Merge remote-tracking branch 'upstream/master'

49 files changed, 2974 insertions, 44 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/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/Makefile.inc b/vorlesungen/slides/Makefile.inc
index 6454463..130fa28 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -9,10 +9,13 @@ 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) $(chapter7) $(chapter8) $(chapter9)

+	$(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 6c102f2..ce63ae7 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -4,36 +4,20 @@
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil

 %

 

-\section{Matrizen-Gruppen}

-% Was sind Symmetrien

-%\folie{7/symmetrien.tex}

-% Algebraische Bedingungen für Matrixgruppen

-%\folie{7/algebraisch.tex}

-% Parametrisierung, Beispiel SO(3)

-%\folie{7/parameter.tex}

-% Mannigfaltigkeiten

-%\folie{7/mannigfaltigkeit.tex}

-% Weitere Beispiele

-% SL_2(R)

-%\folie{7/sl2.tex}

-\folie{7/drehung.tex}

-%\folie{7/drehanim.tex}

-% Semidirekte Produkte SO(2) x R^2, R^+ x R

-%\folie{7/semi.tex}

+%\folie{a/dc/prinzip.tex}

+%\folie{a/dc/effizient.tex}

+%\folie{a/dc/beispiel.tex}

 

-\section{Ableitungen}

-% Kurven in einer Gruppe

-%\folie{7/kurven.tex}

-% Einparameter-Gruppen

-%\folie{7/einparameter.tex}

-% Ableitung einer Einparameter-Gruppe

-%\folie{7/ableitung.tex}

-% Lie-Algebra

-%\folie{7/liealgebra.tex}

-% Kommutator

-%\folie{7/kommutator.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}

 

-\section{Exponentialabbildung}

-% Differentialgleichung für die Exponentialabbildung

-%\folie{7/dg.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}