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-rw-r--r--vorlesungen/slides/2/Makefile.inc14
-rw-r--r--vorlesungen/slides/2/chapter.tex14
-rw-r--r--vorlesungen/slides/2/hilbertraum/adjungiert.tex83
-rw-r--r--vorlesungen/slides/2/hilbertraum/basis.tex65
-rw-r--r--vorlesungen/slides/2/hilbertraum/definition.tex63
-rw-r--r--vorlesungen/slides/2/hilbertraum/energie.tex67
-rw-r--r--vorlesungen/slides/2/hilbertraum/l2.tex61
-rw-r--r--vorlesungen/slides/2/hilbertraum/l2beispiel.tex82
-rw-r--r--vorlesungen/slides/2/hilbertraum/laplace.tex66
-rw-r--r--vorlesungen/slides/2/hilbertraum/plancherel.tex102
-rw-r--r--vorlesungen/slides/2/hilbertraum/qm.tex90
-rw-r--r--vorlesungen/slides/2/hilbertraum/riesz.tex76
-rw-r--r--vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex107
-rw-r--r--vorlesungen/slides/2/hilbertraum/sobolev.tex51
-rw-r--r--vorlesungen/slides/2/hilbertraum/spektral.tex91
-rw-r--r--vorlesungen/slides/2/hilbertraum/sturm.tex58
-rw-r--r--vorlesungen/slides/4/Makefile.inc72
-rw-r--r--vorlesungen/slides/4/chapter.tex62
-rw-r--r--vorlesungen/slides/4/galois/aufloesbarkeit.tex240
-rw-r--r--vorlesungen/slides/4/galois/automorphismus.tex236
-rw-r--r--vorlesungen/slides/4/galois/erweiterung.tex130
-rw-r--r--vorlesungen/slides/4/galois/images/Makefile24
-rw-r--r--vorlesungen/slides/4/galois/images/common.inc178
-rw-r--r--vorlesungen/slides/4/galois/images/wuerfel.pov18
-rw-r--r--vorlesungen/slides/4/galois/images/wuerfel2.pov18
-rw-r--r--vorlesungen/slides/4/galois/konstruktion.tex294
-rw-r--r--vorlesungen/slides/4/galois/quadratur.tex132
-rw-r--r--vorlesungen/slides/4/galois/radikale.tex138
-rw-r--r--vorlesungen/slides/4/galois/sn.tex174
-rw-r--r--vorlesungen/slides/4/galois/winkeldreiteilung.tex188
-rw-r--r--vorlesungen/slides/4/galois/wuerfel.tex128
-rw-r--r--vorlesungen/slides/7/Makefile.inc57
-rw-r--r--vorlesungen/slides/7/ableitung.tex136
-rw-r--r--vorlesungen/slides/7/algebraisch.tex230
-rw-r--r--vorlesungen/slides/7/bch.tex76
-rw-r--r--vorlesungen/slides/7/chapter.tex51
-rw-r--r--vorlesungen/slides/7/dg.tex184
-rw-r--r--vorlesungen/slides/7/drehanim.tex310
-rw-r--r--vorlesungen/slides/7/drehung.tex264
-rw-r--r--vorlesungen/slides/7/einparameter.tex186
-rw-r--r--vorlesungen/slides/7/exponentialreihe.tex24
-rw-r--r--vorlesungen/slides/7/haar.tex84
-rw-r--r--vorlesungen/slides/7/hopf.tex69
-rw-r--r--vorlesungen/slides/7/images/Makefile48
-rw-r--r--vorlesungen/slides/7/images/common.inc140
-rw-r--r--vorlesungen/slides/7/images/commutator.ini16
-rw-r--r--vorlesungen/slides/7/images/commutator.m222
-rw-r--r--vorlesungen/slides/7/images/commutator.pov118
-rw-r--r--vorlesungen/slides/7/images/drehung.inc142
-rw-r--r--vorlesungen/slides/7/images/interpolation.ini8
-rw-r--r--vorlesungen/slides/7/images/interpolation.m54
-rw-r--r--vorlesungen/slides/7/images/interpolation.pov10
-rw-r--r--vorlesungen/slides/7/images/rodriguez.pov236
-rw-r--r--vorlesungen/slides/7/images/test.pov7
-rw-r--r--vorlesungen/slides/7/integration.tex66
-rw-r--r--vorlesungen/slides/7/interpolation.tex112
-rw-r--r--vorlesungen/slides/7/kommutator.tex332
-rw-r--r--vorlesungen/slides/7/kurven.tex208
-rw-r--r--vorlesungen/slides/7/liealgbeispiel.tex78
-rw-r--r--vorlesungen/slides/7/liealgebra.tex170
-rw-r--r--vorlesungen/slides/7/logarithmus.tex82
-rw-r--r--vorlesungen/slides/7/mannigfaltigkeit.tex92
-rw-r--r--vorlesungen/slides/7/parameter.tex214
-rw-r--r--vorlesungen/slides/7/qdreh.tex110
-rw-r--r--vorlesungen/slides/7/quaternionen.tex74
-rw-r--r--vorlesungen/slides/7/semi.tex234
-rw-r--r--vorlesungen/slides/7/sl2.tex484
-rw-r--r--vorlesungen/slides/7/symmetrien.tex290
-rw-r--r--vorlesungen/slides/7/ueberlagerung.tex98
-rw-r--r--vorlesungen/slides/7/vektorlie.tex206
-rw-r--r--vorlesungen/slides/7/zusammenhang.tex99
-rw-r--r--vorlesungen/slides/8/Makefile.inc20
-rw-r--r--vorlesungen/slides/8/amax.tex86
-rw-r--r--vorlesungen/slides/8/chapter.tex21
-rw-r--r--vorlesungen/slides/8/chrind.tex231
-rw-r--r--vorlesungen/slides/8/chrindprop.tex62
-rw-r--r--vorlesungen/slides/8/chroma1.tex56
-rw-r--r--vorlesungen/slides/8/chrwilf.tex115
-rw-r--r--vorlesungen/slides/8/inzidenz.tex4
-rw-r--r--vorlesungen/slides/8/inzidenzd.tex4
-rw-r--r--vorlesungen/slides/8/produkt.tex2
-rw-r--r--vorlesungen/slides/8/spanningtree.tex4
-rw-r--r--vorlesungen/slides/8/subgraph.tex60
-rw-r--r--vorlesungen/slides/8/wavelets/Makefile8
-rw-r--r--vorlesungen/slides/8/wavelets/beispiel.tex44
-rw-r--r--vorlesungen/slides/8/wavelets/dilatation.tex62
-rw-r--r--vorlesungen/slides/8/wavelets/dilbei.tex46
-rw-r--r--vorlesungen/slides/8/wavelets/ev.m97
-rw-r--r--vorlesungen/slides/8/wavelets/fourier.tex86
-rw-r--r--vorlesungen/slides/8/wavelets/frame.tex66
-rw-r--r--vorlesungen/slides/8/wavelets/framekonstanten.tex71
-rw-r--r--vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex78
-rw-r--r--vorlesungen/slides/8/wavelets/funktionen.tex78
-rw-r--r--vorlesungen/slides/8/wavelets/gundh.tex85
-rw-r--r--vorlesungen/slides/8/wavelets/laplacebasis.tex62
-rw-r--r--vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex46
-rw-r--r--vorlesungen/slides/8/wavelets/matrixdilatation.tex39
-rw-r--r--vorlesungen/slides/8/wavelets/vektoren.tex200
-rw-r--r--vorlesungen/slides/8/weitere.tex43
-rw-r--r--vorlesungen/slides/8/wilf.m22
-rw-r--r--vorlesungen/slides/9/Makefile.inc15
-rw-r--r--vorlesungen/slides/9/chapter.tex16
-rw-r--r--vorlesungen/slides/9/parrondo/deformation.tex45
-rw-r--r--vorlesungen/slides/9/parrondo/erwartung.tex81
-rw-r--r--vorlesungen/slides/9/parrondo/kombiniert.tex73
-rw-r--r--vorlesungen/slides/9/parrondo/spiela.tex52
-rw-r--r--vorlesungen/slides/9/parrondo/spielb.tex100
-rw-r--r--vorlesungen/slides/9/parrondo/spielbmod.tex103
-rw-r--r--vorlesungen/slides/9/parrondo/uebersicht.tex17
-rw-r--r--vorlesungen/slides/9/pf/dreieck.tex44
-rw-r--r--vorlesungen/slides/9/pf/folgerungen.tex203
-rw-r--r--vorlesungen/slides/9/pf/positiv.tex64
-rw-r--r--vorlesungen/slides/9/pf/primitiv.tex84
-rw-r--r--vorlesungen/slides/9/pf/trennung.tex99
-rw-r--r--vorlesungen/slides/9/pf/vergleich.tex113
-rw-r--r--vorlesungen/slides/9/pf/vergleich3d.tex26
-rw-r--r--vorlesungen/slides/9/potenz.tex15
-rw-r--r--vorlesungen/slides/Makefile.inc42
-rw-r--r--vorlesungen/slides/test.tex51
119 files changed, 8627 insertions, 3157 deletions
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/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 1ab27fa..5aac429 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -1,36 +1,36 @@
-
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-chapter4 = \
- ../slides/4/ggt.tex \
- ../slides/4/euklidmatrix.tex \
- ../slides/4/euklidbeispiel.tex \
- ../slides/4/euklidtabelle.tex \
- ../slides/4/fp.tex \
- ../slides/4/division.tex \
- ../slides/4/gauss.tex \
- ../slides/4/dh.tex \
- ../slides/4/divisionpoly.tex \
- ../slides/4/euklidpoly.tex \
- ../slides/4/polynomefp.tex \
- ../slides/4/schieberegister.tex \
- ../slides/4/charakteristik.tex \
- ../slides/4/char2.tex \
- ../slides/4/frobenius.tex \
- ../slides/4/qundr.tex \
- ../slides/4/alpha.tex \
- ../slides/4/galois/erweiterung.tex \
- ../slides/4/galois/automorphismus.tex \
- ../slides/4/galois/konstruktion.tex \
- ../slides/4/galois/wuerfel.tex \
- ../slides/4/galois/winkeldreiteilung.tex \
- ../slides/4/galois/quadratur.tex \
- ../slides/4/galois/radikale.tex \
- ../slides/4/galois/aufloesbarkeit.tex \
- ../slides/4/galois/sn.tex \
- ../slides/4/chapter.tex
-
-
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter4 = \
+ ../slides/4/ggt.tex \
+ ../slides/4/euklidmatrix.tex \
+ ../slides/4/euklidbeispiel.tex \
+ ../slides/4/euklidtabelle.tex \
+ ../slides/4/fp.tex \
+ ../slides/4/division.tex \
+ ../slides/4/gauss.tex \
+ ../slides/4/dh.tex \
+ ../slides/4/divisionpoly.tex \
+ ../slides/4/euklidpoly.tex \
+ ../slides/4/polynomefp.tex \
+ ../slides/4/schieberegister.tex \
+ ../slides/4/charakteristik.tex \
+ ../slides/4/char2.tex \
+ ../slides/4/frobenius.tex \
+ ../slides/4/qundr.tex \
+ ../slides/4/alpha.tex \
+ ../slides/4/galois/erweiterung.tex \
+ ../slides/4/galois/automorphismus.tex \
+ ../slides/4/galois/konstruktion.tex \
+ ../slides/4/galois/wuerfel.tex \
+ ../slides/4/galois/winkeldreiteilung.tex \
+ ../slides/4/galois/quadratur.tex \
+ ../slides/4/galois/radikale.tex \
+ ../slides/4/galois/aufloesbarkeit.tex \
+ ../slides/4/galois/sn.tex \
+ ../slides/4/chapter.tex
+
+
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index 3015e7c..0691e39 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -1,31 +1,31 @@
-%
-% chapter.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
-%
-\folie{4/ggt.tex}
-\folie{4/euklidmatrix.tex}
-\folie{4/euklidbeispiel.tex}
-\folie{4/euklidtabelle.tex}
-\folie{4/fp.tex}
-\folie{4/division.tex}
-\folie{4/gauss.tex}
-\folie{4/dh.tex}
-\folie{4/divisionpoly.tex}
-\folie{4/euklidpoly.tex}
-\folie{4/polynomefp.tex}
-\folie{4/alpha.tex}
-\folie{4/schieberegister.tex}
-\folie{4/charakteristik.tex}
-\folie{4/char2.tex}
-\folie{4/frobenius.tex}
-\folie{4/qundr.tex}
-\folie{4/galois/erweiterung.tex}
-\folie{4/galois/automorphismus.tex}
-\folie{4/galois/konstruktion.tex}
-\folie{4/galois/wuerfel.tex}
-\folie{4/galois/winkeldreiteilung.tex}
-\folie{4/galois/quadratur.tex}
-\folie{4/galois/radikale.tex}
-\folie{4/galois/aufloesbarkeit.tex}
-\folie{4/galois/sn.tex}
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{4/ggt.tex}
+\folie{4/euklidmatrix.tex}
+\folie{4/euklidbeispiel.tex}
+\folie{4/euklidtabelle.tex}
+\folie{4/fp.tex}
+\folie{4/division.tex}
+\folie{4/gauss.tex}
+\folie{4/dh.tex}
+\folie{4/divisionpoly.tex}
+\folie{4/euklidpoly.tex}
+\folie{4/polynomefp.tex}
+\folie{4/alpha.tex}
+\folie{4/schieberegister.tex}
+\folie{4/charakteristik.tex}
+\folie{4/char2.tex}
+\folie{4/frobenius.tex}
+\folie{4/qundr.tex}
+\folie{4/galois/erweiterung.tex}
+\folie{4/galois/automorphismus.tex}
+\folie{4/galois/konstruktion.tex}
+\folie{4/galois/wuerfel.tex}
+\folie{4/galois/winkeldreiteilung.tex}
+\folie{4/galois/quadratur.tex}
+\folie{4/galois/radikale.tex}
+\folie{4/galois/aufloesbarkeit.tex}
+\folie{4/galois/sn.tex}
diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
index 3d52b00..ef5902b 100644
--- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -1,120 +1,120 @@
-%
-% aufloesbarkeit.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Auflösbarkeit}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Radikalerweiterung}
-Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$
-einer Radikalerweiterung
-\[
-\Bbbk \subset \Bbbk(\alpha)
-\]
-sind festgelegt durch Wahl von $f(\alpha)$.
-
-\begin{itemize}
-\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt
-\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein
-\end{itemize}
-
-\end{block}}
-\uncover<8->{%
-\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$}
-$\mathbb{Q}\subset \Bbbk$,
-$n$ verschiedene Nullstellen $\mathbb{C}$:
-\[
-\uncover<9->{
-\operatorname{Gal}(\Bbbk/\mathbb{Q})
-\cong
-S_n}
-\uncover<10->{
-\quad
-\text{auflösbar?}}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{\uncover<5->{Galois-Gruppen}}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{1.2}
-
-\uncover<2->{
-\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3});
-\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen};
-}
-
-\node at (0,0) {$\mathbb{Q}$};
-\node at (0,{1*\s}) {$E_1$};
-\node at (0,{2*\s}) {$E_2$};
-\node at (0,{3*\s}) {$E_3$};
-\node at (0,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) {$\Bbbk$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s});
-
-\begin{scope}[xshift=0.5cm]
-\uncover<7->{
-\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3});
-\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe};
-}
-\uncover<5->{
-\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$};
-\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$};
-\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$};
-\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$};
-\node at (1,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$};
-\node at (1,{0.5*\s}) {$\cap\mathstrut$};
-\node at (1,{1.5*\s}) {$\cap\mathstrut$};
-\node at (1,{2.5*\s}) {$\cap\mathstrut$};
-\node at (1,{3.5*\s}) {$\cap\mathstrut$};
-\node at (1,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<6->{
-\begin{scope}[xshift=2.5cm]
-\node at (0,{0*\s}) {$G_n$};
-\node at (0,{1*\s}) {$G_{n-1}$};
-\node at (0,{2*\s}) {$G_{n-2}$};
-\node at (0,{3*\s}) {$G_{n-3}$};
-\node at (0,{5*\s}) {$G_0=\{e\}$};
-\node at (0,{0.5*\s}) {$\cap\mathstrut$};
-\node at (0,{1.5*\s}) {$\cap\mathstrut$};
-\node at (0,{2.5*\s}) {$\cap\mathstrut$};
-\node at (0,{3.5*\s}) {$\cap\mathstrut$};
-\node at (0,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<7->{
-\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$};
-\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$};
-\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$};
-\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch};
-}
-
-\end{scope}
-\end{scope}
-
-
-
-\end{tikzpicture}
-\end{center}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% aufloesbarkeit.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Auflösbarkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Radikalerweiterung}
+Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$
+einer Radikalerweiterung
+\[
+\Bbbk \subset \Bbbk(\alpha)
+\]
+sind festgelegt durch Wahl von $f(\alpha)$.
+
+\begin{itemize}
+\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt
+\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein
+\end{itemize}
+
+\end{block}}
+\uncover<8->{%
+\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$}
+$\mathbb{Q}\subset \Bbbk$,
+$n$ verschiedene Nullstellen $\mathbb{C}$:
+\[
+\uncover<9->{
+\operatorname{Gal}(\Bbbk/\mathbb{Q})
+\cong
+S_n}
+\uncover<10->{
+\quad
+\text{auflösbar?}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{\uncover<5->{Galois-Gruppen}}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{1.2}
+
+\uncover<2->{
+\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3});
+\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen};
+}
+
+\node at (0,0) {$\mathbb{Q}$};
+\node at (0,{1*\s}) {$E_1$};
+\node at (0,{2*\s}) {$E_2$};
+\node at (0,{3*\s}) {$E_3$};
+\node at (0,{4*\s}) {$\vdots\mathstrut$};
+\node at (0,{5*\s}) {$\Bbbk$};
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s});
+
+\begin{scope}[xshift=0.5cm]
+\uncover<7->{
+\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3});
+\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe};
+}
+\uncover<5->{
+\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$};
+\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$};
+\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$};
+\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$};
+\node at (1,{4*\s}) {$\vdots\mathstrut$};
+\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$};
+\node at (1,{0.5*\s}) {$\cap\mathstrut$};
+\node at (1,{1.5*\s}) {$\cap\mathstrut$};
+\node at (1,{2.5*\s}) {$\cap\mathstrut$};
+\node at (1,{3.5*\s}) {$\cap\mathstrut$};
+\node at (1,{4.5*\s}) {$\cap\mathstrut$};
+}
+
+\uncover<6->{
+\begin{scope}[xshift=2.5cm]
+\node at (0,{0*\s}) {$G_n$};
+\node at (0,{1*\s}) {$G_{n-1}$};
+\node at (0,{2*\s}) {$G_{n-2}$};
+\node at (0,{3*\s}) {$G_{n-3}$};
+\node at (0,{5*\s}) {$G_0=\{e\}$};
+\node at (0,{0.5*\s}) {$\cap\mathstrut$};
+\node at (0,{1.5*\s}) {$\cap\mathstrut$};
+\node at (0,{2.5*\s}) {$\cap\mathstrut$};
+\node at (0,{3.5*\s}) {$\cap\mathstrut$};
+\node at (0,{4.5*\s}) {$\cap\mathstrut$};
+}
+
+\uncover<7->{
+\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$};
+\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch};
+
+\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$};
+\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch};
+
+\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$};
+\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch};
+}
+
+\end{scope}
+\end{scope}
+
+
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex
index e59f9b9..6051813 100644
--- a/vorlesungen/slides/4/galois/automorphismus.tex
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -1,118 +1,118 @@
-%
-% automorphismus.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{4pt}
-\setlength{\belowdisplayskip}{4pt}
-\frametitle{Galois-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.40\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{3.0}
-\begin{scope}[xshift=-1.5cm]
-\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut};
-\node at (0,{\s}) {$G$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0);
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0);
-\node at (0,{-0.5*\s}) [left] {$[F:E]$};
-\node at (0,{0.5*\s}) [left] {$[G:F]$};
-\node at (0,0) {$F$};
-\node at (0,{-\s}) {$E$};
-\end{scope}
-\uncover<3->{
-\begin{scope}[xshift=1.8cm]
-\node at (0,{\s+0.1}) [above] {Gruppe\strut};
-\fill (0,{-\s}) circle[radius=0.06];
-\fill (0,0) circle[radius=0.06];
-\fill (0,{\s}) circle[radius=0.06];
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,{-\s}) to[out=100,in=-100] (0,{\s});
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,{-\s}) to[out=80,in=-80] (0,0);
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,0) to[out=80,in=-80] (0,{\s});
-\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$};
-\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$};
-\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$};
-\end{scope}
-\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s});
-\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$};
-}
-\uncover<4->{
-\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s});
-\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<2->{%
-\begin{block}{Automorphismus}
-\vspace{-10pt}
-\[
-\operatorname{Aut}(F)
-=
-\left\{
-f\colon F\to F
-\left|
-\begin{aligned}
-f(x+y)&=f(x)+f(y)\\
-f(xy)&=f(x)f(y)
-\end{aligned}
-\right.
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{block}{Galois-Gruppe}
-Automorphismen, die $E$ festlassen
-\[
-{\color{red}
-\operatorname{Gal}(F/E)
-}
-=
-\left\{
-\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<4->{%
-\begin{block}{Fixkörper}
-$H\subset \operatorname{Aut}(F)$:
-\begin{align*}
-{\color{blue}F^H}
-&=
-\{x\in F\;|\; hx = x\forall h\in H\}
-=\operatorname{Fix}(H)
-\end{align*}
-\end{block}}
-\vspace{-13pt}
-\uncover<5->{%
-\begin{block}{Beispiel}
-\begin{itemize}
-\item<6->
-\(
-\operatorname{Gal}(\mathbb{C}/\mathbb{R})
-=
-\{
-\operatorname{id}_{\mathbb{C}},
-\operatorname{conj}\colon z\mapsto\overline{z}
-\}
-\)
-\item<7->
-\(
-\mathbb{C}^{\operatorname{conj}}
-=
-\mathbb{R}
-\)
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% automorphismus.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{4pt}
+\setlength{\belowdisplayskip}{4pt}
+\frametitle{Galois-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{3.0}
+\begin{scope}[xshift=-1.5cm]
+\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut};
+\node at (0,{\s}) {$G$};
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0);
+\node at (0,{-0.5*\s}) [left] {$[F:E]$};
+\node at (0,{0.5*\s}) [left] {$[G:F]$};
+\node at (0,0) {$F$};
+\node at (0,{-\s}) {$E$};
+\end{scope}
+\uncover<3->{
+\begin{scope}[xshift=1.8cm]
+\node at (0,{\s+0.1}) [above] {Gruppe\strut};
+\fill (0,{-\s}) circle[radius=0.06];
+\fill (0,0) circle[radius=0.06];
+\fill (0,{\s}) circle[radius=0.06];
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+ (0,{-\s}) to[out=100,in=-100] (0,{\s});
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+ (0,{-\s}) to[out=80,in=-80] (0,0);
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+ (0,0) to[out=80,in=-80] (0,{\s});
+\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$};
+\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$};
+\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$};
+\end{scope}
+\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s});
+\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$};
+}
+\uncover<4->{
+\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s});
+\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<2->{%
+\begin{block}{Automorphismus}
+\vspace{-10pt}
+\[
+\operatorname{Aut}(F)
+=
+\left\{
+f\colon F\to F
+\left|
+\begin{aligned}
+f(x+y)&=f(x)+f(y)\\
+f(xy)&=f(x)f(y)
+\end{aligned}
+\right.
+\right\}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{block}{Galois-Gruppe}
+Automorphismen, die $E$ festlassen
+\[
+{\color{red}
+\operatorname{Gal}(F/E)
+}
+=
+\left\{
+\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E
+\right\}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Fixkörper}
+$H\subset \operatorname{Aut}(F)$:
+\begin{align*}
+{\color{blue}F^H}
+&=
+\{x\in F\;|\; hx = x\forall h\in H\}
+=\operatorname{Fix}(H)
+\end{align*}
+\end{block}}
+\vspace{-13pt}
+\uncover<5->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item<6->
+\(
+\operatorname{Gal}(\mathbb{C}/\mathbb{R})
+=
+\{
+\operatorname{id}_{\mathbb{C}},
+\operatorname{conj}\colon z\mapsto\overline{z}
+\}
+\)
+\item<7->
+\(
+\mathbb{C}^{\operatorname{conj}}
+=
+\mathbb{R}
+\)
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex
index 20b278e..6909849 100644
--- a/vorlesungen/slides/4/galois/erweiterung.tex
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -1,65 +1,65 @@
-%
-% erweiterung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Körpererweiterungen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Körpererweiterung}
-$E,F$ Körper: $E\subset F$
-\end{block}
-\uncover<6->{%
-\begin{block}{Vektorraum}
-$F$ ist ein Vektorraum über $E$
-\end{block}}
-\uncover<7->{%
-\begin{block}{Endliche Körpererweiterung}
-$\dim_E F < \infty$
-\end{block}}
-\uncover<8->{%
-\begin{block}{Adjunktion eines $\alpha$}
-$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und
-$\alpha$ enthält.
-\end{block}}
-\uncover<9->{%
-\begin{block}{Algebraische Erweiterung}
-$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von
-$m(X)\in\Bbbk[X]$
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Beispiele}
-\begin{enumerate}
-\item<3->
-$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$
-\item<4->
-$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$
-\item<5->
-$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$
-\end{enumerate}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Grad}
-$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls
-\[
-\dim_E F = n =: [F:E]
-\]
-\uncover<8->{%
-Gleichbedeutend: $\deg m(X) = n$}
-\uncover<10->{%
-\[
-E\subset F\subset G
-\Rightarrow
-[G:E] = [G:F]\cdot [F:E]
-\]
-(in unseren Fällen)}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% erweiterung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Körpererweiterungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Körpererweiterung}
+$E,F$ Körper: $E\subset F$
+\end{block}
+\uncover<6->{%
+\begin{block}{Vektorraum}
+$F$ ist ein Vektorraum über $E$
+\end{block}}
+\uncover<7->{%
+\begin{block}{Endliche Körpererweiterung}
+$\dim_E F < \infty$
+\end{block}}
+\uncover<8->{%
+\begin{block}{Adjunktion eines $\alpha$}
+$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und
+$\alpha$ enthält.
+\end{block}}
+\uncover<9->{%
+\begin{block}{Algebraische Erweiterung}
+$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von
+$m(X)\in\Bbbk[X]$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Beispiele}
+\begin{enumerate}
+\item<3->
+$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$
+\item<4->
+$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$
+\item<5->
+$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$
+\end{enumerate}
+\end{block}}
+\uncover<7->{%
+\begin{block}{Grad}
+$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls
+\[
+\dim_E F = n =: [F:E]
+\]
+\uncover<8->{%
+Gleichbedeutend: $\deg m(X) = n$}
+\uncover<10->{%
+\[
+E\subset F\subset G
+\Rightarrow
+[G:E] = [G:F]\cdot [F:E]
+\]
+(in unseren Fällen)}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
index fd197ce..444944e 100644
--- a/vorlesungen/slides/4/galois/images/Makefile
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -1,12 +1,12 @@
-#
-# Makefile
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all: wuerfel2.png wuerfel.png
-
-wuerfel.png: wuerfel.pov common.inc
- povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
-
-wuerfel2.png: wuerfel2.pov common.inc
- povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all: wuerfel2.png wuerfel.png
+
+wuerfel.png: wuerfel.pov common.inc
+ povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
+
+wuerfel2.png: wuerfel2.pov common.inc
+ povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
index 44ee4c8..6cfcabe 100644
--- a/vorlesungen/slides/4/galois/images/common.inc
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -1,89 +1,89 @@
-//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-#include "textures.inc"
-#include "stones.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.133;
-#declare O = <0, 0, 0>;
-#declare E = <1, 1, 1>;
-#declare a = pow(2, 1/3);
-#declare at = 0.02;
-
-camera {
- location <3, 2, 12>
- look_at E * (a / 2) * 0.93
- right x * imagescale
- up y * imagescale
-}
-
-light_source {
- <11, 20, 16> color White
- area_light <1,0,0> <0,0,1>, 10, 10
- adaptive 1
- jitter
-}
-
-sky_sphere {
- pigment {
- color rgb<1,1,1>
- }
-}
-
-#macro wuerfelgitter(A, AT)
- cylinder { O, <A, 0, 0>, AT }
- cylinder { O, <0, A, 0>, AT }
- cylinder { O, <0, 0, A>, AT }
- cylinder { <A, 0, 0>, <A, A, 0>, AT }
- cylinder { <A, 0, 0>, <A, 0, A>, AT }
- cylinder { <0, A, 0>, <A, A, 0>, AT }
- cylinder { <0, A, 0>, <0, A, A>, AT }
- cylinder { <0, 0, A>, <A, 0, A>, AT }
- cylinder { <0, 0, A>, <0, A, A>, AT }
- cylinder { <A, A, 0>, <A, A, A>, AT }
- cylinder { <A, 0, A>, <A, A, A>, AT }
- cylinder { <0, A, A>, <A, A, A>, AT }
- sphere { <0, 0, 0>, AT }
- sphere { <A, 0, 0>, AT }
- sphere { <0, A, 0>, AT }
- sphere { <0, 0, A>, AT }
- sphere { <A, A, 0>, AT }
- sphere { <A, 0, A>, AT }
- sphere { <0, A, A>, AT }
- sphere { <A, A, A>, AT }
-#end
-
-#macro wuerfel()
- union {
- box { O, E }
- wuerfelgitter(1, 0.5*at)
- texture {
- T_Grnt24
- }
- finish {
- specular 0.9
- metallic
- }
- }
-#end
-
-#macro wuerfel2()
- union {
- wuerfelgitter(a, at)
- pigment {
- color rgb<0.8,0.4,0.4>
- }
- finish {
- specular 0.9
- metallic
- }
- }
-#end
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "textures.inc"
+#include "stones.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.133;
+#declare O = <0, 0, 0>;
+#declare E = <1, 1, 1>;
+#declare a = pow(2, 1/3);
+#declare at = 0.02;
+
+camera {
+ location <3, 2, 12>
+ look_at E * (a / 2) * 0.93
+ right x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <11, 20, 16> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro wuerfelgitter(A, AT)
+ cylinder { O, <A, 0, 0>, AT }
+ cylinder { O, <0, A, 0>, AT }
+ cylinder { O, <0, 0, A>, AT }
+ cylinder { <A, 0, 0>, <A, A, 0>, AT }
+ cylinder { <A, 0, 0>, <A, 0, A>, AT }
+ cylinder { <0, A, 0>, <A, A, 0>, AT }
+ cylinder { <0, A, 0>, <0, A, A>, AT }
+ cylinder { <0, 0, A>, <A, 0, A>, AT }
+ cylinder { <0, 0, A>, <0, A, A>, AT }
+ cylinder { <A, A, 0>, <A, A, A>, AT }
+ cylinder { <A, 0, A>, <A, A, A>, AT }
+ cylinder { <0, A, A>, <A, A, A>, AT }
+ sphere { <0, 0, 0>, AT }
+ sphere { <A, 0, 0>, AT }
+ sphere { <0, A, 0>, AT }
+ sphere { <0, 0, A>, AT }
+ sphere { <A, A, 0>, AT }
+ sphere { <A, 0, A>, AT }
+ sphere { <0, A, A>, AT }
+ sphere { <A, A, A>, AT }
+#end
+
+#macro wuerfel()
+ union {
+ box { O, E }
+ wuerfelgitter(1, 0.5*at)
+ texture {
+ T_Grnt24
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+ }
+#end
+
+#macro wuerfel2()
+ union {
+ wuerfelgitter(a, at)
+ pigment {
+ color rgb<0.8,0.4,0.4>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+ }
+#end
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
index a0466f3..a5db465 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
index a11bab0..ac32b2f 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel2.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-wuerfel2()
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+wuerfel2()
diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex
index b461d44..094b570 100644
--- a/vorlesungen/slides/4/galois/konstruktion.tex
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -1,147 +1,147 @@
-%
-% konstruktion.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Konstruktion mit Zirkel und Lineal}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Strahlensatz}
-\uncover<6->{%
-Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
-kann mit Zirkel und Lineal konstruiert werden.}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Kreis--Gerade}
-Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
-\uncover<13->{%
-$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
-\end{block}}
-\end{column}
-\end{columns}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{0.5}
-\def\t{0.45}
-
-\coordinate (A) at (0,0);
-\coordinate (B) at ({10*\t},0);
-
-\uncover<2->{
- \draw (0,0) -- (30:{10.5*\s});
-}
-
-\uncover<3->{
- \foreach \x in {0,...,10}{
- \fill (30:{\x*\s}) circle[radius=0.03];
- }
- \foreach \x in {0,1,2,3,4,7,8,9}{
- \node at (30:{\x*\s}) [above] {\tiny $\x$};
- }
- \node at (30:{10*\s}) [above right] {$q=10$};
-}
-
-\uncover<4->{
- \foreach \x in {1,...,10}{
- \fill (0:{\x*\t}) circle[radius=0.03];
- \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
- }
-}
-
-\draw (A) -- (0:{10.5*\t});
-\node at (A) [below left] {$A$};
-\node at (B) [below right] {$B$};
-\fill (A) circle[radius=0.05];
-\fill (B) circle[radius=0.05];
-
-\uncover<5->{
- \node at (30:{6*\s}) [above left] {$p=6$};
- \draw[line width=0.2pt] (0,0) -- (0,-0.4);
- \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
- \draw[<->] (0,-0.3) -- ({6*\t},-0.3);
- \node at ({3*\t},-0.4) [below]
- {$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<8->{%
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-%\foreach \x in {8,...,14}{
-% \only<\x>{\node at (4,4) {$\x$};}
-%}
-
-\def\r{4}
-\def\a{50}
-
-\coordinate (A) at ({\r*cos(\a)},0);
-
-\uncover<10->{
- \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
- \fill[color=gray]
- (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
-}
-
-\draw[->] (0,0) -- (95:\r);
-\node at (95:{0.5*\r}) [left] {$c$};
-
-\begin{scope}
- \clip (-1,-0.3) rectangle (4.5,4.1);
- \uncover<10->{
- \draw (-1,0) -- (5,0);
- \draw[->] (0,0) -- (\r,0);
- \draw (0,0) circle[radius=\r];
- \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
- }
-\end{scope}
-
-\uncover<11->{
- \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
-}
-
-\uncover<9->{
- \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
- \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
- \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
-}
-
-\uncover<11->{
- \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
- \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
-}
-
-\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
-\fill[color=blue] (0,0) circle[radius=0.04];
-\fill[color=blue] (A) circle[radius=0.04];
-\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
-
-\uncover<12->{
- \fill[color=white,opacity=0.8]
- ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
- rectangle
- ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
-
- \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
- {$b=\sqrt{c^2-a^2}$};
- \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
- \fill[color=red] (\a:\r) circle[radius=0.05];
- \fill[color=red] (A) circle[radius=0.05];
-}
-
-\end{tikzpicture}
-\end{center}}
-\end{column}
-\end{columns}
-\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}
-Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}
-\end{frame}
+%
+% konstruktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Konstruktion mit Zirkel und Lineal}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Strahlensatz}
+\uncover<6->{%
+Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
+kann mit Zirkel und Lineal konstruiert werden.}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Kreis--Gerade}
+Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
+\uncover<13->{%
+$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{0.5}
+\def\t{0.45}
+
+\coordinate (A) at (0,0);
+\coordinate (B) at ({10*\t},0);
+
+\uncover<2->{
+ \draw (0,0) -- (30:{10.5*\s});
+}
+
+\uncover<3->{
+ \foreach \x in {0,...,10}{
+ \fill (30:{\x*\s}) circle[radius=0.03];
+ }
+ \foreach \x in {0,1,2,3,4,7,8,9}{
+ \node at (30:{\x*\s}) [above] {\tiny $\x$};
+ }
+ \node at (30:{10*\s}) [above right] {$q=10$};
+}
+
+\uncover<4->{
+ \foreach \x in {1,...,10}{
+ \fill (0:{\x*\t}) circle[radius=0.03];
+ \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
+ }
+}
+
+\draw (A) -- (0:{10.5*\t});
+\node at (A) [below left] {$A$};
+\node at (B) [below right] {$B$};
+\fill (A) circle[radius=0.05];
+\fill (B) circle[radius=0.05];
+
+\uncover<5->{
+ \node at (30:{6*\s}) [above left] {$p=6$};
+ \draw[line width=0.2pt] (0,0) -- (0,-0.4);
+ \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
+ \draw[<->] (0,-0.3) -- ({6*\t},-0.3);
+ \node at ({3*\t},-0.4) [below]
+ {$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+%\foreach \x in {8,...,14}{
+% \only<\x>{\node at (4,4) {$\x$};}
+%}
+
+\def\r{4}
+\def\a{50}
+
+\coordinate (A) at ({\r*cos(\a)},0);
+
+\uncover<10->{
+ \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
+ \fill[color=gray]
+ (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
+}
+
+\draw[->] (0,0) -- (95:\r);
+\node at (95:{0.5*\r}) [left] {$c$};
+
+\begin{scope}
+ \clip (-1,-0.3) rectangle (4.5,4.1);
+ \uncover<10->{
+ \draw (-1,0) -- (5,0);
+ \draw[->] (0,0) -- (\r,0);
+ \draw (0,0) circle[radius=\r];
+ \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
+ }
+\end{scope}
+
+\uncover<11->{
+ \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
+}
+
+\uncover<9->{
+ \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
+ \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
+ \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
+}
+
+\uncover<11->{
+ \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
+ \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
+}
+
+\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
+\fill[color=blue] (0,0) circle[radius=0.04];
+\fill[color=blue] (A) circle[radius=0.04];
+\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
+
+\uncover<12->{
+ \fill[color=white,opacity=0.8]
+ ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
+ rectangle
+ ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
+
+ \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
+ {$b=\sqrt{c^2-a^2}$};
+ \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
+ \fill[color=red] (\a:\r) circle[radius=0.05];
+ \fill[color=red] (A) circle[radius=0.05];
+}
+
+\end{tikzpicture}
+\end{center}}
+\end{column}
+\end{columns}
+\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}
+Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
index f9510ba..f5763b9 100644
--- a/vorlesungen/slides/4/galois/quadratur.tex
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -1,66 +1,66 @@
-%
-% quadratur.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Quadratur des Kreises}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{center}
-\uncover<2->{%
-\begin{tikzpicture}[>=latex,thick]
-
-\def\r{2.8}
-\pgfmathparse{sqrt(3.14159)*\r/2}
-\xdef\s{\pgfmathresult}
-
-\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
-\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
-
-\uncover<3->{
- \draw[->,color=red] (0,0) -- (50:\r);
- \fill[color=red] (0,0) circle[radius=0.04];
- \node[color=red] at (50:{0.5*\r}) [below right] {$r$};
-}
-
-\uncover<4->{
- \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
- \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
- \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
- \node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
-}
-
-\uncover<5->{
- \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
- \;\Rightarrow\;
- {\color{blue}l}={\color{red}\sqrt{\pi}r}$};
-}
-
-\end{tikzpicture}}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere ein zu einem Kreis flächengleiches Quadrat
-\end{block}
-\uncover<6->{%
-\begin{block}{Modifizierte Aufgabe}
-Konstruiere eine Strecke, deren Länge Lösung der Gleichung
-$x^2-\pi=0$ ist.
-\end{block}}
-\uncover<7->{%
-\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
-\begin{itemize}
-\item<8-> Lösung in einem Erweiterungskörper
-\item<9-> Lösung ist Nullstelle eines Polynoms
-\item<10-> Lösung ist algebraisch
-\item<11-> $\pi$ ist {\bf nicht} algebraisch
-\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% quadratur.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Quadratur des Kreises}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{center}
+\uncover<2->{%
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.8}
+\pgfmathparse{sqrt(3.14159)*\r/2}
+\xdef\s{\pgfmathresult}
+
+\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
+\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
+
+\uncover<3->{
+ \draw[->,color=red] (0,0) -- (50:\r);
+ \fill[color=red] (0,0) circle[radius=0.04];
+ \node[color=red] at (50:{0.5*\r}) [below right] {$r$};
+}
+
+\uncover<4->{
+ \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
+ \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
+ \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
+ \node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
+}
+
+\uncover<5->{
+ \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
+ \;\Rightarrow\;
+ {\color{blue}l}={\color{red}\sqrt{\pi}r}$};
+}
+
+\end{tikzpicture}}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Aufgabe}
+Konstruiere ein zu einem Kreis flächengleiches Quadrat
+\end{block}
+\uncover<6->{%
+\begin{block}{Modifizierte Aufgabe}
+Konstruiere eine Strecke, deren Länge Lösung der Gleichung
+$x^2-\pi=0$ ist.
+\end{block}}
+\uncover<7->{%
+\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
+\begin{itemize}
+\item<8-> Lösung in einem Erweiterungskörper
+\item<9-> Lösung ist Nullstelle eines Polynoms
+\item<10-> Lösung ist algebraisch
+\item<11-> $\pi$ ist {\bf nicht} algebraisch
+\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
index cb08dca..e9e4ce8 100644
--- a/vorlesungen/slides/4/galois/radikale.tex
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -1,69 +1,69 @@
-%
-% radikale.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Lösung durch Radikale}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Problemstellung}
-Finde Nullstellen eines Polynomes
-\[
-p(X)
-=
-a_nX^n + a_{n-1}X^{n-1}
-+\dots+
-a_1X+a_0
-\]
-$p\in\mathbb{Q}[X]$
-\end{block}
-\uncover<2->{%
-\begin{block}{Radikale}
-Geschachtelte Wurzelausdrücke
-\[
-\sqrt[3]{
--\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-+
-\sqrt[3]{
--\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-\]
-\uncover<3->{(Lösung von $x^3+px+q=0$)}
-\end{block}}
-\uncover<4->{%
-\begin{block}{Lösbar durch Radikale}
-Nullstelle von $p(X)$ ist ein Radikal
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<5->{%
-\begin{block}{Algebraische Formulierung}
-Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$,
-finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart,
-dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{:
-$\Bbbk = \mathbb{Q}[X]/(p)$}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Radikalerweiterung}
-Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften
-\begin{itemize}
-\item<8-> $\alpha$ ist eine Einheitswurzel
-\item<9-> $\alpha^k\in\Bbbk$
-\end{itemize}
-\end{block}}
-\vspace{-5pt}
-\uncover<10->{%
-\begin{block}{Lösbar durch Radikale}
-Radikalerweiterungen
-\[
-\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% radikale.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lösung durch Radikale}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Problemstellung}
+Finde Nullstellen eines Polynomes
+\[
+p(X)
+=
+a_nX^n + a_{n-1}X^{n-1}
++\dots+
+a_1X+a_0
+\]
+$p\in\mathbb{Q}[X]$
+\end{block}
+\uncover<2->{%
+\begin{block}{Radikale}
+Geschachtelte Wurzelausdrücke
+\[
+\sqrt[3]{
+-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
+}
++
+\sqrt[3]{
+-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
+}
+\]
+\uncover<3->{(Lösung von $x^3+px+q=0$)}
+\end{block}}
+\uncover<4->{%
+\begin{block}{Lösbar durch Radikale}
+Nullstelle von $p(X)$ ist ein Radikal
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Algebraische Formulierung}
+Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$,
+finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart,
+dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{:
+$\Bbbk = \mathbb{Q}[X]/(p)$}
+\end{block}}
+\uncover<7->{%
+\begin{block}{Radikalerweiterung}
+Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften
+\begin{itemize}
+\item<8-> $\alpha$ ist eine Einheitswurzel
+\item<9-> $\alpha^k\in\Bbbk$
+\end{itemize}
+\end{block}}
+\vspace{-5pt}
+\uncover<10->{%
+\begin{block}{Lösbar durch Radikale}
+Radikalerweiterungen
+\[
+\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex
index f340825..1cae3fa 100644
--- a/vorlesungen/slides/4/galois/sn.tex
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -1,87 +1,87 @@
-%
-% sn.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Nichtauflösbarkeit von $S_n$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Die symmetrische Gruppe $S_n$}
-Permutationen auf $n$ Elementen
-\[
-\sigma
-=
-\begin{pmatrix}
-1&2&3&\dots&n\\
-\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n)
-\end{pmatrix}
-\]
-\end{block}
-\vspace{-10pt}
-\uncover<2->{%
-\begin{block}{Signum}
-$t(\sigma)=\mathstrut$ Anzahl Transpositionen
-\[
-\operatorname{sgn}(\sigma)
-=
-(-1)^{t(\sigma)}
-=
-\begin{cases}
-\phantom{-}1&\text{$t(\sigma)$ gerade}
-\\
--1&\text{$t(\sigma)$ ungerade}
-\end{cases}
-\]
-Homomorphismus!
-\end{block}}
-\uncover<3->{%
-\begin{block}{Die alternierende Gruppe $A_n$}
-\vspace{-12pt}
-\[
-A_n = \ker \operatorname{sgn}
-=
-\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Normale Untergruppe}
-\begin{itemize}
-\item
-$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$
-\item
-$G/N$ ist wohldefiniert
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<5->{%
-\begin{block}{Einfache Gruppe}
-$G$ einfach $\Leftrightarrow$
-\[
-H\triangleleft G
-\;
-\Rightarrow
-\;
-\text{$H=\{e\}$ oder $H=G$}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<6->{%
-\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$}
-\begin{enumerate}
-\item<7-> Zeigen, dass $A_5$ einfach ist
-\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach
-\end{enumerate}
-\uncover<9->{%
-$\Rightarrow$ i.~A.~keine Lösung der
-einer Polynomgleichung vom Grad $\ge 5$ durch Radikale
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% sn.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Nichtauflösbarkeit von $S_n$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Die symmetrische Gruppe $S_n$}
+Permutationen auf $n$ Elementen
+\[
+\sigma
+=
+\begin{pmatrix}
+1&2&3&\dots&n\\
+\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n)
+\end{pmatrix}
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Signum}
+$t(\sigma)=\mathstrut$ Anzahl Transpositionen
+\[
+\operatorname{sgn}(\sigma)
+=
+(-1)^{t(\sigma)}
+=
+\begin{cases}
+\phantom{-}1&\text{$t(\sigma)$ gerade}
+\\
+-1&\text{$t(\sigma)$ ungerade}
+\end{cases}
+\]
+Homomorphismus!
+\end{block}}
+\uncover<3->{%
+\begin{block}{Die alternierende Gruppe $A_n$}
+\vspace{-12pt}
+\[
+A_n = \ker \operatorname{sgn}
+=
+\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Normale Untergruppe}
+\begin{itemize}
+\item
+$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$
+\item
+$G/N$ ist wohldefiniert
+\end{itemize}
+\end{block}}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Einfache Gruppe}
+$G$ einfach $\Leftrightarrow$
+\[
+H\triangleleft G
+\;
+\Rightarrow
+\;
+\text{$H=\{e\}$ oder $H=G$}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<6->{%
+\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$}
+\begin{enumerate}
+\item<7-> Zeigen, dass $A_5$ einfach ist
+\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach
+\end{enumerate}
+\uncover<9->{%
+$\Rightarrow$ i.~A.~keine Lösung der
+einer Polynomgleichung vom Grad $\ge 5$ durch Radikale
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
index 28c07fe..54b941b 100644
--- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -1,94 +1,94 @@
-%
-% winkeldreiteilung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Winkeldreiteilung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.43\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{5}
-\def\a{25}
-
-\uncover<3->{
- \draw[line width=0.7pt] (\r,0) arc (0:90:\r);
-}
-
-\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
-\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
-
-\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
-
-\uncover<2->{
- \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
- \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
- \node[color=red] at ({0.5*\a}:{0.7*\r})
- {$\displaystyle\frac{\alpha}{3}$};
-}
-
-\uncover<3->{
- \fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
- \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
-
- \fill[color=red] ({\a}:\r) circle[radius=0.05];
- \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
-
- \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
- \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
-}
-
-
-\uncover<4->{
-\node at ({0.5*\r},-0.5) [below] {$\displaystyle
-\cos{\color{blue}\alpha}
-=
-4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
-$};
-}
-
-\uncover<5->{
- \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
- \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.53\textwidth}
-\begin{block}{Aufgabe}
-Teile einen Winkel in drei gleiche Teile
-\end{block}
-\vspace{-2pt}
-\uncover<6->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere $x$ aus $a$ derart, dass
-\[
-p(x)
-=
-x^3-\frac34 x -a = 0
-\]
-\uncover<7->{%
-$a=0$:}
-\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
-\end{block}}
-\vspace{-2pt}
-\uncover<10->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<11->
-$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel
-\item<12->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<13->
-Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% winkeldreiteilung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Winkeldreiteilung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.43\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{5}
+\def\a{25}
+
+\uncover<3->{
+ \draw[line width=0.7pt] (\r,0) arc (0:90:\r);
+}
+
+\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
+\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
+
+\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
+
+\uncover<2->{
+ \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
+ \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
+ \node[color=red] at ({0.5*\a}:{0.7*\r})
+ {$\displaystyle\frac{\alpha}{3}$};
+}
+
+\uncover<3->{
+ \fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
+ \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
+
+ \fill[color=red] ({\a}:\r) circle[radius=0.05];
+ \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
+
+ \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
+ \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
+}
+
+
+\uncover<4->{
+\node at ({0.5*\r},-0.5) [below] {$\displaystyle
+\cos{\color{blue}\alpha}
+=
+4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
+$};
+}
+
+\uncover<5->{
+ \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
+ \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.53\textwidth}
+\begin{block}{Aufgabe}
+Teile einen Winkel in drei gleiche Teile
+\end{block}
+\vspace{-2pt}
+\uncover<6->{%
+\begin{block}{Algebraisierte Aufgabe}
+Konstruiere $x$ aus $a$ derart, dass
+\[
+p(x)
+=
+x^3-\frac34 x -a = 0
+\]
+\uncover<7->{%
+$a=0$:}
+\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
+\end{block}}
+\vspace{-2pt}
+\uncover<10->{%
+\begin{proof}[Unmöglichkeitsbeweis]
+\begin{itemize}
+\item<11->
+$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel
+\item<12->
+$p(x)$ definiert eine Körpererweiterung vom Grad $3$
+\item<13->
+Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
index 907d60a..ada6079 100644
--- a/vorlesungen/slides/4/galois/wuerfel.tex
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -1,64 +1,64 @@
-%
-% wuerfel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Würfelverdoppelung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
-\uncover<2->{
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
-}
-
-\uncover<3->{
- \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
- \node[color=blue] at (0.75,-2.3) [above] {$a$};
-}
-
-\uncover<4->{
- \begin{scope}[yshift=0.03cm]
- \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
- \draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
- \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
- \end{scope}
- \node[color=red] at (0.36,-2.9) [below] {$b$};
-}
-
-\uncover<5->{
-\node at (0,-4) {$
- 2{\color{blue}a}^3={\color{red}b}^3
- \uncover<6->{\;\Rightarrow\;
- \frac{b}{a} = \sqrt[3]{2}}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere einen Würfel mit doppeltem Volumen
-\end{block}
-\uncover<7->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere eine Nullstelle von $p(x)=x^3-2$
-\end{block}}
-\uncover<8->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<9->
-$p(x)$ irreduzibel
-\item<10->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<11->
-Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% wuerfel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Würfelverdoppelung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
+\uncover<2->{
+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
+}
+
+\uncover<3->{
+ \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
+ \node[color=blue] at (0.75,-2.3) [above] {$a$};
+}
+
+\uncover<4->{
+ \begin{scope}[yshift=0.03cm]
+ \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
+ \draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
+ \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
+ \end{scope}
+ \node[color=red] at (0.36,-2.9) [below] {$b$};
+}
+
+\uncover<5->{
+\node at (0,-4) {$
+ 2{\color{blue}a}^3={\color{red}b}^3
+ \uncover<6->{\;\Rightarrow\;
+ \frac{b}{a} = \sqrt[3]{2}}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Aufgabe}
+Konstruiere einen Würfel mit doppeltem Volumen
+\end{block}
+\uncover<7->{%
+\begin{block}{Algebraisierte Aufgabe}
+Konstruiere eine Nullstelle von $p(x)=x^3-2$
+\end{block}}
+\uncover<8->{%
+\begin{proof}[Unmöglichkeitsbeweis]
+\begin{itemize}
+\item<9->
+$p(x)$ irreduzibel
+\item<10->
+$p(x)$ definiert eine Körpererweiterung vom Grad $3$
+\item<11->
+Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 2391099..ffd5091 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -1,22 +1,35 @@
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-chapter5 = \
- ../slides/7/symmetrien.tex \
- ../slides/7/algebraisch.tex \
- ../slides/7/parameter.tex \
- ../slides/7/mannigfaltigkeit.tex \
- ../slides/7/sl2.tex \
- ../slides/7/drehung.tex \
- ../slides/7/drehanim.tex \
- ../slides/7/semi.tex \
- ../slides/7/kurven.tex \
- ../slides/7/einparameter.tex \
- ../slides/7/ableitung.tex \
- ../slides/7/liealgebra.tex \
- ../slides/7/kommutator.tex \
- ../slides/7/dg.tex \
- ../slides/7/chapter.tex
-
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 = \
+ ../slides/7/symmetrien.tex \
+ ../slides/7/algebraisch.tex \
+ ../slides/7/parameter.tex \
+ ../slides/7/mannigfaltigkeit.tex \
+ ../slides/7/sl2.tex \
+ ../slides/7/drehung.tex \
+ ../slides/7/drehanim.tex \
+ ../slides/7/semi.tex \
+ ../slides/7/kurven.tex \
+ ../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/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
index 5a4b94e..12f9084 100644
--- a/vorlesungen/slides/7/ableitung.tex
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -1,68 +1,68 @@
-%
-% ableitung.tex -- Ableitung in der Lie-Gruppe
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Ableitung in der Matrix-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Ableitung in $\operatorname{O}(n)$}
-\uncover<2->{%
-$s \mapsto A(s)\in\operatorname{O}(n)$
-}
-\begin{align*}
-\uncover<3->{I
-&=
-A(s)^tA(s)}
-\\
-\uncover<4->{0
-=
-\frac{d}{ds} I
-&=
-\frac{d}{ds} (A(s)^t A(s))}
-\\
-&\uncover<5->{=
-\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)}
-\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}}
-\uncover<7->{0
-&=
-\dot{A}(0)^t
-+
-\dot{A}(0)}
-\\
-\uncover<8->{\Leftrightarrow
-\qquad
-\dot{A}(0)^t &= -\dot{A}(0)}
-\end{align*}
-\uncover<9->{%
-``Tangentialvektoren'' sind antisymmetrische Matrizen
-}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
-\uncover<2->{%
-$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
-}
-\begin{align*}
-\uncover<3->{1 &= \det A(t)}
-\\
-\uncover<10->{0
-=
-\frac{d}{dt}1
-&=
-\frac{d}{dt} \det A(t)}
-\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}}
-\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)}
-\end{align*}
-\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% ableitung.tex -- Ableitung in der Lie-Gruppe
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ableitung in der Matrix-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{O}(n)$}
+\uncover<2->{%
+$s \mapsto A(s)\in\operatorname{O}(n)$
+}
+\begin{align*}
+\uncover<3->{I
+&=
+A(s)^tA(s)}
+\\
+\uncover<4->{0
+=
+\frac{d}{ds} I
+&=
+\frac{d}{ds} (A(s)^t A(s))}
+\\
+&\uncover<5->{=
+\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)}
+\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}}
+\uncover<7->{0
+&=
+\dot{A}(0)^t
++
+\dot{A}(0)}
+\\
+\uncover<8->{\Leftrightarrow
+\qquad
+\dot{A}(0)^t &= -\dot{A}(0)}
+\end{align*}
+\uncover<9->{%
+``Tangentialvektoren'' sind antisymmetrische Matrizen
+}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
+\uncover<2->{%
+$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
+}
+\begin{align*}
+\uncover<3->{1 &= \det A(t)}
+\\
+\uncover<10->{0
+=
+\frac{d}{dt}1
+&=
+\frac{d}{dt} \det A(t)}
+\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}}
+\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)}
+\end{align*}
+\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex
index fba42cf..31d209a 100644
--- a/vorlesungen/slides/7/algebraisch.tex
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -1,115 +1,115 @@
-%
-% algebraisch.tex -- algebraische Definition der Symmetrien
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Erhaltungsgrössen und Algebra}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Längen und Winkel}
-Längenmessung mit Skalarprodukt
-\begin{align*}
-\|\vec{v}\|^2
-&=
-\langle \vec{v},\vec{v}\rangle
-=
-\vec{v}\cdot \vec{v}
-\uncover<2->{=
-\vec{v}^t\vec{v}}
-\end{align*}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Flächeninhalt/Volumen}
-$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
-\\
-Volumen des Parallelepipeds: $\det V$
-\end{block}}
-\end{column}
-\end{columns}
-%
-\vspace{-7pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Längenerhaltende Transformationen}
-$A\in\operatorname{GL}_n(\mathbb{R})$
-\begin{align*}
-\vec{x}^t\vec{y}
-&=
-(A\vec{x})
-\cdot
-(A\vec{y})
-\uncover<5->{=
-(A\vec{x})^t
-(A\vec{y})}
-\\
-\uncover<6->{
-\vec{x}^tI\vec{y}
-&=
-\vec{x}^tA^tA\vec{y}}
-\uncover<7->{
-\Rightarrow I=A^tA}
-\end{align*}
-\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<9->{%
-\begin{block}{Volumenerhaltende Transformationen}
-$A\in\operatorname{GL}_n(\mathbb{R})$
-\begin{align*}
-\det(V)
-&=
-\det(AV)
-\uncover<10->{=
-\det(A)\det(V)}
-\\
-\uncover<11->{
-1&=\det(A)}
-\end{align*}
-\uncover<10->{
-(Produktsatz für Determinante)
-}
-\end{block}}
-\end{column}
-\end{columns}
-%
-\vspace{-3pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<12->{%
-\begin{block}{Orthogonale Matrizen}
-Längentreue Abbildungen = orthogonale Matrizen:
-\[
-O(n)
-=
-\{
-A \in \operatorname{GL}_n(\mathbb{R})
-\;|\;
-A^tA=I
-\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<13->{%
-\begin{block}{``Spezielle'' Matrizen}
-Volumen-/Orientierungserhaltende Transformationen:
-\[
-\operatorname{SL}_n(\mathbb R)
-=
-\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-
-\end{frame}
-\egroup
+%
+% algebraisch.tex -- algebraische Definition der Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erhaltungsgrössen und Algebra}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Längen und Winkel}
+Längenmessung mit Skalarprodukt
+\begin{align*}
+\|\vec{v}\|^2
+&=
+\langle \vec{v},\vec{v}\rangle
+=
+\vec{v}\cdot \vec{v}
+\uncover<2->{=
+\vec{v}^t\vec{v}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Flächeninhalt/Volumen}
+$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
+\\
+Volumen des Parallelepipeds: $\det V$
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-7pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Längenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\vec{x}^t\vec{y}
+&=
+(A\vec{x})
+\cdot
+(A\vec{y})
+\uncover<5->{=
+(A\vec{x})^t
+(A\vec{y})}
+\\
+\uncover<6->{
+\vec{x}^tI\vec{y}
+&=
+\vec{x}^tA^tA\vec{y}}
+\uncover<7->{
+\Rightarrow I=A^tA}
+\end{align*}
+\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<9->{%
+\begin{block}{Volumenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\det(V)
+&=
+\det(AV)
+\uncover<10->{=
+\det(A)\det(V)}
+\\
+\uncover<11->{
+1&=\det(A)}
+\end{align*}
+\uncover<10->{
+(Produktsatz für Determinante)
+}
+\end{block}}
+\end{column}
+\end{columns}
+%
+\vspace{-3pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Orthogonale Matrizen}
+Längentreue Abbildungen = orthogonale Matrizen:
+\[
+O(n)
+=
+\{
+A \in \operatorname{GL}_n(\mathbb{R})
+\;|\;
+A^tA=I
+\}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<13->{%
+\begin{block}{``Spezielle'' Matrizen}
+Volumen-/Orientierungserhaltende Transformationen:
+\[
+\operatorname{SL}_n(\mathbb R)
+=
+\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
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 0f14a9a..3736e0f 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -1,19 +1,32 @@
-%
-% chapter.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
-%
-\folie{7/symmetrien.tex}
-\folie{7/algebraisch.tex}
-\folie{7/parameter.tex}
-\folie{7/mannigfaltigkeit.tex}
-\folie{7/sl2.tex}
-\folie{7/drehung.tex}
-\folie{7/drehanim.tex}
-\folie{7/semi.tex}
-\folie{7/kurven.tex}
-\folie{7/einparameter.tex}
-\folie{7/ableitung.tex}
-\folie{7/liealgebra.tex}
-\folie{7/kommutator.tex}
-\folie{7/dg.tex}
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{7/symmetrien.tex}
+\folie{7/algebraisch.tex}
+\folie{7/parameter.tex}
+\folie{7/mannigfaltigkeit.tex}
+\folie{7/sl2.tex}
+\folie{7/drehung.tex}
+\folie{7/drehanim.tex}
+\folie{7/semi.tex}
+\folie{7/kurven.tex}
+\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 446b2ab..f9528a4 100644
--- a/vorlesungen/slides/7/dg.tex
+++ b/vorlesungen/slides/7/dg.tex
@@ -1,92 +1,92 @@
-%
-% dg.tex -- Differentialgleichung für die Exponentialabbildung
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Zurück zur Lie-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
-Ableitung von $\gamma(t)$ an der Stelle $t$:
-\begin{align*}
-\dot{\gamma}(t)
-&\uncover<2->{=
-\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
-}
-\\
-&\uncover<3->{=
-\frac{d}{ds}
-\gamma(t+s)
-\bigg|_{s=0}
-}
-\\
-&\uncover<4->{=
-\frac{d}{ds}
-\gamma(t)\gamma(s)
-\bigg|_{s=0}
-}
-\\
-&\uncover<5->{=
-\gamma(t)
-\frac{d}{ds}
-\gamma(s)
-\bigg|_{s=0}
-}
-\uncover<6->{=
-\gamma(t) \dot{\gamma}(0)
-}
-\end{align*}
-\end{block}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Differentialgleichung}
-\vspace{-10pt}
-\[
-\dot{\gamma}(t) = \gamma(t) A
-\quad
-\text{mit}
-\quad
-A=\dot{\gamma}(0)\in LG
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.50\textwidth}
-\uncover<8->{%
-\begin{block}{Lösung}
-Exponentialfunktion
-\[
-\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
-\]
-\end{block}}
-\vspace{-5pt}
-\uncover<9->{%
-\begin{block}{Kontrolle: Tangentialvektor berechnen}
-\vspace{-10pt}
-\begin{align*}
-\frac{d}{dt}e^{At}
-&\uncover<10->{=
-\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!}
-}
-\\
-&\uncover<11->{=
-\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
-}
-\\
-&\uncover<12->{=
-\sum_{k=0} A^k\frac{t^k}{k!}
-A
-}
-\uncover<13->{=
-e^{At} A
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% dg.tex -- Differentialgleichung für die Exponentialabbildung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zurück zur Lie-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
+Ableitung von $\gamma(t)$ an der Stelle $t$:
+\begin{align*}
+\dot{\gamma}(t)
+&\uncover<2->{=
+\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
+}
+\\
+&\uncover<3->{=
+\frac{d}{ds}
+\gamma(t+s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<4->{=
+\frac{d}{ds}
+\gamma(t)\gamma(s)
+\bigg|_{s=0}
+}
+\\
+&\uncover<5->{=
+\gamma(t)
+\frac{d}{ds}
+\gamma(s)
+\bigg|_{s=0}
+}
+\uncover<6->{=
+\gamma(t) \dot{\gamma}(0)
+}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Differentialgleichung}
+%\vspace{-10pt}
+\[
+\dot{\gamma}(t) = \gamma(t) A
+\quad
+\text{mit}
+\quad
+A=\dot{\gamma}(0)\in LG
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<8->{%
+\begin{block}{Lösung}
+Exponentialfunktion
+\[
+\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<9->{%
+\begin{block}{Kontrolle: Tangentialvektor berechnen}
+%\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}e^{At}
+&\uncover<10->{=
+\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!}
+}
+\\
+&\uncover<11->{=
+\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
+}
+\\
+&\uncover<12->{=
+\sum_{k=0} A^k\frac{t^k}{k!}
+A
+}
+\uncover<13->{=
+e^{At} A
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex
index 776617f..ac136f1 100644
--- a/vorlesungen/slides/7/drehanim.tex
+++ b/vorlesungen/slides/7/drehanim.tex
@@ -1,155 +1,155 @@
-%
-% template.tex -- slide template
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
-
-\makeatletter
-\hoffset=-2cm
-\advance\textwidth2cm
-\hsize\textwidth
-\columnwidth\textwidth
-\makeatother
-
-\begin{frame}[t,plain]
-\vspace{-5pt}
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-\fill[color=white] (-4,-4) rectangle (9,4.5);
-
-\def\a{60}
-
-\pgfmathparse{tan(\a)}
-\xdef\T{\pgfmathresult}
-
-\pgfmathparse{-sin(\a)*cos(\a)}
-\xdef\S{\pgfmathresult}
-
-\pgfmathparse{1/cos(\a)}
-\xdef\E{\pgfmathresult}
-
-\def\N{20}
-\pgfmathparse{2*\N}
-\xdef\Nzwei{\pgfmathresult}
-\pgfmathparse{3*\N}
-\xdef\Ndrei{\pgfmathresult}
-
-\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
-\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
-\begin{itemize}
-\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
-as shear mappings and axis rescalings
-\[
-A=
-\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
-\begin{pmatrix}1&s\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\t&1\end{pmatrix}
-\]
-\item Most rotations can be decomposed into a product of
-shear mappings and axis rescalings
-\end{itemize}
-\end{block}
-\end{minipage}};
-
-\foreach \d in {1,2,...,\Ndrei}{
- % Scherung in Y-Richtung
- \ifnum \d>\N
- \pgfmathparse{\T}
- \else
- \pgfmathparse{\T*(\d-1)/(\N-1)}
- \fi
- \xdef\t{\pgfmathresult}
-
- % Scherung in X-Richtung
- \ifnum \d>\Nzwei
- \xdef\s{\S}
- \else
- \ifnum \d<\N
- \xdef\s{0}
- \else
- \ifnum \d=\N
- \xdef\s{0}
- \else
- \pgfmathparse{\S*(\d-\N-1)/(\N-1)}
- \xdef\s{\pgfmathresult}
- \fi
- \fi
- \fi
-
- % Reskalierung der Achsen
- \ifnum \d>\Nzwei
- \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
- \else
- \pgfmathparse{1}
- \fi
- \xdef\e{\pgfmathresult}
-
- % Matrixelemente
- \pgfmathparse{(\e)*((\s)*(\t)+1)}
- \xdef\A{\pgfmathresult}
-
- \pgfmathparse{(\e)*(\s)}
- \xdef\B{\pgfmathresult}
-
- \pgfmathparse{(\t)/(\e)}
- \xdef\C{\pgfmathresult}
-
- \pgfmathparse{1/(\e)}
- \xdef\D{\pgfmathresult}
-
- \only<\d>{
- \node at (5.0,-0.9) [below right] {$
- \begin{aligned}
- t &= \t \\
- s &= \s \\
- d &= \e \\
- D &= \begin{pmatrix}
- \A&\B\\
- \C&\D
- \end{pmatrix}
- \qquad
- \only<60>{\checkmark}
- \end{aligned}
- $};
- }
-
- \begin{scope}
- \clip (-4.05,-4.05) rectangle (4.05,4.05);
- \only<\d>{
- \foreach \x in {-6,...,6}{
- \draw[color=blue,line width=0.5pt]
- \punkt{\x}{-12} -- \punkt{\x}{12};
- }
- \foreach \y in {-12,...,12}{
- \draw[color=darkgreen,line width=0.5pt]
- \punkt{-6}{\y} -- \punkt{6}{\y};
- }
-
- \foreach \r in {1,2,3,4}{
- \draw[color=red] plot[domain=0:359,samples=360]
- ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
- --
- cycle;
- }
- }
- \end{scope}
-
-% \uncover<\d>{
-% \node at (5,4) {\d};
-% }
-}
-
-\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
-\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
-
-\end{tikzpicture}
-\end{center}
-\end{frame}
-\egroup
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
+
+\makeatletter
+\hoffset=-2cm
+\advance\textwidth2cm
+\hsize\textwidth
+\columnwidth\textwidth
+\makeatother
+
+\begin{frame}[t,plain]
+\vspace{-5pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=white] (-4,-4) rectangle (9,4.5);
+
+\def\a{60}
+
+\pgfmathparse{tan(\a)}
+\xdef\T{\pgfmathresult}
+
+\pgfmathparse{-sin(\a)*cos(\a)}
+\xdef\S{\pgfmathresult}
+
+\pgfmathparse{1/cos(\a)}
+\xdef\E{\pgfmathresult}
+
+\def\N{20}
+\pgfmathparse{2*\N}
+\xdef\Nzwei{\pgfmathresult}
+\pgfmathparse{3*\N}
+\xdef\Ndrei{\pgfmathresult}
+
+\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
+\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
+\begin{itemize}
+\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
+as shear mappings and axis rescalings
+\[
+A=
+\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
+\begin{pmatrix}1&s\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\t&1\end{pmatrix}
+\]
+\item Most rotations can be decomposed into a product of
+shear mappings and axis rescalings
+\end{itemize}
+\end{block}
+\end{minipage}};
+
+\foreach \d in {1,2,...,\Ndrei}{
+ % Scherung in Y-Richtung
+ \ifnum \d>\N
+ \pgfmathparse{\T}
+ \else
+ \pgfmathparse{\T*(\d-1)/(\N-1)}
+ \fi
+ \xdef\t{\pgfmathresult}
+
+ % Scherung in X-Richtung
+ \ifnum \d>\Nzwei
+ \xdef\s{\S}
+ \else
+ \ifnum \d<\N
+ \xdef\s{0}
+ \else
+ \ifnum \d=\N
+ \xdef\s{0}
+ \else
+ \pgfmathparse{\S*(\d-\N-1)/(\N-1)}
+ \xdef\s{\pgfmathresult}
+ \fi
+ \fi
+ \fi
+
+ % Reskalierung der Achsen
+ \ifnum \d>\Nzwei
+ \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
+ \else
+ \pgfmathparse{1}
+ \fi
+ \xdef\e{\pgfmathresult}
+
+ % Matrixelemente
+ \pgfmathparse{(\e)*((\s)*(\t)+1)}
+ \xdef\A{\pgfmathresult}
+
+ \pgfmathparse{(\e)*(\s)}
+ \xdef\B{\pgfmathresult}
+
+ \pgfmathparse{(\t)/(\e)}
+ \xdef\C{\pgfmathresult}
+
+ \pgfmathparse{1/(\e)}
+ \xdef\D{\pgfmathresult}
+
+ \only<\d>{
+ \node at (5.0,-0.9) [below right] {$
+ \begin{aligned}
+ t &= \t \\
+ s &= \s \\
+ d &= \e \\
+ D &= \begin{pmatrix}
+ \A&\B\\
+ \C&\D
+ \end{pmatrix}
+ \qquad
+ \only<60>{\checkmark}
+ \end{aligned}
+ $};
+ }
+
+ \begin{scope}
+ \clip (-4.05,-4.05) rectangle (4.05,4.05);
+ \only<\d>{
+ \foreach \x in {-6,...,6}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{\x}{-12} -- \punkt{\x}{12};
+ }
+ \foreach \y in {-12,...,12}{
+ \draw[color=darkgreen,line width=0.5pt]
+ \punkt{-6}{\y} -- \punkt{6}{\y};
+ }
+
+ \foreach \r in {1,2,3,4}{
+ \draw[color=red] plot[domain=0:359,samples=360]
+ ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
+ --
+ cycle;
+ }
+ }
+ \end{scope}
+
+% \uncover<\d>{
+% \node at (5,4) {\d};
+% }
+}
+
+\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
+\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
index e7b4a92..02201d4 100644
--- a/vorlesungen/slides/7/drehung.tex
+++ b/vorlesungen/slides/7/drehung.tex
@@ -1,132 +1,132 @@
-%
-% drehung.tex -- Drehung aus streckungen
-%
-% (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{Drehung aus Streckungen und Scherungen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.38\textwidth}
-\begin{block}{Drehung}
-{\color{blue}Längen}, {\color<2->{blue}Winkel},
-{\color<2->{darkgreen}Orientierung}
-erhalten
-\uncover<2->{
-\[
-\operatorname{SO}(2)
-=
-{\color{blue}\operatorname{O}(2)}
-\cap
-{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
-\]}
-\vspace{-20pt}
-\end{block}
-\uncover<3->{%
-\begin{block}{Zusammensetzung}
-Eine Drehung muss als Zusammensetzung geschrieben werden können:
-\[
-D_{\alpha}
-=
-\begin{pmatrix}
-\cos\alpha & -\sin\alpha\\
-\sin\alpha &\phantom{-}\cos\alpha
-\end{pmatrix}
-=
-DST
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<12->{%
-\begin{block}{Beispiel}
-\vspace{-12pt}
-\[
-D_{60^\circ}
-=
-{\tiny
-\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
-\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
-}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.58\textwidth}
-\uncover<4->{%
-\begin{block}{Ansatz}
-\vspace{-12pt}
-\begin{align*}
-DST
-&=
-\begin{pmatrix}
-c^{-1}&0\\
- 0 &c
-\end{pmatrix}
-\begin{pmatrix}
-1&-s\\
-0&1
-\end{pmatrix}
-\begin{pmatrix}
-1&0\\
-t&1
-\end{pmatrix}
-\\
-&\uncover<5->{=
-\begin{pmatrix}
-c^{-1}&0\\
- 0 &c
-\end{pmatrix}
-\begin{pmatrix}
-1-st&-s\\
- t& 1
-\end{pmatrix}
-}
-\\
-&\uncover<6->{=
-\begin{pmatrix}
-{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
-{\color<9->{blue}ct}&{\color<8->{red}c}
-\end{pmatrix}}
-\uncover<7->{=
-\begin{pmatrix}
-{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
-{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha}
-\end{pmatrix}}
-\end{align*}
-\end{block}}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Koeffizientenvergleich}
-\vspace{-15pt}
-\begin{align*}
-\uncover<8->{
-{\color{red} c}
-&=
-{\color{red}\cos\alpha }}
-&&
-&
-\uncover<9->{
-{\color{blue}
-t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
-\uncover<10->{
-{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
-&
-&\Rightarrow&
-{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
-}
-\\
-\uncover<11->{
-{\color{orange} (1-st)c^{-t}}
-&=
-\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% drehung.tex -- Drehung aus streckungen
+%
+% (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{Drehung aus Streckungen und Scherungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Drehung}
+{\color{blue}Längen}, {\color<2->{blue}Winkel},
+{\color<2->{darkgreen}Orientierung}
+erhalten
+\uncover<2->{
+\[
+\operatorname{SO}(2)
+=
+{\color{blue}\operatorname{O}(2)}
+\cap
+{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
+\]}
+\vspace{-20pt}
+\end{block}
+\uncover<3->{%
+\begin{block}{Zusammensetzung}
+Eine Drehung muss als Zusammensetzung geschrieben werden können:
+\[
+D_{\alpha}
+=
+\begin{pmatrix}
+\cos\alpha & -\sin\alpha\\
+\sin\alpha &\phantom{-}\cos\alpha
+\end{pmatrix}
+=
+DST
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<12->{%
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
+}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\begin{block}{Ansatz}
+%\vspace{-12pt}
+\begin{align*}
+DST
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+ 0 &c
+\end{pmatrix}
+\begin{pmatrix}
+1&-s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&0\\
+t&1
+\end{pmatrix}
+\\
+&\uncover<5->{=
+\begin{pmatrix}
+c^{-1}&0\\
+ 0 &c
+\end{pmatrix}
+\begin{pmatrix}
+1-st&-s\\
+ t& 1
+\end{pmatrix}
+}
+\\
+&\uncover<6->{=
+\begin{pmatrix}
+{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
+{\color<9->{blue}ct}&{\color<8->{red}c}
+\end{pmatrix}}
+\uncover<7->{=
+\begin{pmatrix}
+{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
+{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Koeffizientenvergleich}
+%\vspace{-15pt}
+\begin{align*}
+\uncover<8->{
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }}
+&&
+&
+\uncover<9->{
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
+\uncover<10->{
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+}
+\\
+\uncover<11->{
+{\color{orange} (1-st)c^{-t}}
+&=
+\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
index e9699a6..a32affd 100644
--- a/vorlesungen/slides/7/einparameter.tex
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -1,93 +1,93 @@
-%
-% einparameter.tex -- Einparameter Untergruppen
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Einparameter-Untergruppen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Definition}
-Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
-die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit}
-\[
-\uncover<3->{
-\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
-}
-\]
-\end{block}
-\uncover<4->{%
-\begin{block}{Drehungen}
-Drehmatrizen bilden Einparameter- Untergruppen
-\begin{align*}
-t \mapsto D_{x,t}
-&=
-\begin{pmatrix}
-1&0&0\\
-0&\cos t&-\sin t\\
-0&\sin t& \cos t
-\end{pmatrix}
-\\
-D_{x,t}D_{x,s}
-&=
-D_{x,t+s}
-\end{align*}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<5->{%
-\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
-\[
-\begin{pmatrix}
-1&s\\
-0&1
-\end{pmatrix}
-\begin{pmatrix}
-1&t\\
-0&1
-\end{pmatrix}
-=
-\begin{pmatrix}
-1&s+t\\
-0&1
-\end{pmatrix}
-\]
-\end{block}}
-\vspace{-12pt}
-\uncover<6->{%
-\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
-\[
-\begin{pmatrix}
-e^s&0\\0&e^{-s}
-\end{pmatrix}
-\begin{pmatrix}
-e^t&0\\0&e^{-t}
-\end{pmatrix}
-=
-\begin{pmatrix}
-e^{t+s}&0\\0&e^{-(t+s)}
-\end{pmatrix}
-\]
-\end{block}}
-\vspace{-12pt}
-\uncover<7->{%
-\begin{block}{Gemischt}
-\vspace{-12pt}
-\begin{gather*}
-A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
-\\
-\text{dank}\quad
-\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
-=I
-\end{gather*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% einparameter.tex -- Einparameter Untergruppen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Einparameter-Untergruppen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
+die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit}
+\[
+\uncover<3->{
+\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
+}
+\]
+\end{block}
+\uncover<4->{%
+\begin{block}{Drehungen}
+Drehmatrizen bilden Einparameter- Untergruppen
+\begin{align*}
+t \mapsto D_{x,t}
+&=
+\begin{pmatrix}
+1&0&0\\
+0&\cos t&-\sin t\\
+0&\sin t& \cos t
+\end{pmatrix}
+\\
+D_{x,t}D_{x,s}
+&=
+D_{x,t+s}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
+%\vspace{-12pt}
+\[
+\begin{pmatrix}
+1&s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&t\\
+0&1
+\end{pmatrix}
+=
+\begin{pmatrix}
+1&s+t\\
+0&1
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<6->{%
+\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
+%\vspace{-12pt}
+\[
+\begin{pmatrix}
+e^s&0\\0&e^{-s}
+\end{pmatrix}
+\begin{pmatrix}
+e^t&0\\0&e^{-t}
+\end{pmatrix}
+=
+\begin{pmatrix}
+e^{t+s}&0\\0&e^{-(t+s)}
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<7->{%
+\begin{block}{Gemischt}
+%\vspace{-12pt}
+\begin{gather*}
+A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
+\\
+\text{dank}\quad
+\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
+=I
+\end{gather*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
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/haar.tex b/vorlesungen/slides/7/haar.tex
new file mode 100644
index 0000000..454dd69
--- /dev/null
+++ b/vorlesungen/slides/7/haar.tex
@@ -0,0 +1,84 @@
+%
+% haar.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Haar-Mass}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Invariantes Mass}
+Auf jeder lokalkompakten Gruppe $G$ gibt es ein \only<2->{invariantes }%
+Integral
+\begin{align*}
+\uncover<2->{\text{rechts:}}&&
+\int_G f(g)\,d\mu(g)
+&\uncover<2->{=
+\int_G f(gh)\,d\mu(g)}
+\\
+\uncover<3->{
+\text{links:}&&
+\int_G f(g)\,d\mu(g)
+&=
+\int_G f(hg)\,d\mu(g)}
+\end{align*}
+
+\end{block}
+\uncover<7->{%
+\begin{block}{Modulus-Funktion}
+$\mu$ linksinvariant, dann ist die Rechtsverschiebung ebenfalls
+linksinvariant
+\[
+\int_G f(gh) \, d\mu(g)
+\uncover<8->{
+=
+\int_G f(g) \Delta(h)\, d\mu(g)
+}
+\]
+\uncover<9->{$\Delta(h)$ heisst Modulus-Funktion}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Beispiel: $G=\mathbb{R}$}
+\[
+\int_Gf(g)\,d\mu(g)
+=
+\int_{-\infty}^{\infty} f(x)\,dx
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Beispiel: $\operatorname{SO}(2)$}
+\[
+\int_{\operatorname{SO}(2)}
+f(g)\,d\mu(g)
+=
+\frac{1}{2\pi}
+\int_{0}^{2\pi} f(D_{\alpha})\,d\alpha
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<6->{%
+\begin{block}{Beispiel: $G$ endlich}
+\[
+\int_G f(g)\,d\mu(g) = \frac{1}{|G|}\sum_{g\in G}f(g)
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Unimodular}
+$\Delta(h)=1$ heisst rechtsinvariant = linksinvariant
+\\
+\uncover<11->{%
+$G$ kompakt $\Rightarrow$ unimodular
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/hopf.tex b/vorlesungen/slides/7/hopf.tex
new file mode 100644
index 0000000..a90737f
--- /dev/null
+++ b/vorlesungen/slides/7/hopf.tex
@@ -0,0 +1,69 @@
+%
+% hopf.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Orbit-Räume}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aktion von $\operatorname{SO}(3)$ auf $S^2$}
+\begin{align*}
+S^2 &= \{x\in\mathbb{R}^3\;|\; |x|=1\}
+\\
+\operatorname{SO}(3) \times S^2 &\to S^2: (g, x) \mapsto gx
+\end{align*}
+\uncover<2->{%
+Allgemein: Aktion von $G$ auf $X$
+\begin{align*}
+\text{links:}&&
+G\times X \to X &: (g,x) \mapsto gx
+\\
+\text{rechts:}&&
+X\times G \to X &: (x,g) \mapsto xg
+\end{align*}}
+\end{block}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{block}{Stabilisator}
+Zu $x\in X$ gibt es eine Untergruppe
+\begin{align*}
+G_x = \{g\in G\;|\; gx=x\},
+\end{align*}
+der {\em Stabilisator} von $x$.
+
+\uncover<4->{%
+Der Stabilisator von $v\in S^2$ ist die Gruppe der Drehungen um
+die Achse $v$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Quotient}
+$G$ operiert von rechts auf $X$
+\[
+X/G = \{ xG \;|\; x\in X\}
+\]
+heisst Quotient
+\end{block}}
+\uncover<6->{
+\begin{block}{$\operatorname{SO}(3)/\operatorname{SO}(2)$}
+Wähle $\operatorname{SO}(2)$ als Drehungen um die $z$-Achse:
+\[
+\operatorname{SO}(3) \to S^2
+:
+g \mapsto \text{letzte Spalte von $g$}
+\]
+\uncover<7->{Daher
+\[
+S^2 \cong \operatorname{SO}(3) / \operatorname{SO}(2)
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
index 9de1c34..6f99bc3 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -1,19 +1,29 @@
-#
-# Makefile -- Illustrationen zu
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-# 
-all: rodriguez.jpg
-
-rodriguez.png: rodriguez.pov
- povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
-
-rodriguez.jpg: rodriguez.png
- convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
-
-commutator: commutator.ini commutator.pov common.inc
- povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
-jpg:
- for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
-
-
+#
+# Makefile -- Illustrationen zu
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+# 
+all: rodriguez.jpg test.png
+
+rodriguez.png: rodriguez.pov
+ povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
+
+rodriguez.jpg: rodriguez.png
+ convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
+
+commutator: commutator.ini commutator.pov common.inc
+ povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
+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/common.inc b/vorlesungen/slides/7/images/common.inc
index b028956..0e27c9a 100644
--- a/vorlesungen/slides/7/images/common.inc
+++ b/vorlesungen/slides/7/images/common.inc
@@ -1,70 +1,70 @@
-//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.025;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
- location <18, 15, -50>
- look_at <0.0, 0.5, 0>
- right 16/9 * x * imagescale
- up y * imagescale
-}
-
-light_source {
- <-40, 30, -50> color White
- area_light <1,0,0> <0,0,1>, 10, 10
- adaptive 1
- jitter
-}
-
-sky_sphere {
- pigment {
- color rgb<1,1,1>
- }
-}
-
-#macro arrow(from, to, arrowthickness, c)
-#declare arrowdirection = vnormalize(to - from);
-#declare arrowlength = vlength(to - from);
-union {
- sphere {
- from, 1.1 * arrowthickness
- }
- cylinder {
- from,
- from + (arrowlength - 5 * arrowthickness) * arrowdirection,
- arrowthickness
- }
- cone {
- from + (arrowlength - 5 * arrowthickness) * arrowdirection,
- 2 * arrowthickness,
- to,
- 0
- }
- pigment {
- color c
- }
- finish {
- specular 0.9
- metallic
- }
-}
-#end
-
-#declare l = 1.2;
-
-arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
-arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
-arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
-
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.025;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <18, 15, -50>
+ look_at <0.0, 0.5, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-40, 30, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare l = 1.2;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini
index 44a5ac5..8c2211e 100644
--- a/vorlesungen/slides/7/images/commutator.ini
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -1,8 +1,8 @@
-Input_File_Name=commutator.pov
-Initial_Frame=1
-Final_Frame=60
-Initial_Clock=1
-Final_Clock=60
-Cyclic_Animation=off
-Pause_when_Done=off
-
+Input_File_Name=commutator.pov
+Initial_Frame=1
+Final_Frame=60
+Initial_Clock=1
+Final_Clock=60
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m
index 3f5ea17..5a448db 100644
--- a/vorlesungen/slides/7/images/commutator.m
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -1,111 +1,111 @@
-#
-# commutator.m
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-
-X = [
- 0, 0, 0;
- 0, 0, -1;
- 0, 1, 0
-];
-
-Y = [
- 0, 0, 1;
- 0, 0, 0;
- -1, 0, 0
-];
-
-Z = [
- 0, -1, 0;
- 1, 0, 0;
- 0, 0, 0
-];
-
-function retval = Dx(alpha)
- retval = [
- 1, 0, 0 ;
- 0, cos(alpha), -sin(alpha);
- 0, sin(alpha), cos(alpha)
- ];
-end
-
-function retval = Dy(beta)
- retval = [
- cos(beta), 0, sin(beta);
- 0, 1, 0 ;
- -sin(beta), 0, cos(beta)
- ];
-end
-
-t = 0.9;
-P = Dx(t) * Dy(t)
-Q = Dy(t) * Dx(t)
-P - Q
-(P - Q) * [0;0;1]
-
-function retval = kurven(filename, t)
- retval = -1;
- N = 20;
- fn = fopen(filename, "w");
- fprintf(fn, "//\n");
- fprintf(fn, "// %s\n", filename);
- fprintf(fn, "//\n");
- fprintf(fn, "#macro XYkurve()\n");
- for i = (0:N-1)
- v1 = Dx(t * i / N) * [0;0;1];
- v2 = Dx(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- for i = (0:N-1)
- v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
- v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#end\n");
- fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#macro YXkurve()\n");
- for i = (0:N-1)
- v1 = Dy(t * i / N) * [0;0;1];
- v2 = Dy(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- for i = (0:N-1)
- v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
- v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#end\n");
- fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
- v2(1,1), v2(3,1), v2(2,1));
-
- fclose(fn);
- retval = 0;
-end
-
-function retval = kurve(i)
- n = pi / 180;
- filename = sprintf("f/%04d.inc", i);
- kurven(filename, n * i);
-end
-
-for i = (1:60)
- kurve(i);
-end
+#
+# commutator.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+X = [
+ 0, 0, 0;
+ 0, 0, -1;
+ 0, 1, 0
+];
+
+Y = [
+ 0, 0, 1;
+ 0, 0, 0;
+ -1, 0, 0
+];
+
+Z = [
+ 0, -1, 0;
+ 1, 0, 0;
+ 0, 0, 0
+];
+
+function retval = Dx(alpha)
+ retval = [
+ 1, 0, 0 ;
+ 0, cos(alpha), -sin(alpha);
+ 0, sin(alpha), cos(alpha)
+ ];
+end
+
+function retval = Dy(beta)
+ retval = [
+ cos(beta), 0, sin(beta);
+ 0, 1, 0 ;
+ -sin(beta), 0, cos(beta)
+ ];
+end
+
+t = 0.9;
+P = Dx(t) * Dy(t)
+Q = Dy(t) * Dx(t)
+P - Q
+(P - Q) * [0;0;1]
+
+function retval = kurven(filename, t)
+ retval = -1;
+ N = 20;
+ fn = fopen(filename, "w");
+ fprintf(fn, "//\n");
+ fprintf(fn, "// %s\n", filename);
+ fprintf(fn, "//\n");
+ fprintf(fn, "#macro XYkurve()\n");
+ for i = (0:N-1)
+ v1 = Dx(t * i / N) * [0;0;1];
+ v2 = Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
+ v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#macro YXkurve()\n");
+ for i = (0:N-1)
+ v1 = Dy(t * i / N) * [0;0;1];
+ v2 = Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
+ v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+
+ fclose(fn);
+ retval = 0;
+end
+
+function retval = kurve(i)
+ n = pi / 180;
+ filename = sprintf("f/%04d.inc", i);
+ kurven(filename, n * i);
+end
+
+for i = (1:60)
+ kurve(i);
+end
diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov
index 8229a06..9ae11b9 100644
--- a/vorlesungen/slides/7/images/commutator.pov
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -1,59 +1,59 @@
-//
-// commutator.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-sphere { O, 0.99
- pigment {
- color rgbt<1,1,1,0.5>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-#declare filename = concat("f/", str(clock, -4, 0), ".inc");
-
-#include filename
-
-#declare n1 = vcross(<0,1,0>, finalXY);
-#declare n2 = vcross(<0,1,0>, finalYX);
-
-intersection {
- sphere { O, 1 }
- plane { -n1, 0 }
- plane { n2, 0 }
- pigment {
- color rgb<0,0.4,0.1>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-union {
- XYkurve()
- pigment {
- color Red
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-union {
- YXkurve()
- pigment {
- color Blue
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+sphere { O, 0.99
+ pigment {
+ color rgbt<1,1,1,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+#declare filename = concat("f/", str(clock, -4, 0), ".inc");
+
+#include filename
+
+#declare n1 = vcross(<0,1,0>, finalXY);
+#declare n2 = vcross(<0,1,0>, finalYX);
+
+intersection {
+ sphere { O, 1 }
+ plane { -n1, 0 }
+ plane { n2, 0 }
+ pigment {
+ color rgb<0,0.4,0.1>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ XYkurve()
+ pigment {
+ color Red
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ YXkurve()
+ pigment {
+ color Blue
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
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/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
index 62306f8..07aec19 100644
--- a/vorlesungen/slides/7/images/rodriguez.pov
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -1,118 +1,118 @@
-//
-// rodriguez.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.020;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
- location <8, 15, -50>
- look_at <0.1, 0.475, 0>
- right 16/9 * x * imagescale
- up y * imagescale
-}
-
-light_source {
- <-4, 20, -50> color White
- area_light <1,0,0> <0,0,1>, 10, 10
- adaptive 1
- jitter
-}
-
-sky_sphere {
- pigment {
- color rgb<1,1,1>
- }
-}
-
-#macro arrow(from, to, arrowthickness, c)
-#declare arrowdirection = vnormalize(to - from);
-#declare arrowlength = vlength(to - from);
-union {
- sphere {
- from, 1.1 * arrowthickness
- }
- cylinder {
- from,
- from + (arrowlength - 5 * arrowthickness) * arrowdirection,
- arrowthickness
- }
- cone {
- from + (arrowlength - 5 * arrowthickness) * arrowdirection,
- 2 * arrowthickness,
- to,
- 0
- }
- pigment {
- color c
- }
- finish {
- specular 0.9
- metallic
- }
-}
-#end
-
-#declare K = vnormalize(<0.2,1,0.1>);
-#declare X = vnormalize(<1.1,1,-1.2>);
-#declare O = <0,0,0>;
-
-#declare r = vlength(vcross(K, X)) / vlength(K);
-
-#declare l = 1.0;
-
-arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
-arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
-arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
-
-arrow(O, X, at, Red)
-arrow(O, K, at, Blue)
-
-#macro punkt(H,phi)
- ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
-#end
-
-cone { vdot(K, X) * K, r, O, 0
- pigment {
- color rgbt<0.6,0.6,0.6,0.5>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-
-union {
- #declare phistep = pi / 100;
- #declare phi = 0;
- #while (phi < 2 * pi - phistep/2)
- sphere { punkt(K, phi), at/2 }
- cylinder {
- punkt(X, phi),
- punkt(X, phi + phistep),
- at/2
- }
- #declare phi = phi + phistep;
- #end
- pigment {
- color Orange
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
-#declare Darkgreen = rgb<0,0.5,0>;
-arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
+//
+// rodriguez.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.020;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <8, 15, -50>
+ look_at <0.1, 0.475, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-4, 20, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare K = vnormalize(<0.2,1,0.1>);
+#declare X = vnormalize(<1.1,1,-1.2>);
+#declare O = <0,0,0>;
+
+#declare r = vlength(vcross(K, X)) / vlength(K);
+
+#declare l = 1.0;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
+arrow(O, X, at, Red)
+arrow(O, K, at, Blue)
+
+#macro punkt(H,phi)
+ ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
+#end
+
+cone { vdot(K, X) * K, r, O, 0
+ pigment {
+ color rgbt<0.6,0.6,0.6,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+
+union {
+ #declare phistep = pi / 100;
+ #declare phi = 0;
+ #while (phi < 2 * pi - phistep/2)
+ sphere { punkt(K, phi), at/2 }
+ cylinder {
+ punkt(X, phi),
+ punkt(X, phi + phistep),
+ at/2
+ }
+ #declare phi = phi + phistep;
+ #end
+ pigment {
+ color Orange
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
+#declare Darkgreen = rgb<0,0.5,0>;
+arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
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/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
index 9000160..84bf034 100644
--- a/vorlesungen/slides/7/kommutator.tex
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -1,166 +1,166 @@
-%
-% template.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{Kommutator in $\operatorname{SO}(3)$}
-\vspace{-20pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\t{14.0cm}
-\ifthenelse{\boolean{presentation}}{
-\only<1>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};}
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-\only<59>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}
-}{}
-\only<60>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};}
-\coordinate (A) at (-0.3,3);
-\coordinate (B) at (-1.1,2);
-\coordinate (C) at (-2.1,-1.2);
-\draw[->,color=red,line width=1.4pt]
- (A)
- to[out=-143,in=60]
- (B)
- to[out=-120,in=80]
- (C);
-%\fill[color=red] (B) circle[radius=0.08];
-\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$};
-\coordinate (D) at (0.3,3.2);
-\coordinate (E) at (1.8,2.8);
-\coordinate (F) at (5.2,-0.3);
-\draw[->,color=blue,line width=1.4pt]
- (D)
- to[out=-10,in=157]
- (E)
- to[out=-23,in=120]
- (F);
-\fill[color=blue] (E) circle[radius=0.08];
-\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$};
-\draw[->,color=darkgreen,line width=1.4pt]
- (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6);
-\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$};
-\end{tikzpicture}
-\end{center}
-\end{frame}
-\egroup
+%
+% template.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{Kommutator in $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\t{14.0cm}
+\ifthenelse{\boolean{presentation}}{
+\only<1>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};}
+\only<2>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};}
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+\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};}
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+\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};}
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+\only<28>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};}
+\only<29>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};}
+\only<30>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};}
+\only<31>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};}
+\only<32>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};}
+\only<33>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};}
+\only<34>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};}
+\only<35>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};}
+\only<36>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};}
+\only<37>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};}
+\only<38>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};}
+\only<39>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c39.jpg}};}
+\only<40>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};}
+\only<41>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c41.jpg}};}
+\only<42>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};}
+\only<43>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};}
+\only<44>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};}
+\only<45>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};}
+\only<46>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};}
+\only<47>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c47.jpg}};}
+\only<48>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};}
+\only<49>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};}
+\only<50>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c50.jpg}};}
+\only<51>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};}
+\only<52>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};}
+\only<53>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};}
+\only<54>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};}
+\only<55>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};}
+\only<56>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};}
+\only<57>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};}
+\only<58>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};}
+\only<59>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}
+}{}
+\only<60>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};}
+\coordinate (A) at (-0.3,3);
+\coordinate (B) at (-1.1,2);
+\coordinate (C) at (-2.1,-1.2);
+\draw[->,color=red,line width=1.4pt]
+ (A)
+ to[out=-143,in=60]
+ (B)
+ to[out=-120,in=80]
+ (C);
+%\fill[color=red] (B) circle[radius=0.08];
+\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$};
+\coordinate (D) at (0.3,3.2);
+\coordinate (E) at (1.8,2.8);
+\coordinate (F) at (5.2,-0.3);
+\draw[->,color=blue,line width=1.4pt]
+ (D)
+ to[out=-10,in=157]
+ (E)
+ to[out=-23,in=120]
+ (F);
+\fill[color=blue] (E) circle[radius=0.08];
+\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$};
+\draw[->,color=darkgreen,line width=1.4pt]
+ (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6);
+\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$};
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex
index bca8417..e0690eb 100644
--- a/vorlesungen/slides/7/kurven.tex
+++ b/vorlesungen/slides/7/kurven.tex
@@ -1,104 +1,104 @@
-%
-% kurven.tex -- slide template
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Kurven und Tangenten}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Kurven}
-Kurve in $\mathbb{R}^n$:
-\vspace{-12pt}
-\[
-\gamma
-\colon
-I=[a,b] \to \mathbb{R}^n
-:
-t\mapsto \gamma(t)
-\uncover<2->{
-=
-\begin{pmatrix}
-x_1(t)\\
-x_2(t)\\
-\vdots\\
-x_n(t)
-\end{pmatrix}
-}
-\]
-\vspace{-15pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\coordinate (A) at (1,0.5);
-\coordinate (B) at (4,0.5);
-\coordinate (C) at (2,2.2);
-\coordinate (D) at (5,2);
-\coordinate (E) at ($(C)+(80:2)$);
-
-\draw[color=red,line width=1.4pt]
- (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
-\fill[color=red] (C) circle[radius=0.06];
-\node[color=red] at (C) [left] {$\gamma(t)$};
-
-\uncover<4->{
- \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
- \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
-}
-
-\uncover<2->{
- \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
- \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
-}
-\end{tikzpicture}
-\end{center}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Tangenten}
-Ableitung
-\[
-\frac{d}{dt}\gamma(t)
-=
-\dot{\gamma}(t)
-=
-\begin{pmatrix}
-\dot{x}_1(t)\\
-\dot{x}_2(t)\\
-\vdots\\
-\dot{x}_n(t)
-\end{pmatrix}
-\]
-\uncover<5->{%
-Lineare Approximation:
-\[
-\gamma(t+h)
-=
-\gamma(t)
-+
-\dot{\gamma}(t) \cdot h
-+
-o(h)
-\]}%
-\vspace{-10pt}
-\begin{itemize}
-\item<6->
-Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
-in $\mathbb{R}^n$ liegen
-\item<7->
-Gilt auch für
-\[
-\operatorname{GL}_n(\mathbb{R})
-\uncover<8->{\subset M_n(\mathbb{R})}
-\uncover<9->{ = \mathbb{R}^{n\times n}}
-\]
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% kurven.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kurven und Tangenten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Kurven}
+Kurve in $\mathbb{R}^n$:
+\vspace{-12pt}
+\[
+\gamma
+\colon
+I=[a,b] \to \mathbb{R}^n
+:
+t\mapsto \gamma(t)
+\uncover<2->{
+=
+\begin{pmatrix}
+x_1(t)\\
+x_2(t)\\
+\vdots\\
+x_n(t)
+\end{pmatrix}
+}
+\]
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (1,0.5);
+\coordinate (B) at (4,0.5);
+\coordinate (C) at (2,2.2);
+\coordinate (D) at (5,2);
+\coordinate (E) at ($(C)+(80:2)$);
+
+\draw[color=red,line width=1.4pt]
+ (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
+\fill[color=red] (C) circle[radius=0.06];
+\node[color=red] at (C) [left] {$\gamma(t)$};
+
+\uncover<4->{
+ \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
+ \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
+}
+
+\uncover<2->{
+ \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
+ \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Tangenten}
+Ableitung
+\[
+\frac{d}{dt}\gamma(t)
+=
+\dot{\gamma}(t)
+=
+\begin{pmatrix}
+\dot{x}_1(t)\\
+\dot{x}_2(t)\\
+\vdots\\
+\dot{x}_n(t)
+\end{pmatrix}
+\]
+\uncover<5->{%
+Lineare Approximation:
+\[
+\gamma(t+h)
+=
+\gamma(t)
++
+\dot{\gamma}(t) \cdot h
++
+o(h)
+\]}%
+\vspace{-10pt}
+\begin{itemize}
+\item<6->
+Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
+in $\mathbb{R}^n$ liegen
+\item<7->
+Gilt auch für
+\[
+\operatorname{GL}_n(\mathbb{R})
+\uncover<8->{\subset M_n(\mathbb{R})}
+\uncover<9->{ = \mathbb{R}^{n\times n}}
+\]
+\end{itemize}
+\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/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
index 59c9121..574467b 100644
--- a/vorlesungen/slides/7/liealgebra.tex
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -1,85 +1,85 @@
-%
-% liealgebra.tex -- Lie-Algebra
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Lie-Algebra}
-\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
-\begin{block}{Vektorraum}
-Tangentialvektoren im Punkt $I$:
-\begin{center}
-\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
-\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
-\hline
-\uncover<2->{
-\operatorname{GL}_n(\mathbb{R})
-& beliebige Matrizen
-& M_n(\mathbb{R})
-}
-\\
-\uncover<3->{
-\operatorname{O(n)}
-& antisymmetrische Matrizen
-& \operatorname{o}(n)
-}
-\\
-\uncover<4->{
-\operatorname{SL}_n(\mathbb{R})
-& spurlose Matrizen
-& \operatorname{sl}_2(\mathbb{R})
-}
-\\
-\uncover<5->{
-\operatorname{U(n)}
-& antihermitesche Matrizen
-& \operatorname{u}(n)
-}
-\\
-\uncover<6->{
-\operatorname{SU(n)}
-& spurlose, antihermitesche Matrizen
-& \operatorname{su}(n)
-}
-\end{tabular}
-\end{center}
-\end{block}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.40\textwidth}
-\uncover<7->{%
-\begin{block}{Lie-Klammer}
-Kommutator: $[A,B] = AB-BA$
-\end{block}}
-\uncover<8->{%
-\begin{block}{Nachprüfen}
-$[A,B]\in LG$
-für $A,B\in LG$
-\end{block}}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<9->{%
-\begin{block}{Algebraische Eigenschaften}
-\begin{itemize}
-\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
-\item<11-> Jacobi-Identität
-\[
-[A,[B,C]]+
-[B,[C,A]]+
-[C,[A,B]]
-= 0
-\]
-\end{itemize}
-\vspace{-13pt}
-\uncover<12->{%
-{\usebeamercolor[fg]{title}
-Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% liealgebra.tex -- Lie-Algebra
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra}
+\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
+\begin{block}{Vektorraum}
+Tangentialvektoren im Punkt $I$:
+\begin{center}
+\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
+\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
+\hline
+\uncover<2->{
+\operatorname{GL}_n(\mathbb{R})
+& beliebige Matrizen
+& M_n(\mathbb{R})
+}
+\\
+\uncover<3->{
+\operatorname{O(n)}
+& antisymmetrische Matrizen
+& \operatorname{o}(n)
+}
+\\
+\uncover<4->{
+\operatorname{SL}_n(\mathbb{R})
+& spurlose Matrizen
+& \operatorname{sl}_2(\mathbb{R})
+}
+\\
+\uncover<5->{
+\operatorname{U(n)}
+& antihermitesche Matrizen
+& \operatorname{u}(n)
+}
+\\
+\uncover<6->{
+\operatorname{SU(n)}
+& spurlose, antihermitesche Matrizen
+& \operatorname{su}(n)
+}
+\end{tabular}
+\end{center}
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\uncover<7->{%
+\begin{block}{Lie-Klammer}
+Kommutator: $[A,B] = AB-BA$
+\end{block}}
+\uncover<8->{%
+\begin{block}{Nachprüfen}
+$[A,B]\in LG$
+für $A,B\in LG$
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<9->{%
+\begin{block}{Algebraische Eigenschaften}
+\begin{itemize}
+\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
+\item<11-> Jacobi-Identität
+\[
+[A,[B,C]]+
+[B,[C,A]]+
+[C,[A,B]]
+= 0
+\]
+\end{itemize}
+\vspace{-13pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}
+Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
+}
+\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/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex
index f88042a..077dc9d 100644
--- a/vorlesungen/slides/7/mannigfaltigkeit.tex
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -1,46 +1,46 @@
-%
-% mannigfaltigkeit.tex -- Mannigfaltigkeit
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Mannigfaltigkeit}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{Definition}
-\begin{itemize}
-\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
-\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
-\[
-\varphi_\beta\circ\varphi_\alpha^{-1}
-\colon
-\varphi_\alpha(U_\alpha\cap U_\beta)
-\to
-\varphi_\beta(U_\alpha\cap U_\beta)
-\]
-\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
-\end{itemize}
-\end{block}
-\vspace{-7pt}
-\uncover<5->{%
-\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
-Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
-\end{block}}
-\vspace{-3pt}
-\uncover<6->{%
-\begin{block}{Lie-Gruppe}
-Gruppe und Mannigfaltigkeit
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% mannigfaltigkeit.tex -- Mannigfaltigkeit
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Mannigfaltigkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
+\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
+\[
+\varphi_\beta\circ\varphi_\alpha^{-1}
+\colon
+\varphi_\alpha(U_\alpha\cap U_\beta)
+\to
+\varphi_\beta(U_\alpha\cap U_\beta)
+\]
+\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
+\end{itemize}
+\end{block}
+\vspace{-7pt}
+\uncover<5->{%
+\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
+Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
+\end{block}}
+\vspace{-3pt}
+\uncover<6->{%
+\begin{block}{Lie-Gruppe}
+Gruppe und Mannigfaltigkeit
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex
index afc67c5..f3579a3 100644
--- a/vorlesungen/slides/7/parameter.tex
+++ b/vorlesungen/slides/7/parameter.tex
@@ -1,107 +1,107 @@
-%
-% parameter.tex -- Parametrisierung der Matrizen
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\definecolor{darkyellow}{rgb}{1,0.8,0}
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Drehungen Parametrisieren}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.4\textwidth}
-\begin{block}{Drehung um Achsen}
-\vspace{-12pt}
-\begin{align*}
-\uncover<2->{
-D_{x,\alpha}
-&=
-\begin{pmatrix}
-1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
-\end{pmatrix}
-}
-\\
-\uncover<3->{
-D_{y,\beta}
-&=
-\begin{pmatrix}
-\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta
-\end{pmatrix}
-}
-\\
-\uncover<4->{
-D_{z,\gamma}
-&=
-\begin{pmatrix}
-\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
-\end{pmatrix}
-}
-\intertext{\uncover<5->{beliebige Drehung:}}
-\uncover<5->{
-D
-&=
-D_{x,\alpha}
-D_{y,\beta}
-D_{z,\gamma}
-}
-\end{align*}
-\end{block}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<6->{%
-\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional}
-\uncover<7->{%
-$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
-}
-\\
-\uncover<8->{%
-$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
-}
-\[
-\uncover<9->{
-{\color{red}\vec{x}}
-\mapsto
-}
-\uncover<10->{
-({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
-\cos\omega
-+
-}
-\uncover<11->{
-({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
-+
-}
-\uncover<9->{
-{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x})
-}
-\]
-\vspace{-40pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\uncover<9->{
- \node at (0,0)
- {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}};
- \node[color=red] at (1.6,-0.9) {$\vec{x}$};
- \node[color=blue] at (0.5,2) {$\vec{k}$};
-}
-\uncover<11->{
- \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$};
-}
-\uncover<10->{
- \node[color=yellow] at (2.2,-0.2)
- {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$};
-}
-\end{tikzpicture}
-\end{center}
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-15pt}
-\uncover<5->{%
-{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
-dreidimensionale Gruppe}
-\end{frame}
-\egroup
+%
+% parameter.tex -- Parametrisierung der Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkyellow}{rgb}{1,0.8,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehungen Parametrisieren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.4\textwidth}
+\begin{block}{Drehung um Achsen}
+%\vspace{-12pt}
+\begin{align*}
+\uncover<2->{
+D_{x,\alpha}
+&=
+\begin{pmatrix}
+1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
+\end{pmatrix}
+}
+\\
+\uncover<3->{
+D_{y,\beta}
+&=
+\begin{pmatrix}
+\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta
+\end{pmatrix}
+}
+\\
+\uncover<4->{
+D_{z,\gamma}
+&=
+\begin{pmatrix}
+\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
+\end{pmatrix}
+}
+\intertext{\uncover<5->{beliebige Drehung:}}
+\uncover<5->{
+D
+&=
+D_{x,\alpha}
+D_{y,\beta}
+D_{z,\gamma}
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<6->{%
+\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional}
+\uncover<7->{%
+$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
+}
+\\
+\uncover<8->{%
+$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
+}
+\[
+\uncover<9->{
+{\color{red}\vec{x}}
+\mapsto
+}
+\uncover<10->{
+({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
+\cos\omega
++
+}
+\uncover<11->{
+({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
++
+}
+\uncover<9->{
+{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x})
+}
+\]
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+ \node at (0,0)
+ {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}};
+ \node[color=red] at (1.6,-0.9) {$\vec{x}$};
+ \node[color=blue] at (0.5,2) {$\vec{k}$};
+}
+\uncover<11->{
+ \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$};
+}
+\uncover<10->{
+ \node[color=yellow] at (2.2,-0.2)
+ {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$};
+}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\uncover<5->{%
+{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
+dreidimensionale Gruppe}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/qdreh.tex b/vorlesungen/slides/7/qdreh.tex
new file mode 100644
index 0000000..8ed512a
--- /dev/null
+++ b/vorlesungen/slides/7/qdreh.tex
@@ -0,0 +1,110 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehungen mit Quaternionen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Drehung?}
+Abbildung von $\vec{x}$ mit $\operatorname{Re}\vec{x}=0$:
+\[
+\varrho_{q}
+\colon
+\vec{x}\mapsto q\vec{x}q^{-1} = q\vec{x}\overline{q}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Achse}
+\begin{align*}
+\varrho_q(q)
+&=
+qq\overline{q}
+\uncover<3->{=
+q(qq^{-1})}
+\uncover<4->{=
+q}
+\end{align*}
+\end{block}}
+\uncover<4->{%
+\begin{block}{Norm}
+\begin{align*}
+|\varrho_q(\vec{x})|^2
+&=
+q\vec{x}\overline{q}\overline{(q\vec{x}\overline{q})}
+\uncover<5->{=
+q\vec{x}\overline{q}\overline{\overline{q}}\overline{\vec{x}}\overline{q}
+}
+\\
+&\uncover<6->{=
+q\vec{x}(\overline{q}q)\overline{\vec{x}}\overline{q}}
+\uncover<7->{=
+q(\vec{x}\overline{\vec{x}})\overline{q}}
+\uncover<8->{=
+q\overline{q}|\vec{x}|^2}
+\\
+&\uncover<9->{=
+|\vec{x}|^2}
+\end{align*}
+\uncover<10->{%
+$\Rightarrow$ $\varrho_q\in\operatorname{O}(3)$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Drehung!}
+$\vec{a},\vec{b},\vec{n}$ bilden ein on.~Rechtssystem
+\begin{align*}
+\uncover<12->{
+qa
+&=
+c\vec{a}+s\vec{n}\times \vec{a}}
+\uncover<13->{=
+c\vec{a} + s\vec{b}}
+\\
+\uncover<14->{
+q\vec{a}\overline{q}
+&=
+(c\vec{a}+s\vec{b}) c
+-(c\vec{a}+s\vec{b})\times s\vec{n}}
+\\
+&\uncover<15->{=
+c^2 \vec{a}+ sc\vec{b}
++sc\vec{b} - s^2 \vec{a}}
+\\
+&\uncover<16->{=
+\vec{a} \cos\alpha +\vec{b} \sin\alpha }
+\end{align*}
+\vspace{-5pt}
+\uncover<17->{wegen
+%\vspace{-5pt}
+\[
+\begin{aligned}
+\cos\alpha &= \cos^2\frac{\alpha}2 - \sin^2\frac{\alpha}2 &&=c^2-s^2
+\\
+\sin\alpha &= 2\cos\frac{\alpha}2\sin\frac{\alpha}2&&=2cs
+\end{aligned}\]}
+\end{block}}
+\vspace{-18pt}
+\uncover<18->{%
+\begin{block}{Matrix}
+\[
+D
+=
+\tiny
+\begin{pmatrix}
+1-2(q_2^2+q_3^2)&-2q_0q_3+2q_1q_2&-2q_0q_2+2q_1q_3\\
+ 2q_0q_3+2q_1q_2&1-2(q_1^2+q_3^2)&-2q_0q_1+2q_2q_3\\
+-2q_0q_2+2q_1q_3& 2q_0q_1+2q_2q_3&1-2(q_1^2+q_2^2)
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex
new file mode 100644
index 0000000..f526366
--- /dev/null
+++ b/vorlesungen/slides/7/quaternionen.tex
@@ -0,0 +1,74 @@
+%
+% quaternionen.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quaternionen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Quaternionen}
+$4$-dimensionaler $\mathbb{R}$-Vektorraum
+\[
+\mathbb{H}
+=
+\langle 1,i,j,k\rangle_{\mathbb{R}}
+\]
+mit Rechenregeln
+\[
+i^2=j^2=k^2=ijk=-1
+\]
+$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$
+\begin{itemize}
+\item<2-> Realteil: $\operatorname{Re}x=x_0$
+\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$
+\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$
+\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$
+\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<7->{%
+\begin{block}{Skalarprodukt und Vektorprodukt}
+\begin{align*}
+pq
+&=
+\operatorname{Re}p \operatorname{Re}q
+-
+\operatorname{Im}p\cdot \operatorname{Im}q
+\\
+&\phantom{=}
++
+\operatorname{Re}p\operatorname{Im}q
++
+\operatorname{Im}p\operatorname{Re}q
++
+\operatorname{Im}p\times\operatorname{Im}q
+\end{align*}
+\end{block}}
+\uncover<8->{%
+\begin{block}{Einheitsquaternionen}
+$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$
+\end{block}}
+\uncover<9->{%
+\begin{block}{Polardarstellung}
+\[
+q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2
+\]
+\vspace{-8pt}
+\begin{itemize}
+\item<10->
+Drehmatrix: 9 Parameter, 6 Bedingungen
+\item<11->
+Quaternionen: 4 Parameter, 1 Bedingung
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex
index d74b7d0..cd974c9 100644
--- a/vorlesungen/slides/7/semi.tex
+++ b/vorlesungen/slides/7/semi.tex
@@ -1,117 +1,117 @@
-%
-% semi.tex -- Beispiele: semidirekte Produkte
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Drehung/Skalierung und Verschiebung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Skalierung und Verschiebung}
-Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
-\\
-Wirkung auf $\mathbb{R}$:
-\[
-x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
-\]
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Drehung und Verschiebung}
-Gruppe
-$G=
-\{ (\alpha,\vec{t})
-\;|\;
-\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
-\}$
-Wirkung auf $\mathbb{R}^2$:
-\[
-\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-15pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Verknüpfung}
-\vspace{-15pt}
-\begin{align*}
-(e^{s_1},t_1)(e^{s_2},t_2)x
-&\uncover<4->{=
-(e^{s_1},t_1)(e^{s_2}x+t_2)}
-\\
-&\uncover<5->{=
-e^{s_1+s_2}x + e^{s_1}t_2+t_1}
-\\
-\uncover<6->{
-(e^{s_1},t_1)(e^{s_2},t_2)
-&=
-(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)}
-\end{align*}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Verknüpfung}
-\vspace{-15pt}
-\begin{align*}
-(\alpha_1,\vec{t}_1)
-(\alpha_2,\vec{t}_2)
-\vec{x}
-&\uncover<8->{=
-(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
-\\
-&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
-\\
-\uncover<10->{
-(\alpha_1,\vec{t}_1)
-(\alpha_2,\vec{t}_2)
-&=
-(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-10pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<11->{%
-\begin{block}{Matrixschreibweise}
-\vspace{-12pt}
-\[
-g=(e^s,t) =
-\begin{pmatrix}
-e^s&t\\
-0&1
-\end{pmatrix}
-\quad\text{auf}\quad
-\begin{pmatrix}x\\1\end{pmatrix}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<12->{%
-\begin{block}{Matrixschreibweise}
-\vspace{-12pt}
-\[
-g=(\alpha,\vec{t}) =
-\begin{pmatrix}
-D_{\alpha}&\vec{t}\\
-0&1
-\end{pmatrix}
-\quad\text{auf}\quad
-\begin{pmatrix}\vec{x}\\1\end{pmatrix}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% semi.tex -- Beispiele: semidirekte Produkte
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung/Skalierung und Verschiebung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalierung und Verschiebung}
+Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
+\\
+Wirkung auf $\mathbb{R}$:
+\[
+x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Drehung und Verschiebung}
+Gruppe
+$G=
+\{ (\alpha,\vec{t})
+\;|\;
+\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
+\}$
+Wirkung auf $\mathbb{R}^2$:
+\[
+\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Verknüpfung}
+%\vspace{-15pt}
+\begin{align*}
+(e^{s_1},t_1)(e^{s_2},t_2)x
+&\uncover<4->{=
+(e^{s_1},t_1)(e^{s_2}x+t_2)}
+\\
+&\uncover<5->{=
+e^{s_1+s_2}x + e^{s_1}t_2+t_1}
+\\
+\uncover<6->{
+(e^{s_1},t_1)(e^{s_2},t_2)
+&=
+(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Verknüpfung}
+%\vspace{-15pt}
+\begin{align*}
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+\vec{x}
+&\uncover<8->{=
+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
+\\
+&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
+\\
+\uncover<10->{
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+&=
+(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Matrixschreibweise}
+%\vspace{-12pt}
+\[
+g=(e^s,t) =
+\begin{pmatrix}
+e^s&t\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}x\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Matrixschreibweise}
+%\vspace{-12pt}
+\[
+g=(\alpha,\vec{t}) =
+\begin{pmatrix}
+D_{\alpha}&\vec{t}\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}\vec{x}\\1\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
index 58e87a1..a65b4f6 100644
--- a/vorlesungen/slides/7/sl2.tex
+++ b/vorlesungen/slides/7/sl2.tex
@@ -1,242 +1,242 @@
-%
-% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t,fragile]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{block}{Determinante}
-\[
-A=\begin{pmatrix}
-a&b\\
-c&d
-\end{pmatrix}
-\;\Rightarrow\;
-\det A = ad-bc
-\]
-\end{block}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Dimension}
-\[
-4\; \text{Variablen}
--
-1\; \text{Bedingung}
-=
-3\; \text{Dimensionen}
-\]
-\end{block}
-\end{column}
-\end{columns}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{columns}[t,onlytextwidth]
-\def\s{0.94}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \fill[color=blue!20] (-1,-1) rectangle (1,1);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {4,...,10}{
- \only<\d>{
- \pgfmathparse{1+(\d-4)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
- }
- }{}
- \uncover<11->{
- \xdef\t{1.6}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<3->{%
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- D
- =
- \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {11,...,17}{
- \only<\d>{
- \pgfmathparse{(\d-11)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
- }
- }{}
- \uncover<18->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<11->{
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- S
- =
- \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {18,...,24}{
- \only<\d>{
- \pgfmathparse{(\d-18)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
- }
- }{}
- \uncover<25->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<18->{%
-\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- T
- =
- \begin{pmatrix} 1&0\\t&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\end{columns}}
-\end{frame}
-\egroup
+%
+% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t,fragile]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Determinante}
+\[
+A=\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\;\Rightarrow\;
+\det A = ad-bc
+\]
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Dimension}
+\[
+4\; \text{Variablen}
+-
+1\; \text{Bedingung}
+=
+3\; \text{Dimensionen}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{columns}[t,onlytextwidth]
+\def\s{0.94}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \fill[color=blue!20] (-1,-1) rectangle (1,1);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {4,...,10}{
+ \only<\d>{
+ \pgfmathparse{1+(\d-4)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+ }
+ }{}
+ \uncover<11->{
+ \xdef\t{1.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-\t},{-1/\t}) rectangle (\t,{1/\t});
+ \foreach \x in {-2,...,2}{
+ \draw[color=red,line width=0.3pt]
+ ({\x*\t},-3) -- ({\x*\t},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{\y/\t}) -- (3,{\y/\t});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<3->{%
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ D
+ =
+ \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {11,...,17}{
+ \only<\d>{
+ \pgfmathparse{(\d-11)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+ }
+ }{}
+ \uncover<18->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ ({-1+\t*(-1)},{-1})
+ --
+ ({1+\t*(-1)},{-1})
+ --
+ ({1+\t},{1})
+ --
+ ({-1+\t},{1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<11->{
+ \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ S
+ =
+ \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+ \clip (-2.1,-2.1) rectangle (2.3,2.3);
+ \foreach \x in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+ }
+ \foreach \y in {-2,...,2}{
+ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+ }
+ \ifthenelse{\boolean{presentation}}{
+ \foreach \d in {18,...,24}{
+ \only<\d>{
+ \pgfmathparse{(\d-18)/10}
+ \xdef\t{\pgfmathresult}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+ }
+ }{}
+ \uncover<25->{
+ \xdef\t{0.6}
+ \fill[color=red!40,opacity=0.5]
+ (-1,{\t*(-1)-1})
+ --
+ (1,{\t*1-1})
+ --
+ (1,{\t*1+1})
+ --
+ (-1,{\t*(-1)+1})
+ -- cycle;
+ \foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+ }
+ \foreach \y in {-3,...,3}{
+ \draw[color=red,line width=0.3pt]
+ (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+ }
+ }
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<18->{%
+\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+ \node at (0,-2.1) {$
+ T
+ =
+ \begin{pmatrix} 1&0\\t&1\end{pmatrix}
+ $};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
index 8931a24..35d62d8 100644
--- a/vorlesungen/slides/7/symmetrien.tex
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -1,145 +1,145 @@
-%
-% symmetrien.tex -- Symmetrien
-%
-% (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{Symmetrien}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Diskrete Symmetrien}
-\begin{itemize}
-\item<2->
-Ebenen-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
-{\tiny
-\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\uncover<4->{\!,\;
-\vec{x}
-\mapsto
-\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
-}
-\]
-\vspace{-10pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{10}
-\def\b{50}
-\def\r{2}
-\coordinate (O) at (0,0);
-\coordinate (A) at (\b:\r);
-\coordinate (B) at ({180+2*\a-\b}:\r);
-\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
-\coordinate (N) at (\a:2);
-\coordinate (D) at (\a:{\r*cos(\b-\a)});
-\uncover<3->{
-\clip (-2.5,-0.45) rectangle (2.5,1.95);
-
- \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
- -- cycle;
- \draw[->,color=darkgreen] (O) -- (N);
- \node[color=darkgreen] at (N) [above] {$\vec{n}$};
-
-
- \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
- -- cycle;
- \fill[color=red] (O) circle[radius=0.06];
- \draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
- \fill[color=blue] (C) circle[radius=0.06];
- \draw[color=blue,line width=0.1pt] (A) -- (D);
- \node[color=darkgreen] at (D) [below,rotate=\a]
- {$(\vec{n}\cdot\vec{x})\vec{n}$};
- \draw[color=blue,line width=0.5pt] (A)--(B);
-
- \node[color=blue] at (A) [above right] {$\vec{x}$};
- \node[color=blue] at (B) [above left] {$\vec{x}'$};
-
- \node[color=red] at (O) [below left] {$O$};
-
- \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
- \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
-}
-
-\end{tikzpicture}
-\end{center}
-\vspace{-5pt}
-$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
-\item<5->
-Punkt-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
--
-{\tiny
-\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\]
-\end{itemize}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<6->{%
-\begin{block}{Kontinuierliche Symmetrien}
-\begin{itemize}
-\item<7-> Translation:
-\(
-\vec{x} \mapsto \vec{x} + \vec{t}
-\)
-\item<8-> Drehung:
-\vspace{-3pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{25}
-\def\r{1.3}
-\coordinate (O) at (0,0);
-\begin{scope}
-\clip (-1.1,-0.1) rectangle (2.3,2.3);
-\draw[color=red] (O) circle[radius=2];
-\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
-\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
-\node at ({0.5*\a}:1) {$\alpha$};
-\node at ({90+0.5*\a}:1) {$\alpha$};
-\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
-\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
-\end{scope}
-\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
-\end{tikzpicture}
-\end{center}
-\[
-\uncover<9->{%
-\begin{pmatrix}x\\y\end{pmatrix}
-\mapsto
-\begin{pmatrix}
-{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
-{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
-\end{pmatrix}
-\begin{pmatrix}x\\y\end{pmatrix}
-}
-\]
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<10->{%
-\begin{block}{Definition}
-Längen/Winkel bleiben erhalten
-\\
-\uncover<11->{%
-$\Rightarrow$ $\exists$ Erhaltungsgrösse}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%
+% symmetrien.tex -- Symmetrien
+%
+% (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{Symmetrien}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diskrete Symmetrien}
+\begin{itemize}
+\item<2->
+Ebenen-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+{\tiny
+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\uncover<4->{\!,\;
+\vec{x}
+\mapsto
+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
+}
+\]
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{10}
+\def\b{50}
+\def\r{2}
+\coordinate (O) at (0,0);
+\coordinate (A) at (\b:\r);
+\coordinate (B) at ({180+2*\a-\b}:\r);
+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
+\coordinate (N) at (\a:2);
+\coordinate (D) at (\a:{\r*cos(\b-\a)});
+\uncover<3->{
+\clip (-2.5,-0.45) rectangle (2.5,1.95);
+
+ \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
+ -- cycle;
+ \draw[->,color=darkgreen] (O) -- (N);
+ \node[color=darkgreen] at (N) [above] {$\vec{n}$};
+
+
+ \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
+ -- cycle;
+ \fill[color=red] (O) circle[radius=0.06];
+ \draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
+ \fill[color=blue] (C) circle[radius=0.06];
+ \draw[color=blue,line width=0.1pt] (A) -- (D);
+ \node[color=darkgreen] at (D) [below,rotate=\a]
+ {$(\vec{n}\cdot\vec{x})\vec{n}$};
+ \draw[color=blue,line width=0.5pt] (A)--(B);
+
+ \node[color=blue] at (A) [above right] {$\vec{x}$};
+ \node[color=blue] at (B) [above left] {$\vec{x}'$};
+
+ \node[color=red] at (O) [below left] {$O$};
+
+ \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
+ \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-5pt}
+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
+\item<5->
+Punkt-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+-
+{\tiny
+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Kontinuierliche Symmetrien}
+\begin{itemize}
+\item<7-> Translation:
+\(
+\vec{x} \mapsto \vec{x} + \vec{t}
+\)
+\item<8-> Drehung:
+\vspace{-3pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{25}
+\def\r{1.3}
+\coordinate (O) at (0,0);
+\begin{scope}
+\clip (-1.1,-0.1) rectangle (2.3,2.3);
+\draw[color=red] (O) circle[radius=2];
+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
+\node at ({0.5*\a}:1) {$\alpha$};
+\node at ({90+0.5*\a}:1) {$\alpha$};
+\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
+\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
+\end{scope}
+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
+\end{tikzpicture}
+\end{center}
+\[
+\uncover<9->{%
+\begin{pmatrix}x\\y\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
+\end{pmatrix}
+\begin{pmatrix}x\\y\end{pmatrix}
+}
+\]
+\end{itemize}
+\end{block}}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Definition}
+Längen/Winkel bleiben erhalten
+\\
+\uncover<11->{%
+$\Rightarrow$ $\exists$ Erhaltungsgrösse}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/ueberlagerung.tex b/vorlesungen/slides/7/ueberlagerung.tex
new file mode 100644
index 0000000..426641a
--- /dev/null
+++ b/vorlesungen/slides/7/ueberlagerung.tex
@@ -0,0 +1,98 @@
+%
+% ueberlagerung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$S^3$, $\operatorname{SU}(2)$ und $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\uncover<6->{%
+\begin{block}{Überlagerung}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (0,0);
+\coordinate (B) at (2,0);
+\coordinate (C) at (2,-2);
+\coordinate (D) at (0,-2);
+
+\uncover<7->{
+\node at (A) {$\{\pm 1\}\mathstrut$};
+}
+\uncover<6->{
+\node at (B) {$S^3\mathstrut$};
+\node at ($(B)+(0.1,0)$) [right] {$=\operatorname{SU}(2)\mathstrut$};
+}
+\uncover<7->{
+\node at (C) {$\operatorname{SO}(3)\mathstrut$};
+\node at (D) {$\{I\}\mathstrut$};
+}
+
+\uncover<7->{
+\draw[->,shorten >= 0.3cm,shorten <= 0.5cm] (A) -- (B);
+\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (D);
+\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[->,shorten >= 0.6cm,shorten <= 0.3cm] (D) -- (C);
+}
+
+\end{tikzpicture}
+\end{center}
+\begin{itemize}
+\item<7->
+$\pm q\in S^3$ $\Rightarrow$ $\varrho_{q}=\varrho_{-q}$
+\item<8->
+In der Nähe von $I$ sehen die Gruppen
+$\operatorname{SO}(3)$
+und
+$\operatorname{SU}(2)$
+``gleich'' aus
+\item<9->
+$\operatorname{SU}(2)$ ist geometrisch ``einfacher''
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{block}{Pauli-Matrizen}
+Quaternionen als $2\times 2$-Matrizen schreiben
+\begin{align*}
+1&=\begin{pmatrix}1&0\\0&1\end{pmatrix}=\sigma_0,
+&
+i&=\begin{pmatrix}0&i\\i&0\end{pmatrix}=-i\sigma_1
+\\
+j&=\begin{pmatrix}0&-1\\1&0\end{pmatrix}=-i\sigma_2,
+&
+k&=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=-i\sigma_3
+\end{align*}
+\uncover<2->{%
+erfüllen $i^2=j^2=k^2=ijk=-1$.}
+\end{block}
+\uncover<3->{%
+\begin{block}{$S^3 = \operatorname{SU}(2)$}
+\[
+a+bi+cj+dk
+=
+\begin{pmatrix}
+a+id&-c+bi\\
+c+ib&a-id
+\end{pmatrix}
+=
+A
+\]
+\begin{align*}
+\uncover<4->{
+\det A &= a^2 + b^2 + c^2 + d^2 = 1
+}
+\\
+\uncover<5->{
+A^* &= a - ib - jc - kd
+}
+\end{align*}
+\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/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex
new file mode 100644
index 0000000..6a43cd8
--- /dev/null
+++ b/vorlesungen/slides/7/zusammenhang.tex
@@ -0,0 +1,99 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zusammenhang}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zusammenhängend --- oder nicht}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\ds{2.4}
+\coordinate (A) at (0,0);
+\coordinate (B) at (\ds,0);
+\coordinate (C) at ({2*\ds},0);
+
+\node at (A) {$\operatorname{SO}(n)$};
+\node at (B) {$\operatorname{O}(n)$};
+\node at (C) {$\{\pm 1\}$};
+
+\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B);
+\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C);
+\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$};
+
+\coordinate (A2) at (0,-1.0);
+\coordinate (B2) at (\ds,-1.0);
+\coordinate (C2) at ({2*\ds},-1.0);
+
+\draw[color=blue] (A2) ellipse (1cm and 0.3cm);
+\draw[color=blue] (B2) ellipse (1cm and 0.3cm);
+\node[color=blue] at (C2) {$+1$};
+
+\coordinate (A3) at (0,-1.7);
+\coordinate (B3) at (\ds,-1.7);
+\coordinate (C3) at ({2*\ds},-1.7);
+
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2);
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3);
+
+\draw[color=red] (B3) ellipse (1cm and 0.3cm);
+\node[color=red] at (C3) {$-1$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\begin{block}{Zusammenhangskomponente von $e$}
+$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält:
+\begin{align*}
+\operatorname{SO}(n)&\subset \operatorname{O}(n)
+\\
+\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\}
+ &\subset \operatorname{GL}_n(\mathbb{R})
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenschaften}
+\begin{itemize}
+\item
+{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$,
+dann ist
+\begin{itemize}
+\item
+$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$
+\item
+$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$
+\end{itemize}
+\item
+{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann
+ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$
+$\Rightarrow hG_eh^{-1}\subset G_e$
+\end{itemize}
+\end{block}
+\begin{block}{Quotient}
+$G/G_e$ ist eine diskrete Gruppe
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (0,0);
+\coordinate (B) at (2,0);
+\coordinate (C) at (4,0);
+\node at (A) {$G_e$};
+\node at (B) {$G$};
+\node at (C) {$G/G_e$};
+\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B);
+\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C);
+\end{tikzpicture}
+\end{center}
+\vspace{-7pt}
+$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren
+\end{block}
+\end{column}
+\end{columns}
+\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/Makefile.inc b/vorlesungen/slides/Makefile.inc
index 130fa28..a9d72be 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -1,21 +1,21 @@
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-#
-include ../slides/0/Makefile.inc
-include ../slides/1/Makefile.inc
-include ../slides/2/Makefile.inc
-include ../slides/3/Makefile.inc
-include ../slides/4/Makefile.inc
-include ../slides/5/Makefile.inc
-include ../slides/6/Makefile.inc
-include ../slides/7/Makefile.inc
-include ../slides/8/Makefile.inc
-include ../slides/9/Makefile.inc
-include ../slides/a/Makefile.inc
-
-slides = \
- $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \
- $(chapter5) $(chapter6) $(chapter7) $(chapter8) $(chapter9) \
- $(chaptera)
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+#
+include ../slides/0/Makefile.inc
+include ../slides/1/Makefile.inc
+include ../slides/2/Makefile.inc
+include ../slides/3/Makefile.inc
+include ../slides/4/Makefile.inc
+include ../slides/5/Makefile.inc
+include ../slides/6/Makefile.inc
+include ../slides/7/Makefile.inc
+include ../slides/8/Makefile.inc
+include ../slides/9/Makefile.inc
+include ../slides/a/Makefile.inc
+
+slides = \
+ $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \
+ $(chapter5) $(chapter6) $(chapter7) $(chapter8) $(chapter9) \
+ $(chaptera)
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index ce63ae7..4289c44 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -1,23 +1,28 @@
-%
-% test.tex collection of all slides
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-
-%\folie{a/dc/prinzip.tex}
-%\folie{a/dc/effizient.tex}
-%\folie{a/dc/beispiel.tex}
-
-%\folie{a/ecc/gruppendh.tex}
-%\folie{a/ecc/kurve.tex}
-%\folie{a/ecc/inverse.tex}
-%\folie{a/ecc/operation.tex}
-%\folie{a/ecc/quadrieren.tex}
-%\folie{a/ecc/oakley.tex}
-
-%\folie{a/aes/bytes.tex}
-%\folie{a/aes/sinverse.tex}
-%\folie{a/aes/blocks.tex}
-\folie{a/aes/keys.tex}
-%\folie{a/aes/runden.tex}
-
+%
+% test.tex collection of all slides
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+%\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}
+