin{frame}[t]
 \frametitle{Inzidenz- und Adjazenz-Matrix}
 \vspace{-20pt}
@@ -67,7 +69,7 @@
 \vspace{-15pt}
 \uncover<5->{%
 \begin{block}{Definition}
-\vspace{-20pt}
+%\vspace{-20pt}
 \begin{align*}
 B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\
 B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\
diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex
index 1d8b725..93333bc 100644
--- a/vorlesungen/slides/8/produkt.tex
+++ b/vorlesungen/slides/8/produkt.tex
@@ -56,7 +56,7 @@
 \end{center}
 \vspace{-15pt}
 \begin{block}{Berechne}
-\vspace{-20pt}
+%\vspace{-20pt}
 \begin{align*}
 \uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t
 \end{align*}
diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex
index 425fe1c..62180d9 100644
--- a/vorlesungen/slides/8/spanningtree.tex
+++ b/vorlesungen/slides/8/spanningtree.tex
@@ -3,6 +3,7 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
+\bgroup
 \begin{frame}
 \frametitle{Spannbäume}
 
@@ -121,7 +122,7 @@ Wieviele Spannbäume gibt es?
 \begin{column}{0.56\hsize}
 \uncover<5->{%
 \begin{block}{Laplace-Matrix}
-\vspace{-15pt}
+%\vspace{-15pt}
 \[
 L=
 \tiny
@@ -162,3 +163,4 @@ L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{arr
 \end{columns}
 
 \end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/subgraph.tex b/vorlesungen/slides/8/subgraph.tex
new file mode 100644
index 0000000..f3005f9
--- /dev/null
+++ b/vorlesungen/slides/8/subgraph.tex
@@ -0,0 +1,60 @@
+%
+% subgraph.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\alpha_{\text{max}}$ eines Untergraphen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+$X'$ ein echter Untergraph von $X$ mit Adjazenzmatrix $A'$ und grösstem
+Eigenwert $\alpha_{\text{max}}'$
+\[
+\alpha_{\text{max}}' \le \alpha_{\text{max}}
+\]
+\end{block}
+\uncover<2->{$V'$ die Knoten von $X'$}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<4->
+$f'$ der positive Eigenvektor von $A'$
+\item<5->
+Definiere 
+\[
+g(v)
+=
+\begin{cases}
+f'(v) &\qquad v\in V'\\
+0     &\qquad \text{sonst}
+\end{cases}
+\]
+\item<6-> Skalarprodukte:
+\begin{align*}
+\uncover<7->{\langle f',f'\rangle &= \langle g,g\rangle}
+\\
+\uncover<8->{\langle A'f',f'\rangle &\le \langle Ag,g\rangle}
+\end{align*}
+\item<9-> Vergleich
+\[
+\alpha_{\text{max}}'
+=
+\frac{\langle A'f',f'\rangle}{\langle f',f'\rangle}
+\uncover<10->{\le 
+\frac{\langle Ag,g\rangle}{\langle g,g\rangle}}
+\uncover<11->{\le
+\alpha_{\text{max}}}
+\]
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/Makefile b/vorlesungen/slides/8/wavelets/Makefile
new file mode 100644
index 0000000..3b4a5ce
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/Makefile
@@ -0,0 +1,8 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+vektoren.tex:	ev.m
+	octave ev.m
diff --git a/vorlesungen/slides/8/wavelets/beispiel.tex b/vorlesungen/slides/8/wavelets/beispiel.tex
new file mode 100644
index 0000000..dcc33d4
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/beispiel.tex
@@ -0,0 +1,44 @@
+%
+% beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{
+\node at (0,0) [rotate=-90]
+{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets auf einer Kugel}
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\only<1>{ \bild{0.6}{wavelets-phi-sphere-334.pdf} }
+
+\only<2>{ \bild{0.6}{wavelets-psi-5-sphere-334.pdf} }
+\only<3>{ \bild{0.6}{wavelets-psi-4-sphere-334.pdf} }
+\only<4>{ \bild{0.6}{wavelets-psi-3-sphere-334.pdf} }
+\only<5>{ \bild{0.6}{wavelets-psi-2-sphere-334.pdf} }
+\only<6>{ \bild{0.6}{wavelets-psi-1-sphere-334.pdf} }
+
+\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; }
+\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; }
+\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; }
+\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; }
+\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; }
+\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; }
+
+\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; }
+\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; }
+\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; }
+\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; }
+\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; }
+\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; }
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/dilatation.tex b/vorlesungen/slides/8/wavelets/dilatation.tex
new file mode 100644
index 0000000..881f760
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/dilatation.tex
@@ -0,0 +1,62 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Dilatation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Dilatation in $\mathbb{R}$}
+$f\colon \mathbb{R}\to\mathbb{R}$
+Definition im Ortsraum:
+\[
+(D_af)(x)
+=
+\frac{1}{\sqrt{|a|}}
+f\biggl(\frac{x}{a}\biggr)
+\]
+\uncover<2->{%
+Dilatation im Frequenzraum:
+\[
+\widehat{D_af}(\omega)
+=
+D_{1/a}\hat{f}(\omega)
+\]}
+\uncover<3->{%
+Spektrum wird mit $1/a$ skaliert!}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{``Dilatation'' auf einem Graphen}
+\begin{itemize}
+\item<5-> Dilatation auf dem Graphen gibt es nicht
+\item<6-> Dilatation im Spektrum $\{\lambda_1,\dots,\lambda_n\}$ gibt es nicht
+\item<7-> ``Spektrale Dilatation'' verwenden
+\begin{enumerate}
+\item<8-> Start: $e_k$
+\item<9-> Fourier-Transformation: $\chi^te_k$
+\item<10-> Spektrum skalieren: mit
+$D_{1/a}g$ filtern
+\item<11-> Rücktransformation
+\[
+D_{g,a}e_k
+=
+\chi
+\uncover<12->{\operatorname{diag}(\tilde{D}_{1/a}g(\lambda_*))
+\chi^t e_k}
+\]
+\end{enumerate}
+\end{itemize}
+
+
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/dilbei.tex b/vorlesungen/slides/8/wavelets/dilbei.tex
new file mode 100644
index 0000000..fc66a0a
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/dilbei.tex
@@ -0,0 +1,46 @@
+%
+% beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{
+\node at (0,0) [rotate=-90]
+{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets einer Strecke}
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\only<1>{ \bild{0.6}{wavelets-psi-line-5-10.pdf} }
+\only<2>{ \bild{0.6}{wavelets-psi-line-4-10.pdf} }
+\only<3>{ \bild{0.6}{wavelets-psi-line-3-10.pdf} }
+\only<4>{ \bild{0.6}{wavelets-psi-line-2-10.pdf} }
+\only<5>{ \bild{0.6}{wavelets-psi-line-1-10.pdf} }
+
+\only<6>{ \bild{0.6}{wavelets-phi-line-10.pdf} }
+
+\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; }
+\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; }
+\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; }
+\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; }
+\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; }
+\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; }
+
+
+\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; }
+\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; }
+\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; }
+\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; }
+\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; }
+
+\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; }
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/ev.m b/vorlesungen/slides/8/wavelets/ev.m
new file mode 100644
index 0000000..7f4dd55
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/ev.m
@@ -0,0 +1,97 @@
+#
+# ev.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+L = [
+    2, -1,  0, -1,  0;
+   -1,  4, -1, -1, -1;
+    0, -1,  2,  0, -1;
+   -1, -1,  0,  3, -1;
+    0, -1, -1, -1,  3
+];
+
+[v, lambda] = eig(L);
+
+function knoten(fn, wert, punkt)
+	if (wert > 0) 
+		farbe = sprintf("red!%02d", round(100 * wert));
+	else
+		farbe = sprintf("blue!%02d", round(-100 * wert));
+	end
+	fprintf(fn, "\t\\fill[color=%s] %s circle[radius=0.25];\n",
+		farbe, punkt);
+	fprintf(fn, "\t\\draw %s circle[radius=0.25];\n", punkt);
+endfunction
+
+function vektor(fn, v, name, lambda)
+	fprintf(fn, "\\def\\%s{\n", name);
+	fprintf(fn, "\t\\coordinate (A) at ({0*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (B) at ({1*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (C) at ({2*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (D) at ({0.5*\\a},{-\\b});\n");
+	fprintf(fn, "\t\\coordinate (E) at ({1.5*\\a},{-\\b});\n");
+	fprintf(fn, "\t\\draw (A) -- (B);\n");
+	fprintf(fn, "\t\\draw (A) -- (D);\n");
+	fprintf(fn, "\t\\draw (B) -- (C);\n");
+	fprintf(fn, "\t\\draw (B) -- (D);\n");
+	fprintf(fn, "\t\\draw (B) -- (E);\n");
+	fprintf(fn, "\t\\draw (C) -- (E);\n");
+	fprintf(fn, "\t\\draw (D) -- (E);\n");
+	fprintf(fn, "\t\\node at (-2.8,{-0.5*\\b}) [right] {$\\lambda=%.4f$};\n",
+		round(1000 * abs(lambda)) / 10000);
+	w = v / max(abs(v));
+	knoten(fn, w(1,1), "(A)");
+	knoten(fn, w(2,1), "(B)");
+	knoten(fn, w(3,1), "(C)");
+	knoten(fn, w(4,1), "(D)");
+	knoten(fn, w(5,1), "(E)");
+	fprintf(fn, "}\n");
+endfunction
+
+function punkt(fn, x, wert)
+	fprintf(fn, "({%.4f*\\c},{%.4f*\\d})", x, wert);
+endfunction
+
+function funktion(fn, v, name, lambda)
+	fprintf(fn, "\\def\\%s{\n", name);
+	fprintf(fn, "\t\\draw[color=red,line width=1.4pt]\n\t\t");
+	punkt(fn, -2, v(1,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, -1, v(4,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 0, v(2,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 1, v(5,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 2, v(3,1));
+	fprintf(fn, ";\n");
+	fprintf(fn, "\t\\draw[->] ({-2.1*\\c},0) -- ({2.1*\\c},0);\n");
+	fprintf(fn, "\t\\draw[->] (0,{-1.1*\\d}) -- (0,{1.1*\\d});\n");
+	for x = (-2:2)
+		fprintf(fn, "\t\\fill ({%d*\\c},0) circle[radius=0.05];\n", x);
+	endfor
+	fprintf(fn, "}\n");
+endfunction
+
+fn = fopen("vektoren.tex", "w");
+
+vektor(fn, v(:,1), "vnull", lambda(1,1));
+funktion(fn, v(:,1), "fnull", lambda(1,1));
+
+vektor(fn, v(:,2), "vone", lambda(2,2));
+funktion(fn, v(:,2), "fone", lambda(2,2));
+
+vektor(fn, v(:,3), "vtwo", lambda(3,3));
+funktion(fn, v(:,3), "ftwo", lambda(3,3));
+
+vektor(fn, v(:,4), "vthree", lambda(4,4));
+funktion(fn, v(:,4), "fthree", lambda(4,4));
+
+vektor(fn, v(:,5), "vfour", lambda(5,5));
+funktion(fn, v(:,5), "ffour", lambda(5,5));
+
+fclose(fn);
+
+
diff --git a/vorlesungen/slides/8/wavelets/fourier.tex b/vorlesungen/slides/8/wavelets/fourier.tex
new file mode 100644
index 0000000..3195ec8
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/fourier.tex
@@ -0,0 +1,86 @@
+%
+% fourier.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Fourier-Transformation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Gegeben: Funktion $f$ auf dem Graphen
+\\
+\uncover<2->{%
+Gesucht: Koeffizienten $\hat{f}$ der Darstellung in der Laplace-Basis}
+\end{block}
+\uncover<3->{%
+\begin{block}{Definition $\chi$-Matrix}
+Eigenwerte $0=\lambda_1<\lambda_2\le \dots \le \lambda_n$ von $L$
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}
+\node at (-1.9,0) [left] {$\chi=\mathstrut$};
+\node at (0,0) {$\left(\raisebox{0pt}[1.7cm][1.7cm]{\hspace{3.5cm}}\right)$};
+
+\fill[color=blue!20] (-1.7,-1.7) rectangle (-1.1,1.7);
+\draw[color=blue] (-1.7,-1.7) rectangle (-1.1,1.7);
+\node at (-1.4,0) [rotate=90] {$v_1=\mathstrut$EV zum EW $\lambda_1$\strut};
+
+\fill[color=blue!20] (-1.0,-1.7) rectangle (-0.4,1.7);
+\draw[color=blue] (-1.0,-1.7) rectangle (-0.4,1.7);
+\node at (-0.7,0) [rotate=90] {$v_2=\mathstrut$EV zum EW $\lambda_2$\strut};
+
+\fill[color=blue!20] (1.1,-1.7) rectangle (1.7,1.7);
+\draw[color=blue] (1.1,-1.7) rectangle (1.7,1.7);
+\node at (1.4,0) [rotate=90] {$v_n=\mathstrut$EV zum EW $\lambda_n$\strut};
+
+\node at (0.4,0) {$\dots$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Transformation}
+$L$ symmetrisch
+\\
+\uncover<5->{$\Rightarrow$ 
+Die Eigenvektoren von $L$ können orthonormiert gewählt werden}
+\\
+\uncover<6->{$\Rightarrow$ 
+Koeffizienten können durch Skalarprodukte ermittelt werden:}
+\uncover<7->{%
+\[
+\hat{f}(k)
+=
+\hat{f}(\lambda_k)
+\uncover<8->{=
+\langle v_k, f\rangle
+\quad\Rightarrow\quad
+\hat{f}}
+\uncover<9->{=
+\chi^tf}
+\]}
+\uncover<10->{%
+$\chi$ ist die {\em Fourier-Transformation}}
+\end{block}}
+\uncover<11->{%
+\begin{block}{Rücktransformation}
+Eigenvektoren orthonormiert
+\\
+\uncover<12->{$\Rightarrow$
+$\chi$ orthogonal}
+\uncover<13->{
+\[
+\chi\chi^t = I
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/frame.tex b/vorlesungen/slides/8/wavelets/frame.tex
new file mode 100644
index 0000000..4d0c7d1
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/frame.tex
@@ -0,0 +1,66 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Graph Wavelet Frame}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frame-Vektoren}
+Zu Dilatationsfaktoren $A=\{a_i\,|\,i=1,\dots,N\}$
+konstruiere das Frame
+\begin{align*}
+F=
+\{&D_he_1,\dots,D_he_n,\\
+  &Dg_1e_1,\dots,Dg_1e_n,\\
+  &Dg_2e_1,\dots,Dg_2e_n,\\
+  &\dots\\
+  &Dg_Ne_1,\dots,Dg_Ne_n\}
+\end{align*}
+\uncover<2->{Notation:
+\begin{align*}
+v_{0,k}
+&=
+D_he_k
+\\
+v_{i,k}
+&=
+Dg_ie_k
+\end{align*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Frameoperator}
+\begin{align*}
+\mathcal{T}\colon \mathbb{R}^n\to\mathbb{R}^{nN}
+:
+v
+&\mapsto
+\begin{pmatrix}
+\uncover<4->{\langle D_he_1,v\rangle}\\
+\uncover<4->{\vdots}\\
+\uncover<4->{\langle D_he_n,v\rangle}\\
+\hline
+\uncover<5->{\langle D_{g_1}e_1,v\rangle}\\
+\uncover<5->{\vdots}\\
+\uncover<5->{\langle D_{g_1}e_n,v\rangle}\\
+\hline
+\uncover<6->{\vdots}\\
+\uncover<6->{\vdots}\\
+\hline
+\uncover<7->{\langle D_{g_N}e_1,v\rangle}\\
+\uncover<7->{\vdots}\\
+\uncover<7->{\langle D_{g_N}e_n,v\rangle}
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/framekonstanten.tex b/vorlesungen/slides/8/wavelets/framekonstanten.tex
new file mode 100644
index 0000000..a436536
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/framekonstanten.tex
@@ -0,0 +1,71 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+%\setlength{\abovedisplayskip}{5pt}
+%\setlength{\belowdisplayskip}{5pt}
+\frametitle{Framekonstanten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Menge $\mathcal{F}$ von Vektoren heisst ein Frame,
+falls es Konstanten $A$ und $B$ gibt derart, dass
+\[
+A\|v\|^2
+\le
+\|\mathcal{T}v\|^2
+\sum_{b\in\mathcal{F}} |\langle b,v\rangle|^2
+\le
+B\|v\|^2
+\]
+\uncover<2->{$A>0$ garantiert Invertierbarkeit}
+\end{block}
+\uncover<3->{%
+\begin{block}{$\|\mathcal{T}v\|$ für Graph-Wavelets}
+\begin{align*}
+\|\mathcal{T}v\|^2
+&=
+\sum_k |\langle D_he_k,v\rangle|^2
++
+\sum_{i,k} |\langle D_{g_i}e_k, v\rangle|^2
+\\
+&\uncover<4->{=
+\sum_k |h(\lambda_k) \hat{v}(k)|^2
++
+\sum_{k,i} |g_i(\lambda_k) \hat{v}(k)|^2}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{$A$ und $B$}
+Frame-Norm-Funktion
+\begin{align*}
+f(\lambda)
+&=
+h(\lambda)
++
+\sum_i g_i(\lambda)
+\\
+&\uncover<6->{=
+h(\lambda)
++
+\sum_i g(a_i\lambda)}
+\end{align*}
+\uncover<7->{Abschätzung für Frame-Konstanten
+\begin{align*}
+A&\uncover<8->{=
+\min_{i} f(\lambda_i)}
+\\
+B&\uncover<9->{=
+\max_{i} f(\lambda_i)}
+\end{align*}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex
new file mode 100644
index 0000000..c78e6dd
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex
@@ -0,0 +1,78 @@
+%
+% frequenzlokalisierung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\def\kurve#1#2{
+        \draw[color=#2,line width=1.4pt]
+                plot[domain=0:6.3,samples=400]
+                        ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lokalisierung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Bandpass}
+Gegeben durch $g(\lambda)\ge 0$:
+\begin{align*}
+g(0)                             &= 0\\
+\lim_{\lambda\to\infty}g(\lambda)&= 0
+\end{align*}
+\vspace{-10pt}
+\begin{enumerate}
+\item<3-> Fourier-transformieren
+\item<4-> Amplituden mit $g(\lambda)$ multiplizieren
+\item<5-> Rücktransformieren
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Tiefpass}
+Gegeben durch $h(\lambda)\ge0$:
+\begin{align*}
+h(0)                             &= 1\\
+\lim_{\lambda\to\infty}h(\lambda)&= 0
+\end{align*}
+\vspace{-10pt}
+\begin{enumerate}
+\item<8-> Fourier-Transformation
+\item<9-> Amplituden mit $h(\lambda)$ multiplizieren
+\item<10-> Rücktransformation
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.8]
+
+\uncover<2->{
+\begin{scope}[xshift=-4.5cm]
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+\kurve{3}{red}
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+}
+
+\uncover<7->{
+\begin{scope}[xshift=4.5cm]
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+\draw[color=darkgreen,line width=1.4pt]
+        plot[domain=0:6.3,samples=100]
+                ({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/funktionen.tex b/vorlesungen/slides/8/wavelets/funktionen.tex
new file mode 100644
index 0000000..2e3ae9b
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/funktionen.tex
@@ -0,0 +1,78 @@
+%
+% funktionen.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\knoten#1#2{
+	\draw #1 circle[radius=0.25];
+	\node at #1 {$#2$};
+}
+\def\kante#1#2{
+	\draw[shorten >= 0.25cm,shorten <= 0.25cm] #1 -- #2;
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionen auf einem Graphen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein Graph $G=(V,E)$, eine Funktion auf dem Graphen ist
+\[
+f\colon V \to \mathbb{R} : v\mapsto f(v)
+\]
+Knoten: $V=\{1,\dots,n\}$
+\\
+\uncover<2->{%
+Vektorschreibweise
+\[
+f = \begin{pmatrix}
+f(1)\\f(2)\\\vdots\\f(n)
+\end{pmatrix}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Matrizen}
+Adjazenz-, Grad- und Laplace-Matrix operieren auf Funktionen auf Graphen:
+\[
+L
+=
+\begin{pmatrix*}[r]
+ 2&-1& 0&-1& 0\\
+-1& 4&-1&-1&-1\\
+ 0&-1& 2& 0&-1\\
+-1&-1& 0& 3&-1\\
+ 0&-1&-1&-1& 3\\
+\end{pmatrix*}
+\]
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{2}
+\coordinate (A) at (0,0);
+\coordinate (B) at (\a,0);
+\coordinate (C) at ({2*\a},0);
+\coordinate (D) at ({0.5*\a},{-0.5*sqrt(3)*\a});
+\coordinate (E) at ({1.5*\a},{-0.5*sqrt(3)*\a});
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+\kante{(A)}{(B)}
+\kante{(B)}{(C)}
+\kante{(A)}{(D)}
+\kante{(B)}{(D)}
+\kante{(B)}{(E)}
+\kante{(C)}{(E)}
+\kante{(D)}{(E)}
+\end{tikzpicture}
+\end{center}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/gundh.tex b/vorlesungen/slides/8/wavelets/gundh.tex
new file mode 100644
index 0000000..2d6c677
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/gundh.tex
@@ -0,0 +1,85 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\kurve#1#2{
+        \draw[color=#2,line width=1.4pt]
+                plot[domain=0:6.3,samples=400]
+                        ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Mutterwavelets + Dilatation}
+Eine Menge von Dilatationsfaktoren
+\[
+A= \{a_1,a_2,\dots,a_N\}
+\]
+wählen\uncover<2->{, und mit Funktionen
+\[
+{\color{blue}g_i} = \tilde{D}_{1/a_i}{\color{red}g}
+\]
+die Standardbasisvektoren filtern}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{
+\begin{block}{Vaterwavelets}
+Tiefpass mit Funktion ${\color{darkgreen}h(\lambda)}$,
+Standardbasisvektoren mit ${\color{darkgreen}h}$ filtern:
+\[
+D_{\color{darkgreen}h}e_k
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\kurve{1}{red}
+\uncover<4->{
+\foreach \k in {0,...,4}{
+        \pgfmathparse{0.30*exp(ln(2)*\k)}
+        \xdef\l{\pgfmathresult}
+        \kurve{\l}{blue}
+}
+}
+
+\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$};
+\uncover<4->{
+\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$};
+}
+
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+
+\begin{scope}[xshift=7cm]
+
+\uncover<6->{
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\draw[color=darkgreen,line width=1.4pt]
+        plot[domain=0:6.3,samples=100]
+                ({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/laplacebasis.tex b/vorlesungen/slides/8/wavelets/laplacebasis.tex
new file mode 100644
index 0000000..ced4c09
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/laplacebasis.tex
@@ -0,0 +1,62 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\a{2}
+\def\b{0.8}
+\def\c{1}
+\def\d{0.6}
+\input{../slides/8/wavelets/vektoren.tex}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Laplace-Basis}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\begin{scope}[yshift=-0.4cm,xshift=-5.5cm]
+\fnull
+\end{scope}
+
+\begin{scope}[yshift=-1.8cm,xshift=-5.5cm]
+\fone
+\end{scope}
+
+\begin{scope}[yshift=-3.2cm,xshift=-5.5cm]
+\ftwo
+\end{scope}
+
+\begin{scope}[yshift=-4.6cm,xshift=-5.5cm]
+\fthree
+\end{scope}
+
+\begin{scope}[yshift=-6.0cm,xshift=-5.5cm]
+\ffour
+\end{scope}
+
+\begin{scope}[yshift=0cm]
+\vnull
+\end{scope}
+
+\begin{scope}[yshift=-1.4cm]
+\vone
+\end{scope}
+
+\begin{scope}[yshift=-2.8cm]
+\vtwo
+\end{scope}
+
+\begin{scope}[yshift=-4.2cm]
+\vthree
+\end{scope}
+
+\begin{scope}[yshift=-5.6cm]
+\vfour
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex
new file mode 100644
index 0000000..d6575d0
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex
@@ -0,0 +1,46 @@
+%
+% lokalisierungsvergleich.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lokalisierung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ortsraum}
+Ortsraum$\mathstrut=V$
+\begin{itemize}
+\item<3-> Standardbasis
+\item<5-> lokalisiert in den Knoten
+\item<7-> die meisten $\hat{f}(k)$ gross
+\item<9-> vollständig delokalisiert im Frequenzraum
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Frequenzraum}
+\uncover<2->{Frequenzraum $\mathstrut=\{\lambda_1,\lambda_2,\dots,\lambda_n\}$}
+\begin{itemize}
+\item<4-> Laplace-Basis
+\item<6-> lokalisiert in den Eigenwerten
+\item<8-> die meisten Komponenten gross
+\item<10-> vollständig delokalisiert im Ortsraum
+\end{itemize}
+\end{block}
+\end{column}
+\end{columns}
+\uncover<11->{%
+\begin{block}{Plan}
+Gesucht sind Funktionen auf dem Graphen derart, die
+\begin{enumerate}
+\item<12-> in der Nähe einzelner Knoten konzentriert/lokalisiert sind und
+\item<13-> deren Fourier-Transformation in der Nähe einzelner Eigenwerte 
+konzentriert/lokalisiert ist
+\end{enumerate}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/matrixdilatation.tex b/vorlesungen/slides/8/wavelets/matrixdilatation.tex
new file mode 100644
index 0000000..3536736
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/matrixdilatation.tex
@@ -0,0 +1,39 @@
+%
+% matrixdilatation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Dilatation in Matrixform}
+Dilatationsfaktor $a$, skaliertes Wavelet beim Knoten $k$ mit Spektrum
+$\tilde{D}_{1/a}g$
+\begin{align*}
+D_{g,a}e_k
+&=
+\chi
+\begin{pmatrix}
+g(a\lambda_1)&      0      & \dots  &      0      \\
+      0      &g(a\lambda_2)& \dots  &      0      \\
+   \vdots    &    \vdots   & \ddots &    \vdots   \\
+      0      &      0      & \dots  &g(a\lambda_n)
+\end{pmatrix}
+\chi^t
+e_k
+\intertext{\uncover<2->{``verschmierter'' Standardbasisvektor am Knoten $k$}}
+\uncover<2->{D_he_k
+&=
+\chi
+\begin{pmatrix}
+h(\lambda_1)&     0      & \dots  &     0      \\
+     0      &h(\lambda_2)& \dots  &     0      \\
+  \vdots    &   \vdots   & \ddots &   \vdots   \\
+     0      &     0      & \dots  &h(\lambda_n)
+\end{pmatrix}
+\chi^t
+e_k}
+\end{align*}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/vektoren.tex b/vorlesungen/slides/8/wavelets/vektoren.tex
new file mode 100644
index 0000000..2315d53
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/vektoren.tex
@@ -0,0 +1,200 @@
+\def\vnull{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.0000$};
+	\fill[color=red!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=red!100] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=red!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fnull{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.4472*\d}) --
+		({-1.0000*\c},{0.4472*\d}) --
+		({0.0000*\c},{0.4472*\d}) --
+		({1.0000*\c},{0.4472*\d}) --
+		({2.0000*\c},{0.4472*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vone{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.1586$};
+	\fill[color=blue!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!41] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!41] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fone{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{-0.6533*\d}) --
+		({-1.0000*\c},{-0.2706*\d}) --
+		({0.0000*\c},{-0.0000*\d}) --
+		({1.0000*\c},{0.2706*\d}) --
+		({2.0000*\c},{0.6533*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vtwo{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.3000$};
+	\fill[color=red!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=blue!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\ftwo{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.5000*\d}) --
+		({-1.0000*\c},{-0.5000*\d}) --
+		({0.0000*\c},{-0.0000*\d}) --
+		({1.0000*\c},{-0.5000*\d}) --
+		({2.0000*\c},{0.5000*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vthree{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.4414$};
+	\fill[color=red!41] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=red!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=blue!41] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fthree{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.2706*\d}) --
+		({-1.0000*\c},{-0.6533*\d}) --
+		({0.0000*\c},{0.0000*\d}) --
+		({1.0000*\c},{0.6533*\d}) --
+		({2.0000*\c},{-0.2706*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vfour{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.5000$};
+	\fill[color=red!25] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!100] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!25] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=red!25] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!25] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\ffour{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.2236*\d}) --
+		({-1.0000*\c},{0.2236*\d}) --
+		({0.0000*\c},{-0.8944*\d}) --
+		({1.0000*\c},{0.2236*\d}) --
+		({2.0000*\c},{0.2236*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
diff --git a/vorlesungen/slides/8/weitere.tex b/vorlesungen/slides/8/weitere.tex
new file mode 100644
index 0000000..46a3da0
--- /dev/null
+++ b/vorlesungen/slides/8/weitere.tex
@@ -0,0 +1,43 @@
+%
+% weitere.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Weitere Resultate der spektralen Graphentheorie}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Hoffmann)}
+\[
+\operatorname{chr} X \ge 1 + \frac{\alpha_{\text{max}}}{-\alpha_{\text{min}}}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Satz (Hoffmann)}
+\[
+\operatorname{ind} X \le n \biggl(1-\frac{d_{\text{min}}}{\lambda_{\text{max}}}\biggr)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Korollar}
+Für einen regulären Graphen mit $n$ Knoten gilt
+\begin{align*}
+\operatorname{ind} X
+&\le
+\frac{n}{\displaystyle 1-\frac{d}{\alpha_{\text{min}}}}
+\\
+\operatorname{chr} X
+&\ge
+1-\frac{d}{\alpha_{\text{min}}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wilf.m b/vorlesungen/slides/8/wilf.m
new file mode 100644
index 0000000..49dc161
--- /dev/null
+++ b/vorlesungen/slides/8/wilf.m
@@ -0,0 +1,22 @@
+#
+# wilf.m -- chromatische Zahl für einen Graphen
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+N = 9;
+A = zeros(N,N);
+
+for i = (1:N)
+	j = 1 + rem(i, N)
+	A(i,j) = 1;
+endfor
+for i = (1:3:N-3)
+	j = 1 + rem(i + 2, N)
+	A(i,j) = 1;
+endfor
+
+A(1,3) = 1;
+
+A = A + A'
+
+eig(A)
diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc
index fa6c29b..2257810 100644
--- a/vorlesungen/slides/9/Makefile.inc
+++ b/vorlesungen/slides/9/Makefile.inc
@@ -10,5 +10,20 @@ chapter9 =								\
 	../slides/9/irreduzibel.tex					\
 	../slides/9/stationaer.tex					\
 	../slides/9/pf.tex						\
+	../slides/9/potenz.tex						\
+	../slides/9/pf/positiv.tex					\
+	../slides/9/pf/primitiv.tex					\
+	../slides/9/pf/trennung.tex					\
+	../slides/9/pf/vergleich.tex					\
+	../slides/9/pf/vergleich3d.tex					\
+	../slides/9/pf/dreieck.tex					\
+	../slides/9/pf/folgerungen.tex					\
+	../slides/9/parrondo/uebersicht.tex				\
+	../slides/9/parrondo/erwartung.tex				\
+	../slides/9/parrondo/spiela.tex					\
+	../slides/9/parrondo/spielb.tex					\
+	../slides/9/parrondo/spielbmod.tex				\
+	../slides/9/parrondo/kombiniert.tex				\
+	../slides/9/parrondo/deformation.tex				\
 	../slides/9/chapter.tex
 
diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex
index 9e26587..cbab0f0 100644
--- a/vorlesungen/slides/9/chapter.tex
+++ b/vorlesungen/slides/9/chapter.tex
@@ -10,5 +10,21 @@
 \folie{9/stationaer.tex}
 \folie{9/irreduzibel.tex}
 \folie{9/pf.tex}
+\folie{9/potenz.tex}
 
+\folie{9/pf/positiv.tex}
+\folie{9/pf/primitiv.tex}
+\folie{9/pf/trennung.tex}
+\folie{9/pf/vergleich.tex}
+\folie{9/pf/vergleich3d.tex}
+\folie{9/pf/dreieck.tex}
+\folie{9/pf/folgerungen.tex}
+
+\folie{9/parrondo/uebersicht.tex}
+\folie{9/parrondo/erwartung.tex}
+\folie{9/parrondo/spiela.tex}
+\folie{9/parrondo/spielb.tex}
+\folie{9/parrondo/spielbmod.tex}
+\folie{9/parrondo/kombiniert.tex}
+\folie{9/parrondo/deformation.tex}
 
diff --git a/vorlesungen/slides/9/parrondo/deformation.tex b/vorlesungen/slides/9/parrondo/deformation.tex
new file mode 100644
index 0000000..40d2eb9
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/deformation.tex
@@ -0,0 +1,45 @@
+%
+% deformation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Deformation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Verlustspiele}
+Durch Deformation (Parameter $e$ und $\varepsilon$) kann man
+aus $A_e$ und $B_\varepsilon$ Spiele mit negativer Gewinnerwartung machen
+\uncover<2->{%
+\begin{align*}
+E(X)&=0&&\rightarrow&E(X_e)&<0\\
+E(Y)&=0&&\rightarrow&E(Y_\varepsilon)&<0\\
+\end{align*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Kombiniertes Spiel}
+\uncover<3->{%
+Die Deformation für das Spiel $C$ startet mit Erwartungswert $\frac{18}{709}$}%
+\begin{align*}
+\uncover<4->{E(Z)&=\frac{18}{709}>0}
+&&\uncover<5->{\rightarrow&
+E(Z_*)&>0}
+\end{align*}
+\uncover<6->{Wegen Stetigkeit!}
+\\
+\uncover<5->{Die Deformation ist immer noch ein Gewinnspiel (für Parameter klein genug)}
+\end{block}
+\uncover<7->{%
+\begin{block}{Parrondo-Paradoxon}
+Zufällig zwischen zwei Verlustspielen auswählen kann trotzdem ein
+Gewinnspiel ergeben
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/erwartung.tex b/vorlesungen/slides/9/parrondo/erwartung.tex
new file mode 100644
index 0000000..b58c37f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/erwartung.tex
@@ -0,0 +1,81 @@
+%
+% erwartung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erwartung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zufallsvariable}
+\begin{center}
+\[
+\begin{array}{c|c}
+\text{Werte $X$}&\text{Wahrscheinlichkeit $p$}\\
+\hline
+x_1&p_1=P(X=x_1)\\
+x_2&p_2=P(X=x_2)\\
+\vdots&\vdots\\
+x_n&p_n=P(X=x_n)
+\end{array}
+\]
+\end{center}
+\end{block}
+\uncover<4->{%
+\begin{block}{Einervektoren/-matrizen}
+\[
+U=\begin{pmatrix}
+1&1&\dots&1\\
+1&1&\dots&1\\
+\vdots&\vdots&\ddots&\vdots\\
+1&1&\dots&1
+\end{pmatrix}
+\in
+M_{n\times m}(\Bbbk)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Erwartungswerte}
+\begin{align*}
+E(X)
+&=
+\sum_i x_ip_i
+=
+x^tp
+\uncover<5->{=
+U^t x\odot p}
+\hspace*{3cm}
+\\
+\uncover<2->{E(X^2)
+&=
+\sum_i x_i^2p_i}
+\ifthenelse{\boolean{presentation}}{
+\only<6>{=
+(x\odot x)^tp}}{}
+\uncover<7->{=
+U^t (x\odot x) \odot p}
+\\
+\uncover<3->{E(X^k)
+&=
+\sum_i x_i^kp_i}
+\uncover<8->{=
+U^t x^{\odot k}\odot p}
+\end{align*}
+\uncover<9->{%
+Substitution:
+\begin{align*}
+\uncover<10->{\sum_i &\to U^t}\\
+\uncover<11->{x_i^k &\to x^{\odot k}}
+\end{align*}}%
+\uncover<12->{Kann für Übergangsmatrizen von Markov-Ketten verallgemeinert werden}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/kombiniert.tex b/vorlesungen/slides/9/parrondo/kombiniert.tex
new file mode 100644
index 0000000..5012d06
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/kombiniert.tex
@@ -0,0 +1,73 @@
+%
+% kombiniert.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kombiniertes Spiel $C$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein fairer Münzwurf entscheidet, ob
+Spiel $A$ oder Spiel $B$ gespielt wird
+\end{block}
+\uncover<2->{%
+\begin{block}{Übergangsmatrix}
+Münzwurf $X$
+\begin{align*}
+C
+&=
+P(X=\text{Kopf})\cdot A
++
+P(X=\text{Zahl})\cdot B
+\\
+&\uncover<3->{=
+\begin{pmatrix}
+           0&\frac{3}{8}&\frac{5}{8}\\
+\frac{3}{10}&          0&\frac{3}{8}\\
+\frac{7}{10}&\frac{5}{8}&          0
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-8pt}
+\uncover<4->{%
+\begin{block}{Gewinnerwartung im Einzelspiel}
+\[
+p=\frac13U
+\Rightarrow
+U^t(G\odot C)p
+\uncover<5->{=
+-\frac{1}{30}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Iteriertes Spiel}
+\[
+\overline{p}=C\overline{p}
+\quad
+\uncover<7->{\Rightarrow
+\quad
+\overline{p}=\frac{1}{709}\begin{pmatrix}245\\180\\284\end{pmatrix}}
+\]
+\end{block}}
+\uncover<8->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(Z)
+&=
+U^t (G\odot C) \overline{p}
+\uncover<9->{=
+\frac{18}{709}}
+\end{align*}
+\uncover<10->{$C$ ist ein Gewinnspiel!}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spiela.tex b/vorlesungen/slides/9/parrondo/spiela.tex
new file mode 100644
index 0000000..629586f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spiela.tex
@@ -0,0 +1,52 @@
+%
+% spiela.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spiel $A$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn = Zufallsvariable $X$ mit Werten $\pm 1$
+\begin{align*}
+P(X=\phantom{+}1)
+&=
+\frac12\uncover<2->{+e}
+\\
+P(X=         - 1)
+&=
+\frac12\uncover<2->{-e}
+\end{align*}
+Bernoulli-Experiment mit $p=\frac12\uncover<2->{+e}$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(X)
+&=\uncover<4->{
+P(X=1)\cdot (1)}
+\\
+&\qquad
+\uncover<4->{+
+P(X=-1)\cdot (-1)}
+\\
+&\uncover<5->{=
+\biggl(\frac12+e\biggr)\cdot 1
++
+\biggl(\frac12-e\biggr)\cdot (-1)}
+\\
+&\uncover<6->{=2e}
+\end{align*}
+\uncover<7->{$\Rightarrow$ {\usebeamercolor[fg]{title}Verlustspiel für $e<0$}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spielb.tex b/vorlesungen/slides/9/parrondo/spielb.tex
new file mode 100644
index 0000000..f65564f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielb.tex
@@ -0,0 +1,100 @@
+%
+% spielb.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spiel $B$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des
+aktuellen Kapitals $K$:
+\begin{center}
+\uncover<2->{%
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A0) at (90:2);
+\coordinate (A1) at (210:2);
+\coordinate (A2) at (330:2);
+
+\node at (A0) {$0$};
+\node at (A1) {$1$};
+\node at (A2) {$2$};
+
+\draw (A0) circle[radius=0.4];
+\draw (A1) circle[radius=0.4];
+\draw (A2) circle[radius=0.4];
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2);
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1);
+
+\def\R{1.9}
+\def\r{0.7}
+
+\node at (30:\r) {$\frac{9}{10}$};
+\node at (150:\r) {$\frac1{10}$};
+\node at (270:\r) {$\frac34$};
+
+\node at (30:\R) {$\frac{3}{4}$};
+\node at (150:\R) {$\frac1{4}$};
+\node at (270:\R) {$\frac14$};
+
+\end{tikzpicture}}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Markov-Kette $Y$}
+Übergangsmatrix
+\[
+B=\begin{pmatrix}
+0&\frac14&\frac34\\
+\frac{1}{10}&0&\frac14\\
+\frac{9}{10}&\frac34&0
+\end{pmatrix}
+\]
+\vspace{-10pt}
+
+\uncover<4->{%
+Gewinnmatrix:
+\vspace{-2pt}
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]}
+\end{block}}
+\vspace{-12pt}
+\uncover<5->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+&&&&
+E(Y)
+&=
+U^t(G\odot B)p
+\\
+p&={\textstyle\frac13}U
+&&\Rightarrow&
+E(Y)&={\textstyle\frac1{15}}
+\\
+\overline{p}&={\tiny\frac{1}{13}\begin{pmatrix}5\\2\\6\end{pmatrix}}
+&&\Rightarrow&
+E(Y)&=0
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex
new file mode 100644
index 0000000..66d39bc
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielbmod.tex
@@ -0,0 +1,103 @@
+%
+% spielb.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Modifiziertes Spiel $\tilde{B}$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des
+aktuellen Kapitals $K$:
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A0) at (90:2);
+\coordinate (A1) at (210:2);
+\coordinate (A2) at (330:2);
+
+\node at (A0) {$0$};
+\node at (A1) {$1$};
+\node at (A2) {$2$};
+
+\draw (A0) circle[radius=0.4];
+\draw (A1) circle[radius=0.4];
+\draw (A2) circle[radius=0.4];
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2);
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1);
+
+\def\R{1.9}
+\def\r{0.7}
+
+\node at (30:{0.9*\r}) {\tiny $\frac{9}{10}\uncover<2->{+\varepsilon}$};
+\node at (150:{0.9*\r}) {\tiny $\frac1{10}\uncover<2->{-\varepsilon}$};
+\node at (270:\r) {$\frac34\uncover<2->{-\varepsilon}$};
+
+\node at (30:{1.1*\R}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$};
+\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$};
+\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Markov-Kette $\tilde{Y}$}
+Übergangsmatrix
+\[
+\tilde{B}=
+B\uncover<2->{+\varepsilon F}
+\uncover<3->{=
+B+\varepsilon\begin{pmatrix*}[r]
+0&1&-1\\
+-1&0&1\\
+1&-1&0
+\end{pmatrix*}}
+\]
+\vspace{-12pt}
+
+\uncover<4->{%
+Gewinnmatrix:
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]}
+\end{block}
+\vspace{-12pt}
+\uncover<5->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+\uncover<6->{E(\tilde{Y})
+&=
+U^t(G\odot \tilde{B})p}
+\\
+&\uncover<7->{=
+E(Y) + \varepsilon U^t(G\odot F)p}
+\uncover<8->{=
+{\textstyle\frac1{15}}+2\varepsilon}
+\\
+\uncover<9->{
+\text{rep.}
+&=
+-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2)
+\quad\text{Verlustspiel}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/uebersicht.tex b/vorlesungen/slides/9/parrondo/uebersicht.tex
new file mode 100644
index 0000000..2f3597a
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/uebersicht.tex
@@ -0,0 +1,17 @@
+%
+% uebersicht.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Parrondo-Paradoxon}
+\begin{center}
+\Large
+Zufällige
+Wahl zwischen zwei Verlustspielen = Gewinnspiel?
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/dreieck.tex b/vorlesungen/slides/9/pf/dreieck.tex
new file mode 100644
index 0000000..0a572f3
--- /dev/null
+++ b/vorlesungen/slides/9/pf/dreieck.tex
@@ -0,0 +1,44 @@
+%
+% dreieck.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Verallgemeinerte Dreiecksungleichung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.32\textwidth}
+\begin{block}{Satz}
+\[
+|u+v|\le |u|+|v|
+\]
+Gleichheit wenn lin.~abh.
+\end{block}
+\begin{block}{Satz}
+\[
+\biggl|\sum_i u_i\biggr|
+\le
+\sum_i |u_i|
+\]
+Gleichheit wenn $u_i = \lambda_i u$
+\end{block}
+\begin{block}{Satz}
+\[
+\biggl|\sum_i z_i\biggr|
+\le
+\sum_i |z_i|
+\]
+Gleichheit, wenn $z_i=|z_i|c$, $c\in\mathbb{C}$
+\end{block}
+\end{column}
+\begin{column}{0.68\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/folgerungen.tex b/vorlesungen/slides/9/pf/folgerungen.tex
new file mode 100644
index 0000000..5042c78
--- /dev/null
+++ b/vorlesungen/slides/9/pf/folgerungen.tex
@@ -0,0 +1,203 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Folgerungen für $A>0$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+$u\ge 0$ ein EV zum EW $ \lambda\ne 0$,
+dann ist $u>0$ und $\lambda >0$
+\end{block}
+\uncover<6->{%
+\begin{block}{Satz}
+$v$ ein EV zum EW $\lambda$ mit $|\lambda| = \varrho(A)$,
+dann ist $u=|v|$ mit $u_i=|v_i|$ ein EV mit EW $\varrho(A)$
+\end{block}}
+\uncover<29->{%
+\begin{block}{Satz}
+$v$ ein EV zum EW $\lambda$ mit $|\lambda|=\varrho(A)$,
+dann ist $\lambda=\varrho(A)$
+\end{block}}
+\uncover<46->{%
+\begin{block}{Satz}
+Der \only<57->{verallgemeinerte }Eigenraum zu EW $\varrho(A)$
+ist eindimensional
+\end{block}
+}
+\end{column}
+\ifthenelse{\boolean{presentation}}{
+\only<-6>{
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<3->
+Vergleich: $Au>0$
+\item<4->
+$Au=\lambda u > 0$
+\item<5->
+$\lambda >0$ und $u>0$
+\end{itemize}
+\end{proof}
+\end{column}}
+\only<7-20>{
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis]
+\begin{align*}
+(Au)_i
+&\only<-8>{=
+\sum_j a_{ij}u_j}
+\only<8-9>{=
+\sum_j |a_{ij}v_j|}
+\only<9->{\ge}
+\only<9-10>{
+\biggl|\sum_j a_{ij}v_j\biggr|}
+\only<10>{=}
+\only<10-11>{
+|(Av)_i|}
+\only<11>{=}
+\only<11-12>{
+|\lambda v_i|}
+\only<12>{=}
+\only<12-13>{
+\varrho(A) |v_i|}
+\only<13>{=}
+\uncover<13->{
+\varrho(A) u_i}
+\hspace*{5cm}
+\\
+\uncover<14->{Au&\ge \varrho(A)u}
+\intertext{\uncover<15->{Vergleich}}
+\uncover<16->{A^2u&> \varrho(A)Au}
+\intertext{\uncover<17->{Trennung: $\exists \vartheta >1$ mit}}
+\uncover<18->{A^2u&\ge \vartheta \varrho(A) Au }\\
+\uncover<19->{A^3u&\ge (\vartheta \varrho(A))^2 Au }\\
+\uncover<20->{A^ku&\ge (\vartheta \varrho(A))^{k-1} Au }\\
+\end{align*}
+\end{proof}
+\end{column}}
+\only<21-29>{%
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis, Fortsetzung]
+Abschätzung der Operatornorm:
+\begin{align*}
+\|A^k\|\, |Au|
+\ge
+\|A^{k+1}u\|
+\uncover<22->{
+\ge
+(\vartheta\varrho(A))^k |Au|}
+\end{align*}
+\uncover<23->{Abschätzung des Spektralradius}
+\begin{align*}
+\uncover<24->{\|A^k\| &\ge (\vartheta\varrho(A))^k}
+\\
+\uncover<25->{\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)}
+\\
+\uncover<26->{\lim_{k\to\infty}\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)}
+\\
+\uncover<27->{\varrho(A) &\ge \underbrace{\vartheta}_{>1} \varrho(A)}
+\end{align*}
+\uncover<28->{Widerspruch: $u=v$}
+\end{proof}
+\end{column}}
+\only<30-46>{
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis]
+$u$ ist EV mit EW $\varrho(A)$:
+\[
+Au=\varrho(A)u
+\uncover<31->{\Rightarrow
+\sum_j a_{ij}|v_j| = {\color<38->{red}\varrho(A) |v_i|}}
+\]
+\uncover<33->{Andererseits: $Av=\lambda v$}
+\[
+\uncover<34->{\sum_{j}a_{ij}v_j=\lambda v_i}
+\]
+\uncover<35->{Betrag}
+\begin{align*}
+\uncover<36->{\biggl|\sum_j a_{ij}v_j\biggr|
+&=
+|\lambda v_i|}
+\uncover<37->{=
+{\color<38->{red}\varrho(A) |v_i|}}
+\uncover<39->{=
+\sum_j a_{ij}|v_j|}
+\end{align*}
+\uncover<40->{Dreiecksungleichung: $v_j=|v_j|c, c\in\mathbb{C}$}
+\[
+\uncover<41->{\lambda v = Av}
+\uncover<42->{= Acu}
+\uncover<43->{= c\varrho(A) u}
+\uncover<44->{= \varrho(A)v}
+\]
+\uncover<45->{$\Rightarrow
+\lambda=\varrho(A)
+$}
+\end{proof}
+\end{column}}
+\only<47-57>{
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<48-> $u>0$ ein EV zum EW $\varrho(A)$
+\item<49-> $v$ ein weiterer EV, man darf $v\in\mathbb{R}^n$ annehmen
+\item<50-> Da $u>0$ gibt es $c>0$ mit $u\ge cv$ aber $u\not > cv$
+\item<51-> $u-cv\ge 0$ aber $u-cv\not > 0$
+\item<52-> $A$ anwenden:
+\[
+\begin{array}{ccc}
+\uncover<53->{A(u-cv)}&\uncover<54->{>&0}
+\\
+\uncover<53->{\|}&&
+\\
+\uncover<53->{\varrho(A)(u-cv)}&\uncover<55->{\not>&0}
+\end{array}
+\]
+\uncover<56->{Widerspruch: $v$ existiert nicht}
+\end{itemize}
+\end{proof}
+\end{column}}
+\only<58->{
+\begin{column}{0.48\textwidth}
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<59-> $Au=\varrho(A)u$ und $A^tp^t=\varrho(A)p^t$
+\item<60-> $u>0$ und $p>0$ $\Rightarrow$ $up>0$
+\item<61-> $px=0$, dann ist
+\[
+\uncover<62->{pAx}
+\only<62-63>{=
+(A^tp^t)^t x}
+\only<63-64>{=
+\varrho(A) (p^t)^t x}
+\uncover<64->{=
+\varrho(A) px}
+\uncover<65->{= 0}
+\]
+\uncover<66->{also ist $\{x\in\mathbb{R}^n\;|\; px=0\}$
+invariant}
+\item<67-> Annahme: $v\in \mathcal{E}_{\varrho(A)}$
+\item<68-> Dann muss es einen EV zum EW $\varrho(A)$ in 
+$\mathcal{E}_{\varrho(A)}$ geben
+\item<69-> Widerspruch: der Eigenraum ist eindimensional
+\end{itemize}
+\end{proof}
+\end{column}}
+}{
+\begin{column}{0.48\textwidth}
+\begin{block}{}
+\usebeamercolor[fg]{title}
+Beweise: Buch Abschnitt 9.3 
+\end{block}
+\end{column}
+}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/positiv.tex b/vorlesungen/slides/9/pf/positiv.tex
new file mode 100644
index 0000000..d7e833d
--- /dev/null
+++ b/vorlesungen/slides/9/pf/positiv.tex
@@ -0,0 +1,64 @@
+%
+% positiv.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Positive und nichtnegative Matrizen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Positive Matrix\strut}
+Eine Matrix $A$ heisst positiv, wenn
+\[
+a_{ij} > 0\quad\forall i,j
+\]
+Man schreibt $A>0\mathstrut$
+\end{block}
+\uncover<2->{%
+\begin{block}{Relation $>\mathstrut$}
+Man schreibt $A>B$ wenn $A-B > 0\mathstrut$
+\end{block}}
+\uncover<5->{%
+\begin{block}{Wahrscheinlichkeitsmatrix}
+\[
+W=\begin{pmatrix}
+0.7&0.2&0.1\\
+0.2&0.6&0.1\\
+0.1&0.2&0.8
+\end{pmatrix}
+\]
+Spaltensumme$\mathstrut=1$, Zeilensumme$\mathstrut=?$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Nichtnegative Matrix\strut}
+Eine Matrix $A$ heisst nichtnegativ, wenn
+\[
+a_{ij} \ge 0\quad\forall i,j
+\]
+Man schreibt $A\ge 0\mathstrut$
+\end{block}}
+\uncover<4->{%
+\begin{block}{Relation $\ge\mathstrut$}
+Man schreibt $A\ge B$ wenn $A-B \ge 0\mathstrut$
+\end{block}}
+\uncover<6->{%
+\begin{block}{Permutationsmatrix}
+\[
+P=\begin{pmatrix}
+0&0&1\\
+1&0&0\\
+0&1&0
+\end{pmatrix}
+\]
+Genau eine $1$ in jeder Zeile/Spalte
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/primitiv.tex b/vorlesungen/slides/9/pf/primitiv.tex
new file mode 100644
index 0000000..961b1d5
--- /dev/null
+++ b/vorlesungen/slides/9/pf/primitiv.tex
@@ -0,0 +1,84 @@
+%
+% primitiv.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Primitive Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+$A\ge 0$ heisst primitiv, wenn es ein $n>0$ gibt mit $A^n>0$
+\end{block}
+\uncover<9->{%
+\begin{block}{Intuition}
+\begin{itemize}
+\item<10->
+Markov-Ketten: $a_{ij} > 0$ bedeutet, $i$ von $j$ aus erreichbar.
+\item<11->
+Band: {\em alle} Verbindung mit allen Nachbarn
+\item<12->
+$n$-te Potenz: Pfade der Länge $n$ 
+\item<13->
+Durchmesser: wenn $n>\text{Durchmesser des Zustandsdiagramms}$,
+dann ist $A^n>0$
+\end{itemize}
+\end{block}
+}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Beispiel: Reduzible W'keitsmatrix}
+\vspace{-5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\fill[color=gray!40] (-1,0) rectangle (0,1);
+\fill[color=gray!40] (0,-1) rectangle (1,0);
+\draw[line width=0.3pt] (0,-1) -- (0,1);
+\draw[line width=0.3pt] (-1,0) -- (1,0);
+%\draw (-1,-1) rectangle (1,1);
+\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$};
+\node at (-1.3,0) [left] {$\mathstrut W=$};
+\node at (0.5,0.5) {$0$};
+\node at (-0.5,-0.5) {$0$};
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+
+$\Rightarrow$ $W$ ist nicht primitiv
+\end{block}}
+\uncover<3->{%
+\begin{block}{Beispiel: Bandmatrix}
+\centering
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+\clip (-1,-1) rectangle (1,1);
+\foreach \n in {3,...,8}{
+	\pgfmathparse{0.3*(\n-2)}
+	\xdef\x{\pgfmathresult}
+	\only<\n>{
+		\fill[color=gray!40]
+			({-1.2-\x},1) -- (1,{-1.2-\x}) -- (1,{-0.8+\x})
+			-- ({-0.8+\x},1) -- cycle;
+	}
+}
+\fill[color=gray] (-1.2,1) -- (1,-1.2) -- (1,-0.8) -- (-0.8,1) -- cycle;
+\end{scope}
+\foreach \n in {2,...,8}{
+	\uncover<\n>{
+		\pgfmathparse{int(\n-2)}
+		\xdef\k{\pgfmathresult}
+		\node at (-1.3,0) [left] {$\mathstrut B^{\k}=$};
+	}
+}
+\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$};
+\end{tikzpicture}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/trennung.tex b/vorlesungen/slides/9/pf/trennung.tex
new file mode 100644
index 0000000..9c85849
--- /dev/null
+++ b/vorlesungen/slides/9/pf/trennung.tex
@@ -0,0 +1,99 @@
+%
+% trennung.tex -- slide template
+%
+% (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{Trennung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (u) at (3.5,4.5);
+\coordinate (v) at (2.5,2);
+\coordinate (va) at ({(3.5/2.5)*2.5},{(3.5/2.5)*2});
+
+\uncover<3->{
+\fill[color=darkgreen!20] (0,0) rectangle (5.3,5.3);
+\node[color=darkgreen] at (1.5,4.9) {$u\not\ge w$};
+\node[color=darkgreen] at (4.4,0.6) {$u\not\ge w$};
+}
+
+\uncover<5->{
+\begin{scope}
+\clip (0,0) rectangle (5.3,5.3);
+\draw[color=darkgreen] (0,0) -- ($3*(v)$);
+\end{scope}
+
+\node[color=darkgreen] at ($1.2*(va)$)
+	[below,rotate={atan(2/2.5)}] {$(1+\mu)v$};
+}
+
+\uncover<2->{
+	\fill[color=red!20] (0,0) rectangle (u);
+}
+
+\fill[color=red] (u) circle[radius=0.08];
+\node[color=red] at (u) [above right] {$u$};
+
+\uncover<4->{
+	\fill[color=blue!40,opacity=0.5] (0,0) rectangle (v);
+}
+
+\uncover<2->{
+	\fill[color=blue] (v) circle[radius=0.08];
+	\node[color=blue] at (v) [above] {$v$};
+}
+
+\uncover<4->{
+	\draw[color=blue] (0,0) -- (va);
+
+	\fill[color=blue] (va) circle[radius=0.08];
+	\node[color=blue] at (va) [above left] {$(1+\varepsilon)v$};
+}
+
+\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}];
+\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}];
+
+\uncover<2->{
+	\draw[->,color=red] (3.0,-0.2) -- (3.0,1.5);
+	\node[color=red] at (3.0,-0.2) [below]
+		{$\{w\in\mathbb{R}^n\;|\; w<u\}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+$u>v\ge 0$\uncover<4->{, dann gibt es $\varepsilon>0$ mit
+\[
+u\ge (1+\varepsilon)v
+\]}%
+\uncover<5->{und für $\mu>\varepsilon$ ist 
+\[
+u \not\ge (1+\mu)v 
+\]}
+\uncover<6->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<7->
+$u>v$ $\Rightarrow$ $u_i/v_i>1$ falls $v_i>0$
+\item<8->
+\[
+\vartheta = \min_{v_i\ne 0} \frac{u_i}{v_i} > 1
+\]
+\uncover<9->{$\varepsilon = \vartheta - 1$}
+\end{itemize}
+\end{proof}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/vergleich.tex b/vorlesungen/slides/9/pf/vergleich.tex
new file mode 100644
index 0000000..c1a1f7a
--- /dev/null
+++ b/vorlesungen/slides/9/pf/vergleich.tex
@@ -0,0 +1,113 @@
+%
+% vergleich.tex -- slide template
+%
+% (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{Vergleich}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\a{1.2}    \def\b{0.35}
+\def\c{0.5}    \def\d{1.25}
+\def\r{4}
+
+\coordinate (u) at (3.5,0);
+\coordinate (v) at (2.5,0);
+
+\coordinate (Au) at ({3.5*\a},{3.5*\c});
+\coordinate (Av) at ({2.5*\a},{2.5*\c});
+
+\uncover<2->{
+	\begin{scope}
+	\clip (0,0) rectangle (5,5);
+	\fill[color=red!20] (0,0) circle[radius=4];
+	\end{scope}
+	\node[color=red] at (0,4) [below right] {$\mathbb{R}^n$};
+
+	\fill[color=blue!40,opacity=0.5] (0,0) -- ({\a*\r},{\c*\r})
+		-- plot[domain=0:90,samples=100]
+		({\r*(\a*cos(\x)+\b*sin(\x))},{\r*(\c*cos(\x)+\d*sin(\x))})
+		-- ({\b*\r},{\d*\r}) -- cycle;
+	\node[color=blue] at ({\r*\b},{\r*\d}) [below right] {$A\mathbb{R}^n$};
+}
+
+\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}];
+\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}];
+
+\uncover<3->{
+	\fill[color=darkgreen!30,opacity=0.5]
+		(0,0) rectangle ({3.5*\a},{3.5*\c});
+	\draw[color=white,line width=0.7pt]
+		({3.5*\a},0) -- ({3.5*\a},{3.5*\c}) -- (0,{3.5*\c});
+}
+
+\uncover<2->{
+	\draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\a},{\r*\c});
+	\draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\b},{\r*\d});
+
+	\draw[->,color=red,line width=1.4pt] (0,0) -- (4,0);
+	\draw[->,color=red,line width=1.4pt] (0,0) -- (0,4);
+}
+
+\draw[color=darkgreen,line width=2pt] (u) -- (v);
+\fill[color=darkgreen] (u) circle[radius=0.08];
+\fill[color=darkgreen] (v) circle[radius=0.08];
+
+\node[color=darkgreen] at (u) [below right] {$u$};
+\node[color=darkgreen] at (v) [below left] {$v$};
+\node[color=darkgreen] at ($0.5*(u)+0.5*(v)$) [above] {$v\le u$};
+
+\uncover<3->{
+	\draw[color=darkgreen,line width=2pt] (Au) -- (Av);
+	\fill[color=darkgreen] (Au) circle[radius=0.08];
+	\fill[color=darkgreen] (Av) circle[radius=0.08];
+
+	\node[color=darkgreen] at (Au) [above left] {$Au$};
+	\node[color=darkgreen] at (Av) [above left] {$Av$};
+
+	\node[color=darkgreen] at ($0.5*(Au)+0.5*(Av)$)
+		[below,rotate={atan(\c/\a)}] {$Av<Au$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+$u\ge v\ge 0$ \uncover<2->{und $A > 0$}\uncover<3->{ $\Rightarrow$ $Au>Av$}
+\end{block}
+\uncover<4->{%
+\begin{block}{intuitiv}
+$A>0$ befördert $\ge$ zu $>$
+\end{block}}
+\uncover<5->{%
+\begin{proof}[Beweis]
+$d=u-v\ge 0$
+\begin{align*}
+(Ad)_i
+\uncover<6->{=
+\sum_{j}
+\underbrace{a_{ij}}_{>0}d_j}
+\uncover<7->{>
+0}
+\uncover<8->{\quad\Rightarrow\quad
+Au > Av}
+\end{align*}
+\uncover<7->{da mindestens ein $d_j>0$ ist}
+\end{proof}}
+\uncover<9->{%
+\begin{block}{Korollar}
+$A>0$ und $d\ge 0$ $\Rightarrow$ $Ad > 0$
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/vergleich3d.tex b/vorlesungen/slides/9/pf/vergleich3d.tex
new file mode 100644
index 0000000..1c019a6
--- /dev/null
+++ b/vorlesungen/slides/9/pf/vergleich3d.tex
@@ -0,0 +1,26 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Vergleich}
+
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.57\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.38\textwidth}
+\begin{block}{Satz}
+$u\ge v\ge 0$ $\Rightarrow$ $Au>Av$
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/potenz.tex b/vorlesungen/slides/9/potenz.tex
new file mode 100644
index 0000000..2c3afa3
--- /dev/null
+++ b/vorlesungen/slides/9/potenz.tex
@@ -0,0 +1,15 @@
+%
+% potenz.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzmethode}
+\begin{center}
+\includegraphics[width=0.9\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/positiv.pdf}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 17c8a28..4289c44 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,9 +3,26 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\folie{7/mannigfaltigkeit.tex}
-\folie{7/haar.tex}
-\folie{7/quaternionen.tex}
-\folie{7/qdreh.tex}
-\folie{7/ueberlagerung.tex}
-\folie{7/hopf.tex}
+%\folie{9/google.tex}
+%\folie{9/markov.tex}
+%\folie{9/stationaer.tex}
+%\folie{9/irreduzibel.tex}
+%\folie{9/pf.tex}
+
+%\folie{9/pf/positiv.tex}
+%\folie{9/pf/primitiv.tex}
+%\folie{9/pf/trennung.tex}
+%\folie{9/pf/vergleich.tex}
+%\folie{9/pf/vergleich3d.tex}
+%\folie{9/pf/dreieck.tex}
+%\folie{9/pf/folgerungen.tex}
+%\folie{9/potenz.tex}
+
+\folie{9/parrondo/erwartung.tex}
+%\folie{9/parrondo/uebersicht.tex}
+\folie{9/parrondo/spiela.tex}
+\folie{9/parrondo/spielb.tex}
+\folie{9/parrondo/spielbmod.tex}
+\folie{9/parrondo/kombiniert.tex}
+\folie{9/parrondo/deformation.tex}
+
diff --git a/vorlesungen/stream/countdown.html b/vorlesungen/stream/countdown.html
index 940e269..12f99ac 100644
--- a/vorlesungen/stream/countdown.html
+++ b/vorlesungen/stream/countdown.html
@@ -17,7 +17,7 @@ color: #990000;
 <body> 
 <div id="demo"></div> 
 <script> 
-var deadline = new Date("Mar 29, 2021 17:00:00").getTime(); 
+var deadline = new Date("May 17, 2021 17:00:00").getTime(); 
 var x = setInterval(function() { 
 var now = new Date().getTime(); 
 var t = deadline - now;