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-rw-r--r--vorlesungen/slides/10/ableitung-exp.tex60
-rw-r--r--vorlesungen/slides/10/intro.tex45
-rw-r--r--vorlesungen/slides/10/matrix-dgl.tex83
-rw-r--r--vorlesungen/slides/10/n-zu-1.tex63
-rw-r--r--vorlesungen/slides/10/potenzreihenmethode.tex91
-rw-r--r--vorlesungen/slides/10/repetition.tex119
-rw-r--r--vorlesungen/slides/10/so2.tex141
-rw-r--r--vorlesungen/slides/10/taylor.tex216
-rw-r--r--vorlesungen/slides/10/template.tex21
-rw-r--r--vorlesungen/slides/10/vektorfelder.mp361
-rw-r--r--vorlesungen/slides/10/vektorfelder.tex82
-rw-r--r--vorlesungen/slides/4/Makefile.inc62
-rw-r--r--vorlesungen/slides/4/chapter.tex53
-rw-r--r--vorlesungen/slides/4/galois/aufloesbarkeit.tex120
-rw-r--r--vorlesungen/slides/4/galois/automorphismus.tex118
-rw-r--r--vorlesungen/slides/4/galois/erweiterung.tex65
-rw-r--r--vorlesungen/slides/4/galois/images/Makefile12
-rw-r--r--vorlesungen/slides/4/galois/images/common.inc89
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-rw-r--r--vorlesungen/slides/4/galois/images/wuerfel.pov9
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-rw-r--r--vorlesungen/slides/4/galois/images/wuerfel2.pov9
-rw-r--r--vorlesungen/slides/4/galois/konstruktion.tex147
-rw-r--r--vorlesungen/slides/4/galois/quadratur.tex66
-rw-r--r--vorlesungen/slides/4/galois/radikale.tex69
-rw-r--r--vorlesungen/slides/4/galois/sn.tex87
-rw-r--r--vorlesungen/slides/4/galois/winkeldreiteilung.tex94
-rw-r--r--vorlesungen/slides/4/galois/wuerfel.tex64
-rw-r--r--vorlesungen/slides/5/potenzreihenmethode.tex2
-rw-r--r--vorlesungen/slides/6/Makefile.inc18
-rw-r--r--vorlesungen/slides/6/chapter.tex16
-rw-r--r--vorlesungen/slides/6/darstellungen/charakter.tex108
-rw-r--r--vorlesungen/slides/6/darstellungen/definition.tex59
-rw-r--r--vorlesungen/slides/6/darstellungen/irreduzibel.tex43
-rw-r--r--vorlesungen/slides/6/darstellungen/schur.tex45
-rw-r--r--vorlesungen/slides/6/darstellungen/skalarprodukt.tex39
-rw-r--r--vorlesungen/slides/6/darstellungen/summe.tex82
-rw-r--r--vorlesungen/slides/6/darstellungen/zyklisch.tex77
-rw-r--r--vorlesungen/slides/6/permutationen/matrizen.tex75
-rw-r--r--vorlesungen/slides/7/Makefile.inc22
-rw-r--r--vorlesungen/slides/7/ableitung.tex68
-rw-r--r--vorlesungen/slides/7/algebraisch.tex115
-rw-r--r--vorlesungen/slides/7/chapter.tex19
-rw-r--r--vorlesungen/slides/7/dg.tex92
-rw-r--r--vorlesungen/slides/7/drehanim.tex155
-rw-r--r--vorlesungen/slides/7/drehung.tex132
-rw-r--r--vorlesungen/slides/7/einparameter.tex93
-rw-r--r--vorlesungen/slides/7/images/Makefile19
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-rw-r--r--vorlesungen/slides/7/images/common.inc70
-rw-r--r--vorlesungen/slides/7/images/commutator.ini8
-rw-r--r--vorlesungen/slides/7/images/commutator.m111
-rw-r--r--vorlesungen/slides/7/images/commutator.pov59
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-rw-r--r--vorlesungen/slides/7/images/rodriguez.pov118
-rw-r--r--vorlesungen/slides/7/kommutator.tex166
-rw-r--r--vorlesungen/slides/7/kurven.tex104
-rw-r--r--vorlesungen/slides/7/liealgebra.tex85
-rw-r--r--vorlesungen/slides/7/mannigfaltigkeit.tex46
-rw-r--r--vorlesungen/slides/7/parameter.tex107
-rw-r--r--vorlesungen/slides/7/semi.tex117
-rw-r--r--vorlesungen/slides/7/sl2.tex242
-rw-r--r--vorlesungen/slides/7/symmetrien.tex145
-rw-r--r--vorlesungen/slides/Makefile.inc38
-rw-r--r--vorlesungen/slides/a/Makefile.inc25
-rw-r--r--vorlesungen/slides/a/aes/blocks.tex193
-rw-r--r--vorlesungen/slides/a/aes/bytes.tex96
-rw-r--r--vorlesungen/slides/a/aes/keys.tex36
-rw-r--r--vorlesungen/slides/a/aes/runden.tex47
-rw-r--r--vorlesungen/slides/a/aes/sinverse.tex15
-rw-r--r--vorlesungen/slides/a/chapter.tex23
-rw-r--r--vorlesungen/slides/a/dc/beispiel.tex54
-rw-r--r--vorlesungen/slides/a/dc/effizient.tex65
-rw-r--r--vorlesungen/slides/a/dc/naiv.txt2
-rw-r--r--vorlesungen/slides/a/dc/prinzip.tex86
-rw-r--r--vorlesungen/slides/a/ecc/gruppendh.tex51
-rw-r--r--vorlesungen/slides/a/ecc/inverse.tex48
-rw-r--r--vorlesungen/slides/a/ecc/kurve.tex56
-rw-r--r--vorlesungen/slides/a/ecc/oakley.tex85
-rw-r--r--vorlesungen/slides/a/ecc/oakley1.txt14
-rw-r--r--vorlesungen/slides/a/ecc/oakley2.txt16
-rw-r--r--vorlesungen/slides/a/ecc/oakley3.txt17
-rw-r--r--vorlesungen/slides/a/ecc/oakley4.txt17
-rw-r--r--vorlesungen/slides/a/ecc/operation.tex68
-rw-r--r--vorlesungen/slides/a/ecc/prime1.txt5
-rw-r--r--vorlesungen/slides/a/ecc/prime2.txt8
-rw-r--r--vorlesungen/slides/a/ecc/primes13
-rw-r--r--vorlesungen/slides/a/ecc/quadrieren.tex59
-rw-r--r--vorlesungen/slides/slides.tex24
-rw-r--r--vorlesungen/slides/test.tex44
150 files changed, 6109 insertions, 99 deletions
diff --git a/vorlesungen/slides/10/ableitung-exp.tex b/vorlesungen/slides/10/ableitung-exp.tex
new file mode 100644
index 0000000..10ce191
--- /dev/null
+++ b/vorlesungen/slides/10/ableitung-exp.tex
@@ -0,0 +1,60 @@
+%
+% ableitung-exp.tex -- Ableitung von exp(x)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ %\frametitle{Ableitung von $\exp(x)$}
+ %\vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Ableitung von $\exp(at)$}
+ \begin{align*}
+ \frac{d}{dt} \exp(at)
+ &=
+ \frac{d}{dt} \sum_{k=0}^{\infty} a^k \frac{t^k}{k!}
+ \\
+ &\uncover<2->{
+ = \sum_{k=0}^{\infty} a^k\frac{kt^{k-1}}{k(k-1)!}
+ }
+ \\
+ &\uncover<3->{
+ = a \sum_{k=1}^{\infty}
+ a^{k-1}\frac{t^{k-1}}{(k-1)!}
+ }
+ \\
+ &\uncover<4->{
+ = a \exp(at)
+ }
+ \end{align*}
+ \end{block}
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<5->{
+ \begin{block}{Ableitung von $\exp(At)$}
+ \begin{align*}
+ \frac{d}{dt} \exp(At)
+ &=
+ \frac{d}{dt} \sum_{k=0}^{\infty} A^k \frac{t^k}{k!}
+ \\
+ &=
+ \sum_{k=0}^{\infty} A^k\frac{kt^{k-1}}{k(k-1)!}
+ \\
+ &=
+ A \sum_{k=1}^{\infty} A^{k-1}\frac{t^{k-1}}{(k-1)!}
+ \\
+ &=
+ A \exp(At)
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/intro.tex b/vorlesungen/slides/10/intro.tex
new file mode 100644
index 0000000..276bf49
--- /dev/null
+++ b/vorlesungen/slides/10/intro.tex
@@ -0,0 +1,45 @@
+%
+% intro.tex -- Repetition Lie-Gruppen und -Algebren
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+% \frametitle{Repetition}
+% \vspace{-20pt}
+ \begin{block}{Offene Fragen}
+ \begin{itemize}[<+->]
+ \item Woher kommt die Exponentialfunktion?
+ \begin{fleqn}
+ \[
+ \exp(At)
+ =
+ 1
+ + At
+ + A^2\frac{t^2}{2}
+ + A^3\frac{t^3}{3!}
+ + \ldots
+ \]
+ \end{fleqn}
+ \item Wie löst man eine Matrix-DGL?
+ \begin{fleqn}
+ \[
+ \dot\gamma(t) = A\gamma(t),
+ \qquad
+ \gamma(t) \in G \subset M_n
+ \]
+ \end{fleqn}
+ \item Lie-Gruppen und Lie-Algebren
+ \item Was bedeutet $\exp(At)$?
+ \end{itemize}
+ \end{block}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/matrix-dgl.tex b/vorlesungen/slides/10/matrix-dgl.tex
new file mode 100644
index 0000000..ae68fb1
--- /dev/null
+++ b/vorlesungen/slides/10/matrix-dgl.tex
@@ -0,0 +1,83 @@
+%
+% matrix-dgl.tex -- Matrix-Differentialgleichungen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{1.~Ordnung mit Skalaren}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Aufgabe}
+ Sei $a, x(t), x_0 \in \mathbb R$,
+ \[
+ \dot x(t) = ax(t),
+ \quad
+ x(0) = x_0
+ \]
+ \end{block}
+ \begin{block}{Potenzreihen-Ansatz}
+ Sei $a_k \in \mathbb R$,
+ \[
+ x(t) = a_0 + a_1t + a_2t^2 + a_3t^3 \ldots
+ \]
+ \end{block}
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Lösung}
+ Einsetzen in DGL, Koeffizientenvergleich liefert
+ \[ x(t) = \exp(at) \, x_0, \]
+ wobei
+ \begin{align*}
+ \exp(at)
+ &= 1 + at + \frac{a^2t^2}{2} + \frac{a^3t^3}{3!} + \ldots \\
+ &{\color{gray}(= e^{at}.)}
+ \end{align*}
+ \end{block}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{1.~Ordnung mit Matrizen}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Aufgabe}
+ Sei $A \in M_n$, $x(t), x_0 \in \mathbb R^n$,
+ \[
+ \dot x(t) = Ax(t),
+ \quad
+ x(0) = x_0
+ \]
+ \end{block}
+ \begin{block}{Potenzreihen-Ansatz}
+ Sei $A_k \in \mathbb M_n$,
+ \[
+ x(t) = A_0 + A_1t + A_2t^2 + A_3t^3 \ldots
+ \]
+ \end{block}
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Lösung}
+ Einsetzen in DGL, Koeffizientenvergleich liefert
+ \[ x(t) = \exp(At) \, x_0, \]
+ wobei
+ \[
+ \exp(At)
+ = 1 + At + \frac{A^2t^2}{2} + \frac{A^3t^3}{3!} + \ldots
+ \]
+ \end{block}
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/n-zu-1.tex b/vorlesungen/slides/10/n-zu-1.tex
new file mode 100644
index 0000000..09475ad
--- /dev/null
+++ b/vorlesungen/slides/10/n-zu-1.tex
@@ -0,0 +1,63 @@
+%
+% n-zu-1.tex -- Umwandlend einer DGL n-ter Ordnung in ein System 1. Ordnung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ %\frametitle{Reicht $1.$ Ordnung?}
+ %\vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Beispiel: DGL 3.~Ordnung} \vspace*{-1ex}
+ \begin{align*}
+ x^{(3)} + a_2 \ddot x + a_1 \dot x + a_0 x = 0 \\
+ \Rightarrow
+ x^{(3)} = -a_2 \ddot x - a_1 \dot x - a_0 x
+ \end{align*}
+ \end{block}
+ }
+ \uncover<2->{
+ \begin{block}{Ziel: Nur noch 1.~Ableitungen}
+ Einführen neuer Variablen:
+ \begin{align*}
+ x_0 &\coloneqq x &
+ x_1 &\coloneqq \dot x &
+ x_2 &\coloneqq \ddot x
+ \end{align*}
+ System von Gleichungen 1.~Ordnung
+ \begin{align*}
+ \dot x_0 &= x_1 \\
+ \dot x_1 &= x_2 \\
+ \dot x_2 &= -a_2 x_2 - a_1 x_1 - a_0 x_0
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \uncover<3->{
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Als Vektor-Gleichung} \vspace*{-1ex}
+ \begin{align*}
+ \frac{d}{dt}
+ \begin{pmatrix} x_0 \\ x_1 \\ x_2 \end{pmatrix}
+ = \begin{pmatrix}
+ 0 & 1 & 0 \\
+ 0 & 0 & 1 \\
+ -a_0 & -a_1 & -a_2
+ \end{pmatrix}
+ \begin{pmatrix} x_0 \\ x_1 \\ x_2 \end{pmatrix}
+ \end{align*}
+
+ \uncover<4->{Geht für jede lineare Differentialgleichung!}
+
+ \end{block}
+ \end{column}
+ }
+ \end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/10/potenzreihenmethode.tex b/vorlesungen/slides/10/potenzreihenmethode.tex
new file mode 100644
index 0000000..1715134
--- /dev/null
+++ b/vorlesungen/slides/10/potenzreihenmethode.tex
@@ -0,0 +1,91 @@
+%
+% potenzreihenmethode.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Bearbeitet durch Roy Seitz
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzreihenmethode}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Lineare Differentialgleichung}
+\begin{align*}
+x'&=ax&&\Rightarrow&x'-ax&=0
+\\
+x(0)&=C
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Potenzreihenansatz}
+\begin{align*}
+x(t)
+&=
+a_0+ a_1t + a_2t^2 + \dots
+\\
+x(0)&=a_0=C
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösung}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcr}
+\uncover<3->{ x'(t)}
+ \uncover<5->{
+ &=&\phantom{(} a_1\phantom{\mathstrut-aa_0)}
+ &+& 2a_2\phantom{\mathstrut-aa_1)}t
+ &+& 3a_3\phantom{\mathstrut-aa_2)}t^2
+ &+& 4a_4\phantom{\mathstrut-aa_3)}t^3
+ &+& \dots}\\
+\uncover<3->{-ax(t)}
+ \uncover<6->{
+ &=&\mathstrut-aa_0 \phantom{)}
+ &-& aa_1\phantom{)}t
+ &-& aa_2\phantom{)}t^2
+ &-& aa_3\phantom{)}t^3
+ &-& \dots}\\[2pt]
+\hline
+\\[-10pt]
+\uncover<3->{0}
+ \uncover<7->{
+ &=&(a_1-aa_0)
+ &+& (2a_2-aa_1)t
+ &+& (3a_3-aa_2)t^2
+ &+& (4a_4-aa_3)t^3
+ &+& \dots}\\
+\end{array}
+\]
+\begin{align*}
+\uncover<4->{
+a_0&=C}\uncover<8->{,
+\quad
+a_1=aa_0=aC}\uncover<9->{,
+\quad
+a_2=\frac12a^2C}\uncover<10->{,
+\quad
+a_3=\frac16a^3C}\uncover<11->{,
+\ldots,
+a_k=\frac1{k!}a^kC}
+\hspace{3cm}
+\\
+\uncover<4->{
+\Rightarrow x(t) &= C}\uncover<8->{+Cat}\uncover<9->{ + C\frac12(at)^2}
+\uncover<10->{ + C \frac16(at)^3}
+\uncover<11->{ + \dots+C\frac{1}{k!}(at)^k+\dots}
+\ifthenelse{\boolean{presentation}}{
+\only<12>{
+=
+C\sum_{k=0}^\infty \frac{(at)^k}{k!}}
+}{}
+\uncover<13->{=
+C\exp(at)}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/10/repetition.tex b/vorlesungen/slides/10/repetition.tex
new file mode 100644
index 0000000..7c007ca
--- /dev/null
+++ b/vorlesungen/slides/10/repetition.tex
@@ -0,0 +1,119 @@
+%
+% repetition.tex -- Repetition Lie-Gruppen und -Algebren
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Repetition}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Lie-Gruppe}
+ Kontinuierliche Matrix-Gruppe $G$ mit bestimmter Eigenschaft
+ \end{block}
+ }
+ \uncover<3->{
+ \begin{block}{Ein-Parameter-Untergruppe}
+ Darstellung der Lie-Gruppe $G$:
+ \[
+ \gamma \colon \mathbb R \to G
+ : \quad
+ t \mapsto \gamma(t),
+ \]
+ so dass
+ \[ \gamma(s + t) = \gamma(t) \gamma(s). \]
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<2->{
+ \begin{block}{Beispiel}
+ Volumen-erhaltende Abbildungen:
+ \[ \gSL2R= \{A \in M_2 \,|\, \det(A) = 1\} .\]
+ \begin{align*}
+ \uncover<4->{ \gamma_x(t) }
+ &
+ \uncover<4->{= \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} }
+ \\
+ \uncover<5->{ \gamma_y(t) }
+ &
+ \uncover<5->{= \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} }
+ \\
+ \uncover<6->{ \gamma_h(t)}
+ &
+ \uncover<6->{= \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} }
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Repetition}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Lie-Algebra aus Lie-Gruppe}
+ Ableitungen der Ein-Parameter-Untergruppen:
+ \begin{align*}
+ G &\to \mathcal A \\
+ \gamma &\mapsto \dot\gamma(0)
+ \end{align*}
+ \uncover<3->{
+ Lie-Klammer als Produkt:
+ \[ [A, B] = AB - BA \in \mathcal A \]
+ }
+ \end{block}
+ }
+ \uncover<7->{\vspace*{-4ex}
+ \begin{block}{Lie-Gruppe aus Lie-Algebra}
+ Lösung der Differentialgleichung:
+ \[
+ \dot\gamma(t) = A\gamma(t)
+ \quad \text{mit} \quad
+ A = \dot\gamma(0)
+ \]
+ \[
+ \Rightarrow \gamma(t) = \exp(At)
+ \]
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<2->{
+ \begin{block}{Beispiel}
+ Lie-Algebra von \gSL2R:
+ \[ \asl2R = \{ A \in M_2 \,|\, \Spur(A) = 0 \} \]
+ \end{block}
+ }
+ \begin{align*}
+ \uncover<4->{ X(t) }
+ &
+ \uncover<4->{= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} }
+ \\
+ \uncover<5->{ Y(t) }
+ &
+ \uncover<5->{= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} }
+ \\
+ \uncover<6->{ H(t) }
+ &
+ \uncover<6->{= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} }
+ \end{align*}
+
+ \end{column}
+ \end{columns}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/so2.tex b/vorlesungen/slides/10/so2.tex
new file mode 100644
index 0000000..dcbcdc8
--- /dev/null
+++ b/vorlesungen/slides/10/so2.tex
@@ -0,0 +1,141 @@
+%
+% so2.tex -- Illustration of so(2) -> SO(2)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Von der Lie-Gruppe zur -Algebra}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Lie-Gruppe}
+ Darstellung von \gSO2:
+ \begin{align*}
+ \mathbb R
+ &\to
+ \gSO2
+ \\
+ t
+ &\mapsto
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \uncover<2->{
+ \begin{block}{Ableitung am neutralen Element}
+ \begin{align*}
+ \frac{d}{d t}
+ &
+ \left.
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \right|_{ t = 0}
+ \\
+ =
+ &
+ \begin{pmatrix} -\sin0 & -\cos0 \\ \phantom-\cos0 & -\sin0 \end{pmatrix}
+ =
+ \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<3->{
+ \begin{block}{Lie-Algebra}
+ Darstellung von \aso2:
+ \begin{align*}
+ \mathbb R
+ &\to
+ \aso2
+ \\
+ t
+ &\mapsto
+ \begin{pmatrix}
+ 0 & -t \\
+ t & \phantom-0
+ \end{pmatrix}
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Von der Lie-Algebra zur -Gruppe}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \uncover<1->{
+ \begin{block}{Differentialgleichung}
+ Gegeben:
+ \[
+ J
+ =
+ \dot\gamma(0) = \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+ \]
+ Gesucht:
+ \[ \dot \gamma (t) = J \gamma(t) \qquad \gamma \in \gSO2 \]
+ \[ \Rightarrow \gamma(t) = \exp(Jt) \gamma(0) = \exp(Jt) \]
+ \end{block}
+ }
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \uncover<2->{
+ \begin{block}{Lie-Algebra}
+ Potenzen von $J$:
+ \begin{align*}
+ J^2 &= -I &
+ J^3 &= -J &
+ J^4 &= I &
+ \ldots
+ \end{align*}
+ \end{block}
+ }
+ \end{column}
+ \end{columns}
+\uncover<3->{
+ Folglich:
+ \begin{align*}
+ \exp(Jt)
+ &= I + Jt
+ + J^2\frac{t^2}{2!}
+ + J^3\frac{t^3}{3!}
+ + J^4\frac{t^4}{4!}
+ + J^5\frac{t^5}{5!}
+ + \ldots \\
+ &= \begin{pmatrix}
+ \vspace*{3pt}
+ 1 - \frac{t^2}{2} + \frac{t^4}{4!} - \ldots
+ &
+ -t + \frac{t^3}{3!} - \frac{t^5}{5!} + \ldots
+ \\
+ t - \frac{t^3}{3!} + \frac{t^5}{5!} - \ldots
+ &
+ 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \ldots
+ \end{pmatrix}
+ =
+ \begin{pmatrix}
+ \cos t & -\sin t \\
+ \sin t & \phantom-\cos t
+ \end{pmatrix}
+ \end{align*}
+ }
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
new file mode 100644
index 0000000..8c71965
--- /dev/null
+++ b/vorlesungen/slides/10/taylor.tex
@@ -0,0 +1,216 @@
+%
+% taylor.tex -- Repetition Taylot-Reihen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Beispiel $\sin(x)$}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
+ \begin{block}{Taylor-Approximationen von $\sin(x)$}
+ \begin{align*}
+ p_{
+ \only<1>{0}
+ \only<2>{1}
+ \only<3>{2}
+ \only<4>{3}
+ \only<5>{4}
+ \only<6>{5}
+ \only<7->{n}
+ }(x)
+ &=
+ \uncover<1->{0}
+ \uncover<2->{+ x}
+ \uncover<3->{+ 0 \frac{x^2}{2!}}
+ \uncover<4->{- 1 \frac{x^3}{3!}}
+ \uncover<5->{+ 0 \frac{x^4}{4!}}
+ \uncover<6->{+ 1 \frac{x^5}{5!}}
+ \uncover<7->{+ \ldots}
+ \uncover<8->{
+ = \sum_{k=0}^{n/2} (-1)^{2k + 1}\frac{x^{2k+1}}{(2k+1)!}
+ }
+ \end{align*}
+ \end{block}
+ \begin{center}
+ \begin{tikzpicture}[>=latex,thick,scale=1.3]
+ \draw[->] (-5.0, 0.0) -- (5.0,0.0) coordinate[label=$x$];
+ \draw[->] ( 0.0,-1.5) -- (0.0,1.5);
+ \clip (-5,-1.5) rectangle (5,1.5);
+ \draw[domain=-4:4, samples=50, smooth, blue]
+ plot ({\x}, {sin(180/3.1415968*\x)})
+ node[above right] {$\sin(x)$};
+ \uncover<1|handout:0>{
+ \draw[domain=-4:4, samples=2, smooth, red]
+ plot ({\x}, {0})
+ node[above right] {$p_0(x)$};}
+ \uncover<2|handout:0>{
+ \draw[domain=-1.5:1.5, samples=2, smooth, red]
+ plot ({\x}, {\x})
+ node[below right] {$p_1(x)$};}
+ \uncover<3|handout:0>{
+ \draw[domain=-1.5:1.5, samples=2, smooth, red]
+ plot ({\x}, {\x})
+ node[below right] {$p_2(x)$};}
+ \uncover<4>{
+ \draw[domain=-3:3, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6})
+ node[above right] {$p_3(x)$};}
+ \uncover<5|handout:0>{
+ \draw[domain=-3:3, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6})
+ node[above right] {$p_4(x)$};}
+ \uncover<6|handout:0>{
+ \draw[domain=-3.9:3.9, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120})
+ node[below right] {$p_5(x)$};}
+ \uncover<7|handout:0>{
+ \draw[domain=-3.9:3.9, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120})
+ node[below right] {$p_6(x)$};}
+ \uncover<8-|handout:0>{
+ \draw[domain=-4:4, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120 -
+ \x*\x*\x*\x*\x*\x*\x/5040})
+ node[above right] {$p_7(x)$};}
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Taylor-Reihen}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
+ \begin{block}{Polynom-Approximationen von $f(t)$}
+ \begin{align*}
+ p_n(t)
+ &=
+ f(0)
+ \uncover<2->{ + f' (0) t }
+ \uncover<3->{ + f''(0)\frac{t^2}{2} }
+ \uncover<4->{ + \ldots + f^{(n)}(0) \frac{t^n}{n!} }
+ \uncover<5->{ = \sum_{k=0}^{n} f^{(k)} \frac{t^k}{k!} }
+ \end{align*}
+ \end{block}
+ \uncover<6->{
+ \begin{block}{Erste $n$ Ableitungen von $f(0)$ und $p_n(0)$ sind gleich!}}
+ \begin{align*}
+ \uncover<6->{ p'_n(t) }
+ &
+ \uncover<7->{
+ = f'(0)
+ + f''(0)t
+ + \mathcal O(t^2)
+ }
+ &\uncover<8->{\Rightarrow}&&
+ \uncover<8->{p'_n(0) = f'(0)}
+ \\
+ \uncover<9->{ p''_n(t) }
+ &
+ \uncover<10->{
+ = f''(0)
+ + \mathcal O(t)
+ }
+ &\uncover<11->{\Rightarrow}&&
+ \uncover<11->{ p''_n(0) = f''(0) }
+ \end{align*}
+ \end{block}
+ \uncover<12->{
+ \begin{block}{Für alle praktisch relevanten Funktionen $f(t)$ gilt:}
+ \begin{align*}
+ \lim_{n\to \infty} p_n(t)
+ =
+ f(t)
+ \end{align*}
+ \end{block}
+ }
+\end{frame}
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Beispiel $e^t$}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
+ \begin{block}{Taylor-Approximationen von $e^{at}$}
+ \begin{align*}
+ p_{
+ \only<1>{0}
+ \only<2>{1}
+ \only<3>{2}
+ \only<4>{3}
+ \only<5>{4}
+ \only<6>{5}
+ \only<7->{n}
+ }(t)
+ &=
+ 1
+ \uncover<2->{+ a t}
+ \uncover<3->{+ a^2 \frac{t^2}{2}}
+ \uncover<4->{+ a^3 \frac{t^3}{3!}}
+ \uncover<5->{+ a^4 \frac{t^4}{4!}}
+ \uncover<6->{+ a^5 \frac{t^5}{5!}}
+ \uncover<7->{+ a^6 \frac{t^6}{6!}}
+ \uncover<8->{+ \ldots
+ = \sum_{k=0}^{n} a^k \frac{t^k}{k!}}
+ \\
+ &
+ \uncover<9->{= \exp(at)}
+ \end{align*}
+ \end{block}
+ \begin{center}
+ \begin{tikzpicture}[>=latex,thick,scale=1.3]
+ \draw[->] (-4.0, 0.0) -- (4.0,0.0) coordinate[label=$t$];
+ \draw[->] ( 0.0,-0.5) -- (0.0,2.5);
+ \clip (-3,-0.5) rectangle (3,2.5);
+ \draw[domain=-4:1, samples=50, smooth, blue]
+ plot ({\x}, {exp(\x)})
+ node[above right] {$\exp(t)$};
+ \uncover<1|handout:0>{
+ \draw[domain=-4:4, samples=12, smooth, red]
+ plot ({\x}, {1})
+ node[below right] {$p_0(t)$};}
+ \uncover<2|handout:0>{
+ \draw[domain=-4:1.5, samples=10, smooth, red]
+ plot ({\x}, {1 + \x})
+ node[below right] {$p_1(t)$};}
+ \uncover<3|handout:0>{
+ \draw[domain=-4:1, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2})
+ node[below right] {$p_2(t)$};}
+ \uncover<4>{
+ \draw[domain=-4:1, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6})
+ node[below right] {$p_3(t)$};}
+ \uncover<5|handout:0>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24})
+ node[below left] {$p_4(t)$};}
+ \uncover<6|handout:0>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120})
+ node[below left] {$p_5(t)$};}
+ \uncover<7|handout:0>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120
+ + \x*\x*\x*\x*\x*\x/720})
+ node[below left] {$p_6(t)$};}
+ \uncover<8-|handout:0>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120
+ + \x*\x*\x*\x*\x*\x/720
+ + \x*\x*\x*\x*\x*\x*\x/5040})
+ node[below left] {$p_7(t)$};}
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/10/template.tex b/vorlesungen/slides/10/template.tex
new file mode 100644
index 0000000..50f0a3b
--- /dev/null
+++ b/vorlesungen/slides/10/template.tex
@@ -0,0 +1,21 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Template}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/10/vektorfelder.mp b/vorlesungen/slides/10/vektorfelder.mp
new file mode 100644
index 0000000..e63b2d5
--- /dev/null
+++ b/vorlesungen/slides/10/vektorfelder.mp
@@ -0,0 +1,361 @@
+%
+% Stroemungsfelder linearer Differentialgleichungen
+%
+% (c) 2015 Prof Dr Andreas Mueller, Hochschule Rapperswil
+% 2021-04-14, Roy Seitz, Copied for SeminarMatrizen
+%
+verbatimtex
+\documentclass{book}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{txfonts}
+\begin{document}
+etex;
+
+input TEX;
+
+TEXPRE("%&latex" & char(10) &
+"\documentclass{book}" &
+"\usepackage{times}" &
+"\usepackage{amsmath}" &
+"\usepackage{amssymb}" &
+"\usepackage{amsfonts}" &
+"\usepackage{txfonts}" &
+"\begin{document}");
+TEXPOST("\end{document}");
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% so(2)
+%
+beginfig(1)
+
+% Scaling parameter
+numeric unit;
+unit := 150;
+
+% Some points
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+% % Draw circles
+% for x = 0.2 step 0.2 until 1.4:
+% path p;
+% p = (x,0);
+% for a = 5 step 5 until 355:
+% p := p--(x*cosd(a), x*sind(a));
+% endfor;
+% p := p--cycle;
+% pickup pencircle scaled 1pt;
+% draw p scaled unit withcolor red;
+% endfor;
+
+% Define DGL
+def dglField(expr x, y) =
+ %(-0.5 * (x + y), -0.5 * (y - x))
+ (-y, x)
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(1)
+%
+beginfig(2)
+
+numeric unit;
+unit := 150;
+
+z0 = ( 0, 0);
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 0, 1);
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(2)
+%
+beginfig(3)
+
+numeric unit;
+unit := 150;
+
+z0 = ( 0, 0);
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 1, 0.5);
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(3)
+%
+beginfig(4)
+
+numeric unit;
+unit := 150;
+
+z0 = ( 0, 0);
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 3, 0.25);
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(4)
+%
+beginfig(5)
+
+numeric unit;
+unit := 150;
+
+z0 = ( 0, 0);
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 7, 0.125);
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+%
+% Vektorfeld in der Ebene mit Lösungskurve
+% Euler(5)
+%
+beginfig(6)
+
+numeric unit;
+unit := 150;
+
+z0 = ( 0, 0);
+z1 = (-1.5, 0) * unit;
+z2 = ( 1.5, 0) * unit;
+z3 = ( 0, -1.5) * unit;
+z4 = ( 0, 1.5) * unit;
+
+pickup pencircle scaled 1pt;
+drawarrow (z1 shifted (-10,0))--(z2 shifted (10,0));
+drawarrow (z3 shifted (0,-10))--(z4 shifted (0,10));
+label.top(btex $x_1$ etex, z2 shifted (10,0));
+label.rt(btex $x_2$ etex, z4 shifted (0,10));
+
+def dglField(expr x, y) =
+ (-y, x)
+enddef;
+
+def dglFieldp(expr z) =
+ dglField(xpart z, ypart z)
+enddef;
+
+def curve(expr z, l, s) =
+ path p;
+ p := z;
+ for t = 0 step 1 until l:
+ p := p--((point (length p) of p) shifted (s * dglFieldp(point (length p) of p)));
+ endfor;
+ draw p scaled unit withcolor red;
+enddef;
+
+pair A;
+A := (1, 0);
+draw A scaled unit withpen pencircle scaled 8bp withcolor red;
+curve(A, 99, 0.01);
+
+% Draw arrows for each grid point
+pickup pencircle scaled 0.5pt;
+for x = -1.5 step 0.1 until 1.55:
+ for y = -1.5 step 0.1 until 1.55:
+ drawarrow ((x, y) * unit)
+ --(((x,y) * unit) shifted (8 * dglField(x,y)))
+ withcolor blue;
+ endfor;
+endfor;
+
+endfig;
+
+
+end;
diff --git a/vorlesungen/slides/10/vektorfelder.tex b/vorlesungen/slides/10/vektorfelder.tex
new file mode 100644
index 0000000..3ba7cda
--- /dev/null
+++ b/vorlesungen/slides/10/vektorfelder.tex
@@ -0,0 +1,82 @@
+%
+% iterativ.tex -- Iterative Approximation in \dot x = J x
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Als Strömungsfeld}
+ \vspace{-20pt}
+ \begin{columns}[t,onlytextwidth]
+ \begin{column}{0.48\textwidth}
+ \vfil
+ \only<1|handout:0>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-1.pdf}
+ }
+ \only<2|handout:0>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-2.pdf}
+ }
+ \only<3>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-3.pdf}
+ }
+ \only<4|handout:0>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-4.pdf}
+ }
+ \only<5|handout:0>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-5.pdf}
+ }
+ \only<6-|handout:0>{
+ \includegraphics[width=\linewidth,keepaspectratio]
+ {../slides/10/vektorfelder-6.pdf}
+ }
+ \vfil
+ \end{column}
+ \begin{column}{0.48\textwidth}
+ \begin{block}{Differentialgleichung}
+ \[
+ \dot x(t) = J x(t)
+ \quad
+ J = \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+ \quad
+ x_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
+ \]
+ \end{block}
+
+ \only<2|handout:0>{
+ Nach einem Schritt der Länge $t$:
+ \[
+ x(t) = x_0 + \dot x t = x_0 + Jx_0t = (1 + Jt)x_0
+ \]
+ }
+
+ \only<3|handout:0>{
+ Nach zwei Schritten der Länge $t/2$:
+ \[
+ x(t) = \left(1 + \frac{Jt}{2}\right)^2x_0
+ \]
+ }
+
+ \only<4->{
+ Nach n Schritten der Länge $t/n$:
+ \[
+ x(t) = \left(1 + \frac{Jt}{n}\right)^nx_0
+ \]
+ }
+ \only<6->{
+ \[
+ \lim_{n\to\infty}\left(1 + \frac{At}{n}\right)^n = \exp(At)
+ \]
+ }
+ \end{column}
+ \end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 6616f56..1ab27fa 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -1,26 +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/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 6872018..3015e7c 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -1,22 +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}
+%
+% 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
new file mode 100644
index 0000000..3d52b00
--- /dev/null
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex
new file mode 100644
index 0000000..e59f9b9
--- /dev/null
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex
new file mode 100644
index 0000000..20b278e
--- /dev/null
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
new file mode 100644
index 0000000..fd197ce
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -0,0 +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
diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
new file mode 100644
index 0000000..44ee4c8
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -0,0 +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
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.png b/vorlesungen/slides/4/galois/images/wuerfel.png
new file mode 100644
index 0000000..ff6fc14
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.png
Binary files differ
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
new file mode 100644
index 0000000..a0466f3
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -0,0 +1,9 @@
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.png b/vorlesungen/slides/4/galois/images/wuerfel2.png
new file mode 100644
index 0000000..68919cc
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.png
Binary files differ
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
new file mode 100644
index 0000000..a11bab0
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -0,0 +1,9 @@
+//
+// 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
new file mode 100644
index 0000000..b461d44
--- /dev/null
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
new file mode 100644
index 0000000..f9510ba
--- /dev/null
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
new file mode 100644
index 0000000..cb08dca
--- /dev/null
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex
new file mode 100644
index 0000000..f340825
--- /dev/null
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
new file mode 100644
index 0000000..28c07fe
--- /dev/null
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
new file mode 100644
index 0000000..907d60a
--- /dev/null
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/5/potenzreihenmethode.tex b/vorlesungen/slides/5/potenzreihenmethode.tex
index 0c3503d..12d3fa5 100644
--- a/vorlesungen/slides/5/potenzreihenmethode.tex
+++ b/vorlesungen/slides/5/potenzreihenmethode.tex
@@ -79,7 +79,7 @@ a_k=\frac1{k!}a^kC}
\\
\uncover<4->{
\Rightarrow y(x) &= C}\uncover<8->{+Cax}\uncover<9->{ + C\frac12(ax)^2}
-\uncover<10->{ + C \frac16(ac)^3}
+\uncover<10->{ + C \frac16(ax)^3}
\uncover<11->{ + \dots+C\frac{1}{k!}(ax)^k+\dots}
\ifthenelse{\boolean{presentation}}{
\only<12>{
diff --git a/vorlesungen/slides/6/Makefile.inc b/vorlesungen/slides/6/Makefile.inc
new file mode 100644
index 0000000..b46d6b6
--- /dev/null
+++ b/vorlesungen/slides/6/Makefile.inc
@@ -0,0 +1,18 @@
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter6 = \
+ ../slides/6/permutationen/matrizen.tex \
+ \
+ ../slides/6/darstellungen/definition.tex \
+ ../slides/6/darstellungen/charakter.tex \
+ ../slides/6/darstellungen/summe.tex \
+ ../slides/6/darstellungen/irreduzibel.tex \
+ ../slides/6/darstellungen/schur.tex \
+ ../slides/6/darstellungen/skalarprodukt.tex \
+ ../slides/6/darstellungen/zyklisch.tex \
+ \
+ ../slides/6/chapter.tex
+
diff --git a/vorlesungen/slides/6/chapter.tex b/vorlesungen/slides/6/chapter.tex
new file mode 100644
index 0000000..37f442d
--- /dev/null
+++ b/vorlesungen/slides/6/chapter.tex
@@ -0,0 +1,16 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+
+\folie{6/permutationen/matrizen.tex}
+
+\folie{6/darstellungen/definition.tex}
+\folie{6/darstellungen/charakter.tex}
+\folie{6/darstellungen/summe.tex}
+\folie{6/darstellungen/irreduzibel.tex}
+\folie{6/darstellungen/schur.tex}
+\folie{6/darstellungen/skalarprodukt.tex}
+\folie{6/darstellungen/zyklisch.tex}
+
diff --git a/vorlesungen/slides/6/darstellungen/charakter.tex b/vorlesungen/slides/6/darstellungen/charakter.tex
new file mode 100644
index 0000000..ea90b6d
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/charakter.tex
@@ -0,0 +1,108 @@
+%
+% chrakter.tex -- Charakter einer Darstellung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakter einer Darstellung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Definition}
+$\varrho\colon G\to\operatorname{GL}_n(\mathbb{C})$ eine Darstellung.
+\\
+Der {\em Charakter} von $\varrho$ ist die Abbildung
+\[
+\chi_{\varrho}
+\colon
+G\to \mathbb{C}^n
+:
+g\mapsto \chi_{\varrho}(g)=\operatorname{Spur}\varrho(g)
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Eigenschaften}
+\begin{enumerate}
+\item
+$\chi_{\varrho}(e) = n$
+\item<6->
+$\chi_{\varrho}(g^{-1}) = \overline{\chi_{\varrho}(g)}$
+\item<15->
+$\chi_{\varrho}(hgh^{-1}) = \chi_{\varrho}(g)$
+\end{enumerate}
+\uncover<21->{%
+Aus 3. folgt, dass Charaktere {\em Klassenfunktionen} sind}
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<2->{%
+\begin{block}{Begründung}
+\begin{enumerate}
+\item<3->
+$\chi_{\varrho}(e)
+=
+\operatorname{Spur}\varrho(e)
+\uncover<4->{=
+\operatorname{Spur}I_n}
+\uncover<5->{=
+n}
+$
+\item<6->
+$g$ hat endliche Ordnung, d.~h.~$g^k=e$
+\\
+\uncover<7->{%
+$\lambda_i$ in der Jordan-NF erfüllen $\lambda_i^k=1$}
+\\
+$\uncover<8->{\Rightarrow|\lambda_i|=1}
+\uncover<9->{\Rightarrow \lambda_i^{-1} = \overline{\lambda_i}}$
+\begin{align*}
+\uncover<10->{
+\llap{$\chi_{\varrho}(g^{-1})$}
+&=
+\operatorname{Spur}(\varrho(g^{-1}))}
+\uncover<11->{=
+\sum_{i} n_i\overline{\lambda_i}}
+\\[-4pt]
+&\uncover<12->{=
+\overline{
+\sum_{i} n_i\lambda_i
+}}
+\uncover<13->{=
+\operatorname{Spur}\varrho(g)}
+\uncover<14->{=
+\chi_{\varrho}(g)}
+\end{align*}
+\item<16->
+Durch Nachrechnen:
+\begin{align*}
+\chi_{\varrho}(hgh^{-1})
+&\uncover<17->{=
+\operatorname{Spur}
+(
+\varrho(h)
+\varrho(g)
+\varrho(h^{-1})
+)}
+\\
+&\uncover<18->{=
+\operatorname{Spur}
+(
+\varrho(h^{-1})
+\varrho(h)
+\varrho(g)
+)}
+\\
+&\uncover<19->{=
+\operatorname{Spur}\varrho(g)}
+\uncover<20->{=
+\chi_{\varrho}(g)}
+\end{align*}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/definition.tex b/vorlesungen/slides/6/darstellungen/definition.tex
new file mode 100644
index 0000000..9d93e7f
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/definition.tex
@@ -0,0 +1,59 @@
+%
+% definition.tex -- Definition einer Darstellung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Darstellung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+$G$ eine Gruppe, $V$ ein $\Bbbk$-Vektorraum.
+\\
+\uncover<2->{%
+Ein Homomorphismus
+\[
+\varrho
+\colon
+G\to \operatorname{GL}(V)
+\]
+heisst {\em $n$-dimensionale Darstellung} der Gruppe $G$.}
+\end{block}
+\uncover<3->{%
+\begin{block}{Idee}
+Algebra und Analysis in $\operatorname{GL}_n(\Bbbk)$ nutzen, um
+mehr über $G$ herauszufinden
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Beispiel $S_n$}
+$S_n$ die symmetrische Gruppe,
+$\sigma\mapsto A_{\tilde{f}}$ die
+Abbildung auf die zugehörige Permutationsmatrix
+ist eine $n$-dimensionale Darstellung von $S_n$
+\end{block}}
+\uncover<5->{%
+\begin{block}{Beispiel Matrizengruppe}
+Eine Matrizengruppe $G$ ist eine Teilmenge von $M_n(\Bbbk)$.
+\\
+\uncover<6->{%
+$g\in G \Rightarrow g^{-1}\in G$, daher $G\subset\operatorname{GL}_n(\Bbbk)$}
+\\
+\uncover<7->{%
+Die Einbettung
+\[
+G\to\operatorname{GL}_n(\Bbbk)
+:
+g \mapsto g
+\]
+ist eine Darstellung}\uncover<8->{, die sog.~{\em reguläre Darstellung}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/irreduzibel.tex b/vorlesungen/slides/6/darstellungen/irreduzibel.tex
new file mode 100644
index 0000000..6a6991e
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/irreduzibel.tex
@@ -0,0 +1,43 @@
+%
+% irreduzibel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Irreduzible Darstellungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Darstellung $\varrho\colon G\to\operatorname{GL}(V)$ heisst
+irreduzibel, wenn es keine Zerlegung von $\varrho$ in zwei
+Darstellungen $\varrho_i\colon G\to\operatorname{GL}(U_i)$ ($i=1,2$)
+gibt derart, dass $\varrho = \varrho_1\oplus\varrho_2$
+\end{block}
+\begin{block}{Isomorphe Darstellungen}
+$\varrho_i$ sind {\em isomorphe} Darstellungen in $V_i$ wenn es
+$f\colon V_1\overset{\cong}{\to} V_2$ gibt mit
+\begin{align*}
+f \circ \varrho_i(g)\circ f^{-1} &= \varrho_2(g)
+\\
+f \circ \varrho_i(g)\phantom{\mathstrut\circ f^{-1}}&= \varrho_2(g)\circ f
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Lemma von Schur}
+$\varrho_i$ zwei irreduzible Darstellungen und $f$ so, dass
+$f\circ \varrho_1(g)=\varrho_2(g)\circ f$ für alle $g$.
+Dann gilt
+\begin{enumerate}
+\item $\varrho_i$ nicht isomorph $\Rightarrow$ $f=0$
+\item $V_1=V_2$ $\Rightarrow$ $f=\lambda I$
+\end{enumerate}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/schur.tex b/vorlesungen/slides/6/darstellungen/schur.tex
new file mode 100644
index 0000000..69ce9ee
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/schur.tex
@@ -0,0 +1,45 @@
+%
+% schur.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Folgerungen aus Schurs Lemma}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Mittelung einer Abbildung}
+$h\colon V_1\to V_2$
+\[
+h^G = \frac{1}{|G|} \sum_{g\in G} \varrho_2(g)^{-1} \circ f \circ \varrho_1(g)
+\]
+\begin{enumerate}
+\item $\varrho_i$ nicht isomorph $\Rightarrow$ $h^G=0$
+\item $V_1=V_2$, $h^G = \frac1n\operatorname{Spur}h$
+\end{enumerate}
+\end{block}
+\begin{block}{Matrixelemente für $\varrho_i$ nicht isomorph}
+$\varrho_i$ nicht isomorph, dann ist
+\[
+\frac{1}{|G|} \sum_{g\in G} \varrho_1(g^{-1})_{kl}\varrho_2(g)_{uv}=0
+\]
+für alle $k,l,u,v$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrixelemente $V_1=V_2$, $\varrho_i$ iso}
+F¨r $k=v$ und $l=u$ gilt
+\[
+\frac{1}{|G|} \sum_{g\in G} \varrho_1(g^{-1})_{kl} \varrho_2(g)_{uv}
+=
+\frac1n
+\]
+und $=0$ sonst
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/skalarprodukt.tex b/vorlesungen/slides/6/darstellungen/skalarprodukt.tex
new file mode 100644
index 0000000..653bdce
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/skalarprodukt.tex
@@ -0,0 +1,39 @@
+%
+% skalarprodukt.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Skalarprodukt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition des Skalarproduktes}
+$\varphi$, $\psi$ komplexe Funktionen auf $G$:
+\[
+\langle \varphi,\psi\rangle
+=
+\frac{1}{|G|} \sum_{g\in G} \overline{\varphi(g)} \psi(g)
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+\begin{enumerate}
+\item
+$\chi$ der Charakter einer irrediziblen Darstellung
+$\Rightarrow$ $\langle \chi,\chi\rangle=1$.
+\item
+$\chi$ und $\chi'$ Charaktere nichtisomorpher Darstellungen
+$\Rightarrow$
+$\langle \chi,\chi'\rangle=0$
+\end{enumerate}
+D.~h.~Charaktere irreduzibler Darstellungen sind orthonormiert
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/summe.tex b/vorlesungen/slides/6/darstellungen/summe.tex
new file mode 100644
index 0000000..9152e1f
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/summe.tex
@@ -0,0 +1,82 @@
+%
+% Summe.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Direkte Summe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Gegeben}
+Gegeben zwei Darstellungen
+\begin{align*}
+\varrho_1&\colon G \to \mathbb{C}^{n_1}
+\\
+\varrho_2&\colon G \to \mathbb{C}^{n_2}
+\end{align*}
+\end{block}
+\vspace{-12pt}
+\begin{block}{Direkte Summe der Darstellungen}
+\vspace{-12pt}
+\begin{align*}
+\varrho_1\oplus\varrho_2
+&\colon
+G\to \mathbb{C}^{n_1+n_2} = \mathbb{C}^{n_1}\times\mathbb{C}^{n_2}
+=:
+\mathbb{C}^{n_1}\oplus\mathbb{C}^{n_2}
+\\
+&\colon g\mapsto (\varrho_1(g),\varrho_2(g))
+\end{align*}
+\end{block}
+\vspace{-12pt}
+\begin{block}{Charakter}
+\vspace{-12pt}
+\begin{align*}
+\chi_{\varrho_1\oplus\varrho_2}(g)
+&=
+\operatorname{Spur}(\varrho_1\oplus\varrho_2)(g)
+\\
+&=
+\operatorname{Spur}{\varrho_1(g)}
++
+\operatorname{Spur}{\varrho_1(g)}
+\\
+&=
+\chi_{\varrho_1}(g)
++
+\chi_{\varrho_2}(g)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Tensorprodukt}
+$n_1\times n_2$-dimensionale
+Darstellung $\varrho_1\otimes\varrho_2$ mit Matrix
+\[
+\begin{pmatrix}
+\varrho_1(g)_{11} \varrho_2(g)
+ &\dots
+ &\varrho_1(g)_{1n_1} \varrho_2(g)\\
+\vdots&\ddots&\vdots\\
+\varrho_1(g)_{n_11} \varrho_2(g)
+ &\dots
+ &\varrho_1(g)_{n_1n_1} \varrho_2(g)
+\end{pmatrix}
+\]
+Die ``Einträge'' sind $n_2\times n_2$-Blöcke
+\end{block}
+\begin{block}{Darstellungsring}
+Die Menge der Darstellungen $R(G)$ einer Gruppe hat
+einer Ringstruktur mit $\oplus$ und $\otimes$
+\\
+$\Rightarrow$
+Algebra zum Studium der möglichen Darstellungen von $G$ verwenden
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/darstellungen/zyklisch.tex b/vorlesungen/slides/6/darstellungen/zyklisch.tex
new file mode 100644
index 0000000..6e36d1d
--- /dev/null
+++ b/vorlesungen/slides/6/darstellungen/zyklisch.tex
@@ -0,0 +1,77 @@
+%
+% zyklisch.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beispiel: Zyklische Gruppen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Gruppe}
+\(
+C_n = \mathbb{Z}/n\mathbb{Z}
+\)
+\end{block}
+\begin{block}{Darstellungen von $C_n$}
+Gegeben durch $\varrho_k(1)=e^{2\pi i k/n}$,
+\[
+\varrho_k(l) = e^{2\pi ikl/n}
+\]
+\end{block}
+\vspace{-10pt}
+\begin{block}{Charaktere}
+\vspace{-10pt}
+\[
+\chi_k(l) = e^{2\pi ikl/n}
+\]
+haben Skalarprodukte
+\[
+\langle \chi_k,\chi_{k'}\rangle
+=
+\begin{cases}
+1&\quad k= k'\\
+0&\quad\text{sonst}
+\end{cases}
+\]
+Die Darstellungen $\chi_k$ sind nicht isomorph
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Orthonormalbasis}
+Die Funktionen $\chi_k$ bilden eine Orthonormalbasis von $L^2(C_n)$
+\end{block}
+\vspace{-4pt}
+\begin{block}{Analyse einer Darstellung}
+$\varrho\colon C_n\to \mathbb{C}^n$ eine Darstellung,
+$\chi_\varrho$ der Charakter lässt zerlegen:
+\begin{align*}
+c_k
+&=
+\langle \chi_k, \chi\rangle = \frac{1}{n} \sum_{l} \chi_k(l) e^{-2\pi ilk/n}
+\\
+\chi(l)
+&=
+\sum_{k} c_k \chi_k
+=
+\sum_{k} c_k e^{2\pi ikl/n}
+\end{align*}
+\end{block}
+\vspace{-13pt}
+\begin{block}{Fourier-Theorie}
+\vspace{-3pt}
+\begin{center}
+\begin{tabular}{>{$}l<{$}l}
+C_n&Diskrete Fourier-Theorie\\
+U(1)&Fourier-Reihen\\
+\mathbb{R}&Fourier-Integral
+\end{tabular}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/6/permutationen/matrizen.tex b/vorlesungen/slides/6/permutationen/matrizen.tex
new file mode 100644
index 0000000..346993d
--- /dev/null
+++ b/vorlesungen/slides/6/permutationen/matrizen.tex
@@ -0,0 +1,75 @@
+%
+% matrizen.tex -- Darstellung der Permutationen als Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Permutationsmatrizen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Permutationsabbildung}
+$\sigma\in S_n$ eine Permutation, definiere
+\[
+f
+\colon
+e_i \mapsto e_{\sigma(i)}
+\]
+($e_i$ Standardbasisvektor)
+\end{block}
+\begin{block}{Lineare Abbildung}
+$f$ kann erweitert werden zu einer linearen Abbildung
+\[
+\tilde{f}
+\colon
+\Bbbk^n \to \Bbbk^n
+:
+\sum_{k=1}^n a_i e_i
+\mapsto
+\sum_{k=1}^n a_i f(e_i)
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Permutationsmatrix}
+Matrix $A_{\tilde{f}}$ der linearen Abbildung $\tilde{f}$
+hat die Matrixelemente
+\[
+a_{ij}
+=
+\begin{cases}
+1&\qquad i=\sigma(j)\\
+0&\qquad\text{sonst}
+\end{cases}
+\]
+\end{block}
+\vspace{-10pt}
+\begin{block}{Beispiel}
+\vspace{-20pt}
+\[
+\begin{pmatrix}
+1&2&3&4\\
+3&2&4&1
+\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+0&0&0&1\\
+0&1&0&0\\
+1&0&0&0\\
+0&0&1&0
+\end{pmatrix}
+\]
+\end{block}
+\vspace{-10pt}
+\begin{block}{Homomorphismus}
+Die Abbildung
+$S_n\to\operatorname{GL}(\Bbbk)\colon \sigma \mapsto A_{\tilde{f}}$
+ist ein Homomorphismus
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
new file mode 100644
index 0000000..2391099
--- /dev/null
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -0,0 +1,22 @@
+#
+# 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
+
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
new file mode 100644
index 0000000..5a4b94e
--- /dev/null
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex
new file mode 100644
index 0000000..fba42cf
--- /dev/null
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
new file mode 100644
index 0000000..0f14a9a
--- /dev/null
+++ b/vorlesungen/slides/7/chapter.tex
@@ -0,0 +1,19 @@
+%
+% 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}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex
new file mode 100644
index 0000000..446b2ab
--- /dev/null
+++ b/vorlesungen/slides/7/dg.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex
new file mode 100644
index 0000000..776617f
--- /dev/null
+++ b/vorlesungen/slides/7/drehanim.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
new file mode 100644
index 0000000..e7b4a92
--- /dev/null
+++ b/vorlesungen/slides/7/drehung.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
new file mode 100644
index 0000000..e9699a6
--- /dev/null
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
new file mode 100644
index 0000000..9de1c34
--- /dev/null
+++ b/vorlesungen/slides/7/images/Makefile
@@ -0,0 +1,19 @@
+#
+# 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
+
+
diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg
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diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc
new file mode 100644
index 0000000..b028956
--- /dev/null
+++ b/vorlesungen/slides/7/images/common.inc
@@ -0,0 +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)
+
diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini
new file mode 100644
index 0000000..44a5ac5
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -0,0 +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
+
diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m
new file mode 100644
index 0000000..3f5ea17
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov
new file mode 100644
index 0000000..8229a06
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -0,0 +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
+ }
+}
+
diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg
new file mode 100644
index 0000000..5c49700
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.jpg
Binary files differ
diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png
new file mode 100644
index 0000000..6d9e9e4
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.png
Binary files differ
diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
new file mode 100644
index 0000000..62306f8
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -0,0 +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)
diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
new file mode 100644
index 0000000..9000160
--- /dev/null
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -0,0 +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}};}
+\only<2>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};}
+\only<3>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};}
+\only<4>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};}
+\only<5>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};}
+\only<6>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};}
+\only<7>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};}
+\only<8>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};}
+\only<9>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};}
+\only<10>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};}
+\only<11>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};}
+\only<12>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};}
+\only<13>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};}
+\only<14>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};}
+\only<15>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};}
+\only<16>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};}
+\only<17>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};}
+\only<18>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};}
+\only<19>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};}
+\only<20>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};}
+\only<21>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};}
+\only<22>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};}
+\only<23>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};}
+\only<24>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};}
+\only<25>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};}
+\only<26>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};}
+\only<27>{\node at (0,0) {
+\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};}
+\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
new file mode 100644
index 0000000..bca8417
--- /dev/null
+++ b/vorlesungen/slides/7/kurven.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
new file mode 100644
index 0000000..59c9121
--- /dev/null
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex
new file mode 100644
index 0000000..f88042a
--- /dev/null
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex
new file mode 100644
index 0000000..afc67c5
--- /dev/null
+++ b/vorlesungen/slides/7/parameter.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex
new file mode 100644
index 0000000..d74b7d0
--- /dev/null
+++ b/vorlesungen/slides/7/semi.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
new file mode 100644
index 0000000..58e87a1
--- /dev/null
+++ b/vorlesungen/slides/7/sl2.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
new file mode 100644
index 0000000..8931a24
--- /dev/null
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc
index 4bf9348..130fa28 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -1,17 +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/8/Makefile.inc
-include ../slides/9/Makefile.inc
-
-slides = \
- $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \
- $(chapter5) $(chapter8) $(chapter9)
+#
+# 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/a/Makefile.inc b/vorlesungen/slides/a/Makefile.inc
new file mode 100644
index 0000000..0c7ab0b
--- /dev/null
+++ b/vorlesungen/slides/a/Makefile.inc
@@ -0,0 +1,25 @@
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chaptera = \
+ ../slides/a/dc/prinzip.tex \
+ ../slides/a/dc/effizient.tex \
+ ../slides/a/dc/beispiel.tex \
+ \
+ ../slides/a/ecc/gruppendh.tex \
+ ../slides/a/ecc/kurve.tex \
+ ../slides/a/ecc/inverse.tex \
+ ../slides/a/ecc/operation.tex \
+ ../slides/a/ecc/quadrieren.tex \
+ ../slides/a/ecc/oakley.tex \
+ \
+ ../slides/a/aes/bytes.tex \
+ ../slides/a/aes/sinverse.tex \
+ ../slides/a/aes/blocks.tex \
+ ../slides/a/aes/keys.tex \
+ ../slides/a/aes/runden.tex \
+ \
+ ../slides/a/chapter.tex
+
diff --git a/vorlesungen/slides/a/aes/blocks.tex b/vorlesungen/slides/a/aes/blocks.tex
new file mode 100644
index 0000000..9e95a86
--- /dev/null
+++ b/vorlesungen/slides/a/aes/blocks.tex
@@ -0,0 +1,193 @@
+%
+% blocks.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\s{0.4}
+\def\punkt#1#2{({#1*\s},{(3-#2)*\s})}
+\def\feld#1#2#3{
+ \fill[color=#3] \punkt{(#1-0.5)}{(#2+0.5)}
+ rectangle \punkt{(#1+0.5)}{(#2-0.5)};
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Blocks}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Blocks}
+$4\times k$ Matrizen mit $k=4,\dots,8$
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\xdef\s{0.4}
+\foreach \i in {0,...,31}{
+ \pgfmathparse{mod(\i,4)}
+ \xdef\y{\pgfmathresult}
+ \pgfmathparse{int(\i/4)}
+ \xdef\x{\pgfmathresult}
+ \node at \punkt{\x}{\y} {\tiny $\i$};
+}
+\foreach \x in {-0.5,0.5,...,7.5}{
+ \draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+}
+\foreach \y in {-0.5,0.5,...,3.5}{
+ \draw \punkt{-0.5}{\y} -- \punkt{7.5}{\y};
+}
+\end{tikzpicture}
+\end{center}
+\uncover<2->{%
+Spalten sind $4$-dimensionale $\mathbb{F}_{2^8}$-Vektoren
+}
+\end{block}
+\uncover<3->{%
+\begin{block}{Zeilenshift}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\xdef\s{0.35}
+
+\begin{scope}
+ \feld{0}{3}{red!20}
+ \feld{0}{2}{red!20}
+ \feld{0}{1}{red!20}
+ \feld{0}{0}{red!20}
+
+ \feld{1}{3}{red!10}
+ \feld{1}{2}{red!10}
+ \feld{1}{1}{red!10}
+ \feld{1}{0}{red!10}
+
+ \feld{2}{3}{yellow!20}
+ \feld{2}{2}{yellow!20}
+ \feld{2}{1}{yellow!20}
+ \feld{2}{0}{yellow!20}
+
+ \feld{3}{3}{yellow!10}
+ \feld{3}{2}{yellow!10}
+ \feld{3}{1}{yellow!10}
+ \feld{3}{0}{yellow!10}
+
+ \feld{4}{3}{darkgreen!20}
+ \feld{4}{2}{darkgreen!20}
+ \feld{4}{1}{darkgreen!20}
+ \feld{4}{0}{darkgreen!20}
+
+ \feld{5}{3}{darkgreen!10}
+ \feld{5}{2}{darkgreen!10}
+ \feld{5}{1}{darkgreen!10}
+ \feld{5}{0}{darkgreen!10}
+
+ \feld{6}{3}{blue!20}
+ \feld{6}{2}{blue!20}
+ \feld{6}{1}{blue!20}
+ \feld{6}{0}{blue!20}
+
+ \feld{7}{3}{blue!10}
+ \feld{7}{2}{blue!10}
+ \feld{7}{1}{blue!10}
+ \feld{7}{0}{blue!10}
+
+ \foreach \x in {-0.5,0.5,...,7.5}{
+ \draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+ }
+ \foreach \y in {-0.5,0.5,...,3.5}{
+ \draw \punkt{-0.5}{\y} -- \punkt{7.5}{\y};
+ }
+\end{scope}
+
+\begin{scope}[xshift=3.5cm]
+ \feld{0}{0}{red!20}
+ \feld{1}{1}{red!20}
+ \feld{2}{2}{red!20}
+ \feld{3}{3}{red!20}
+
+ \feld{1}{0}{red!10}
+ \feld{2}{1}{red!10}
+ \feld{3}{2}{red!10}
+ \feld{4}{3}{red!10}
+
+ \feld{2}{0}{yellow!20}
+ \feld{3}{1}{yellow!20}
+ \feld{4}{2}{yellow!20} \feld{5}{3}{yellow!20}
+
+ \feld{3}{0}{yellow!10}
+ \feld{4}{1}{yellow!10}
+ \feld{5}{2}{yellow!10}
+ \feld{6}{3}{yellow!10}
+
+ \feld{4}{0}{darkgreen!20}
+ \feld{5}{1}{darkgreen!20}
+ \feld{6}{2}{darkgreen!20}
+ \feld{7}{3}{darkgreen!20}
+
+ \feld{5}{0}{darkgreen!10}
+ \feld{6}{1}{darkgreen!10}
+ \feld{7}{2}{darkgreen!10}
+ \feld{0}{3}{darkgreen!10}
+
+ \feld{6}{0}{blue!20}
+ \feld{7}{1}{blue!20}
+ \feld{0}{2}{blue!20}
+ \feld{1}{3}{blue!20}
+
+ \feld{7}{0}{blue!10}
+ \feld{0}{1}{blue!10}
+ \feld{1}{2}{blue!10}
+ \feld{2}{3}{blue!10}
+
+ \foreach \x in {-0.5,0.5,...,7.5}{
+ \draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+ }
+ \foreach \y in {-0.5,0.5,...,3.5}{
+ \draw \punkt{-0.5}{\y} -- \punkt{7.5}{\y};
+ }
+
+ \node at \punkt{-1.5}{1.5} {$\rightarrow$};
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<4->{%
+\begin{block}{Spalten mischen}
+Lineare Operation auf Spaltenvektoren mit Matrix
+\begin{align*}
+C&=\begin{pmatrix}
+\texttt{02}_{16}&\texttt{03}_{16}&\texttt{01}_{16}&\texttt{01}_{16}\\
+\texttt{01}_{16}&\texttt{02}_{16}&\texttt{03}_{16}&\texttt{01}_{16}\\
+\texttt{01}_{16}&\texttt{01}_{16}&\texttt{02}_{16}&\texttt{03}_{16}\\
+\texttt{03}_{16}&\texttt{01}_{16}&\texttt{01}_{16}&\texttt{02}_{16}
+\end{pmatrix}
+\\
+\uncover<5->{
+\det C
+&=
+\texttt{0a}_{16}
+}
+\uncover<6->{
+\ne 0}
+\uncover<7->{
+\quad\Rightarrow\quad \exists C^{-1}
+}
+\end{align*}
+\end{block}}
+\uncover<8->{%
+\begin{block}{Als Polynommultiplikation}
+Spalten = Polynome in $\mathbb{F}_{2^8}[Z]/(Z^4-1)$,
+\\
+\uncover<9->{%
+$C=\mathstrut$ Multiplikation mit
+\[
+c(Z) = \texttt{03}_{16}Z^3 + Z^2 + Z + \texttt{02}_{16}
+\]
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/aes/bytes.tex b/vorlesungen/slides/a/aes/bytes.tex
new file mode 100644
index 0000000..e873e9a
--- /dev/null
+++ b/vorlesungen/slides/a/aes/bytes.tex
@@ -0,0 +1,96 @@
+%
+% bytes.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Bytes}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Endlicher Körper}
+1 Byte = 8 bits: $\mathbb{F}_{2^8}$
+mit Minimalpolynom:
+\[
+m(X) = X^8+X^4+X^3+X+1
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Inverse $a^{-1}$}
+Mit dem euklidischen Algorithmus
+\[
+\begin{aligned}
+sa+tm&=1
+&&\Rightarrow&
+\uncover<3->{
+a^{-1} &= s}
+\\
+&
+&&&
+\uncover<4->{
+\overline{a}
+&=
+\begin{cases}
+a^{-1}&\; a\ne 0\\
+0 &\; a = 0
+\end{cases}}
+\end{aligned}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Vektorraum}
+$\mathbb{R}_{2^8}$
+ist ein $8$-dimensionaler $\mathbb{F}_2$-Vektorraum
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{S-Box}
+$S\colon a\mapsto A\overline{a}+q$ mit
+\begin{align*}
+\only<1-7>{\phantom{\mathstrut^{-1}}A}
+\ifthenelse{\boolean{presentation}}{}{\only<8>{A^{-1}}}
+&=\only<1-7>{\begin{pmatrix}
+1&0&0&0&1&1&1&1\\
+1&1&0&0&0&1&1&1\\
+1&1&1&0&0&0&1&1\\
+1&1&1&1&0&0&0&1\\
+1&1&1&1&1&0&0&0\\
+0&1&1&1&1&1&0&0\\
+0&0&1&1&1&1&1&0\\
+0&0&0&1&1&1&1&1
+\end{pmatrix}}
+\ifthenelse{\boolean{presentation}}{}{
+\only<8->{
+\begin{pmatrix}
+0&0&1&0&0&1&0&1\\
+1&0&0&1&0&0&1&0\\
+0&1&0&0&1&0&0&1\\
+1&0&1&0&0&1&0&0\\
+0&1&0&1&0&0&1&0\\
+0&0&1&0&1&0&0&1\\
+1&0&0&1&0&1&0&0\\
+0&1&0&0&1&0&1&0
+\end{pmatrix}}
+}
+\\
+q&=X^7+X^6+X+1
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Inverse $S$-Box}
+\vspace{-10pt}
+\[
+S^{-1}(b) = \overline{A^{-1}(b-q)}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/aes/keys.tex b/vorlesungen/slides/a/aes/keys.tex
new file mode 100644
index 0000000..d2ab712
--- /dev/null
+++ b/vorlesungen/slides/a/aes/keys.tex
@@ -0,0 +1,36 @@
+%
+% keys.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Schlüsselerzeugung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/keys.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Algorithmus}
+\begin{enumerate}
+\item<2->
+Startblock: begebener Schlüssel
+\item<3->
+Zeilenpermutation:
+$\pi=\mathstrut$ Multiplikation mit $Z^3=Z^{-1}$
+\item<4-> $S$-Box
+\item<5-> $r_i$: Addition einer Konstanten
+\[
+r_i = (\texttt{02}_{16})^{i-1}
+\]
+\end{enumerate}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/aes/runden.tex b/vorlesungen/slides/a/aes/runden.tex
new file mode 100644
index 0000000..570b577
--- /dev/null
+++ b/vorlesungen/slides/a/aes/runden.tex
@@ -0,0 +1,47 @@
+%
+% runden.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$n$ Runden}
+\vspace{-23pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Verschlüsselung}
+In Runde $i=0,\dots,n-1$
+\begin{enumerate}
+\item<2-> Wende die $S$-Box auf alle Bytes des Blocks an
+\item<3-> Führe den Zeilenschift durch
+\item<4-> Mische die Spalten
+\item<5-> Berechne den Schlüsselblock $i$ ($i=0$: ursprünglicher Schlüssel)
+\item<6-> Addiere (XOR) den Rundenschlüssel
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Entschlüsselung}
+In Runde $i=0,\dots,n-1$
+\begin{enumerate}
+\item<8-> Addiere den Rundenschlüssel $n-1-i$
+\item<9-> Invertiere Spaltenmischung (mit $C^{-1}$)
+\item<10-> Invertiere den Zeilenshift
+\item<11-> Wende $S^{-1}$ an auf jedes Byte
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<12->{%
+\begin{block}{Charakteristika}
+\begin{itemize}
+\item<13-> Invertierbar
+\item<14-> Skalierbar: beliebig grosse Blöcke (Vielfache von 32\,bit)
+\item<15-> Keine ``magischen'' Schritte
+\end{itemize}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/aes/sinverse.tex b/vorlesungen/slides/a/aes/sinverse.tex
new file mode 100644
index 0000000..059100e
--- /dev/null
+++ b/vorlesungen/slides/a/aes/sinverse.tex
@@ -0,0 +1,15 @@
+%
+% sinverse.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Inverse $S$-Box}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/sbox.pdf}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/chapter.tex b/vorlesungen/slides/a/chapter.tex
new file mode 100644
index 0000000..78eec84
--- /dev/null
+++ b/vorlesungen/slides/a/chapter.tex
@@ -0,0 +1,23 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+
+\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}
+
diff --git a/vorlesungen/slides/a/dc/beispiel.tex b/vorlesungen/slides/a/dc/beispiel.tex
new file mode 100644
index 0000000..4c99e9e
--- /dev/null
+++ b/vorlesungen/slides/a/dc/beispiel.tex
@@ -0,0 +1,54 @@
+%
+% beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\u#1#2{\uncover<#1->{#2}}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beispiel}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Berechne $1291^{17}\in\mathbb{F}_{2027}$
+\end{block}
+\uncover<2->{%
+\begin{block}{Exponent}
+\vspace{-10pt}
+\[
+17 = 2^4 + 1
+=
+\texttt{10001}_2
+=
+\texttt{0x11}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Divide-and-Conquor}
+\begin{center}
+\begin{tabular}{|>{$}r<{$}>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|}
+\hline
+i&2^i& a^{2^i} & n & n_i & m \\
+\hline
+0& 1& 1291 & 17 & \u{4}{1}&\u{5}{ 1291}\\
+1& 2& \u{6}{ 487}& \u{7}{8}& \u{8}{0}& \u{9}{\color{gray}1291}\\
+2& 4&\u{10}{ 10}&\u{11}{4}&\u{12}{0}&\u{13}{\color{gray}1291}\\
+3& 8&\u{14}{ 100}&\u{15}{2}&\u{16}{0}&\u{17}{\color{gray}1291}\\
+4& 16&\u{18}{1892}&\u{19}{1}&\u{20}{1}&\u{21}{ 37}\\
+\hline
+\end{tabular}
+\end{center}
+\end{block}}
+\uncover<22->{%
+\begin{block}{Resultat}
+\(1291^{17} \equiv 37\mod 2027\)
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/dc/effizient.tex b/vorlesungen/slides/a/dc/effizient.tex
new file mode 100644
index 0000000..327ee7e
--- /dev/null
+++ b/vorlesungen/slides/a/dc/effizient.tex
@@ -0,0 +1,65 @@
+%
+% effizient.tex -- Effiziente Berechnung der Potenz
+%
+% (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{Effiziente Berechnung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Prinzip}
+\begin{enumerate}
+\item<3-> {\color{red}Bits mit Shift isolieren}
+\item<4-> {\color{blue}Laufend reduzieren}
+\item<5-> {\color{darkgreen}effizient quadrieren}
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Algorithmus}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<3->{
+\fill[color=red!20] (2.3,-2.44) rectangle (3.8,-1.98);
+\fill[color=red!20] (1.45,-3.88) rectangle (3.2,-3.42);
+}
+\uncover<4->{
+\fill[color=blue!20] (2.15,-2.94) rectangle (3.7,-2.48);
+}
+\uncover<5->{
+\fill[color=darkgreen!20] (1.45,-4.37) rectangle (3.8,-3.91);
+}
+\node at (0,0) [below right] {\begin{minipage}{6cm}\obeylines
+{\tt int potenz(int $a$, int $n$) \{}\\
+\hspace*{0.7cm}{\tt int m = 1;}\\
+\hspace*{0.7cm}{\tt int q = $a$;}\\
+\uncover<2->{%
+\hspace*{0.7cm}{\tt while ($n$ > 0) \{}\\
+\uncover<3->{%
+\hspace*{1.4cm}{\tt if (0x1 \& $n$) \{}\\
+\uncover<4->{%
+\hspace*{2.1cm}{\tt m *= q;}\\
+}%
+\hspace*{1.4cm}{\tt \}}\\
+\hspace*{1.4cm}{\tt $n$ >{}>= 1;}\\
+}%
+\uncover<5->{%
+\hspace*{1.4cm}{\tt q = sqr(q);}\\
+}%
+\hspace*{0.7cm}{\tt \}}\\
+}%
+\hspace*{0.7cm}{\tt return m;}\\
+{\tt \}}
+\end{minipage}};
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/dc/naiv.txt b/vorlesungen/slides/a/dc/naiv.txt
new file mode 100644
index 0000000..bf5569d
--- /dev/null
+++ b/vorlesungen/slides/a/dc/naiv.txt
@@ -0,0 +1,2 @@
+int m = 1, i = 0;
+while (i++ < n) { m *= a; }
diff --git a/vorlesungen/slides/a/dc/prinzip.tex b/vorlesungen/slides/a/dc/prinzip.tex
new file mode 100644
index 0000000..c75af61
--- /dev/null
+++ b/vorlesungen/slides/a/dc/prinzip.tex
@@ -0,0 +1,86 @@
+%
+% prinzip.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzieren $\mod p$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Berechne $a^n\in\mathbb{F}_p$ für grosses $n$
+\end{block}
+\uncover<2->{%
+\begin{block}{Mengengerüst}
+\(
+\log_2 n > 2000
+\)
+\\
+\uncover<3->{%
+RSA mit $N=pq$: Exponenten sind $e,d$, $e$ klein, aber
+\(
+ed\equiv 1 \mod \varphi(N)
+\)}
+\end{block}}
+\uncover<4->{%
+\begin{block}{Naive Idee}
+\verbatiminput{../slides/a/dc/naiv.txt}
+Laufzeit: $O(n) \uncover<5->{= O(2^{\log_2n})}$%
+\uncover<5->{, d.~h.~exponentiell in der Bitlänge von $n$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Idee 1: Exponent binär schreiben}
+\vspace{-12pt}
+\[
+n = n_k2^k + n_{k-1}2^{k-1} + \dots +n_12^1 + n_02^0
+\]
+\end{block}}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Idee 2: Potenzgesetze}
+\vspace{-12pt}
+\[
+a^n
+=
+a^{n_k2^k}
+a^{n_{k-1}2^k}
+\dots
+a^{n_12^1}
+a^{n_02^0}
+\uncover<8->{=
+\prod_{n_i = 1}
+a^{2^i}}
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<9->{%
+\begin{block}{Idee 3: Quadrieren}
+\vspace{-10pt}
+\begin{align*}
+a^{2^i}
+&=
+a^{2\cdot 2^{i-1}}
+\uncover<10->{=
+(a^{2^{i-1}})^2}
+\\
+&\uncover<11->{=
+(\dots(a\underbrace{\mathstrut^2)^2\dots)^2}_{\displaystyle i}}
+\end{align*}
+\end{block}}
+\vspace{-18pt}
+\uncover<12->{%
+\begin{block}{Laufzeit}
+Multiplikationen: $\le 2 \cdot(\log_2(n) - 1)$
+\\
+\uncover<13->{Worst case Laufzeit: $O(\log_2 n)$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/gruppendh.tex b/vorlesungen/slides/a/ecc/gruppendh.tex
new file mode 100644
index 0000000..13d85c8
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/gruppendh.tex
@@ -0,0 +1,51 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Diffie-Hellmann verallgemeinern}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diffie-Hellman in $\mathbb{F}_p$\strut}
+\begin{enumerate}
+\item<2-> Parteien einigen sich auf $g\in \mathbb{F}_p$, $g\ne 0$, $g\ne 1$
+\item<3-> $A$ und $B$ wählen Exponenten $a,b\in \mathbb{N}$
+\item<4-> Parteien tauschen $u=g^a$ und $v=g^b$ aus
+\item<5-> Parteien berechnen $v^a$ und $u^b$
+\[
+v^a = (g^b)^a = g^{ab} =(g^a)^b = u^b
+\]
+gemeinsamer privater Schlüssel
+\end{enumerate}
+\end{block}
+\uncover<11->{%
+{\usebeamercolor[fg]{title}Spezialfall:} $G=\mathbb{F}_p^*$
+}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Diffie-Hellmann in $G$\strut}
+\begin{enumerate}
+\item<7-> Parteien einigen sich auf $g\in G$, $g\ne e$
+\item<8-> $A$ und $B$ wählen Exponenten $a,b\in \mathbb{N}$
+\item<9-> Parteien tauschen $u=g^a$ und $v=g^b$ aus
+\item<10-> Parteien berechnen $v^a$ und $u^b$
+\[
+v^a = (g^b)^a = g^{ab} =(g^a)^b = u^b
+\]
+gemeinsamer privater Schlüssel
+\end{enumerate}
+\end{block}}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}Idee:} Wähle effizient zu berechnende, ``grosse''
+Gruppen, mit ``komplizierter'' Multiplikation
+}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/inverse.tex b/vorlesungen/slides/a/ecc/inverse.tex
new file mode 100644
index 0000000..c50f698
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/inverse.tex
@@ -0,0 +1,48 @@
+%
+% inverse.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Involution/Inverse}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{In speziellen Koordinaten}
+\vspace{-12pt}
+\[
+v^2 = u^3+Au+B
+\]
+\uncover<2->{invariant unter $v\mapsto -v$}%
+\\
+\uncover<3->{{\color{red}geht nicht in $\mathbb{F}_2$}}
+\end{block}
+\uncover<4->{%
+\begin{block}{Allgemein}
+\vspace{-12pt}
+\begin{align*}
+Y^2+XY &= X^3 + aX+b
+\\
+\uncover<5->{%
+Y(Y+X) &= X^3 + aX + b}
+\end{align*}
+\uncover<6->{invariant unter}
+\begin{align*}
+\uncover<7->{X&\mapsto X,& Y&\mapsto -X-Y}
+\\
+\uncover<8->{&&\Rightarrow X+Y&\mapsto -Y}
+\end{align*}
+\uncover<9->{Spezialfall $\mathbb{F}_2$: $Y\leftrightarrow X+Y$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/kurve.tex b/vorlesungen/slides/a/ecc/kurve.tex
new file mode 100644
index 0000000..04d15f8
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/kurve.tex
@@ -0,0 +1,56 @@
+%
+% kurve.tex -- elliptische Kurven
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Elliptische Kurven}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\uncover<5->{%
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Allgemein}
+mit $a,b\in\Bbbk$
+\[
+Y^2 + XY = X^3 + aX + b
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Spezielle Parametrisierung}
+\vspace{-10pt}
+\begin{align*}
+Y^2 + XY + \frac14X^2
+&=
+X^3 + \frac14X^2 + aX + b
+\\
+\uncover<3->{
+(Y+\frac12X)^2
+&=
+X^3 + \frac14X^2 + aX + b
+}\\
+\uncover<4->{
+v^2
+&=
+u^3+Au+B}
+\end{align*}
+\uncover<4->{mit
+\[
+v=Y+{\textstyle\frac12}X,
+\qquad
+u=X-\frac1{12}
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/oakley.tex b/vorlesungen/slides/a/ecc/oakley.tex
new file mode 100644
index 0000000..6980c10
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/oakley.tex
@@ -0,0 +1,85 @@
+%
+% oakley.tex -- Oakley Gruppen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Oakley-Gruppen}
+\only<1>{%
+\small
+\verbatiminput{../slides/a/ecc/oakley1.txt}
+$\approx 1.55252\cdot 10^{231}$
+}
+\only<2>{%
+\begin{block}{$\mathbb{F}_p$}
+Endlicher Körper mit $p = $
+\verbatiminput{../slides/a/ecc/prime1.txt}
+\end{block}
+}
+\only<3>{%
+\small
+\verbatiminput{../slides/a/ecc/oakley2.txt}
+}
+\only<4>{%
+\begin{block}{$\mathbb{F}_p$}
+Endlicher Körper mit $p = $
+\verbatiminput{../slides/a/ecc/prime2.txt}
+$\approx 1.7977\cdot 10^{308}$
+\end{block}
+}
+\only<5>{%
+\small
+\verbatiminput{../slides/a/ecc/oakley3.txt}
+}
+\only<6>{%
+\begin{block}{Oakley Gruppe 3}
+\begin{align*}
+m(x) &= x^{155} + x^{62} + 1
+\\
+a &= 0
+\\
+b &= \texttt{0x07338f}
+\\
+g_x &= 0x7b = x^6 + x^5 + x^4 + x^3 + x + 1
+\\
+&=
+x^{18}+x^{17}+x^{16}
++
+x^{13}+x^{12}
++
+x^{9}+x^{8}+x^{7}
++
+x^{3}+x^{1}+x^{1}+1
+\\
+|G|&=45671926166590716193865565914344635196769237316 = 4.5672\cdot 10^{46}
+\\
+\log_2|G|&=155\,\text{bit}
+\end{align*}
+\end{block}}
+\only<7>{%
+\small
+\verbatiminput{../slides/a/ecc/oakley4.txt}
+}
+\only<8>{%
+\begin{block}{Oakley Gruppe 4}
+\begin{align*}
+m(x) &= x^{185} + x^{69} + 1
+\\
+a &= 0
+\\
+b &= \texttt{0x1ee9} = x^{12} + x^{11}+x^{10}+x^9 + x^7+x^6+x^5 + x^3+1
+\\
+g_x &= \texttt{0x18} = x^4+x^3
+\\
+|G| &= 49039857307708443467467104857652682248052385001045053116
+\\
+&= 4.9040\cdot 10^{55}
+\\
+\log_2|G| &= 185
+\end{align*}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/oakley1.txt b/vorlesungen/slides/a/ecc/oakley1.txt
new file mode 100644
index 0000000..4cc31ae
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/oakley1.txt
@@ -0,0 +1,14 @@
+6.1 First Oakley Default Group
+
+ Oakley implementations MUST support a MODP group with the following
+ prime and generator. This group is assigned id 1 (one).
+
+ The prime is: 2^768 - 2 ^704 - 1 + 2^64 * { [2^638 pi] + 149686 }
+ Its hexadecimal value is
+
+ FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
+ 29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
+ EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
+ E485B576 625E7EC6 F44C42E9 A63A3620 FFFFFFFF FFFFFFFF
+
+ The generator is: 2.
diff --git a/vorlesungen/slides/a/ecc/oakley2.txt b/vorlesungen/slides/a/ecc/oakley2.txt
new file mode 100644
index 0000000..ddb2d2a
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/oakley2.txt
@@ -0,0 +1,16 @@
+6.2 Second Oakley Group
+
+ IKE implementations SHOULD support a MODP group with the following
+ prime and generator. This group is assigned id 2 (two).
+
+ The prime is 2^1024 - 2^960 - 1 + 2^64 * { [2^894 pi] + 129093 }.
+ Its hexadecimal value is
+
+ FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
+ 29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
+ EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
+ E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
+ EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE65381
+ FFFFFFFF FFFFFFFF
+
+ The generator is 2 (decimal)
diff --git a/vorlesungen/slides/a/ecc/oakley3.txt b/vorlesungen/slides/a/ecc/oakley3.txt
new file mode 100644
index 0000000..ab2c78f
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/oakley3.txt
@@ -0,0 +1,17 @@
+6.3 Third Oakley Group
+
+ IKE implementations SHOULD support a EC2N group with the following
+ characteristics. This group is assigned id 3 (three). The curve is
+ based on the Galois Field GF[2^155]. The field size is 155. The
+ irreducible polynomial for the field is:
+ u^155 + u^62 + 1.
+ The equation for the elliptic curve is:
+ y^2 + xy = x^3 + ax^2 + b.
+
+ Field Size: 155
+ Group Prime/Irreducible Polynomial:
+ 0x0800000000000000000000004000000000000001
+ Group Generator One: 0x7b
+ Group Curve A: 0x0
+ Group Curve B: 0x07338f
+ Group Order: 0X0800000000000000000057db5698537193aef944
diff --git a/vorlesungen/slides/a/ecc/oakley4.txt b/vorlesungen/slides/a/ecc/oakley4.txt
new file mode 100644
index 0000000..3ec20cc
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/oakley4.txt
@@ -0,0 +1,17 @@
+6.4 Fourth Oakley Group
+
+ IKE implementations SHOULD support a EC2N group with the following
+ characteristics. This group is assigned id 4 (four). The curve is
+ based on the Galois Field GF[2^185]. The field size is 185. The
+ irreducible polynomial for the field is:
+ u^185 + u^69 + 1. The
+ equation for the elliptic curve is:
+ y^2 + xy = x^3 + ax^2 + b.
+
+ Field Size: 185
+ Group Prime/Irreducible Polynomial:
+ 0x020000000000000000000000000000200000000000000001
+ Group Generator One: 0x18
+ Group Curve A: 0x0
+ Group Curve B: 0x1ee9
+ Group Order: 0X01ffffffffffffffffffffffdbf2f889b73e484175f94ebc
diff --git a/vorlesungen/slides/a/ecc/operation.tex b/vorlesungen/slides/a/ecc/operation.tex
new file mode 100644
index 0000000..61ef95d
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/operation.tex
@@ -0,0 +1,68 @@
+%
+% operation.tex -- Gruppen-Operation auf einer elliptischen Kurve
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Gruppenoperation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+\end{center}
+\vspace{-23pt}
+\uncover<8->{%
+\begin{block}{Verifizieren}
+\begin{enumerate}
+\item<9-> Assoziativ?
+\item<10-> Neutrales Element $\mathstrut=\infty$
+\item<11-> Involution = Inverse?
+\end{enumerate}
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{block}{Gerade}
+$g_1,g_2\in G$, $t\in \Bbbk$
+\begin{align*}
+g(t)
+&=
+tg_1+(1-t)g_2
+\\
+\uncover<2->{
+\begin{pmatrix}X(t)\\Y(t)\end{pmatrix}
+&=
+t\begin{pmatrix}x_1\\y_1\end{pmatrix}
++
+(1-t)\begin{pmatrix}x_2\\y_2\end{pmatrix}
+\in\Bbbk^2
+}
+\end{align*}
+\end{block}
+\vspace{-13pt}
+\uncover<3->{%
+\begin{block}{3. Schnittpunkt}
+$g(t)$ einsetzen in die elliptische Kurve
+\[
+p(t)
+=
+Y(t)^2+X(t)Y(t)-X(t)^3-aX(t)-b=0
+\]
+\vspace{-12pt}
+\begin{enumerate}
+\item<4->
+kubisches Polynom mit Nullstellen $t=0,1$
+\item<5->
+$p(t) $ ist durch $t(t-1)$ teilbar
+\item<6->
+$p(t) = t(t-1)(Jt+K)=0
+\uncover<7->{\Rightarrow t=-K/J$}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/prime1.txt b/vorlesungen/slides/a/ecc/prime1.txt
new file mode 100644
index 0000000..eb4515d
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/prime1.txt
@@ -0,0 +1,5 @@
+ 15 52518 09230 07089 35130 91813 12584
+81755 63133 40494 34514 31320 23511 94902 96623 99491 02107
+25866 94538 76591 64244 29100 07680 28886 42291 50803 71891
+80463 42632 72761 30312 82983 74438 08208 90196 28850 91706
+91316 59317 53674 69551 76311 98433 71637 22100 72105 77919
diff --git a/vorlesungen/slides/a/ecc/prime2.txt b/vorlesungen/slides/a/ecc/prime2.txt
new file mode 100644
index 0000000..13458fb
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/prime2.txt
@@ -0,0 +1,8 @@
+ 1797 69313
+48623 15907 70839 15679 37874 53197 86029 60487 56011 70644
+44236 84197 18021 61585 19368 94783 37958 64925 54150 21805
+65485 98050 36464 40548 19923 91000 50792 87700 33558 16639
+22955 31362 39076 50873 57599 14822 57486 25750 07425 30207
+74477 12589 55095 79377 78424 44242 66173 34727 62929 93876
+68709 20560 60502 70810 84290 76929 32019 12819 44676 27007
+
diff --git a/vorlesungen/slides/a/ecc/primes b/vorlesungen/slides/a/ecc/primes
new file mode 100644
index 0000000..3feea29
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/primes
@@ -0,0 +1,13 @@
+#! /bin/bash
+#
+# primes
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+bc <<EOF
+ibase=16
+FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245E485B576625E7EC6F44C42E9A63A3620FFFFFFFFFFFFFFFF
+
+FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7EDEE386BFB5A899FA5AE9F24117C4B1FE649286651ECE65381FFFFFFFFFFFFFFFF
+
+EOF
diff --git a/vorlesungen/slides/a/ecc/quadrieren.tex b/vorlesungen/slides/a/ecc/quadrieren.tex
new file mode 100644
index 0000000..942c73b
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/quadrieren.tex
@@ -0,0 +1,59 @@
+%
+% quadrieren.tex -- Quadrieren
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quadrieren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Problem}
+\( g = g_1 = g_2 \)
+$\Rightarrow$
+Tangente
+\\
+\uncover<2->{{\color{red}ohne Analysis!}}
+\end{block}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<3->{%
+\begin{block}{Lösung}
+Finde $h\in G$ derart, dass
+\begin{align*}
+g(t)
+&=
+tg + (1-t)h
+\\
+\uncover<4->{%
+\begin{pmatrix}X(t)\\Y(t)\end{pmatrix}
+&=
+t\begin{pmatrix}x_g\\y_g\end{pmatrix}
++(1-t) \begin{pmatrix}x_h\\y_h\end{pmatrix}
+}
+\end{align*}
+\uncover<5->{eingesetzt
+\[
+p(t)
+=
+Y(t)^2+X(t)Y(t)-X(t)^3-aX(t)-b
+=
+0
+\]}%
+\uncover<6->{%
+Nullstellen $0$ (doppelt) und $1$ hat:}
+\[
+\uncover<7->{p(t) = c(t^3-t)}
+\]
+\uncover<8->{Koeffizientenvergleich: einfachere Gleichungen für $x_h$ und $y_h$}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex
index b606375..6c24e22 100644
--- a/vorlesungen/slides/slides.tex
+++ b/vorlesungen/slides/slides.tex
@@ -47,15 +47,15 @@
\titel
\input{5/chapter.tex}
-%\title[Permutationen]{Permutationen}
-%\section{Permutationen}
-%\titel
-%\input{6/chapter.tex}
+\title[Permutationen]{Permutationen}
+\section{Permutationen}
+\titel
+\input{6/chapter.tex}
-%\title[Matrizengruppen]{Matrizengruppen}
-%\section{Matrizengruppen}
-%\titel
-%\input{7/chapter.tex}
+\title[Matrizengruppen]{Matrizengruppen}
+\section{Matrizengruppen}
+\titel
+\input{7/chapter.tex}
\title[Graphen]{Graphen}
\section{Graphen}
@@ -67,10 +67,10 @@
\titel
\input{9/chapter.tex}
-%\title[Krypto]{Anwendungen in Kryptographie und Codierungstheorie}
-%\section{Krypto}
-%\titel
-%\input{a/chapter.tex}
+\title[Krypto]{Anwendungen in Kryptographie und Codierungstheorie}
+\section{Krypto}
+\titel
+\input{a/chapter.tex}
%\title[Homologie]{Homologie}
%\section{Homologie}
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index d079a05..ce63ae7 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -1,21 +1,23 @@
-%
-% test.tex collection of all slides
-%
-% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-%\folie{5/verzerrung.tex}
-%\folie{5/plan.tex}
-%\folie{5/planbeispiele.tex}
-%\folie{5/approximation.tex}
-
-% XXX Visualisierung Cayley-Hamilton-Produkte
-% XXX \folie{5/chvisual.tex}
-
-% XXX stone weierstrass incomplete
-%\folie{5/stoneweierstrass.tex}
-%\folie{5/swbeweis.tex}
-
-% XXX polynome auf dem spektrum
-% XXX Motiviation für *-Operation
-%\folie{5/normal.tex}
-\folie{5/normalbeispiel34.tex}
+%
+% 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}
+