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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/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index c857fec..cbd4dfe 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -17,5 +17,19 @@ chapter2 = \ ../slides/2/frobeniusanwendung.tex \ ../slides/2/quotient.tex \ ../slides/2/quotientv.tex \ + ../slides/2/hilbertraum/definition.tex \ + ../slides/2/hilbertraum/l2beispiel.tex \ + ../slides/2/hilbertraum/basis.tex \ + ../slides/2/hilbertraum/plancherel.tex \ + ../slides/2/hilbertraum/l2.tex \ + ../slides/2/hilbertraum/riesz.tex \ + ../slides/2/hilbertraum/rieszbeispiel.tex \ + ../slides/2/hilbertraum/adjungiert.tex \ + ../slides/2/hilbertraum/spektral.tex \ + ../slides/2/hilbertraum/sturm.tex \ + ../slides/2/hilbertraum/laplace.tex \ + ../slides/2/hilbertraum/qm.tex \ + ../slides/2/hilbertraum/energie.tex \ + ../slides/2/hilbertraum/sobolev.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 49e656a..d3714c3 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -15,3 +15,17 @@ \folie{2/frobeniusanwendung.tex} \folie{2/quotient.tex} \folie{2/quotientv.tex} +\folie{2/hilbertraum/definition.tex} +\folie{2/hilbertraum/l2beispiel.tex} +\folie{2/hilbertraum/basis.tex} +\folie{2/hilbertraum/plancherel.tex} +\folie{2/hilbertraum/l2.tex} +\folie{2/hilbertraum/riesz.tex} +\folie{2/hilbertraum/rieszbeispiel.tex} +\folie{2/hilbertraum/adjungiert.tex} +\folie{2/hilbertraum/spektral.tex} +\folie{2/hilbertraum/sturm.tex} +\folie{2/hilbertraum/laplace.tex} +\folie{2/hilbertraum/qm.tex} +\folie{2/hilbertraum/energie.tex} +\folie{2/hilbertraum/sobolev.tex} diff --git a/vorlesungen/slides/2/hilbertraum/adjungiert.tex b/vorlesungen/slides/2/hilbertraum/adjungiert.tex new file mode 100644 index 0000000..da41576 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/adjungiert.tex @@ -0,0 +1,83 @@ +% +% adjungiert.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Adjungierter Operator} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> +$A\colon H\to L$ lineare Abbildung zwischen Hilberträumen, $y\in L$ +\item<3-> +\[ +H\to\mathbb{C} +: +x\mapsto \langle y, Ax\rangle_L +\] +ist eine lineare Abbildung $H\to\mathbb{C}$ +\item<4-> +Nach dem Darstellungssatz gibt es $v\in H$ mit +\[ +\langle y,Ax\rangle_L = \langle v,x\rangle_H +\quad +\forall x\in H +\] +\end{itemize} +\uncover<5->{% +Die Abbildung +\[ +L\to H +: +y\mapsto v =: A^*y +\] +heisst {\em adjungierte Abbildung}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Endlichdimensional (Matrizen)} +\[ +A^* = \overline{A}^t +\] +\end{block}} +\vspace{-8pt} +\uncover<7->{% +\begin{block}{Selbstabbildungen} +Für Operatoren $A\colon H\to H$ ist $A^*\colon H\to H$ +\[ +\langle x,Ay\rangle += +\langle A^*x, y\rangle +\quad +\forall x,y\in H +\] +\end{block}} +\vspace{-8pt} +\uncover<9->{% +\begin{block}{Selbstadjungierte Operatoren} +\[ +A=A^* +\uncover<10->{\;\Leftrightarrow\; +\langle x,Ay \rangle += +\langle A^*x,y \rangle} +\uncover<11->{= +\langle Ax,y \rangle} +\] +\uncover<12->{Matrizen: +\begin{itemize} +\item<13-> hermitesch +\item<14-> für reelle Hilberträume: symmetrisch +\end{itemize}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/basis.tex b/vorlesungen/slides/2/hilbertraum/basis.tex new file mode 100644 index 0000000..022fa07 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/basis.tex @@ -0,0 +1,65 @@ +% +% basis.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hilbert-Basis} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Menge $\mathcal{B}=\{b_k|k>0\}$ ist eine Hilbertbasis, wenn +\begin{itemize} +\item<2-> $\mathcal{B}$ ist orthonormiert: $\langle b_k,b_l\rangle=\delta_{kl}$ +\item<3-> Der Unterraum $\langle b_k|k>0\rangle\subset H$ ist +dicht: +Jeder Vektor von $H$ kann beliebig genau durch Linearkombinationen von $b_k$ +approximiert werden. +\end{itemize} +\uncover<4->{% +Ein Hilbertraum mit einer Hilbertbasis heisst {\em separabel}} +\end{block} +\uncover<5->{% +\begin{block}{Endlichdimensional} +Der Algorithmus bricht nach endlich vielen Schritten ab. +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Konstruktion} +Iterativ: $\mathcal{B}_0=\emptyset$ +\begin{enumerate} +\item<7-> $V_k = \langle \mathcal{B}_k \rangle$ +\item<8-> Wenn $V_k\ne H$, wähle einen Vektor +\begin{align*} +x\in V_k^{\perp} +&= +\{ +x\in H\;|\; x\perp V_k +\} +\\ +&= +\{x\in H\;|\; +x\perp y\;\forall y\in V_k +\} +\end{align*} +\item<9-> $b_{k+1} = x/\|x\|$ +\[ +\mathcal{B}_{k+1} = \mathcal{B}_k\cup \{b_{k+1}\} +\] +\end{enumerate} +\uncover<10->{% +Wenn $H$ separabel ist, dann ist +\[ +\mathcal{B} = \bigcup_{k} \mathcal{B}_k +\] +eine Hilbertbasis für $H$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/definition.tex b/vorlesungen/slides/2/hilbertraum/definition.tex new file mode 100644 index 0000000..d101637 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/definition.tex @@ -0,0 +1,63 @@ +% +% definition.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hilbertraum --- Definition} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\mathbb{C}$-Hilbertraum $H$} +\begin{enumerate} +\item<2-> $\mathbb{C}$-Vektorraum, muss nicht endlichdimensional sein +\item<3-> Sesquilineares Skalarprodukt +\[ +\langle \cdot,\cdot\rangle +\colon H \to \mathbb{C}: (x,y) \mapsto \langle x,y\rangle +\] +Dazugehörige Norm: +\[ +\|x\| = \sqrt{\langle x,x\rangle} +\] +\item<4-> Vollständigkeit: jede Cauchy-Folge konvergiert +\end{enumerate} +\uncover<5->{% +Ohne Vollständigkeit: {\em Prähilbertraum}} +\end{block} +\uncover<6->{% +\begin{block}{$\mathbb{R}$-Hilbertraum} +Vollständiger $\mathbb{R}$-Vektorraum mit bilinearem Skalarprodukt +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Vollständigkeit} +\begin{itemize} +\item<8-> $(x_n)_{n\in\mathbb{N}}$ ist eine Cauchy-Folge: +Für alle $\varepsilon>0$ gibt es $N>0$ derart, dass +\[ +\| x_n-x_m\| < \varepsilon\quad\forall n,m>N +\] +\item<9-> Grenzwert existiert: $\exists x\in H$ derart, dass es für alle +$\varepsilon >0$ ein $N>0$ gibt derart, dass +\[ +\|x_n-x\|<\varepsilon\quad\forall n>N +\] +\end{itemize} +\end{block}} +\uncover<10->{% +\begin{block}{Cauchy-Schwarz-Ungleichung} +\[ +|\langle x,y\rangle| +\le \|x\| \cdot \|y\| +\] +Gleichheit für linear abhängige $x$ und $y$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/energie.tex b/vorlesungen/slides/2/hilbertraum/energie.tex new file mode 100644 index 0000000..202a7c5 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/energie.tex @@ -0,0 +1,67 @@ +% +% energie.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Energie --- Zeitentwicklung --- Schrödinger} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.30\textwidth} +\uncover<2->{% +\begin{block}{Totale Energie} +Hamilton-Funktion +\begin{align*} +H +&= +\frac12mv^2 + V(x) +\\ +&= +\frac{p^2}{2m} + V(x) +\end{align*} +\end{block}} +\uncover<3->{% +\begin{block}{Quantisierungsregel} +\begin{align*} +\text{Variable}&\to \text{Operator} +\\ +x_k & \to x_k +\\ +p_k & \to \frac{\hbar}{i} \frac{\partial}{\partial x_k} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.66\textwidth} +\uncover<4->{% +\begin{block}{Energie-Operator} +\[ +H += +-\frac{\hbar^2}{2m}\Delta + V(x) +\] +\end{block}} +\uncover<5->{% +\begin{block}{Eigenwertgleichung} +\[ +-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t) = E\psi(x,t) +\] +Zeitunabhängige Schrödingergleichung +\end{block}} +\uncover<6->{% +\begin{block}{Zeitabhängigkeit = Schrödingergleichung} +\[ +-\frac{\hbar}{i} +\frac{\partial}{\partial t} +\psi(x,t) += +-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t) +\] +\uncover<7->{Eigenwertgleichung durch Separation von $t$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/l2.tex b/vorlesungen/slides/2/hilbertraum/l2.tex new file mode 100644 index 0000000..bd744ab --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/l2.tex @@ -0,0 +1,61 @@ +% +% l2.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$L^2$-Hilbertraum} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> +Vektorraum: Funktionen +\[ +f\colon [a,b] \to \mathbb{C} +\] +\item<3-> +Sesquilineares Skalarprodukt +\[ +\langle f,g\rangle += +\int_a^b \overline{f(x)}\, g(x) \,dx +\] +\item<4-> +Norm: +\[ +\|f\|^2 = \int_a^b |f(x)|^2\,dx +\] +\item<5-> +Vollständigkeit? +\uncover<6->{$\rightarrow$ +Lebesgue Konvergenz-Satz} +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Vollständigkeit} +\begin{itemize} +\item +Funktioniert nicht für Riemann-Integral +\item<8-> +Erweiterung des Integrals auf das sogenannte Lebesgue-Integral (nach +Henri Lebesgue) +\item<9-> +Abzählbare Mengen spielen keine Rolle $\rightarrow$ Nullmengen +\item<10-> +Funktionen $\rightarrow$ Klassen von Funktionen, die sich auf einer Nullmenge +unterscheiden +\item<11-> +Konvergenz-Satz von Lebesgue $\rightarrow$ es funktioniert +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/l2beispiel.tex b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex new file mode 100644 index 0000000..3ae44af --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex @@ -0,0 +1,82 @@ +% +% l2beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beispiele: $\mathbb{R},\mathbb{R}^2,\dots,\mathbb{R}^n,\dots,l^2$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> Quadratsummierbare Folgen von komplexen Zahlen +\[ +l^2 += +\biggl\{ +(x_k)_{k\in\mathbb{N}}\,\bigg|\, \sum_{k=0}^\infty |x_k|^2 < \infty +\biggr\} +\] +\item<3-> Skalarprodukt: +\begin{align*} +\langle x,y\rangle +&= +\sum_{k=0}^\infty \overline{x}_ky_k, +& +\uncover<4->{\|x\|^2 = \sum_{k=0}^\infty |x_k|^2} +\end{align*} +\item<5-> Vollständigkeit, +Konvergenz: Cauchy-Schwarz-Ungleichung +\[ +\biggl| +\sum_{k=0}^\infty \overline{x}_ky_k +\biggr| +\le +\sum_{k=0}^\infty |x_k|^2 +\sum_{l=0}^\infty |y_l|^2 +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Standardbasisvektoren} +\begin{align*} +e_i +&= +(0,\dots,0,\underset{\underset{\textstyle i}{\textstyle\uparrow}}{1},0,\dots) +\\ +\uncover<7->{(e_i)_k &= \delta_{ik}} +\end{align*} +\uncover<8->{sind orthonormiert: +\begin{align*} +\langle e_i,e_j\rangle +&= +\sum_k \overline{\delta}_{ik}\delta_{jk} +\uncover<9->{= +\delta_{ij}} +\end{align*}} +\end{block}} +\vspace{-16pt} +\uncover<10->{% +\begin{block}{Analyse} +$x_k$ kann mit Skalarprodukten gefunden werden: +\begin{align*} +\hat{x}_i += +\langle e_i,x\rangle +&\uncover<11->{= +\sum_{k=0}^\infty \overline{\delta}_{ik} x_k} +\uncover<12->{= +x_i} +\end{align*} +\uncover<13->{(Fourier-Koeffizienten)} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/laplace.tex b/vorlesungen/slides/2/hilbertraum/laplace.tex new file mode 100644 index 0000000..8f6b196 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/laplace.tex @@ -0,0 +1,66 @@ +% +% laplace.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Höhere Dimension} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Problem} +Gegeben: $\Omega\subset\mathbb{R}^n$ ein Gebiet +\\ +Gesucht: Lösungen von $\Delta u=0$ mit $u_{|\partial\Omega}=0$ +\end{block} +\uncover<2->{% +\begin{block}{Funktionen} +Hilbertraum $H$ der Funktionen $f:\overline{\Omega}\to\mathbb{C}$ +mit $f_{|\partial\Omega}=0$ +\end{block}} +\uncover<3->{% +\begin{block}{Skalarprodukt} +\[ +\langle f,g\rangle += +\int_{\Omega} \overline{f}(x) g(x)\,d\mu(x) +\] +\end{block}} +\uncover<4->{% +\begin{block}{Laplace-Operator} +\[ +\Delta \psi = \operatorname{div}\operatorname{grad}\psi +\] +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<5->{% +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,\Delta g\rangle +&\uncover<6->{= +\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)} +\\ +&\uncover<7->{= +\int_{\partial\Omega} +\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)} +\\ +&\uncover<7->{\qquad +- +\int_{\Omega} +\operatorname{grad}\overline{f}(x)\cdot \operatorname{grad}g(x) +\,d\mu(x)} +\\ +&\uncover<8->{=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)} +\\ +&\uncover<9->{= +\langle \Delta f,g\rangle} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex new file mode 100644 index 0000000..73dd46b --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -0,0 +1,102 @@ +% +% plancherel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Plancherel-Gleichung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hilbertraum mit Hilbert-Basis} +$H$ Hilbertraum mit Hilbert-Basis +$\mathcal{B}=\{b_k\;|\; k>0\}$, $x\in H$ +\end{block} +\uncover<2->{% +\begin{block}{Analyse: Fourier-Koeffizienten} +\begin{align*} +a_k = \hat{x}_k &=\langle b_k, x\rangle +\\ +\uncover<3->{\hat{x}&=\mathcal{F}x} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Synthese: Fourier-Reihe} +\begin{align*} +\tilde{x} +&= +\sum_k a_k b_k +\uncover<5->{= +\sum_k \langle x,b_k\rangle b_k} +\end{align*} +\end{block}} +\vspace{-6pt} +\uncover<6->{% +\begin{block}{Analyse von $\tilde{x}$} +\begin{align*} +\langle b_l,\tilde{x}\rangle +&= +\biggl\langle +b_l,\sum_{k}\langle b_k,x\rangle b_k +\biggr\rangle +\uncover<7->{= +\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle} +\uncover<8->{= +\sum_k \langle b_k,x\rangle\delta_{kl}} +\uncover<9->{= +\langle b_l,x\rangle} +\uncover<10->{= +\hat{x}_l} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Plancherel-Gleichung} +\begin{align*} +\|\tilde{x}\|^2 +&= +\langle \tilde{x},\tilde{x}\rangle += +\biggl\langle +\sum_k \hat{x}_kb_k, +\sum_l \hat{x}_lb_l +\biggr\rangle +\\ +&\uncover<12->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle} +\uncover<13->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl}} +\\ +\uncover<14->{ +\|\tilde{x}\|^2 +&= +\sum_k |\hat{x}_k|^2} +\uncover<15->{= +\|\hat{x}\|_{l^2}^2} +\uncover<16->{= +\|\mathcal{F}x\|_{l^2}^2} +\end{align*} +\end{block}} +\vspace{-12pt} +\uncover<17->{% +\begin{block}{Isometrie} +\begin{align*} +\mathcal{F} +\colon +H \to l^2 +\colon +x\mapsto \hat{x} +\end{align*} +\uncover<18->{Alle separablen Hilberträume sind isometrisch zu $l^2$ via +%Fourier-Transformation +$\mathcal{F}$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/qm.tex b/vorlesungen/slides/2/hilbertraum/qm.tex new file mode 100644 index 0000000..a108121 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/qm.tex @@ -0,0 +1,90 @@ +% +% qm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Anwendung: Quantenmechanik} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zustände (Wellenfunktion)} +$L^2$-Funktionen auf $\mathbb{R}^3$ +\[ +\psi\colon\mathbb{R}^3\to\mathbb{C} +\] +\end{block} +\vspace{-6pt} +\uncover<2->{% +\begin{block}{Wahrscheinlichkeitsinterpretation} +\[ +|\psi(x)|^2 = \left\{ +\begin{minipage}{4.6cm}\raggedright +Wahrscheinlichkeitsdichte für Position $x$ des Teilchens +\end{minipage}\right. +\] +\end{block}} +\vspace{-6pt} +\uncover<3->{% +\begin{block}{Skalarprodukt} +\[ +\langle\psi,\psi\rangle += +\int_{\mathbb{R}^3} |\psi(x)|^2\,dx = 1 +\] +\end{block}} +\vspace{-6pt} +\uncover<4->{% +\begin{block}{Messgrösse $A$} +Selbstadjungierter Operator $A$ +\\ +\uncover<5->{$\rightarrow$ +Hilbertbasis $|i\rangle$ von EV von $A$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Überlagerung} +\begin{align*} +|\psi\rangle +&= +\sum_i +w_i|i\rangle +\\ +\uncover<7->{\langle \psi|\psi\rangle +&= +\sum_i |w_i|^2 \qquad\text{(Plancherel)}} +\end{align*} +\uncover<8->{% +$|w_i|^2=|\langle \psi|i\rangle|^2$ Wahrscheinlichkeit für Zustand $|i\rangle$ +} +\end{block}} +\uncover<9->{% +\begin{block}{Erwartungswert} +\begin{align*} +E(A) +&\uncover<10->{= +\sum_i |w_i|^2 \alpha_i} +\uncover<11->{= +\sum_i \overline{w}_i\alpha_i w_i } +\hspace{5cm} +\\ +&\only<12>{= +\sum_{i,j} \overline{w}_j\alpha_i w_i \langle j|i\rangle} +\uncover<13->{= +\sum_{i} \overline{w}_j\langle j| \sum_i \alpha_i w_i |i\rangle} +\\ +&\uncover<14->{= +\sum_{i,j} \overline{w}_j w_i \langle j| +A|i\rangle} +\uncover<15->{= +\langle \psi| A |\psi\rangle} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/riesz.tex b/vorlesungen/slides/2/hilbertraum/riesz.tex new file mode 100644 index 0000000..437fb3c --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/riesz.tex @@ -0,0 +1,76 @@ +% +% riesz.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Darstellungssatz von Riesz} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Dualraum} +$V$ ein Vektorraum, $V^*$ der Raum aller Linearformen +\[ +f\colon V\to \mathbb{C} +\] +\end{block} +\uncover<3->{% +\begin{block}{Beispiel: $l^\infty$} +$l^\infty=\text{beschränkte Folgen in $\mathbb{C}$}$, +Linearformen: +\begin{align*} +\uncover<4->{ +f(x) +&= +\sum_{i=0}^\infty f_ix_i} +\\ +\uncover<5->{ +\|f\| +&= +\sup_{\|x\|_{\infty}\le 1} +|f(x)|} +\uncover<6->{= +\sum_{k\in\mathbb{N}} |f_k|} +\\ +\uncover<7->{ +\Rightarrow +l^{\infty*} +&= +l^1} +\uncover<9->{\qquad(\ne l^2)} +\\ +\uncover<8->{ +&=\{\text{summierbare Folgen in $\mathbb{C}$}\} +} +\end{align*} + +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiel: $\mathbb{C}^n$} +${\mathbb{C}^n}^* = \mathbb{C}^n$ +\end{block}} +\uncover<10->{% +\begin{theorem}[Riesz] +Zu einer stetigen Linearform $f\colon H\to\mathbb{C}$ gibt es $v\in H$ mit +\[ +f(x) = \langle v,x\rangle +\quad\forall x\in H +\] +und $\|f\| = \|v\|$ +\end{theorem}} +\uncover<11->{% +\begin{block}{Dualraum von $H$} +$H^*=H$ +\end{block}}% +\uncover<12->{% +Der Hilbertraum ist die ``intuitiv richtige, unendlichdimensionale'' +Verallgemeinerung von $\mathbb{C}^n$} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex new file mode 100644 index 0000000..de9383f --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex @@ -0,0 +1,107 @@ +% +% rieszbeispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Linearform auf $L^2$-Funktionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Linearform auf $\mathbb{C}^n$} +\begin{align*} +{\color{blue}x}&=\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}, +& +f({\color{blue}x}) +&= +\begin{pmatrix}f_1&f_2&\dots&f_n\end{pmatrix} {\color{blue}x} +\\ +\uncover<2->{ +{\color{red}v}&= +\rlap{$ +\begin{pmatrix} +\overline{f}_1&\overline{f}_2&\dots&\overline{f}_n +\end{pmatrix}^t +\uncover<3->{\;\Rightarrow\; +f({\color{blue}x})=\langle {\color{red}v},{\color{blue}x}\rangle} +$}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Linearform auf $L^2([a,b])$} +\begin{align*} +{\color{red}x}&\in L^2([a,b]) +\\ +\uncover<5->{ +f&\colon L^2([a,b]) \to \mathbb{C} +: {\color{red}x} \mapsto f({\color{red}x})} +\intertext{\uncover<6->{Riesz-Darstellungssatz: $\exists {\color{blue}v}\in L^2([a,b])$}} +\uncover<7->{f({\color{red}x}) +&= +\int_a^b {\color{blue}\overline{v}(t)}{\color{red}x(t)}\,dt} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope}[xshift=-3.5cm] +\def\s{0.058} +\foreach \n in {0,...,5}{ +\uncover<3->{ + \draw[color=red,line width=3pt] + ({\n+\s},{1/(\n+0.5)}) -- ({\n+\s},0); + \node[color=red] at ({\n},{-0.2+1/(\n+0.5)}) + [above right] {$v_\n\mathstrut$}; +} + \draw[color=blue,line width=3pt] + ({\n-\s},{0.4+0.55*sin(200*\n)+0.25*\n}) -- ({\n-\s},0); + \node[color=blue] at ({\n},{-0.2+0.4+0.55*sin(200*\n)+0.25*\n}) + [above left] {$x_\n\mathstrut$}; +} +\draw[->] (-0.6,0) -- (6,0) coordinate[label={$n$}]; +\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}]; +\foreach \n in {0,...,5}{ + \fill (\n,0) circle[radius=0.08]; + \node at (\n,0) [below] {$\n$\strut}; +} +\node at (5.6,0) [below] {$\cdots$\strut}; +\end{scope} +\uncover<4->{ +\begin{scope}[xshift=3.5cm] +\uncover<7->{ +\fill[color=red!40,opacity=0.5] + plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}) + -- + (5,0) -- (0,0) -- cycle; +} +\fill[color=blue!40,opacity=0.5] + plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}) + -- (5,0) -- (0,0) -- cycle; +\uncover<7->{ +\draw[color=red,line width=1.4pt] + plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}); +\node[color=red] at (0,2) [right] {$x(t)$}; +} + +\draw[color=blue,line width=1.4pt] + plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}); +\node[color=blue] at (4.5,2) [right]{$v(t)$}; + +\draw[->] (-0.6,0) -- (6.0,0) coordinate[label={$t$}]; +\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}]; +\draw (0.0,-0.1) -- (0.0,0.1); +\node at (0.0,0) [below] {$a$\strut}; +\draw (5.0,-0.1) -- (5.0,0.1); +\node at (5.0,0) [below] {$b$\strut}; +\end{scope} +} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/sobolev.tex b/vorlesungen/slides/2/hilbertraum/sobolev.tex new file mode 100644 index 0000000..828d34d --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sobolev.tex @@ -0,0 +1,51 @@ +% +% sobolev.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sobolev-Raum} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Vektorrraum $W$} +Funktionen $f\colon \Omega\to\mathbb{C}$ +\begin{itemize} +\item<2-> +$f\in L^2(\Omega)$ +\item<3-> +$\nabla f\in L^2(\Omega)$ +\item<4-> +homogene Randbedingungen: +$f_{|\partial \Omega}=0$ +\end{itemize} +\end{block} +\uncover<5->{% +\begin{block}{Skalarprodukt} +\begin{align*} +\langle f,g\rangle_W +&\uncover<6->{= +\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)} +\\ +&\uncover<7->{\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)} +\\ +&\uncover<8->{=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<9->{% +\begin{block}{Vollständigkeit} +\dots +\end{block}} +\uncover<10->{% +\begin{block}{Anwendung} +``Ein Hilbertraum für jedes partielle Differentialgleichungsproblem'' +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/spektral.tex b/vorlesungen/slides/2/hilbertraum/spektral.tex new file mode 100644 index 0000000..b561b69 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/spektral.tex @@ -0,0 +1,91 @@ +% +% spektral.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spektraltheorie für selbstadjungierte Operatoren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Voraussetzungen} +\begin{itemize} +\item +Hilbertraum $H$ +\item +$A\colon H\to H$ linear +\end{itemize} +\end{block} +\uncover<2->{% +\begin{block}{Eigenwerte} +$x\in H$ ein EV von $A$ zum EW $\lambda\ne 0$ +\begin{align*} +\uncover<3->{\langle x,x\rangle +&= +\frac1{\lambda} +\langle x,\lambda x\rangle} +\uncover<3->{= +\frac1{\lambda} +\langle x,Ax\rangle} +\\ +&\uncover<4->{= +\frac1{\lambda} +\langle Ax,x\rangle} +\uncover<5->{= +\frac{\overline{\lambda}}{\lambda} +\langle x,x\rangle} +\\ +\uncover<6->{\frac{\overline{\lambda}}{\lambda}&=1 +\quad\Rightarrow\quad +\overline{\lambda} = \lambda} +\uncover<7->{\quad\Rightarrow\quad +\lambda\in\mathbb{R}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Orthogonalität} +$u,v$ EV zu EW $\mu,\lambda\in \mathbb{R}\setminus\{0\}$, $\overline{\mu}=\mu\ne\lambda$ +\begin{align*} +\uncover<9->{ +\langle u,v\rangle +&= +\frac{1}{\mu} +\langle \mu u,v\rangle} +\uncover<10->{= +\frac{1}{\mu} +\langle Au,v\rangle} +\\ +&\uncover<11->{= +\frac{1}{\mu} +\langle u,Av\rangle} +\uncover<12->{= +\frac{1}{\mu} +\langle u,\lambda v\rangle} +\uncover<13->{= +\frac{\lambda}{\mu} +\langle u,v\rangle} +\\ +\uncover<14->{\Rightarrow +\; +0 +&= +\underbrace{\biggl(\frac{\lambda}{\mu}-1\biggr)}_{\displaystyle \ne 0} +\langle u,v\rangle} +\uncover<15->{\;\Rightarrow\; +\langle u,v\rangle = 0} +\end{align*} +\uncover<16->{EV zu verschiedenen EW sind orthogonal} +\end{block}} +\end{column} +\end{columns} +\uncover<17->{% +\begin{block}{Spektralsatz} +Es gibt eine Hilbertbasis von $H$ aus Eigenvektoren von $A$ +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex new file mode 100644 index 0000000..a6865ab --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sturm.tex @@ -0,0 +1,58 @@ +% +% sturm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sturm-Liouville-Problem} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wellengleichung} +Saite mit variabler Massedichte führt auf die DGL +\[ +-y''(t) + q(t) y(t) = \lambda y(t), +\quad +q(t) > 0 +\] +mit Randbedingungen $y(0)=y(1)=0$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Sturm-Liouville-Operator} +\[ +A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p +\] +auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten +\[ +f(0)=f(1)=0 +\] +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,Ag \rangle +&\uncover<4->{= +\langle f,-D^2 g\rangle + \langle f,qg\rangle += +- +\int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt ++\langle f,qg\rangle} +\\ +&\uncover<5->{=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} ++\int_0^1 \overline{f}'(t)g'(t)\,dt ++\langle f,qg\rangle} +\uncover<6->{=-\int_0^1 \overline{f}''(t)g(t)\,dt ++\langle qf,g\rangle} +\\ +&\uncover<7->{=\langle Af,g\rangle} +\end{align*} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index ad1081e..5aac429 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -17,6 +17,20 @@ chapter4 = \ ../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 a10712a..0691e39 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -16,3 +16,16 @@ \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/char2.tex b/vorlesungen/slides/4/char2.tex new file mode 100644 index 0000000..2b5709a --- /dev/null +++ b/vorlesungen/slides/4/char2.tex @@ -0,0 +1,48 @@ +% +% char2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Charakteristik 2} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Plus und Minus} +\[ +x+x = 2x = 0 +\uncover<2->{\Rightarrow +-x=x} +\] +\end{block} +\uncover<3->{% +\begin{block}{Quadrieren} +In $\mathbb{F}_2$ ist $2=0$, d.h +\[ +(x+y)^2 += +x^2 + 2xy + y^2 +\uncover<4->{= +x^2 + y^2} +\] +für alle $x,y\in\Bbbk$ +\end{block}} +\uncover<6->{% +\begin{block}{Frobenius-Automorphismus} +\[ +(x+y)^{2^n} = x^{2^n}+y^{2^n} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Pascal-Dreieck} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial2.pdf} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/charakteristik.tex b/vorlesungen/slides/4/charakteristik.tex new file mode 100644 index 0000000..a0d6d3e --- /dev/null +++ b/vorlesungen/slides/4/charakteristik.tex @@ -0,0 +1,71 @@ +% +% charakteristisk.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Primkörper und Charakteristik} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Primkörper} +$1\in\Bbbk$ +\begin{enumerate} +\item<2-> +$n\cdot 1\ne 0\;\forall n\in\mathbb{N}$\uncover<3->{: +$\Rightarrow$ +$\mathbb{Z}\subset \Bbbk$} +\uncover<4->{% +$\Rightarrow$ +$\mathbb{Q}\subset \Bbbk$} +\item<5-> +$\{n\mathbb{Z}\;|\; +\text{$n\cdot 1 = 0$ in $\Bbbk$}\} += +p\mathbb{Z}$ +\uncover<6->{ +$\Rightarrow$ +$\mathbb{F}_p\subset \Bbbk$} +\end{enumerate} +\end{block} +\uncover<7->{% +\begin{block}{Primkörper} +Der Primkörper $\operatorname{Prim}(\Bbbk)$ +eines Körpers $\Bbbk$ ist der kleinste in $\Bbbk$ +enthaltene Körper +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Charakteristik} +\vspace{-10pt} +\[ +\operatorname{char}(\Bbbk) += +\begin{cases} +\uncover<9->{p&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{F}_p}\\ +\uncover<10->{0&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{Q}} +\end{cases} +\] +\vspace{-10pt} +\end{block}} +\uncover<11->{% +\begin{block}{Vektorraum} +$\Bbbk$ ist ein Vektorraum über $\operatorname{Prim}(\Bbbk)$ +durch Einschränkung der Multiplikation auf $\operatorname{Prim}(\Bbbk)$ +(Körperstruktur vergessen) +\end{block}} +\uncover<12->{% +\begin{block}{Endliche Körper} +\begin{itemize} +\item<13-> +Endliche Körper haben immer Charakteristik $p\ne 0$ +\item<14-> +$\Bbbk$ ist eine endlichdimensionaler $\mathbb{F}_p$-Vektorraum +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex index be5b3ca..c63afec 100644 --- a/vorlesungen/slides/4/euklidmatrix.tex +++ b/vorlesungen/slides/4/euklidmatrix.tex @@ -18,7 +18,7 @@ a_k = b_kq_k + r_k \;\Rightarrow\; \left\{ \begin{aligned} -a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\ +a_{k+1} &= b_k = \phantom{a_k-q_k}b_k \\ b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k \end{aligned} \right.} diff --git a/vorlesungen/slides/4/frobenius.tex b/vorlesungen/slides/4/frobenius.tex new file mode 100644 index 0000000..56fd78f --- /dev/null +++ b/vorlesungen/slides/4/frobenius.tex @@ -0,0 +1,54 @@ +% +% frobenius.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Frobenius-Automorphismus} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +$\operatorname{Prim}(\Bbbk) = \mathbb{F}_p$ +\uncover<2->{% +\begin{block}{Binomial-Koeffizienten} +\vspace{-10pt} +\begin{align*} +\binom{p}{k} +&= +\frac{ +{\color{red}p}\cdot(p-1)\cdot(p-2)\cdot\dots\cdot (p-k+1) +}{ +1\cdot2\cdot3\cdot\dots\cdot k +} +\intertext{{\color{red}$p$} wird nicht gekürzt wegen} +\uncover<3->{1&\not\equiv 0 \mod p}\\ +\uncover<3->{2&\not\equiv 0 \mod p}\\ +\uncover<3->{ &\phantom{a}\vdots}\\ +\uncover<3->{k&\not\equiv 0 \mod p} +\end{align*} +\vspace{-10pt} +\end{block}} +\vspace{-5pt} +\uncover<4->{% +\begin{block}{Frobenius-Authomorphismus} +\vspace{-10pt} +\begin{align*} +\uncover<5->{(x+y)^{p\phantom{\mathstrut^n}} +&= +x^{p\phantom{\mathstrut}^n}+y^{p\phantom{mathstrut^n}}} +\\ +\uncover<6->{(x+y)^{p^n} &= x^{p^n}+y^{p^n}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Pascal-Dreieck} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial5.pdf} +\end{center} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex new file mode 100644 index 0000000..ef5902b --- /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..6051813 --- /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..6909849 --- /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..444944e --- /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..6cfcabe --- /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 Binary files differnew file mode 100644 index 0000000..ff6fc14 --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel.png diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov new file mode 100644 index 0000000..a5db465 --- /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 Binary files differnew file mode 100644 index 0000000..68919cc --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel2.png diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov new file mode 100644 index 0000000..ac32b2f --- /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..094b570 --- /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..f5763b9 --- /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..e9e4ce8 --- /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..1cae3fa --- /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..54b941b --- /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..ada6079 --- /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/4/qundr.tex b/vorlesungen/slides/4/qundr.tex new file mode 100644 index 0000000..a6f89bd --- /dev/null +++ b/vorlesungen/slides/4/qundr.tex @@ -0,0 +1,138 @@ +% +% qundr.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.8,0,0} +\definecolor{darkblue}{rgb}{0,0,0.8} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (ll) at (-6,-3.6); +\coordinate (lr) at (6,-3.6); +\coordinate (ur) at (6,3.6); +\coordinate (ul) at (-6,3.6); + +\def\d{0.6} +\def\D{0.5} + +\coordinate (q) at (0,{-2.25+\d}); +\coordinate (r) at (-1.5,{\d+\D}); +\coordinate (a) at (1.5,{\d-\D}); +\coordinate (c) at (0,{2.25+\d}); + +\coordinate (m1) at ($0.5*(q)+0.5*(r)$); +\coordinate (m2) at ($0.5*(q)+0.5*(a)$); +\coordinate (m3) at ($0.5*(c)+0.5*(r)$); +\coordinate (m4) at ($0.5*(c)+0.5*(a)$); + +\def\t{1.5} +\coordinate (M1) at ($(m1)+\t*(m1)-\t*(m4)$); +\coordinate (M2) at ($(m2)+\t*(m2)-\t*(m3)$); +\coordinate (M4) at ($(m4)+\t*(m4)-\t*(m1)$); +\coordinate (M3) at ($(m3)+\t*(m3)-\t*(m2)$); + +\begin{scope} +\clip (ll) rectangle (ur); + +\uncover<3->{ + \fill[color=blue!30] + ($0.9*(m1)+0.1*(M1)+(-6,0)$) -- ($0.9*(m1)+0.1*(M1)$) + -- (M4) -- (ul) -- cycle; +} + +\uncover<4->{ + \fill[color=red!60,opacity=0.5] + ($0.9*(m2)+0.1*(M2)$) -- ($0.9*(m2)+0.1*(M2)+(6,0)$) + -- (ur) -- (M3) -- cycle; +} + +\uncover<2->{ + \fill[color=darkgreen!60,opacity=0.5] + ($1.09*(m3)-0.09*(M3)$) -- ($1.09*(m3)-0.09*(M3)+(-6,0)$) + -- (ll) -- (M2) -- cycle; +} + +\uncover<6->{ + \fill[color=gray,opacity=0.5] + ({6-0.1},{\d+0.22}) rectangle ({6-2.4},{\d+0.62}); + \node[color=yellow] at (6,\d) [above left] {überabzählbar\strut}; + + \fill[color=gray,opacity=0.5] + ({-6+0.1},{\d-0.15}) rectangle ({-6+1.75},{\d-0.55}); + \node[color=yellow] at (-6,\d) [below right] {abzählbar\strut}; + + \draw[color=yellow,line width=2pt] (-7,\d) -- (7,\d); +} + +\end{scope} + +\node at (q) {$\mathbb{Q}$\strut}; +\node at ($(q)+(0,-0.2)$) [below] {Primkörper}; + +\uncover<3->{ + \node at (r) {$\mathbb{R}$\strut}; + \node at (r) [left] {$\text{reelle Zahlen}=\mathstrut$}; + \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (r); + \node at ($0.5*(q)+0.5*(r)$) + [below,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$}; + \node[color=blue] at (ul) + [above right] {topologische Vervollständigung}; +} + +\uncover<4->{ + \node at (a) {$\mathbb{A}$\strut}; + \node at (a) [right] {$\mathstrut = \text{algebraische Zahlen}$}; + \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (a); + \node at ($0.5*(q)+0.5*(a)$) + [below,rotate={atan((2.25-\D)/1.5)}] {index $\infty$}; + \node[color=red] at (ur) + [above left] {algebraische Vervollständigung}; +} + +\uncover<5->{ + \node at (c) {$\mathbb{C}$\strut}; + \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (r) -- (c); + \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (a) -- (c); + \node at ($(c)+(0,0.2)$) [above] {komplexe Zahlen}; + \node at ($0.5*(r)+0.5*(c)$) + [above,rotate={atan((2.25-\D)/1.5)}] {index 2}; + \node at ($0.5*(a)+0.5*(c)$) + [above,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$}; +} + +\uncover<3->{ + \node[color=darkblue] at (ul) [below right] + {\begin{minipage}{0.3\textwidth}\raggedright + Grenzwerte von Cauchy-Folgen in $\mathbb{Q}$ hinzufügen + \end{minipage}}; +} + +\uncover<4->{ + \node[color=darkred] at (ur) [below left] + {\begin{minipage}{0.3\textwidth}\raggedleft + Nullstellen von Polynomen in $\mathbb{Q}[X]$ hinzufügen + \end{minipage}}; +} + +\uncover<2->{ + \node[color=darkgreen] at (ll) [above right] + {\begin{minipage}{0.4\textwidth}\raggedright + \begin{block}{Archimedische Eigenschaft} + Für $a>b >0$ gibt es $n\in\mathbb{N}$ mit + $n\cdot b > a$ + \end{block} + \end{minipage}}; + + \node[color=darkgreen] at (ll) [below right] + {geordneter Körper, nötig für die Definition von Cauchy-Folgen}; +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 4ca3de4..5b849ec 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,6 +5,8 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter5 = \ + ../slides/5/plan.tex \ + ../slides/5/planbeispiele.tex \ ../slides/5/verzerrung.tex \ ../slides/5/motivation.tex \ ../slides/5/charpoly.tex \ @@ -27,6 +29,8 @@ chapter5 = \ \ ../slides/5/spektrum.tex \ ../slides/5/normal.tex \ + ../slides/5/normalbeispiel.tex \ + ../slides/5/normalbeispiel34.tex \ ../slides/5/unitaer.tex \ \ ../slides/5/konvergenzradius.tex \ @@ -36,9 +40,12 @@ chapter5 = \ ../slides/5/satzvongelfand.tex \ \ ../slides/5/stoneweierstrass.tex \ + ../slides/5/swbeweis.tex \ ../slides/5/potenzreihenmethode.tex \ ../slides/5/logarithmusreihe.tex \ ../slides/5/exponentialfunktion.tex \ ../slides/5/hyperbolisch.tex \ + \ + ../slides/5/approximation.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/approximation.tex b/vorlesungen/slides/5/approximation.tex new file mode 100644 index 0000000..a35bae7 --- /dev/null +++ b/vorlesungen/slides/5/approximation.tex @@ -0,0 +1,56 @@ +% +% approximation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\begin{frame}[t] +\frametitle{Approximation einer reellen Funktion} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.5\textwidth} +\begin{block}{Gegeben} +Eine stetige Funktion $f\colon[a,b]\to\mathbb{R}$ +\end{block} +\end{column} +\begin{column}{0.5\textwidth} +\uncover<2->{% +\begin{block}{Gesucht} +Approximationspolynome $p_n\to f$ gleichmässig auf $[a,b]$ +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Lösungsmöglichkeiten} +\vspace{-3pt} +\begin{center} +\renewcommand{\arraystretch}{1.3} +\begin{tabular}{|p{4.2cm}|l|} +\hline +Familie&Approximationspolynom für $[a,b]=[0,1]$ +\\ +\hline +\uncover<4->{% +\raggedright +Lagrange-Interpolationspolynom} +&\uncover<5->{% +$\displaystyle\begin{aligned} +l(x)&=(x-x_0)(x-x_1)\dots(x-x_n),\quad x_k = \frac{k}{n} +\\ +p_n(x)&= \sum_{k=0}^n f(x_k)\frac{l(x)}{x-x_k} +\end{aligned}$} +\\ +\hline\uncover<6->{% +\raggedright +Approximation mit Bernstein-Polynomen} +&\uncover<7->{$\displaystyle \begin{aligned} +B_{k,n}(t) &= \frac{1}{(b-a)^n}\binom{n}{k}(t-a)^k(b-t)^{n-k} +\\ +B_n(f)(t) &= \sum_{k=0}^n B_{k,n}(t) \cdot f\biggl(\frac{k}{n}\biggr) +\end{aligned}$} +\\ +\hline +\end{tabular} +\end{center} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg Binary files differindex 9cb789c..bebc36f 100644 --- a/vorlesungen/slides/5/beispiele/kombiniert.jpg +++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov index c187d08..d17adb7 100644 --- a/vorlesungen/slides/5/beispiele/kombiniert.pov +++ b/vorlesungen/slides/5/beispiele/kombiniert.pov @@ -18,5 +18,6 @@ ebene(k21, k22, gruen2) arrow(O, j11, at, orange1) arrow(O, j12, at, orange1) arrow(O, k11, at, gruen1) +gerade(k11, gruen1) ebene(j11, j12, orange1) diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 96eea29..cdf2ea5 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,6 +3,8 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % +\folie{5/plan.tex} +\folie{5/planbeispiele.tex} \folie{5/verzerrung.tex} \folie{5/motivation.tex} \folie{5/charpoly.tex} @@ -28,9 +30,13 @@ \folie{5/Aiteration.tex} \folie{5/satzvongelfand.tex} \folie{5/stoneweierstrass.tex} +\folie{5/swbeweis.tex} \folie{5/potenzreihenmethode.tex} \folie{5/logarithmusreihe.tex} \folie{5/exponentialfunktion.tex} \folie{5/hyperbolisch.tex} \folie{5/spektrum.tex} \folie{5/normal.tex} +\folie{5/normalbeispiel.tex} +\folie{5/normalbeispiel34.tex} +\folie{5/approximation.tex} diff --git a/vorlesungen/slides/5/normalbeispiel.tex b/vorlesungen/slides/5/normalbeispiel.tex new file mode 100644 index 0000000..e130c15 --- /dev/null +++ b/vorlesungen/slides/5/normalbeispiel.tex @@ -0,0 +1,108 @@ +% +% normalbeispiel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkred}{rgb}{0.8,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beispiele für normale Matrizen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\uncover<3->{% +\begin{block}{Symmetrisch und Antisymmetrisch} +$A\in M_n(\mathbb{C})$ +\begin{align*} +A&=\pm A^t &&\Rightarrow &AA^* &=A\overline{A^t} =\pm A\overline{A} +\\ + & && & &=\pm\overline{A}A =\overline{A^t}A +\\ + & && & &=A^*A +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<4->{% +\begin{block}{Orthogonal} +$A\in M_n(\mathbb{R})\;\Rightarrow\; A^*=A^t$ +\begin{align*} +AA^t&=I &&\Rightarrow& AA^*&=AA^t=I\\ + & && & &=A^tA=A^*A +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\uncover<1->{% +\begin{block}{Hermitesch und Antihermitesch} +$A\in M_n(\mathbb{C})$ +\begin{align*} +A&=\pm A^* &&\Rightarrow &AA^* &=\pm A^2=A^*A +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<2->{% +\begin{block}{Unitär} +$A\in M_n(\mathbb{C})$ +\begin{align*} +AA^*&=I &&\Rightarrow& AA^*=I=A^*A +\end{align*} +\end{block}} +\end{column} +\end{columns} +%\uncover<5->{% +%\begin{block}{Weitere} +%$N\in M_n(\mathbb{C})$ nilpotent, $N^k=0$\uncover<11->{ +%$\Rightarrow$ +%normal für $l=k-l\Rightarrow l=\frac{k}{2}$} +%\uncover<6->{% +%\[ +%\left. +%\begin{aligned} +%A &=N^l+(N^t)^{k-l} +%\\ +%A^t&=(N^t)^l+N^{k-1} +%\end{aligned} +%\right\} +%\uncover<7->{% +%\Rightarrow +%\left\{ +%\begin{aligned} +%\mathstrut +%A^t A +%&\only<8>{= +%((N^t)^l+N^{k-l}) (N^l+(N^t)^{k-l})} +%\uncover<9->{= +%{\color<10>{darkgreen}(N^t)^lN^l} +%\only<9>{+ +%{\color{orange}(N^t)^k}} +%+ +%{\color<10>{darkred}N^{k-l}(N^t)^{k-l}} +%\only<9>{+ +%{\color{orange}N^k}}} +%\\ +%\mathstrut +%A A^t +%&\only<8>{= +%(N^l+(N^t)^{k-l})((N^t)^l+N^{k-l})} +%\uncover<9->{= +%{\color<10>{darkred}N^l(N^t)^l} +%+ +%\only<9>{{\color{orange}N^k} +%+ +%{\color{orange}(N^t)^k} +%+} +%{\color<10>{darkgreen}(N^t)^{k-l}N^{k-l}}} +%\end{aligned} +%\right.} +%\hspace{20cm} +%\]} +%\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/normalbeispiel34.tex b/vorlesungen/slides/5/normalbeispiel34.tex new file mode 100644 index 0000000..f2647b0 --- /dev/null +++ b/vorlesungen/slides/5/normalbeispiel34.tex @@ -0,0 +1,80 @@ +% +% normalbeispiel34.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.8,0,0} +\begin{frame}[t] +\frametitle{Beispiele normaler Matrizen für $n=3$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\begin{align*} +A +&= +\begin{pmatrix} +\alpha&\beta & 0 \\ + 0 &\alpha&\beta \\ +\beta & 0 &\alpha +\end{pmatrix}, +\; +A^t= +\begin{pmatrix} +\alpha& 0 &\beta \\ +\beta &\alpha& 0 \\ + 0 &\beta &\alpha +\end{pmatrix} +& +\uncover<2->{% +&\Rightarrow\left\{ +\begin{aligned} +AA^t&=\begin{pmatrix} +\alpha^2+\beta^2 & \alpha\beta & \alpha\beta \\ +\alpha\beta & \alpha^2+\beta^2 & \alpha\beta \\ +\alpha\beta & \alpha\beta & \alpha^2+\beta^2 +\end{pmatrix} +\\ +&\phantom{ooooooooooooooo}\| +\\ +A^tA&=\begin{pmatrix} +\alpha^2+\beta^2 & \alpha\beta & \alpha\beta \\ +\alpha\beta & \alpha^2+\beta^2 & \alpha\beta \\ +\alpha\beta & \alpha\beta & \alpha^2+\beta^2 +\end{pmatrix} +\end{aligned}\right.} +\\ +\uncover<3->{ +A&=\alpha I + \beta O}\uncover<4->{, O=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}\in \operatorname{O}(3)} +& +\uncover<5->{ +&\Rightarrow +\left\{ +\begin{aligned} +AA^*&= \alpha^2I^2 + \beta^2 +\ifthenelse{\boolean{presentation}}{ \only<6->{I} }{} \only<-5>{OO^*} ++ \alpha\beta(O+O^*)\\ +A^*A&= \alpha^2I^2 + \beta^2 +\ifthenelse{\boolean{presentation}}{ \only<6->{I} }{} \only<-5>{O^*O} ++ \alpha\beta(O^*+O) +\end{aligned} +\right.} +\\ +\uncover<7->{A&=U+V^*,\text{normal}}\uncover<10->{\text{, } +{\color{darkgreen}UV}={\color{darkgreen}VU}} +& +&\uncover<8->{\Rightarrow +\left\{ +\begin{aligned} +AA^* &= UU^* + {\color<9->{darkgreen}UV} + {\color<9->{darkred}V^*U^*} + V^*V +\\ +A^*A &= U^*U + {\color<9->{darkred}U^*V^*} + {\color<9->{darkgreen}VU} + VV^* +\end{aligned} +\right.} +\end{align*} +\end{column} +\begin{column}{0.49\textwidth} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/plan.tex b/vorlesungen/slides/5/plan.tex new file mode 100644 index 0000000..23b1b93 --- /dev/null +++ b/vorlesungen/slides/5/plan.tex @@ -0,0 +1,198 @@ +% +% plan.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.5,0} +\definecolor{darkred}{rgb}{0.8,0.0,0} +\begin{frame}[t] +\frametitle{Was ist $f(A)$?} +\vspace{-5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<7->{ + \fill[color=blue!20] (-1.5,0.7) rectangle (11.5,3.8); +} + +\uncover<4->{ + \fill[color=darkgreen!20] (-1.5,-0.7) rectangle (11.5,0.7); +} + +\uncover<12->{ + \fill[color=darkred!20] (-1.5,-0.7) rectangle (11.5,-3.8); +} + +\begin{scope}[xshift=-1cm] +\node at (0,0) [left] {$A$}; +\end{scope} + +%\foreach \x in {1,...,20}{ +% \only<\x>{ \node at (-1,3) {\x}}; +%} + +% +% Blauer Ast +% + +\uncover<2->{ + \draw[->,color=blue,shorten <= 0.3cm, shorten >= 0.0cm] + (-1.2,0) -- (0,1.3); + + \begin{scope}[xshift=0cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{R}[X]\\ + f(A)&=p(A) + \end{aligned} + $}; + \end{scope} +} + +\uncover<7->{ + \draw[->,color=blue] (1.8,2.1) -- (3.6,3); + + \begin{scope}[xshift=3.6cm,yshift=3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3cm}\raggedright + $f$ durch $p_n\in\mathbb{R}[X]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<8->{ + \draw[->,color=blue] (7.3,3) -- (9.5,1.9); + + \begin{scope}[xshift=7.6cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=blue] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] {$\displaystyle f(A) = \lim_{n\to\infty}p_n(A)$}; + \end{scope} +} + +\uncover<9->{ + \node[color=blue] at (3.6,1.6) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf Spektrum $\operatorname{Sp}(A)\subset\mathbb{R}$ + \end{minipage}}; +} + +\uncover<11->{ + \node[color=blue] at (-1.5,3.8) [below right] + {$A$ symmetrisch: $A=A^*$}; +} +\uncover<10->{ + \node[color=blue] at (11.5,3.8) [below left] {$A$ diagonalisierbar}; +} + +% +% Roter Ast +% + +\uncover<12->{ + \draw[->,color=darkred,shorten <= 0.3cm, shorten >= 0.0cm] (-1.2,0) -- (0,-1.3); + + \begin{scope}[xshift=0cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{C}[Z,\overline{Z}]\\ + f(A)&=p(A,A^*) + \end{aligned}$}; + \end{scope} +} + +\uncover<13->{ + \node[color=darkred] at (1.7,-2.1) [below left] + {Für $|Z|^2 = Z\overline{Z}$}; +} + +\uncover<14->{ + \draw[->,color=darkred] (1.8,-2.1) -- (3.6,-3); + + \begin{scope}[xshift=3.6cm,yshift=-3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3.5cm}\raggedright + $f$ durch $q_n\in\mathbb{C}[Z,\overline{Z}]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<15->{ + \draw[->,color=darkred] (7.3,-3) -- (9.5,-1.85); + + \begin{scope}[xshift=7.6cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=darkred] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] + {$\displaystyle f(A) = \lim_{n\to\infty}q_n(A,A^*)$}; + \end{scope} +} + +\uncover<16->{ + \node[color=darkred] at (3.6,-1.8) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf $\operatorname{Sp}(A)\cup\operatorname{Sp}(A^*)$ + \end{minipage}}; +} + +\uncover<17->{ + \node[color=darkred] at (11.5,-3.8) [above left] {% + \begin{minipage}{3.5cm}\raggedleft + nur sinnvoll definiert wenn + $AA^*=A^*A$ + \end{minipage}}; +} + +\uncover<18->{ + \node[color=darkred] at (-1.5,-3.8) [above right] + {$A$ normal: $AA^*=A^*A$}; +} + +% +% Grüner Ast +% + +\uncover<3->{ + \draw[->,color=darkgreen,shorten <= 0.0cm, shorten >= 0.0cm] + (-1,0) -- (0,0); + + \begin{scope}[xshift=0cm,yshift=0cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(z)=\sum_{k=0}^\infty a_kz^k$}; + \end{scope} +} + +\uncover<5->{ + \node[color=darkgreen] at (5.9,0) [above] {$f(z)$ analytisch!}; +} +\uncover<6->{ + \node[color=darkgreen] at (5.9,0) [below] + {$\varrho(A)<\text{Konvergenzradius}$}; +} + +\uncover<4->{ + \draw[->,color=darkgreen] (2.9,0) -- (8.5,0); + + \begin{scope}[xshift=8.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(A)=\sum_{k=0}^\infty a_kA^k$}; + \end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/planbeispiele.tex b/vorlesungen/slides/5/planbeispiele.tex new file mode 100644 index 0000000..7b98a95 --- /dev/null +++ b/vorlesungen/slides/5/planbeispiele.tex @@ -0,0 +1,103 @@ +% +% planbeispiele.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.8,0,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{frame}[t] +\frametitle{Beispiele} +\vspace{-15pt} +\begin{columns}[t] +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=blue!20} +\setbeamercolor{block title}{bg=blue!20} +\uncover<2->{% +\begin{block}{$A$ diagonal, $\operatorname{Sp}(A)\subset\mathbb{R}$\strut} +Beispiele: +\begin{align*} +f(x) +&= +x^k, +\\ +f(x)&= +\sqrt{x}, +\sqrt[k]{x} +\\ +f(x)&=|x| +\end{align*} +\vspace{43pt} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkgreen!20} +\setbeamercolor{block title}{bg=darkgreen!20} +\uncover<1->{% +\begin{block}{$f(z)$ analytisch\strut} +Beispiele: +\begin{align*} +e^z +&= +\sum_{k=0}^\infty \frac{z^k}{k!} +\\ +\cos z +&= +\sum_{k=0}^\infty (-1)^k\frac{z^{2k}}{2k!} +\\ +\sin z +&= +\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkred!20} +\setbeamercolor{block title}{bg=darkred!20} +\uncover<3->{% +\begin{block}{$A$ normal, $AA^*=A^*A$\strut} +Beispiele: +\begin{align*} +f(z)&=\sqrt{z\overline{z}}=|z| +\end{align*} +\vspace{76pt} +\end{block}} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t] +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=blue!20} +\setbeamercolor{block title}{bg=blue!20} +\uncover<5->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{blue}diagonalisierbare} +Matrizen $A\in M_n(\mathbb{R})$ +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkgreen!20} +\setbeamercolor{block title}{bg=darkgreen!20} +\uncover<4->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{darkgreen}jedes} $A\in M_n(\mathbb{C})$ +\vspace{14pt} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkred!20} +\setbeamercolor{block title}{bg=darkred!20} +\uncover<6->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{darkred}normale} +Matrizen $A\in M_n(\mathbb{C})$ +\end{block}} +\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/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex index 3f9cab5..e2e9e30 100644 --- a/vorlesungen/slides/5/stoneweierstrass.tex +++ b/vorlesungen/slides/5/stoneweierstrass.tex @@ -3,9 +3,64 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] -\frametitle{Stone-Weierstrass} - -TODO XXX - +\frametitle{Allgemeiner Approximationssatz} +\vspace{-20pt} +\begin{columns}[t] +\begin{column}{0.5\textwidth} +\begin{theorem}[Stone-Weierstrass, $\mathbb{R}$] +$A$ eine {\color{darkgreen}$\mathbb{R}$}-Algebra +von stetigen Funktionen auf einem +%abgeschlossenen und beschränkten +kompakten +Definitionsgebiet $D\subset {\color{darkgreen}\mathbb{R}}$, +\begin{itemize} +\item<2-> konstante Funktion $c\in A$, +\item<3-> für $d_1,d_2\in D$ gibt es ein $s\in A$ mit +$s(d_1)\ne s(d_2)$. +\end{itemize} +\uncover<4->{% +Dann lässt sich jede stetige Funktion durch Funktionen aus $A$ +approximieren} +\end{theorem} +\uncover<5->{ +\begin{block}{Anwendung} +\uncover<6->{$A={\color{darkgreen}\mathbb{R}}[X]$}\uncover<7->{, +$s(X)=X$}\uncover<8->{, +jede stetige Funktion kann durch +Polynome in $X$ approximiert werden} +\end{block}} +\end{column} +\begin{column}{0.5\textwidth} +\uncover<9->{% +\begin{theorem}[Stone-Weierstrass, $\mathbb{C}$] +$A$ eine {\color<10->{red}$\mathbb{C}$}-Algebra von stetigen Funktionen +auf einem +%abgeschlossenen und beschränkten +kompakten +Definitionsgebiet $D\subset {\color<10->{red}\mathbb{C}}$, +\begin{itemize} +\item konstante Funktion $c\in A$, +\item für $d_1,d_2\in D$ gibt es ein $s\in A$ mit +$s(d_1)\ne s(d_2)$. +\only<11->{ +\item {\color{red}$f\in A\Rightarrow \overline{f}\in A$} +} +\end{itemize} +Dann lässt sich jede stetige Funktion durch Funktionen aus $A$ +approximieren +\end{theorem}} +\vspace{-5pt} +\uncover<12->{% +\begin{block}{Anwendung} +$A={\color{red}\mathbb{C}}[Z,\overline{Z}]$\uncover<13->{, +$s(Z{\color{red},\overline{Z}})=Z$}\uncover<14->{, +jede stetige Funktion +lässt sich durch Polynome in $Z{\color{red},\overline{Z}}$ approximieren} +\end{block}} +\end{column} +\end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/5/swbeweis.tex b/vorlesungen/slides/5/swbeweis.tex new file mode 100644 index 0000000..927322b --- /dev/null +++ b/vorlesungen/slides/5/swbeweis.tex @@ -0,0 +1,56 @@ +% +% swbeweis.tex +% +% (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{Beweisidee Stone-Weierstrass} +\vspace{-15pt} +\begin{columns}[t] +\begin{column}{0.5\textwidth} +\begin{enumerate} +\item<1-> +$\exists$ eine monoton wachsende Folge von Polynomen $u_n(t)\to \sqrt{t}$ +gleichmässig auf $[0,1]\subset{\color{darkgreen}\mathbb{R}}$ +\item<2-> +$f\in A$, dann kann man $|f| = \sqrt{f^2}$ beliebig genau approximieren +durch Funktionen +in $A$ +\item<3-> +$f,g\in A$, dann kann +\begin{align*} +\max(a,b)&={\textstyle\frac12}(f+g+|f-g|)\\ +\min(a,b)&={\textstyle\frac12}(f+g-|f-g|) +\end{align*} +in $A$ beliebig genau approximiert werden. +\end{enumerate} +\end{column} +\begin{column}{0.5\textwidth} +\begin{enumerate} +\setcounter{enumi}{3} +\item<4-> +Für $x,y\in D$ und $\alpha,\beta\in\mathbb{R}$ gibt es $f\in A$ mit +$f(x)=\alpha$ und $f(y)=\beta$ +\item<5-> +Zu +$f\colon D\to\mathbb{R}$ stetig und $x\in D$ gibt es $g\in A$ mit $g(x)=f(x)$ +und $g(y) \le f(y)+\varepsilon$ für $y\ne x$ +\item<6-> +Für $f$ gibt es endlich viele Approximationen $g_i$ mit Punkten $x_i$ +wie in Schritt~4. +Dann ist $\max_i g_i$ eine Approximation von $f$, die beliebig genau in +$A$ approximiert werden kann. +\end{enumerate} +\end{column} +\end{columns} + +\vspace{10pt} +\uncover<7->{% +Schritt~2 braucht in {\color{red}$\mathbb{C}$} die komplex Konjugierte: +$|f|^2=f\overline{f}$} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/Makefile.inc b/vorlesungen/slides/6/Makefile.inc new file mode 100644 index 0000000..bc6882a --- /dev/null +++ b/vorlesungen/slides/6/Makefile.inc @@ -0,0 +1,32 @@ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter6 = \ + ../slides/6/punktgruppen/ebene.tex \ + ../slides/6/punktgruppen/semidirekt.tex \ + ../slides/6/punktgruppen/c.tex \ + ../slides/6/punktgruppen/d.tex \ + ../slides/6/punktgruppen/p.tex \ + ../slides/6/punktgruppen/chemie.tex \ + ../slides/6/punktgruppen/aufspaltung.tex \ + \ + ../slides/6/produkte/frei.tex \ + ../slides/6/produkte/direkt.tex \ + \ + ../slides/6/normalteiler/normal.tex \ + ../slides/6/normalteiler/konjugation.tex \ + \ + ../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..e1711d7 --- /dev/null +++ b/vorlesungen/slides/6/chapter.tex @@ -0,0 +1,30 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% + +\folie{6/punktgruppen/ebene.tex} +\folie{6/punktgruppen/semidirekt.tex} +\folie{6/punktgruppen/c.tex} +\folie{6/punktgruppen/d.tex} +\folie{6/punktgruppen/p.tex} +\folie{6/punktgruppen/chemie.tex} +\folie{6/punktgruppen/aufspaltung.tex} + +\folie{6/produkte/frei.tex} +\folie{6/produkte/direkt.tex} + +\folie{6/normalteiler/normal.tex} +\folie{6/normalteiler/konjugation.tex} + +\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..91d8a18 --- /dev/null +++ b/vorlesungen/slides/6/darstellungen/irreduzibel.tex @@ -0,0 +1,47 @@ +% +% 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} +\uncover<2->{% +\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) +\\ +\uncover<3->{% +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} +\uncover<4->{% +\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<5-> $\varrho_i$ nicht isomorph $\Rightarrow$ $f=0$ +\item<6-> $V_1=V_2,\varrho_1=\varrho_2$ $\Rightarrow$ $f=\lambda I$ +\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..144de4c --- /dev/null +++ b/vorlesungen/slides/6/darstellungen/schur.tex @@ -0,0 +1,47 @@ +% +% 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} +$h\colon V_1\to V_2$ +\[ +h^G = \frac{1}{|G|} \sum_{g\in G} \varrho_2(g)^{-1} \circ h \circ \varrho_1(g) +\] +\begin{enumerate} +\item<2-> $\varrho_i$ nicht isomorph $\Rightarrow$ $h^G=0$ +\item<3-> $V_1=V_2,\varrho_1=\varrho_2$, $h^G = \frac1n\operatorname{Spur}h$ +\end{enumerate} +\end{block} +\uncover<4->{% +\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} +\uncover<5->{% +\begin{block}{Matrixelemente $V_1=V_2$, $\varrho_i$ iso} +Für $k=v$ 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..46cc8e9 --- /dev/null +++ b/vorlesungen/slides/6/darstellungen/skalarprodukt.tex @@ -0,0 +1,42 @@ +% +% 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} +\uncover<2->{% +\begin{block}{Satz} +\begin{enumerate} +\item +$\chi$ der Charakter einer irrediziblen Darstellung +$\Rightarrow$ $\langle \chi,\chi\rangle=1$. +\item<3-> +$\chi$ und $\chi'$ Charaktere nichtisomorpher Darstellungen +$\Rightarrow$ +$\langle \chi,\chi'\rangle=0$ +\end{enumerate} +\uncover<4->{% +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..b0d193f --- /dev/null +++ b/vorlesungen/slides/6/darstellungen/summe.tex @@ -0,0 +1,89 @@ +% +% 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} +\uncover<2->{% +\begin{block}{Direkte Summe der Darstellungen} +%\vspace{-12pt} +\begin{align*} +\varrho_1\oplus\varrho_2 +&\colon +G\to \mathbb{C}^{n_1+n_2} +\only<3|handout:0>{ += \mathbb{C}^{n_1}\times\mathbb{C}^{n_2}} +\uncover<4->{=: +\mathbb{C}^{n_1}\oplus\mathbb{C}^{n_2}} +\hspace*{5cm} +\\ +&\colon g\mapsto (\varrho_1(g),\varrho_2(g)) +\end{align*} +\end{block}} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Charakter} +%\vspace{-12pt} +\begin{align*} +\chi_{\varrho_1\oplus\varrho_2}(g) +&= +\operatorname{Spur}(\varrho_1\oplus\varrho_2)(g) +\\ +&\uncover<6->{= +\operatorname{Spur}{\varrho_1(g)} ++ +\operatorname{Spur}{\varrho_1(g)}} +\\ +&\uncover<7->{= +\chi_{\varrho_1}(g) ++ +\chi_{\varrho_2}(g)} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Tensorprodukt} +$n_1\times n_2$-dimensionale +Darstellung $\varrho_1\otimes\varrho_2$ mit Matrix +\[ +\begin{pmatrix} +\varrho_1(g)_{11} \varrho_2(g) + &\dots + &\varrho_1(g)_{1n_1} \varrho_2(g)\\ +\vdots&\ddots&\vdots\\ +\varrho_1(g)_{n_11} \varrho_2(g) + &\dots + &\varrho_1(g)_{n_1n_1} \varrho_2(g) +\end{pmatrix} +\] +\uncover<9->{Die ``Einträge'' sind $n_2\times n_2$-Blöcke} +\end{block}} +\uncover<10->{% +\begin{block}{Darstellungsring} +Die Menge der Darstellungen $R(G)$ einer Gruppe hat +einer Ringstruktur mit $\oplus$ und $\otimes$ +\\ +\uncover<11->{$\Rightarrow$ +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..312d0e8 --- /dev/null +++ b/vorlesungen/slides/6/darstellungen/zyklisch.tex @@ -0,0 +1,84 @@ +% +% 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} +\uncover<2->{% +\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} +\uncover<3->{ +\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} +\uncover<5->{% +\begin{block}{Orthonormalbasis} +Die Funktionen $\chi_k$ bilden eine Orthonormalbasis von $L^2(C_n)$ +\end{block}} +\vspace{-4pt} +\uncover<6->{% +\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} +\\ +\uncover<7->{ +\chi(l) +&= +\sum_{k} c_k \chi_k += +\sum_{k} c_k e^{2\pi ikl/n} +} +\end{align*} +\end{block}} +\vspace{-13pt} +\uncover<8->{% +\begin{block}{Fourier-Theorie} +\vspace{-3pt} +\begin{center} +\begin{tabular}{>{$}l<{$}l} +\uncover<9->{C_n&Diskrete Fourier-Theorie}\\ +\uncover<10->{U(1)&Fourier-Reihen}\\ +\uncover<11->{\mathbb{R}&Fourier-Integral} +\end{tabular} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/normalteiler/konjugation.tex b/vorlesungen/slides/6/normalteiler/konjugation.tex new file mode 100644 index 0000000..70ce01f --- /dev/null +++ b/vorlesungen/slides/6/normalteiler/konjugation.tex @@ -0,0 +1,77 @@ +% +% konjugation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Konjugation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{``Basiswechsel''} +In der Gruppe $\operatorname{GL}_n(\Bbbk)$ +\[ +A' = TAT^{-1} +\] +$T\in\operatorname{GL}_n(\Bbbk)$ +\\ +$A$ und $A'$ sind ``gleichwertig'' +\end{block} +\uncover<2->{% +\begin{block}{Definition} +$g_1,g_2\in G$ sind {\em konjugiert}, wenn es +$h\in G$ gibt mit +\[ +g_1 = hg_2h^{-1} +\] +\end{block}} +\uncover<3->{% +\begin{block}{Beispiel} +Konjugierte Elemente in $\operatorname{GL}_n(\Bbbk)$ haben die +gleiche Spur und Determinante +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Konjugationsklasse} +Die Konjugationsklasse von $g$ ist +\[ +\llbracket g\rrbracket += +\{h\in G\;|\; \text{$h$ konjugiert zu $g$}\} +\] +\end{block}} +\vspace{-7pt} +\uncover<5->{% +\begin{block}{Klassenzerlegung} +\begin{align*} +G +&= +\{e\} +\cup +\llbracket g_1\rrbracket +\cup +\llbracket g_2\rrbracket +\cup +\dots +\\ +&\uncover<6->{= +C_e\cup C_1 \cup C_2\cup\dots} +\end{align*} +\end{block}} +\vspace{-7pt} +\uncover<7->{% +\begin{block}{Klassenfunktionen} +Funktionen, die auf Konjugationsklassen konstant sind +\end{block}} +\uncover<8->{% +\begin{block}{Beispiele} +Spur, Determinante +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/normalteiler/normal.tex b/vorlesungen/slides/6/normalteiler/normal.tex new file mode 100644 index 0000000..42336b9 --- /dev/null +++ b/vorlesungen/slides/6/normalteiler/normal.tex @@ -0,0 +1,79 @@ +% +% normal.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Normalteiler} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Gegeben} +Eine Gruppe $G$ mit Untergruppe $N\subset G$ +\end{block} +\uncover<2->{% +\begin{block}{Bedingung} +Welche Eigenschaft muss $N$ zusätzlich haben, +damit +\[ +G/N += +\{ gN \;|\; g\in G\} +\] +eine Gruppe wird. + +\uncover<3->{Wähle Repräsentaten $g_1N=g_2N$} +\uncover<4->{% +\begin{align*} +g_1g_2N +&\uncover<5->{= +g_1g_2NN} +\uncover<6->{= +g_1g_2Ng_2^{-1}g_2N} +\\ +&\uncover<7->{= +g_1(g_2Ng_2^{-1})g_2N} +\\ +&\uncover<8->{\stackrel{?}{=} g_1Ng_2N} +\end{align*}} +\uncover<9->{Funktioniert nur wenn $g_2Ng_2^{-1}=N$ ist} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<10->{% +\begin{block}{Universelle Eigenschaft} +Ist $\varphi\colon G\to G'$ ein Homomorphismus mit $\varphi(N)=\{e\}$% +\uncover<11->{, dann gibt es einen Homomorphismus $G/N\to G'$:} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (N) at (-2.5,0); +\coordinate (G) at (0,0); +\coordinate (quotient) at (2.5,0); +\coordinate (Gprime) at (0,-2.5); +\coordinate (e) at (-2.5,-2.5); +\node at (N) {$N$}; +\node at (e) {$\{e\}$}; +\node at (G) {$G$}; +\node at (Gprime) {$G'$}; +\node at (quotient) {$G/N$}; +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (N) -- (G); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (N) -- (e); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (e) -- (Gprime); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (G) -- (Gprime); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (G) -- (quotient); +\uncover<11->{ +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm,color=red] (quotient) -- (Gprime); +\node[color=red] at ($0.5*(quotient)+0.5*(Gprime)$) [below right] {$\exists!$}; +} +\node at ($0.5*(quotient)$) [above] {$\pi$}; +\node at ($0.5*(Gprime)$) [left] {$\varphi$}; +\end{tikzpicture} +\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..d40c396 --- /dev/null +++ b/vorlesungen/slides/6/permutationen/matrizen.tex @@ -0,0 +1,79 @@ +% +% 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} +\uncover<2->{% +\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} +\uncover<3->{% +\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} +\uncover<4->{% +\begin{block}{Beispiel} +\vspace{-10pt} +\[ +\begin{pmatrix} +1&2&3&4\\ +3&2&4&1 +\end{pmatrix} +\mapsto +\begin{pmatrix} +0&0&0&1\\ +0&1&0&0\\ +1&0&0&0\\ +0&0&1&0 +\end{pmatrix} +\] +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\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/6/produkte/direkt.tex b/vorlesungen/slides/6/produkte/direkt.tex new file mode 100644 index 0000000..c851335 --- /dev/null +++ b/vorlesungen/slides/6/produkte/direkt.tex @@ -0,0 +1,66 @@ +% +% direkt.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Direktes Produkt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Zwei Gruppen $H_1$ und $H_2$ +\\ +Gruppe $G=H_1\times H_2$ mit +\begin{itemize} +\item<2-> Elemente $(h_1,h_2)\in H_1\times H_2$ +\item<3-> Neutrales Element $(e_1,e_2)$ +\item<4-> Inverses Elemente $(h_1,h_2)^{-1}=(h_1^{-1},h_2^{-1})$ +\end{itemize} +heisst {\em direktes Produkt} +\end{block} +\uncover<5->{% +\begin{block}{Vertauschbarkeit} +Das direkte Produkt ist ein Produkt, in dem Elemente von $H_1$ und +$H_2$ vollständig vertauschbar sind +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Universelle Eigenschaft} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (S) at (0,2.5); +\coordinate (H1) at (-2.5,0); +\coordinate (H2) at (2.5,0); + +\node at (H1) {$H_1$}; +\node at (H2) {$H_2$}; +\node at (0,0) {$H_1\times H_2$}; +\node at (S) {$S$}; + +\draw[->,shorten >= 0.25cm,shorten <= 0.8cm] (0,0) -- (H1); +\draw[->,shorten >= 0.25cm,shorten <= 0.8cm] (0,0) -- (H2); + +\draw[->,shorten >= 0.25cm,shorten <= 0.25cm] (S) -- (H1); +\draw[->,shorten >= 0.25cm,shorten <= 0.25cm] (S) -- (H2); + +\node at ($0.5*(S)+0.5*(H1)$) [above left] {$f_1$}; +\node at ($0.5*(S)+0.5*(H2)$) [above right] {$f_2$}; + +\uncover<7->{ +\draw[->,shorten >= 0.25cm,shorten <= 0.25cm,color=red] (S) -- (0,0); +\node[color=red] at ($0.36*(S)$) [left] {$f_1\times f_2$}; +\node[color=red] at ($0.36*(S)$) [right] {$\exists!$}; +} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/produkte/frei.tex b/vorlesungen/slides/6/produkte/frei.tex new file mode 100644 index 0000000..6c23e6b --- /dev/null +++ b/vorlesungen/slides/6/produkte/frei.tex @@ -0,0 +1,79 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Freie Gruppen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Gruppe aus Symbolen} +Erzeugende Elemente $\{a,b,c,\dots\}$ +\\ +\uncover<2->{% +Wörter = +Folgen von Symbolen $a$, $a^{-1}$, $b$, $b^{-1}$} +\\ +\uncover<3->{ +{\em freie Gruppe}: +\begin{align*} +F&=\langle a,b,c,\dots\rangle +\\ +&= +\{\text{Wörter}\} +/\text{Kürzungsregel} +\end{align*}} +\vspace{-10pt} +\begin{itemize} +\item<4-> neutrales Element: $e = \text{leere Symbolfolge}$ +\item<5-> Gruppenoperation: Verkettung +\item<6-> Kürzungsregel: +\begin{align*} +xx^{-1}&\to e, +& +x^{-1}x&\to e +\end{align*} +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Universelle Eigenschaft} +$g_i\in G$, dann gibt es genau einen Homomorphismus +\[ +\varphi +\colon +\langle g_i| 1\le i\le k\rangle +\to +G +\] +\end{block}} +\vspace{-10pt} +\uncover<8->{% +\begin{block}{Quotient einer freien Gruppe} +Jede endliche Gruppe ist Quotient einer freien Gruppe +\[ +N +\xhookrightarrow{} +\langle g_i\rangle +\twoheadrightarrow +G +\] +oder +\[ +G = \langle g_i\rangle / N +\] +\end{block}} +\vspace{-10pt} +\uncover<11->{% +\begin{block}{Maximal nichtkommutativ} +Die freie Gruppe ist die ``maximal nichtkommutative'' Gruppe +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/WasserstoffAufspaltung.pdf b/vorlesungen/slides/6/punktgruppen/WasserstoffAufspaltung.pdf Binary files differnew file mode 100644 index 0000000..56cbf7b --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/WasserstoffAufspaltung.pdf diff --git a/vorlesungen/slides/6/punktgruppen/aufspaltung.tex b/vorlesungen/slides/6/punktgruppen/aufspaltung.tex new file mode 100644 index 0000000..633f700 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/aufspaltung.tex @@ -0,0 +1,15 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Aufspaltung} +\begin{center} +\includegraphics[width=0.66\textwidth]{../slides/6/punktgruppen/WasserstoffAufspaltung.pdf} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/c.tex b/vorlesungen/slides/6/punktgruppen/c.tex new file mode 100644 index 0000000..80790b1 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/c.tex @@ -0,0 +1,49 @@ +% +% c.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehgruppen} +\vspace{-25pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.33\textwidth} +\begin{block}{$C_n$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/cn.jpg} +\end{center} +\begin{itemize} +\item Eine $n$-zählige Achse +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<2->{% +\begin{block}{$C_{nv}$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/cnv.jpg} +\end{center} +\begin{itemize} +\item Eine $n$-zählige Achse +\item $n$ dazu senkrechte Symmetrieebenen +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<3->{% +\begin{block}{$C_{nh}$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/cnh.jpg} +\end{center} +\begin{itemize} +\item Eine $n$-zählige Achse +\item Eine dazu senkrechte Spiegelebene +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/chemie.tex b/vorlesungen/slides/6/punktgruppen/chemie.tex new file mode 100644 index 0000000..7f8b7a8 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/chemie.tex @@ -0,0 +1,63 @@ +% +% chemie.tex -- Anwendung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Anwendung: Energieniveaus eines Atoms} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Schrödingergleichung} +Partielle Differentialgleichung für die Wellenfunktion +eines Teilchens im Potential $V(x)$ +\[ +-\frac{\hbar^2}{2m}\Delta \Psi ++ +V(x)\Psi += +E\Psi +\] +$V(x)$ = Potential der Atomkerne eines Molekuls +\end{block} +\uncover<2->{% +\begin{block}{Symmetrien} +$g\in\operatorname{O}(3)$ wirkt auf $V$ und $\Psi$ +\begin{align*} +(g\cdot V)(x) &= V(g\cdot x) +\\ +(g\cdot \Psi)(x) &= \Psi(g\cdot x) +\end{align*} +Symmetrie von $V$: $g\cdot V=V$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Lösungen} +Eigenfunktionen $\Psi$ zum Eigenwert $E$ +\[ +g\cdot V=V +\Rightarrow +g\cdot \Psi +\text{ Lösung} +\] +mit gleichem Eigenwert! +\end{block}} +\uncover<4->{% +\begin{block}{Eigenräume} +Die Symmetriegruppe $G\subset \operatorname{O}(3)$ eines Moleküls +operiert auf dem Eigenraum +\end{block}} +\uncover<5->{% +\begin{block}{Externe Felder} +Externe Felder zerstören die Symmetrie +$\Rightarrow$ +die Energieniveaus/Spektrallinien spalten sich auf +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/d.tex b/vorlesungen/slides/6/punktgruppen/d.tex new file mode 100644 index 0000000..9dd0a7a --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/d.tex @@ -0,0 +1,53 @@ +% +% d.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Diedergruppen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.33\textwidth} +\begin{block}{$D_n$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/dn.jpg} +\end{center} +\vspace{-8pt} +\begin{itemize} +\item $C_n$ Achse +\item $n$ $C_2$ Achse senkrecht dazu +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<2->{% +\begin{block}{$D_{nd}$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/dnd.jpg} +\end{center} +\vspace{-8pt} +\begin{itemize} +\item $D_n$ Achse +\item $n$ winkelhalbierende Spiegelebenen der $C_2$-Achsen +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<3->{% +\begin{block}{$D_{nh}$} +\begin{center} +\includegraphics[width=\textwidth]{../slides/6/punktgruppen/images/dnh.jpg} +\end{center} +\vspace{-8pt} +\begin{itemize} +\item $D_n$ Achse +\item Spiegelbene senkrecht dazu +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/ebene.tex b/vorlesungen/slides/6/punktgruppen/ebene.tex new file mode 100644 index 0000000..3b715e4 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/ebene.tex @@ -0,0 +1,79 @@ +% +% ebene.tex -- Punktgruppen in der Ebene +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Punktgruppen in der Ebene} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zyklische Gruppen} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{40} +\def\r{2} +\def\R{2.5} +\fill[color=blue!20] (0,0) -- (0:{1.1*\R}) arc (0:\a:{1.1*\R}) -- cycle; +\node[color=blue] at ({0.5*\a}:{0.8*\r}) {$\displaystyle\frac{2\pi}n$}; +\fill (0,0) circle[radius=0.08]; +\draw[color=red] (0:\r) -- (0:\R) + -- ({1*\a}:\r) -- ({1*\a}:\R) + -- ({2*\a}:\r) -- ({2*\a}:\R) + -- ({3*\a}:\r) -- ({3*\a}:\R) + -- ({4*\a}:\r) -- ({4*\a}:\R) + -- ({5*\a}:\r) -- ({5*\a}:\R) + -- ({6*\a}:\r) -- ({6*\a}:\R) + -- ({7*\a}:\r) -- ({7*\a}:\R) + -- ({8*\a}:\r) %-- ({8*\a}:\R) +; +\end{tikzpicture} +\end{center} +\[ +C_n += +\{\text{Drehungen um Winkel $2\pi/n$}\} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Diedergruppen} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{40} +\def\r{2} +\def\R{2.5} +\fill[color=blue!20] (0,0) -- (0:{1.1*\R}) arc (0:\a:{1.1*\R}) -- cycle; +\node[color=blue] at ({0.5*\a}:{0.8*\r}) {$\displaystyle\frac{2\pi}n$}; +\fill (0,0) circle[radius=0.08]; +\draw[color=red] (0:\r) -- ({0.5*\a}:\R) + -- ({1*\a}:\r) -- ({1.5*\a}:\R) + -- ({2*\a}:\r) -- ({2.5*\a}:\R) + -- ({3*\a}:\r) -- ({3.5*\a}:\R) + -- ({4*\a}:\r) -- ({4.5*\a}:\R) + -- ({5*\a}:\r) -- ({5.5*\a}:\R) + -- ({6*\a}:\r) -- ({6.5*\a}:\R) + -- ({7*\a}:\r) -- ({7.5*\a}:\R) + -- ({8*\a}:\r) %-- ({8.5*\a}:\R) +; +\end{tikzpicture} +\end{center} +\begin{align*} +D_n +&= +\langle\text{Spiegelung}, +\text{Drehungen}\rangle +\\ +&= +C_2 +\ltimes +C_n +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/images/Makefile b/vorlesungen/slides/6/punktgruppen/images/Makefile new file mode 100644 index 0000000..e909884 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/Makefile @@ -0,0 +1,40 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: cn.jpg cnv.jpg cnh.jpg dn.jpg dnd.jpg dnh.jpg + +cn.png: common.inc cn.pov + povray +A0.1 -W1920 -H1080 -Ocn.png cn.pov +cn.jpg: cn.png + convert -extract 1050x1050+450+4 cn.png cn.jpg + +cnv.png: common.inc cnv.pov + povray +A0.1 -W1920 -H1080 -Ocnv.png cnv.pov +cnv.jpg: cnv.png + convert -extract 1050x1050+450+4 cnv.png cnv.jpg + +cnh.png: common.inc cnh.pov + povray +A0.1 -W1920 -H1080 -Ocnh.png cnh.pov +cnh.jpg: cnh.png + convert -extract 1050x1050+450+4 cnh.png cnh.jpg + +dn.png: common.inc dn.pov + povray +A0.1 -W1920 -H1080 -Odn.png dn.pov +dn.jpg: dn.png + convert -extract 1050x1050+450+4 dn.png dn.jpg + +dnd.png: common.inc dnd.pov + povray +A0.1 -W1920 -H1080 -Odnd.png dnd.pov +dnd.jpg: dnd.png + convert -extract 1050x1050+450+4 dnd.png dnd.jpg + +dnh.png: common.inc dnh.pov + povray +A0.1 -W1920 -H1080 -Odnh.png dnh.pov +dnh.jpg: dnh.png + convert -extract 1050x1050+450+4 dnh.png dnh.jpg + + + + diff --git a/vorlesungen/slides/6/punktgruppen/images/cn.jpg b/vorlesungen/slides/6/punktgruppen/images/cn.jpg Binary files differnew file mode 100644 index 0000000..4ea4e92 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cn.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/cn.pov b/vorlesungen/slides/6/punktgruppen/images/cn.pov new file mode 100644 index 0000000..39d65be --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cn.pov @@ -0,0 +1,10 @@ +// +// cn.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.4,0.6,0.6,0.5,0.8,-0.6,0.0) +Vachse() diff --git a/vorlesungen/slides/6/punktgruppen/images/cnh.jpg b/vorlesungen/slides/6/punktgruppen/images/cnh.jpg Binary files differnew file mode 100644 index 0000000..72181e8 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cnh.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/cnh.pov b/vorlesungen/slides/6/punktgruppen/images/cnh.pov new file mode 100644 index 0000000..65d27a4 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cnh.pov @@ -0,0 +1,11 @@ +// +// cnh.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.6,0.8,0.6,0.6,0.8,-0.6,0.0) +Vachse() +Hebene() diff --git a/vorlesungen/slides/6/punktgruppen/images/cnv.jpg b/vorlesungen/slides/6/punktgruppen/images/cnv.jpg Binary files differnew file mode 100644 index 0000000..fd81513 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cnv.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/cnv.pov b/vorlesungen/slides/6/punktgruppen/images/cnv.pov new file mode 100644 index 0000000..a87e075 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/cnv.pov @@ -0,0 +1,11 @@ +// +// cnv.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.4,0.6,0.6,0.5,0.8,-0.6,0.5) +Vachse() +Vebene() diff --git a/vorlesungen/slides/6/punktgruppen/images/common.inc b/vorlesungen/slides/6/punktgruppen/images/common.inc new file mode 100644 index 0000000..ffd9e79 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/common.inc @@ -0,0 +1,200 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.22; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <3, 3.2, -10> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-21, 20, -50> color 0.7*White + area_light <10,0,0> <0,0,10>, 10, 10 + adaptive 1 + jitter +} + +light_source { + <8, 80, -5> color 0.6*White + area_light <10,0,0> <0,0,10>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.0 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end +#declare r = 1.2; + +arrow(< -r, 0, 0 >, < r, 0, 0 >, at, Gray) +arrow(< 0, 0, -r >, < 0, 0, r >, at, Gray) +arrow(< 0, -r, 0 >, < 0, r, 0 >, at, Gray) + +#macro kranzpunkt(r, winkel, h) + < r * cos(winkel), h, r * sin(winkel) > +#end + +#declare N = 13; +#declare h = 0.6; + +#macro deckel(r, R, scherwinkel, h) + #declare phi = 0; + #declare phistep = 2 * pi / N; + #while (phi < (2 * pi) - phistep/2) + triangle { + <0, h, 0>, + kranzpunkt(r, phi, h), + kranzpunkt(R, phi + scherwinkel, h) + } + triangle { + <0, h, 0>, + kranzpunkt(R, phi + scherwinkel, h) + kranzpunkt(r, phi + phistep, h) + } + #declare phi = phi + phistep; + #end +#end + + +#macro mantel(roben, Roben, hoben, runten, Runten, hunten, scherwinkel) + #declare phi = 0; + #declare phistep = 2 * pi / N; + #while (phi < 2 * pi - phistep/2) + triangle { + kranzpunkt(runten, phi, hunten), + kranzpunkt(Runten, phi + scherwinkel, hunten), + kranzpunkt(roben, phi, hoben) + } + triangle { + kranzpunkt(Runten, phi + scherwinkel, hunten), + kranzpunkt(Roben, phi + scherwinkel, hoben), + kranzpunkt(roben, phi, hoben) + } + triangle { + kranzpunkt(Runten, phi + scherwinkel, hunten), + kranzpunkt(runten, phi + phistep, hunten), + kranzpunkt(Roben, phi + scherwinkel, hoben) + } + triangle { + kranzpunkt(runten, phi + phistep, hunten), + kranzpunkt(roben, phi + phistep, hoben), + kranzpunkt(Roben, phi + scherwinkel, hoben) + } + #declare phi = phi + phistep; + #end +#end + +#declare scherwinkel = function(scherfaktor) { (scherfaktor * 2 * pi / N) }; + +#macro koerper(roben, Roben, hoben, runten, Runten, hunten, scherfaktor) +mesh { + deckel(roben, Roben, scherwinkel(scherfaktor), hoben) + deckel(runten, Runten, scherwinkel(scherfaktor), hunten) + mantel(roben, Roben, hoben, runten, Runten, hunten, scherwinkel(scherfaktor)) + pigment { + color Gray + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro Hvektor(a) + <cos(a*2*pi/N),0,sin(a*2*pi/N)> +#end + +#declare VachseFarbe = rgb<1,0.6,0>; +#declare HachseFarbe = rgb<0.8,0.2,0.8>; +#declare VebeneFarbe = rgbf<0.2,0.8,1.0,0.7>; +#declare HebeneFarbe = rgbf<0.2,0.4,0.2,0.7>; + +#macro ebene(richtung, farbe) +intersection { + cylinder { <0, -1, 0>, <0, 1, 0>, 1.0 } + plane { vnormalize(richtung), 0.003 } + plane { -vnormalize(richtung), 0.003 } + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro Vebene() + ebene(Hvektor(-1.25), VebeneFarbe) +#end + +#macro Hebene() + ebene(<0,1,0>, HebeneFarbe) +#end + +#macro achse(richtung, farbe) + cylinder { 1.1 * vnormalize(richtung), + -1.1 * vnormalize(richtung), + 1.5 * at + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Vachse() + achse(<0,1,0>, VachseFarbe) +#end + +#macro Hachse() + achse(Hvektor(-1.5), HachseFarbe) +#end diff --git a/vorlesungen/slides/6/punktgruppen/images/dn.jpg b/vorlesungen/slides/6/punktgruppen/images/dn.jpg Binary files differnew file mode 100644 index 0000000..f895d44 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dn.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/dn.pov b/vorlesungen/slides/6/punktgruppen/images/dn.pov new file mode 100644 index 0000000..36eed3e --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dn.pov @@ -0,0 +1,12 @@ +// +// dn.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.5,0.7,0.6,0.6,0.8,0,0.0) +koerper(0.6,0.8,0,0.5,0.7,-0.6,1.0) +Vachse() +Hachse() diff --git a/vorlesungen/slides/6/punktgruppen/images/dnd.jpg b/vorlesungen/slides/6/punktgruppen/images/dnd.jpg Binary files differnew file mode 100644 index 0000000..089e24f --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dnd.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/dnd.pov b/vorlesungen/slides/6/punktgruppen/images/dnd.pov new file mode 100644 index 0000000..f0ec115 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dnd.pov @@ -0,0 +1,13 @@ +// +// dnd.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.5,0.7,0.6,0.6,0.8,0,0.25) +koerper(0.6,0.8,0,0.5,0.7,-0.6,0.75) +Vachse() +Hachse() +ebene(Hvektor(2.25), VebeneFarbe) diff --git a/vorlesungen/slides/6/punktgruppen/images/dnh.jpg b/vorlesungen/slides/6/punktgruppen/images/dnh.jpg Binary files differnew file mode 100644 index 0000000..c62dbbb --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dnh.jpg diff --git a/vorlesungen/slides/6/punktgruppen/images/dnh.pov b/vorlesungen/slides/6/punktgruppen/images/dnh.pov new file mode 100644 index 0000000..6f14271 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/images/dnh.pov @@ -0,0 +1,13 @@ +// +// dnh.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" + +koerper(0.5,0.7,0.6,0.6,0.8,0,0.5) +koerper(0.6,0.8,0,0.5,0.7,-0.6,0.5) +Vachse() +Hachse() +Hebene() diff --git a/vorlesungen/slides/6/punktgruppen/p.tex b/vorlesungen/slides/6/punktgruppen/p.tex new file mode 100644 index 0000000..ea51e93 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/p.tex @@ -0,0 +1,38 @@ +% +% p.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Platonische Körper} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.33\textwidth} +\begin{block}{$T = T_h \cap \operatorname{SO(3)}$} +\begin{center} +\includegraphics[width=0.8\textwidth]{../slides/6/punktgruppen/toi/T.jpg} +\end{center} +\end{block} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<2->{% +\begin{block}{$O = O_h \cap \operatorname{SO(3)}$} +\begin{center} +\includegraphics[width=0.8\textwidth]{../slides/6/punktgruppen/toi/O.jpg} +\end{center} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\uncover<3->{% +\begin{block}{$I = I_h \cap \operatorname{SO(3)}$} +\begin{center} +\includegraphics[width=0.8\textwidth]{../slides/6/punktgruppen/toi/I.jpg} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/semidirekt.tex b/vorlesungen/slides/6/punktgruppen/semidirekt.tex new file mode 100644 index 0000000..69c1173 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/semidirekt.tex @@ -0,0 +1,80 @@ +% +% semidirekt.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Semidirektes Produkt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gegeben $H$ eine Gruppe, eine abelsche Gruppe $A$, +$\vartheta\colon H\to\operatorname{Aut}(A)$. +\[ +G += +G\ltimes A += +\{(h,a) \;|\; h\in H,a\in A\} +\] +heisst {\em semidirektes Produkt}. +\begin{itemize} +\item<2-> +Neutrales Element: $(e,0)$ +\item<3-> +Gruppenoperation +\[ +(h_1,a_1)\cdot(h_2,a_2) += +(h_1h_2, a_1 + \vartheta(h_1)a_2) +\] +\item<4-> +Inverse: +$(h,a)^{-1} += +(h^{-1},-\vartheta(h)^{-1}a) +$ +\uncover<5->{% +Kontrolle: +\begin{align*} +&\phantom{\mathstrut=\mathstrut} +(h,a)\cdot (h^{-1},-\vartheta(h)^{-1}a) +\\ +&\uncover<6->{=(hh^{-1},a-\vartheta(h)\vartheta(h)^{-1}a)} +\uncover<7->{=(e,0)} +\end{align*}} +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Drehungen und Spiegelungen von $\mathbb{R}^2$} +Spiegelung: $C_2$ +Drehungen der: $\operatorname{SO}(2)$ +Drehungen und Spiegelungen: +$C_2\ltimes \operatorname{SO}(2)=O(2)$ +\end{block}} +\uncover<9->{% +\begin{block}{Drehungen und Translationen} +Drehungen: $H=\operatorname{SO}(2)$ +\\ +Translationen: $A=\mathbb{R}^2$ +\\ +Bewegungen der Ebene: $\operatorname{SO}(2)\ltimes \mathbb{R}^2$ +\end{block}} +\uncover<10->{% +\begin{block}{Dopplereffekt und Laufzeit} +Dopplereffekt: $\mathbb{R}^+$ (Skalierung) +\\ +Laufzeit: $\mathbb{R}$ (Verschiebung) +\\ +Skalierung und Verschiebung: $\mathbb{R}^+\ltimes \mathbb{R}$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/punktgruppen/toi/I.jpg b/vorlesungen/slides/6/punktgruppen/toi/I.jpg Binary files differnew file mode 100644 index 0000000..70d2c17 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/toi/I.jpg diff --git a/vorlesungen/slides/6/punktgruppen/toi/O.jpg b/vorlesungen/slides/6/punktgruppen/toi/O.jpg Binary files differnew file mode 100644 index 0000000..45307c5 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/toi/O.jpg diff --git a/vorlesungen/slides/6/punktgruppen/toi/T.jpg b/vorlesungen/slides/6/punktgruppen/toi/T.jpg Binary files differnew file mode 100644 index 0000000..f710696 --- /dev/null +++ b/vorlesungen/slides/6/punktgruppen/toi/T.jpg diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc new file mode 100644 index 0000000..ffd5091 --- /dev/null +++ b/vorlesungen/slides/7/Makefile.inc @@ -0,0 +1,35 @@ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/7/symmetrien.tex \ + ../slides/7/algebraisch.tex \ + ../slides/7/parameter.tex \ + ../slides/7/mannigfaltigkeit.tex \ + ../slides/7/sl2.tex \ + ../slides/7/drehung.tex \ + ../slides/7/drehanim.tex \ + ../slides/7/semi.tex \ + ../slides/7/kurven.tex \ + ../slides/7/einparameter.tex \ + ../slides/7/ableitung.tex \ + ../slides/7/liealgebra.tex \ + ../slides/7/liealgbeispiel.tex \ + ../slides/7/vektorlie.tex \ + ../slides/7/kommutator.tex \ + ../slides/7/bch.tex \ + ../slides/7/dg.tex \ + ../slides/7/interpolation.tex \ + ../slides/7/exponentialreihe.tex \ + ../slides/7/logarithmus.tex \ + ../slides/7/zusammenhang.tex \ + ../slides/7/quaternionen.tex \ + ../slides/7/qdreh.tex \ + ../slides/7/ueberlagerung.tex \ + ../slides/7/hopf.tex \ + ../slides/7/haar.tex \ + ../slides/7/integration.tex \ + ../slides/7/chapter.tex + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex new file mode 100644 index 0000000..12f9084 --- /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..31d209a --- /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/bch.tex b/vorlesungen/slides/7/bch.tex new file mode 100644 index 0000000..0148dc4 --- /dev/null +++ b/vorlesungen/slides/7/bch.tex @@ -0,0 +1,76 @@ +% +% bch.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Baker-Campbell-Hausdorff-Formel} +$g(t),h(t)\in G +\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit } +g(t)=\exp At, h(t)=\exp Bt}$ +\uncover<3->{% +\begin{align*} +g(t) +&= +I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots, +& +h(t) +&= +I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots +\end{align*}} +\uncover<5->{% +\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{-1}h(t)^{-1}$} +\begin{align*} +\uncover<6->{c(t) +&= +\biggl( + {\color<7,9-11,13-15,19-21>{red}I} + + {\color<8,16-19>{red}A}t + + \frac{{\color<12>{red}A^2}t^2}{2!} + + \dots +\biggr) +\biggl( + {\color<7,8,10-12,14-15,17-18,21>{red}I} + + {\color<9,16,19-20>{red}B}t + + \frac{{\color<13>{red}B^2}t^2}{2!} + + \dots +\biggr) +\exp(-{\color<10,14,17,19,21>{red}A}t) +\exp(-{\color<11,15,18,20-21>{red}B}t) +} +\\ +&\uncover<7->{={\color<7>{red}I}} +\uncover<8->{+t( + \uncover<8->{ {\color<8>{red}A}} + \uncover<9->{+ {\color<9>{red}B}} + \uncover<10->{- {\color<10>{red}A}} + \uncover<11->{- {\color<11>{red}B}} +)} +\uncover<12->{+\frac{t^2}{2!}( + \uncover<12->{ {\color<12>{red}A^2}} + \uncover<13->{+ {\color<13>{red}B^2}} + \uncover<14->{+ {\color<14>{red}A^2}} + \uncover<15->{+ {\color<15>{red}B^2}} +)} +\\ +&\phantom{\mathstrut=I} +\uncover<12->{+t^2( + \uncover<16->{ {\color<16>{red}AB}} + \uncover<17->{- {\color<17>{red}A^2}} + \uncover<18->{- {\color<18>{red}AB}} + \uncover<19->{- {\color<19>{red}BA}} + \uncover<20->{- {\color<20>{red}B^2}} + \uncover<21->{+ {\color<21>{red}AB}} +)} +\uncover<22->{+t^3(\dots)+\dots} +\\ +&\uncover<23->{= +I + \frac{t^2}{2}[A,B] + o(t^3) +} +\end{align*}} +\end{block} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex new file mode 100644 index 0000000..3736e0f --- /dev/null +++ b/vorlesungen/slides/7/chapter.tex @@ -0,0 +1,32 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\folie{7/drehanim.tex} +\folie{7/semi.tex} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/liealgbeispiel.tex} +\folie{7/vektorlie.tex} +\folie{7/kommutator.tex} +\folie{7/bch.tex} +\folie{7/dg.tex} +\folie{7/interpolation.tex} +\folie{7/exponentialreihe.tex} +\folie{7/logarithmus.tex} +\folie{7/zusammenhang.tex} +\folie{7/quaternionen.tex} +\folie{7/qdreh.tex} +\folie{7/ueberlagerung.tex} +\folie{7/hopf.tex} +\folie{7/haar.tex} +\folie{7/integration.tex} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex new file mode 100644 index 0000000..f9528a4 --- /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..ac136f1 --- /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..02201d4 --- /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..a32affd --- /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/exponentialreihe.tex b/vorlesungen/slides/7/exponentialreihe.tex new file mode 100644 index 0000000..b1aeda6 --- /dev/null +++ b/vorlesungen/slides/7/exponentialreihe.tex @@ -0,0 +1,24 @@ +% +% exponentialreihe.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Exponentialreihe} +\begin{align*} +h(s) &= \exp(tA_0 + sB) = \sum_{k=0}^\infty \frac{1}{k!} (tA_0 + sB)^k +\\ +&= +I + (tA_0 + sB) + \frac{1}{2!}(t^2A_0^2 + ts(A_0B + BA_0) + s^2B^2) ++ \frac{1}{3!}(t^3A_0^3 + t^2s(A_0^2B + A_0BA_0 + BA_0^2) + \dots) ++ \dots +\\ +\frac{dg(s)}{ds} +&= +B + \frac1{2!}t(A_0B+BA_0) + \frac{1}{3!}t^2(A_0^2B+A_0BA_0+BA_0^2) + \dots +\end{align*} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/haar.tex b/vorlesungen/slides/7/haar.tex new file mode 100644 index 0000000..454dd69 --- /dev/null +++ b/vorlesungen/slides/7/haar.tex @@ -0,0 +1,84 @@ +% +% haar.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Haar-Mass} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Invariantes Mass} +Auf jeder lokalkompakten Gruppe $G$ gibt es ein \only<2->{invariantes }% +Integral +\begin{align*} +\uncover<2->{\text{rechts:}}&& +\int_G f(g)\,d\mu(g) +&\uncover<2->{= +\int_G f(gh)\,d\mu(g)} +\\ +\uncover<3->{ +\text{links:}&& +\int_G f(g)\,d\mu(g) +&= +\int_G f(hg)\,d\mu(g)} +\end{align*} + +\end{block} +\uncover<7->{% +\begin{block}{Modulus-Funktion} +$\mu$ linksinvariant, dann ist die Rechtsverschiebung ebenfalls +linksinvariant +\[ +\int_G f(gh) \, d\mu(g) +\uncover<8->{ += +\int_G f(g) \Delta(h)\, d\mu(g) +} +\] +\uncover<9->{$\Delta(h)$ heisst Modulus-Funktion} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Beispiel: $G=\mathbb{R}$} +\[ +\int_Gf(g)\,d\mu(g) += +\int_{-\infty}^{\infty} f(x)\,dx +\] +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Beispiel: $\operatorname{SO}(2)$} +\[ +\int_{\operatorname{SO}(2)} +f(g)\,d\mu(g) += +\frac{1}{2\pi} +\int_{0}^{2\pi} f(D_{\alpha})\,d\alpha +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{Beispiel: $G$ endlich} +\[ +\int_G f(g)\,d\mu(g) = \frac{1}{|G|}\sum_{g\in G}f(g) +\] +\end{block}} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Unimodular} +$\Delta(h)=1$ heisst rechtsinvariant = linksinvariant +\\ +\uncover<11->{% +$G$ kompakt $\Rightarrow$ unimodular +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/hopf.tex b/vorlesungen/slides/7/hopf.tex new file mode 100644 index 0000000..a90737f --- /dev/null +++ b/vorlesungen/slides/7/hopf.tex @@ -0,0 +1,69 @@ +% +% hopf.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Orbit-Räume} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aktion von $\operatorname{SO}(3)$ auf $S^2$} +\begin{align*} +S^2 &= \{x\in\mathbb{R}^3\;|\; |x|=1\} +\\ +\operatorname{SO}(3) \times S^2 &\to S^2: (g, x) \mapsto gx +\end{align*} +\uncover<2->{% +Allgemein: Aktion von $G$ auf $X$ +\begin{align*} +\text{links:}&& +G\times X \to X &: (g,x) \mapsto gx +\\ +\text{rechts:}&& +X\times G \to X &: (x,g) \mapsto xg +\end{align*}} +\end{block} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Stabilisator} +Zu $x\in X$ gibt es eine Untergruppe +\begin{align*} +G_x = \{g\in G\;|\; gx=x\}, +\end{align*} +der {\em Stabilisator} von $x$. + +\uncover<4->{% +Der Stabilisator von $v\in S^2$ ist die Gruppe der Drehungen um +die Achse $v$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Quotient} +$G$ operiert von rechts auf $X$ +\[ +X/G = \{ xG \;|\; x\in X\} +\] +heisst Quotient +\end{block}} +\uncover<6->{ +\begin{block}{$\operatorname{SO}(3)/\operatorname{SO}(2)$} +Wähle $\operatorname{SO}(2)$ als Drehungen um die $z$-Achse: +\[ +\operatorname{SO}(3) \to S^2 +: +g \mapsto \text{letzte Spalte von $g$} +\] +\uncover<7->{Daher +\[ +S^2 \cong \operatorname{SO}(3) / \operatorname{SO}(2) +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile new file mode 100644 index 0000000..6f99bc3 --- /dev/null +++ b/vorlesungen/slides/7/images/Makefile @@ -0,0 +1,29 @@ +# +# Makefile -- Illustrationen zu +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: rodriguez.jpg test.png + +rodriguez.png: rodriguez.pov + povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov + +rodriguez.jpg: rodriguez.png + convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg + +commutator: commutator.ini commutator.pov common.inc + povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini +jpg: + for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done + +dreibein/timestamp: interpolation.m + octave interpolation.m + touch dreibein/timestamp + +test.png: test.pov drehung.inc dreibein/d025.inc dreibein/timestamp + povray +A0.1 -W1080 -H1080 -Otest.png test.pov + +dreibein/d025.inc: dreibein/timestamp + +animation: + povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg Binary files differnew file mode 100644 index 0000000..b2dbdb2 --- /dev/null +++ b/vorlesungen/slides/7/images/c/c01.jpg diff --git a/vorlesungen/slides/7/images/c/c02.jpg b/vorlesungen/slides/7/images/c/c02.jpg Binary files differnew file mode 100644 index 0000000..9b45ba3 --- /dev/null +++ b/vorlesungen/slides/7/images/c/c02.jpg diff --git a/vorlesungen/slides/7/images/c/c03.jpg 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+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..8c2211e --- /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..5a448db --- /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..9ae11b9 --- /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/drehung.inc b/vorlesungen/slides/7/images/drehung.inc new file mode 100644 index 0000000..c9b4bb7 --- /dev/null +++ b/vorlesungen/slides/7/images/drehung.inc @@ -0,0 +1,142 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.23; +#declare O = <0, 0, 0>; +#declare at = 0.02; + +camera { + location <8.5, 2, 6.5> + look_at <0, 0, 0> + right x * imagescale + up y * imagescale +} + +//light_source { +// <-14, 20, -50> color White +// area_light <1,0,0> <0,0,1>, 10, 10 +// adaptive 1 +// jitter +//} + +light_source { + <41, 20, 10> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.0 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end +#declare r = 1.0; + +arrow(< -r-0.2, 0.0, 0 >, < r+0.2, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -r-0.2>, < 0.0, 0.0, r+0.2 >, at, Gray) +arrow(< 0.0, -r-0.2, 0 >, < 0.0, r+0.2, 0.0 >, at, Gray) + +#declare farbeX = rgb<1.0,0.2,0.6>; +#declare farbeY = rgb<0.0,0.8,0.4>; +#declare farbeZ = rgb<0.4,0.6,1.0>; + +#declare farbex = rgb<1.0,0.0,0.0>; +#declare farbey = rgb<0.0,0.6,0.0>; +#declare farbez = rgb<0.0,0.0,1.0>; + +#macro quadrant(X, Y, Z) + intersection { + sphere { O, 0.5 } + plane { -X, 0 } + plane { -Y, 0 } + plane { -Z, 0 } + pigment { + color rgb<1.0,0.6,0.2> + } + finish { + specular 0.95 + metallic + } + } + arrow(O, X, 1.1*at, farbex) + arrow(O, Y, 1.1*at, farbey) + arrow(O, Z, 1.1*at, farbez) +#end + +#macro drehung(X, Y, Z) +// intersection { +// sphere { O, 0.5 } +// plane { -X, 0 } +// plane { -Y, 0 } +// plane { -Z, 0 } +// pigment { +// color Gray +// } +// finish { +// specular 0.95 +// metallic +// } +// } + arrow(O, 1.1*X, 0.9*at, farbeX) + arrow(O, 1.1*Y, 0.9*at, farbeY) + arrow(O, 1.1*Z, 0.9*at, farbeZ) +#end + +#macro achse(H) + cylinder { H, -H, at + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } + } + cylinder { 0.003 * H, -0.003 * H, 1 + pigment { + color rgbt<0.6,0.4,0.2,0.5> + } + finish { + specular 0.95 + metallic + } + } +#end diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini new file mode 100644 index 0000000..f07c079 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.ini @@ -0,0 +1,8 @@ +Input_File_Name=interpolation.pov +Initial_Frame=0 +Final_Frame=50 +Initial_Clock=0 +Final_Clock=50 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m new file mode 100644 index 0000000..31554e8 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.m @@ -0,0 +1,54 @@ +# +# interpolation.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 50; +global A; +global B; + +A = (pi / 2) * [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; +g0 = expm(A) + +B = (pi / 2) * [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; +g1 = expm(B) + +function retval = g(t) + global A; + global B; + retval = expm((1-t)*A+t*B); +endfunction + +function dreibein(fn, M, funktion) + fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n", + funktion, + M(1,1), M(3,1), M(2,1), + M(1,2), M(3,2), M(2,2), + M(1,3), M(3,3), M(2,3)); +endfunction + +G = g1 * inverse(g0); +[V, lambda] = eig(G); +H = real(V(:,3)); + +D = logm(g1*inverse(g0)); + +for i = (0:N) + filename = sprintf("dreibein/d%03d.inc", i); + fn = fopen(filename, "w"); + t = i/N; + dreibein(fn, g(t), "quadrant"); + dreibein(fn, expm(t*D)*g0, "drehung"); + fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1)); + fclose(fn); +endfor + diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov new file mode 100644 index 0000000..71e0257 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.pov @@ -0,0 +1,10 @@ +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" + +#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc"); +#include filename + diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg Binary files differnew file mode 100644 index 0000000..5c49700 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.jpg diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png Binary files differnew file mode 100644 index 0000000..6d9e9e4 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.png diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov new file mode 100644 index 0000000..07aec19 --- /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/images/test.pov b/vorlesungen/slides/7/images/test.pov new file mode 100644 index 0000000..5707be1 --- /dev/null +++ b/vorlesungen/slides/7/images/test.pov @@ -0,0 +1,7 @@ +// +// test.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" +#include "dreibein/d025.inc" diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex new file mode 100644 index 0000000..525e6de --- /dev/null +++ b/vorlesungen/slides/7/integration.tex @@ -0,0 +1,66 @@ +% +% integration.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Invariante Integration} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Koordinatenwechsel} +Die Koordinatentransformation +$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$ +hat die Ableitungsmatrix +\[ +t_{ij} += +\frac{\partial y_i}{\partial x_j} +\] +\uncover<2->{% +$n$-faches Integral +\begin{gather*} +\int\dots\int +h(f(x)) +\det +\biggl( +\frac{\partial y_i}{\partial x_j} +\biggr) +\,dx_1\,\dots dx_n +\\ += +\int\dots\int +h(y) +\,dy_1\,\dots dy_n +\end{gather*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{auf einer Lie-Gruppe} +Koordinatenwechsel sind Multiplikationen mit einer +Matrix $g\in G$ +\end{block}} +\uncover<4->{% +\begin{block}{Volumenelement in $I$} +Man muss nur das Volumenelement in $I$ in einem beliebigen +Koordinatensystem definieren: +\[ +dV = dy_1\,\dots\,dy_n +\] +\end{block}} +\uncover<5->{% +\begin{block}{Volumenelement in $g$} +\[ +\text{``\strut}g\cdot dV\text{\strut''} += +\det(g) \, dy_1\,\dots\,dy_n +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex new file mode 100644 index 0000000..249ee26 --- /dev/null +++ b/vorlesungen/slides/7/interpolation.tex @@ -0,0 +1,112 @@ +% +% interpolation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Interpolation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen +$g_0\in\operatorname{SO}(3)$ +und +$g_1\in\operatorname{SO}(3)$: +\[ +g_0=g(0) +\quad\wedge\quad +g_1=g(1) +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Lösung} +$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$ +\begin{align*} +\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)} +\\ +\uncover<5->{A(t)^t +&=(1-t)A_0^t + tA_1^t} +\\ +&\uncover<6->{= +-(1-t)A_0 - t A_1} +\uncover<7->{= +-A(t)} +\\ +\uncover<9->{\Rightarrow +g(t) &= \exp A(t) \in \operatorname{SO}(3)} +\\ +&\uncover<10->{\ne +\exp (\log(g_1g_0^{-1})t) g_0} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Animation} +\centering +\ifthenelse{\boolean{presentation}}{ +\bild{12}{i00} +\bild{13}{i01} +\bild{14}{i02} +\bild{15}{i03} +\bild{16}{i04} +\bild{17}{i05} +\bild{18}{i06} +\bild{19}{i07} +\bild{20}{i08} +\bild{21}{i09} +\bild{22}{i10} +\bild{23}{i11} +\bild{24}{i12} +\bild{25}{i13} +\bild{26}{i14} +\bild{27}{i15} +\bild{28}{i16} +\bild{29}{i17} +\bild{30}{i18} +\bild{31}{i19} +\bild{32}{i20} +\bild{33}{i21} +\bild{34}{i22} +\bild{35}{i23} +\bild{36}{i24} +\bild{37}{i25} +\bild{38}{i26} +\bild{39}{i27} +\bild{40}{i28} +\bild{41}{i29} +\bild{42}{i30} +\bild{43}{i31} +\bild{44}{i32} +\bild{45}{i33} +\bild{46}{i34} +\bild{47}{i35} +\bild{48}{i36} +\bild{49}{i37} +\bild{50}{i38} +\bild{51}{i39} +\bild{52}{i40} +\bild{53}{i41} +\bild{54}{i42} +\bild{55}{i43} +\bild{56}{i44} +\bild{57}{i45} +\bild{58}{i46} +\bild{59}{i47} +\bild{60}{i48} +\bild{61}{i49} +\bild{62}{i50} +}{ +\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png} +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex new file mode 100644 index 0000000..84bf034 --- /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) { 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+\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..e0690eb --- /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/liealgbeispiel.tex b/vorlesungen/slides/7/liealgbeispiel.tex new file mode 100644 index 0000000..a17de40 --- /dev/null +++ b/vorlesungen/slides/7/liealgbeispiel.tex @@ -0,0 +1,78 @@ +% +% liealgbeispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra Beispiele} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{sl}_2(\mathbb{R})$} +Spurlose Matrizen: +\[ +\operatorname{sl}_2(\mathbb{R}) += +\{A\in M_n(\mathbb{R})\;|\; \operatorname{Spur}A=0\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $[A,B]\in \operatorname{sl}_2(\mathbb{R})$: +\begin{align*} +\operatorname{Spur}([A,B]) +&= +\operatorname{Spur}(AB-BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(AB) +\\ +&=0 +\end{align*} +$\Rightarrow$ $\operatorname{sl}_2(\mathbb{R})$ ist eine Lie-Algebra +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{so}(n)$} +Antisymmetrische Matrizen: +\[ +\operatorname{so}(n) += +\{A\in M_n(\mathbb{R}) +\;|\; +A=-A^t +\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $A,B\in \operatorname{so}(n)$ +\begin{align*} +[A,B]^t +&= +(AB-BA)^t +\\ +&= +B^tA^t - A^tB^t +\\ +&= +(-B)(-A)-(-A)(-B) +\\ +&= +BA-AB += +-(AB-BA) +\\ +&= +-[A,B] +\end{align*} +$\Rightarrow$ $\operatorname{so}(n)$ ist eine Lie-Algebra +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex new file mode 100644 index 0000000..574467b --- /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/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex new file mode 100644 index 0000000..58065d7 --- /dev/null +++ b/vorlesungen/slides/7/logarithmus.tex @@ -0,0 +1,82 @@ +% +% logarithmus.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmus} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\frac{d}{dx}\log(1+x) +&= \frac{1}{1+x} +\\ +\uncover<2->{ +\Rightarrow\quad +\log (1+x) +&= +\int_0^x \frac{1}{1+t}\,dt} +\end{align*} +\begin{align*} +\uncover<3->{\frac{1}{1+t} +&= +1-t+t^2-t^3+\dots} +\\ +\uncover<4->{\log(1+x) +&=\int_0^x +1-t+t^2-t^3+\dots +\,dt} +\\ +&\only<5>{= +x-\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}4 + \dots} +\uncover<6->{= +\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}} +\\ +\uncover<7->{\log (I+A) +&= +\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Konvergenzradius} +Polstelle bei $x=-1$ +\( +\varrho =1 +\) +\end{block}} +\vspace{-5pt} +\begin{block}{\uncover<9->{Alternative: Spektraltheorie}} +\uncover<9->{ +Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$ +definiert:} +\vspace{-15pt} +\uncover<8->{ +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1); +} +\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}]; +\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}]; +\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1]; +\draw[color=blue] (1,0) circle[radius=1]; +\uncover<9->{ + \draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0); +} +\fill (1,0) circle[radius=0.08]; +\node at (2.3,1.9) {$\mathbb{C}$}; +\node at (1,0) [below] {$1$}; +\end{tikzpicture} +\end{center}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex new file mode 100644 index 0000000..077dc9d --- /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..f3579a3 --- /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/qdreh.tex b/vorlesungen/slides/7/qdreh.tex new file mode 100644 index 0000000..8ed512a --- /dev/null +++ b/vorlesungen/slides/7/qdreh.tex @@ -0,0 +1,110 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen mit Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Drehung?} +Abbildung von $\vec{x}$ mit $\operatorname{Re}\vec{x}=0$: +\[ +\varrho_{q} +\colon +\vec{x}\mapsto q\vec{x}q^{-1} = q\vec{x}\overline{q} +\] +\end{block} +\uncover<2->{% +\begin{block}{Achse} +\begin{align*} +\varrho_q(q) +&= +qq\overline{q} +\uncover<3->{= +q(qq^{-1})} +\uncover<4->{= +q} +\end{align*} +\end{block}} +\uncover<4->{% +\begin{block}{Norm} +\begin{align*} +|\varrho_q(\vec{x})|^2 +&= +q\vec{x}\overline{q}\overline{(q\vec{x}\overline{q})} +\uncover<5->{= +q\vec{x}\overline{q}\overline{\overline{q}}\overline{\vec{x}}\overline{q} +} +\\ +&\uncover<6->{= +q\vec{x}(\overline{q}q)\overline{\vec{x}}\overline{q}} +\uncover<7->{= +q(\vec{x}\overline{\vec{x}})\overline{q}} +\uncover<8->{= +q\overline{q}|\vec{x}|^2} +\\ +&\uncover<9->{= +|\vec{x}|^2} +\end{align*} +\uncover<10->{% +$\Rightarrow$ $\varrho_q\in\operatorname{O}(3)$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Drehung!} +$\vec{a},\vec{b},\vec{n}$ bilden ein on.~Rechtssystem +\begin{align*} +\uncover<12->{ +qa +&= +c\vec{a}+s\vec{n}\times \vec{a}} +\uncover<13->{= +c\vec{a} + s\vec{b}} +\\ +\uncover<14->{ +q\vec{a}\overline{q} +&= +(c\vec{a}+s\vec{b}) c +-(c\vec{a}+s\vec{b})\times s\vec{n}} +\\ +&\uncover<15->{= +c^2 \vec{a}+ sc\vec{b} ++sc\vec{b} - s^2 \vec{a}} +\\ +&\uncover<16->{= +\vec{a} \cos\alpha +\vec{b} \sin\alpha } +\end{align*} +\vspace{-5pt} +\uncover<17->{wegen +%\vspace{-5pt} +\[ +\begin{aligned} +\cos\alpha &= \cos^2\frac{\alpha}2 - \sin^2\frac{\alpha}2 &&=c^2-s^2 +\\ +\sin\alpha &= 2\cos\frac{\alpha}2\sin\frac{\alpha}2&&=2cs +\end{aligned}\]} +\end{block}} +\vspace{-18pt} +\uncover<18->{% +\begin{block}{Matrix} +\[ +D += +\tiny +\begin{pmatrix} +1-2(q_2^2+q_3^2)&-2q_0q_3+2q_1q_2&-2q_0q_2+2q_1q_3\\ + 2q_0q_3+2q_1q_2&1-2(q_1^2+q_3^2)&-2q_0q_1+2q_2q_3\\ +-2q_0q_2+2q_1q_3& 2q_0q_1+2q_2q_3&1-2(q_1^2+q_2^2) +\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex new file mode 100644 index 0000000..f526366 --- /dev/null +++ b/vorlesungen/slides/7/quaternionen.tex @@ -0,0 +1,74 @@ +% +% quaternionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quaternionen} +$4$-dimensionaler $\mathbb{R}$-Vektorraum +\[ +\mathbb{H} += +\langle 1,i,j,k\rangle_{\mathbb{R}} +\] +mit Rechenregeln +\[ +i^2=j^2=k^2=ijk=-1 +\] +$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$ +\begin{itemize} +\item<2-> Realteil: $\operatorname{Re}x=x_0$ +\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$ +\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$ +\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$ +\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<7->{% +\begin{block}{Skalarprodukt und Vektorprodukt} +\begin{align*} +pq +&= +\operatorname{Re}p \operatorname{Re}q +- +\operatorname{Im}p\cdot \operatorname{Im}q +\\ +&\phantom{=} ++ +\operatorname{Re}p\operatorname{Im}q ++ +\operatorname{Im}p\operatorname{Re}q ++ +\operatorname{Im}p\times\operatorname{Im}q +\end{align*} +\end{block}} +\uncover<8->{% +\begin{block}{Einheitsquaternionen} +$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$ +\end{block}} +\uncover<9->{% +\begin{block}{Polardarstellung} +\[ +q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2 +\] +\vspace{-8pt} +\begin{itemize} +\item<10-> +Drehmatrix: 9 Parameter, 6 Bedingungen +\item<11-> +Quaternionen: 4 Parameter, 1 Bedingung +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex new file mode 100644 index 0000000..cd974c9 --- /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..a65b4f6 --- /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..35d62d8 --- /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/7/ueberlagerung.tex b/vorlesungen/slides/7/ueberlagerung.tex new file mode 100644 index 0000000..426641a --- /dev/null +++ b/vorlesungen/slides/7/ueberlagerung.tex @@ -0,0 +1,98 @@ +% +% ueberlagerung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$S^3$, $\operatorname{SU}(2)$ und $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\uncover<6->{% +\begin{block}{Überlagerung} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (2,-2); +\coordinate (D) at (0,-2); + +\uncover<7->{ +\node at (A) {$\{\pm 1\}\mathstrut$}; +} +\uncover<6->{ +\node at (B) {$S^3\mathstrut$}; +\node at ($(B)+(0.1,0)$) [right] {$=\operatorname{SU}(2)\mathstrut$}; +} +\uncover<7->{ +\node at (C) {$\operatorname{SO}(3)\mathstrut$}; +\node at (D) {$\{I\}\mathstrut$}; +} + +\uncover<7->{ +\draw[->,shorten >= 0.3cm,shorten <= 0.5cm] (A) -- (B); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (D); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C); +\draw[->,shorten >= 0.6cm,shorten <= 0.3cm] (D) -- (C); +} + +\end{tikzpicture} +\end{center} +\begin{itemize} +\item<7-> +$\pm q\in S^3$ $\Rightarrow$ $\varrho_{q}=\varrho_{-q}$ +\item<8-> +In der Nähe von $I$ sehen die Gruppen +$\operatorname{SO}(3)$ +und +$\operatorname{SU}(2)$ +``gleich'' aus +\item<9-> +$\operatorname{SU}(2)$ ist geometrisch ``einfacher'' +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\begin{block}{Pauli-Matrizen} +Quaternionen als $2\times 2$-Matrizen schreiben +\begin{align*} +1&=\begin{pmatrix}1&0\\0&1\end{pmatrix}=\sigma_0, +& +i&=\begin{pmatrix}0&i\\i&0\end{pmatrix}=-i\sigma_1 +\\ +j&=\begin{pmatrix}0&-1\\1&0\end{pmatrix}=-i\sigma_2, +& +k&=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=-i\sigma_3 +\end{align*} +\uncover<2->{% +erfüllen $i^2=j^2=k^2=ijk=-1$.} +\end{block} +\uncover<3->{% +\begin{block}{$S^3 = \operatorname{SU}(2)$} +\[ +a+bi+cj+dk += +\begin{pmatrix} +a+id&-c+bi\\ +c+ib&a-id +\end{pmatrix} += +A +\] +\begin{align*} +\uncover<4->{ +\det A &= a^2 + b^2 + c^2 + d^2 = 1 +} +\\ +\uncover<5->{ +A^* &= a - ib - jc - kd +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/vektorlie.tex b/vorlesungen/slides/7/vektorlie.tex new file mode 100644 index 0000000..621a832 --- /dev/null +++ b/vorlesungen/slides/7/vektorlie.tex @@ -0,0 +1,206 @@ +% +% viktorlie.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Vektorprodukt als Lie-Algebra} +%\vspace{-10pt} +\centering +\begin{tikzpicture}[>=latex,thick] +\arraycolsep=2.4pt +\def\Ax{0} +\def\Ux{4.1} +\def\Kx{7.2} +\def\Rx{13.1} + +\def\Lx{2.2} +\def\Ly{0} +\def\Lz{-2.2} + +\fill[color=red!20] (\Ax,{\Lx-1.55}) rectangle ({\Ux-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Ux,{\Lx-1.55}) rectangle ({\Kx-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Kx,{\Lx-1.55}) rectangle ({\Rx},{\Lx+0.55}); + +\fill[color=darkgreen!20] (\Ax,{\Ly-1.55}) rectangle ({\Ux-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Ux,{\Ly-1.55}) rectangle ({\Kx-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Kx,{\Ly-1.55}) rectangle ({\Rx},{\Ly+0.55}); + +\fill[color=blue!20] (\Ax,{\Lz-1.55}) rectangle ({\Ux-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Ux,{\Lz-1.55}) rectangle ({\Kx-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Kx,{\Lz-1.55}) rectangle ({\Rx},{\Lz+0.55}); + +\coordinate (A) at (\Ax,3.2); +\coordinate (Ax) at (\Ax,\Lx); +\coordinate (Ay) at (\Ax,\Ly); +\coordinate (Az) at (\Ax,\Lz); + +\node at (A) [right] + {\usebeamercolor[fg]{title}Drehmatrix, $\operatorname{SO}(n)$\strut}; + +\node at (Ax) [right] {$\displaystyle\tiny +D_{x,\alpha}=\begin{pmatrix} +1&0&0\\ +0&\cos\alpha&-\sin\alpha\\ +0&\sin\alpha&\cos\alpha +\end{pmatrix}$}; + +\node at (Ay) [right] {$\displaystyle\tiny +D_{y,\alpha}=\begin{pmatrix} +\cos\alpha&0&\sin\alpha\\ +0&1&0\\ +-\sin\alpha&0&\cos\alpha +\end{pmatrix}$}; + +\node at (Az) [right] {$\displaystyle\tiny +D_{z,\alpha}=\begin{pmatrix} +\cos\alpha&-\sin\alpha&0\\ +\sin\alpha&\cos\alpha&0\\ +0&0&1 +\end{pmatrix}$}; + +\coordinate (U) at (\Ux,3.2); +\coordinate (Ux) at (\Ux,\Lx); +\coordinate (Uy) at (\Ux,\Ly); +\coordinate (Uz) at (\Ux,\Lz); +\coordinate (Ex) at (\Ux,{\Lx-1}); +\coordinate (Ey) at (\Ux,{\Ly-1}); +\coordinate (Ez) at (\Ux,{\Lz-1}); + +\uncover<2->{ +\node at (U) [right] + {\usebeamercolor[fg]{title}Ableitung, $\operatorname{so}(n)$\strut}; + +\node at (Ux) [right] {$\displaystyle\tiny +U_x=\begin{pmatrix*}[r] +0&0&0\\ +0&0&-1\\ +0&1&0 +\end{pmatrix*} +$}; + +\node at (Uy) [right] {$\displaystyle\tiny +U_y=\begin{pmatrix*}[r] +0&0&1\\ +0&0&0\\ +-1&0&0 +\end{pmatrix*} +$}; + +\node at (Uz) [right] {$\displaystyle\tiny +U_z=\begin{pmatrix*}[r] +0&-1&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix*} +$}; +} + +\uncover<9->{ +\node at (Ex) [right] {$\displaystyle +\, e_x = \tiny\begin{pmatrix}1\\0\\0\end{pmatrix} +$}; + +\node at (Ey) [right] {$\displaystyle +\, e_y = \tiny\begin{pmatrix}0\\1\\0\end{pmatrix} +$}; + +\node at (Ez) [right] {$\displaystyle +\, e_z = \tiny\begin{pmatrix}0\\0\\1\end{pmatrix} +$}; +} + +\coordinate (K) at (\Kx,3.2); +\coordinate (Kx) at (\Kx,\Lx); +\coordinate (Ky) at (\Kx,\Ly); +\coordinate (Kz) at (\Kx,\Lz); +\coordinate (Vx) at (\Kx,{\Lx-1}); +\coordinate (Vy) at (\Kx,{\Ly-1}); +\coordinate (Vz) at (\Kx,{\Lz-1}); + +\uncover<3->{ +\node at (K) [right] + {\usebeamercolor[fg]{title}Kommutator\strut}; + +\node at (Kx) [right] {$\displaystyle +\begin{aligned} +[U_y,U_z] &\uncover<4->{= +{\tiny +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +0&1&0 +\end{pmatrix}} +\uncover<5->{\mathstrut- +\tiny +\begin{pmatrix} +0&0&0\\ +0&0&1\\ +0&0&0 +\end{pmatrix}}} +\uncover<6->{=U_x} +\end{aligned} +$}; +} + +\uncover<7->{ +\node at (Ky) [right] {$\displaystyle +\begin{aligned} +[U_z,U_x] &= +{\tiny +\begin{pmatrix} +0&0&1\\ +0&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +1&0&0 +\end{pmatrix}} +=U_y +\end{aligned} +$}; +} + +\uncover<8->{ +\node at (Kz) [right] {$\displaystyle +\begin{aligned} +[U_x,U_y] &= +{\tiny +\begin{pmatrix} +0&0&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&1&0\\ +0&0&0\\ +0&0&0 +\end{pmatrix}} +=U_z +\end{aligned} +$}; +} + +\uncover<10->{ +\node at (Vx) [right] {$\displaystyle \phantom{]}e_y\times e_z = e_x$}; +} + +\uncover<11->{ +\node at (Vy) [right] {$\displaystyle \phantom{]}e_z\times e_x = e_y$}; +} + +\uncover<12->{ +\node at (Vz) [right] {$\displaystyle \phantom{]}e_x\times e_y = e_z$}; +} + +\end{tikzpicture} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex new file mode 100644 index 0000000..6a43cd8 --- /dev/null +++ b/vorlesungen/slides/7/zusammenhang.tex @@ -0,0 +1,99 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zusammenhängend --- oder nicht} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\ds{2.4} +\coordinate (A) at (0,0); +\coordinate (B) at (\ds,0); +\coordinate (C) at ({2*\ds},0); + +\node at (A) {$\operatorname{SO}(n)$}; +\node at (B) {$\operatorname{O}(n)$}; +\node at (C) {$\{\pm 1\}$}; + +\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B); +\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C); +\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$}; + +\coordinate (A2) at (0,-1.0); +\coordinate (B2) at (\ds,-1.0); +\coordinate (C2) at ({2*\ds},-1.0); + +\draw[color=blue] (A2) ellipse (1cm and 0.3cm); +\draw[color=blue] (B2) ellipse (1cm and 0.3cm); +\node[color=blue] at (C2) {$+1$}; + +\coordinate (A3) at (0,-1.7); +\coordinate (B3) at (\ds,-1.7); +\coordinate (C3) at ({2*\ds},-1.7); + +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2); +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3); + +\draw[color=red] (B3) ellipse (1cm and 0.3cm); +\node[color=red] at (C3) {$-1$}; + +\end{tikzpicture} +\end{center} +\end{block} +\begin{block}{Zusammenhangskomponente von $e$} +$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält: +\begin{align*} +\operatorname{SO}(n)&\subset \operatorname{O}(n) +\\ +\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\} + &\subset \operatorname{GL}_n(\mathbb{R}) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Eigenschaften} +\begin{itemize} +\item +{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$, +dann ist +\begin{itemize} +\item +$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$ +\item +$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$ +\end{itemize} +\item +{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann +ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$ +$\Rightarrow hG_eh^{-1}\subset G_e$ +\end{itemize} +\end{block} +\begin{block}{Quotient} +$G/G_e$ ist eine diskrete Gruppe +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (4,0); +\node at (A) {$G_e$}; +\node at (B) {$G$}; +\node at (C) {$G/G_e$}; +\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B); +\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C); +\end{tikzpicture} +\end{center} +\vspace{-7pt} +$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index d46dc7f..6ac5665 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -28,5 +28,25 @@ chapter8 = \ ../slides/8/tokyo/bahn0.tex \ ../slides/8/tokyo/bahn1.tex \ ../slides/8/tokyo/bahn2.tex \ + ../slides/8/chrind.tex \ + ../slides/8/chrindprop.tex \ + ../slides/8/chroma1.tex \ + ../slides/8/amax.tex \ + ../slides/8/subgraph.tex \ + ../slides/8/chrwilf.tex \ + ../slides/8/weitere.tex \ + ../slides/8/wavelets/funktionen.tex \ + ../slides/8/wavelets/laplacebasis.tex \ + ../slides/8/wavelets/vektoren.tex \ + ../slides/8/wavelets/fourier.tex \ + ../slides/8/wavelets/lokalisierungsvergleich.tex \ + ../slides/8/wavelets/frequenzlokalisierung.tex \ + ../slides/8/wavelets/dilatation.tex \ + ../slides/8/wavelets/matrixdilatation.tex \ + ../slides/8/wavelets/gundh.tex \ + ../slides/8/wavelets/dilbei.tex \ + ../slides/8/wavelets/frame.tex \ + ../slides/8/wavelets/framekonstanten.tex \ + ../slides/8/wavelets/beispiel.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/amax.tex b/vorlesungen/slides/8/amax.tex new file mode 100644 index 0000000..951400a --- /dev/null +++ b/vorlesungen/slides/8/amax.tex @@ -0,0 +1,86 @@ +% +% amax.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\alpha_{\text{max}}$ und $d$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Definition} +$\alpha_{\text{max}}$ ist der grösste Eigenwert der Adjazenzmatrix +\end{block} +\uncover<2->{ +\begin{block}{Fakten} +\begin{itemize} +\item<3-> +Der Eigenwert $\alpha_{\text{max}}$ ist einfach +\item<4-> +Es gibt einen positiven Eigenvektor $f$ zum Eigenwert $\alpha_{\text{max}}$ +\item<5-> +$f$ maximiert +\[ +\frac{\langle Af,f\rangle}{\langle f,f\rangle} += +\alpha_{\text{max}} +\] +\end{itemize} +Herkunft: Perron-Frobenius-Theorie positiver Matrizen (nächste Woche) +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<6->{% +\begin{block}{Mittlerer Grad} +\[ +\overline{d} += +\frac1{n} \sum_{v} \operatorname{deg}(v) +\le +\alpha_{\text{max}} +\le +d +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{proof}[Beweis] +\begin{itemize} +\item Konstante Funktion $1$ anstelle von $f$: +\[ +\frac{\langle A1,1\rangle}{\langle 1,1\rangle} +\uncover<8->{= +\frac{\sum_v \operatorname{deg}(v)}{n}} +\uncover<9->{= +\overline{d}} +\uncover<10->{\le +\alpha_{\text{max}}} +\] +\item<11-> Komponenten von $Af$ summieren: +\begin{align*} +\uncover<12->{ +\alpha_{\text{max}} +f(v) &= (Af)(v)}\uncover<13->{ = \sum_{u\sim v} f(u)} +\\ +\uncover<14->{\alpha_{\text{max}} +\sum_{v}f(v) +&= +\sum_v +\operatorname{deg}(v) f(v)} +\\ +&\uncover<15->{\le +d\sum_v f(v)} +\; +\uncover<16->{\Rightarrow +\; +\alpha_{\text{max}} \le d} +\end{align*} +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 6a0b13f..69b7231 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -30,3 +30,24 @@ \folie{8/tokyo/bahn1.tex} \folie{8/tokyo/bahn2.tex} +\folie{8/chrind.tex} +\folie{8/chrindprop.tex} +\folie{8/chroma1.tex} +\folie{8/amax.tex} +\folie{8/subgraph.tex} +\folie{8/chrwilf.tex} +\folie{8/weitere.tex} + +\folie{8/wavelets/funktionen.tex} +\folie{8/wavelets/laplacebasis.tex} +\folie{8/wavelets/fourier.tex} +\folie{8/wavelets/lokalisierungsvergleich.tex} +\folie{8/wavelets/frequenzlokalisierung.tex} +\folie{8/wavelets/dilatation.tex} +\folie{8/wavelets/matrixdilatation.tex} +\folie{8/wavelets/gundh.tex} +\folie{8/wavelets/frame.tex} +\folie{8/wavelets/dilbei.tex} +\folie{8/wavelets/framekonstanten.tex} +\folie{8/wavelets/beispiel.tex} + diff --git a/vorlesungen/slides/8/chrind.tex b/vorlesungen/slides/8/chrind.tex new file mode 100644 index 0000000..bd406ab --- /dev/null +++ b/vorlesungen/slides/8/chrind.tex @@ -0,0 +1,231 @@ +% +% chrind.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Chromatische Zahl und Unabhängigkeitszahl} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Chromatische Zahl} +$\operatorname{chr}(G)=\mathstrut$ +minimale Anzahl Farben, die zum Einfärben eines Graphen $G$ nötig sind derart, +dass benachbarte Knoten verschiedene Farben haben. +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\only<1>{ + \fill[color=white] (PA) circle[radius=\r]; + \fill[color=white] (PB) circle[radius=\r]; + \fill[color=white] (PC) circle[radius=\r]; + \fill[color=white] (PD) circle[radius=\r]; + \fill[color=white] (PE) circle[radius=\r]; + \fill[color=white] (QA) circle[radius=\r]; + \fill[color=white] (QB) circle[radius=\r]; + \fill[color=white] (QC) circle[radius=\r]; + \fill[color=white] (QD) circle[radius=\r]; + \fill[color=white] (QE) circle[radius=\r]; +} + +\only<2->{ + \fill[color=blau] (PA) circle[radius=\r]; + \fill[color=rot] (PB) circle[radius=\r]; + \fill[color=rot] (PC) circle[radius=\r]; + \fill[color=gruen] (PD) circle[radius=\r]; + \fill[color=gruen] (PE) circle[radius=\r]; + + \fill[color=rot] (QA) circle[radius=\r]; + \fill[color=blau] (QB) circle[radius=\r]; + \fill[color=gruen] (QC) circle[radius=\r]; + \fill[color=rot] (QD) circle[radius=\r]; + \fill[color=blau] (QE) circle[radius=\r]; +} + +\draw (PA) circle[radius=\r]; +\draw (PB) circle[radius=\r]; +\draw (PC) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QA) circle[radius=\r]; +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QD) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{chr} G = 3$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Unabhängigkeitszahl} +$\operatorname{ind}(G)=\mathstrut$ +maximale Anzahl nicht benachbarter Knoten +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} +\definecolor{gelb}{rgb}{0,0,1} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\foreach \n in {1,...,7}{ + \only<\n>{\node[color=white] at (1,2.9) {$\n$};} +} + +\fill[color=white] (PA) circle[radius=\r]; +\fill[color=white] (PB) circle[radius=\r]; +\fill[color=white] (PC) circle[radius=\r]; +\fill[color=white] (PD) circle[radius=\r]; +\fill[color=white] (PE) circle[radius=\r]; +\fill[color=white] (QA) circle[radius=\r]; +\fill[color=white] (QB) circle[radius=\r]; +\fill[color=white] (QC) circle[radius=\r]; +\fill[color=white] (QD) circle[radius=\r]; +\fill[color=white] (QE) circle[radius=\r]; + +\only<4->{ + \fill[color=rot] (QA) circle[radius={1.5*\r}]; + \fill[color=rot!40] (QB) circle[radius=\r]; + \fill[color=rot!40] (QE) circle[radius=\r]; + \fill[color=rot!40] (PA) circle[radius=\r]; +} + +\only<5->{ + \fill[color=blau] (PB) circle[radius={1.5*\r}]; + \fill[color=blau!40] (PD) circle[radius=\r]; + \fill[color=blau!40] (PE) circle[radius=\r]; + \fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r]; +} + +\only<6->{ + \fill[color=gruen] (PC) circle[radius={1.5*\r}]; + \fill[color=gruen!40] (QC) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r]; +} + +\only<7->{ + \fill[color=gelb] (QD) circle[radius={1.5*\r}]; + \fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r]; +} + +\only<-3|handout:0>{ + \draw (QA) circle[radius=\r]; +} +\only<4->{ + \draw (QA) circle[radius={1.5*\r}]; +} + +\only<-4|handout:0>{ + \draw (PB) circle[radius=\r]; +} +\only<5->{ + \draw (PB) circle[radius={1.5*\r}]; +} + +\only<-5|handout:0>{ + \draw (PC) circle[radius=\r]; +} +\only<6->{ + \draw (PC) circle[radius={1.5*\r}]; +} + +\only<-6|handout:0>{ + \draw (QD) circle[radius=\r]; +} +\only<7->{ + \draw (QD) circle[radius={1.5*\r}]; +} + +\draw (PA) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\only<4|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 1$}; +} +\only<5|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 2$}; +} +\only<6|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 3$}; +} +\only<7->{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 4$}; +} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chrindprop.tex b/vorlesungen/slides/8/chrindprop.tex new file mode 100644 index 0000000..094588c --- /dev/null +++ b/vorlesungen/slides/8/chrindprop.tex @@ -0,0 +1,62 @@ +% +% chrindprop.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang zwischen $\operatorname{chr}G$ und $\operatorname{ind}G$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit $n$ Knoten, dann gilt +\[ +\operatorname{chr}G +\cdot +\operatorname{ind}G +\ge n +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +Peterson-Graph $K$ hat $n=10$ Knoten: +\[ +\operatorname{chr}(K) +\cdot +\operatorname{ind}(K) += +3\cdot 4 +\ge +10 += +n +\] +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> eine minimale Färbung hat $\operatorname{chr}(G)$ Farben +\item<5-> Sie teilt die Knoten in $\operatorname{chr}(G)$ +gleichfarbige Mengen auf +\item<6-> Jede einfarbige Menge von Knoten ist unabhängig, d.~h.~sie +besteht aus Knoten, die nicht miteinander verbunden sind. +\item<7-> Jede einfarbige Menge enthält höchstens $\operatorname{ind}(G)$ +\item<8-> Die Gesamtzahl der Knoten ist +\[ +n\uncover<9->{=\sum_{\text{Farbe}}\underbrace{|V_{\text{Farbe}}|}_{\le \operatorname{ind}(G)}} +\uncover<10->{\le +\operatorname{chr}(G) +\cdot +\operatorname{ind}(G)} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chroma1.tex b/vorlesungen/slides/8/chroma1.tex new file mode 100644 index 0000000..6a55704 --- /dev/null +++ b/vorlesungen/slides/8/chroma1.tex @@ -0,0 +1,56 @@ +% +% chroma1.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Schranke für $\operatorname{chr}(G)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt +\[ +\operatorname{chr}(G) \le d + 1 +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<3-> +Peterson-Graph $G$: maximaler Grad ist $d=3$, aber +\[ +\operatorname{chr}(G) += +3 +< d+1=4 +\] +\item<4-> +Voller Graph $V$: maximaler Grad ist $d=n-1$, +\[ +\operatorname{chr}(V) = n = d+1 +\] +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<4->{% +\begin{proof}[Beweis] +Mit vollständiger Induktion, d.~h.~Annahme: Graphen mit $<n$ Knoten und +maximalem Grad $d$ lassen sich mit höchstens $d+1$ Farben färben. +\begin{itemize} +\item<5-> $X$ ein Graph mit $n$ Knoten +\item<6-> entferne den Knoten $v\in X$, $X'=X\setminus\{v\}$ +\item<7-> $X'$ lässt sich mit höchstens $d+1$ Farben einfärben +\item<8-> $v$ hat höchstens $d$ Nachbarn, die höchsten $d$ verschiedene +Farben haben +\item<9-> Es bleibt eine Farbe für $v$ +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chrwilf.tex b/vorlesungen/slides/8/chrwilf.tex new file mode 100644 index 0000000..7edb10e --- /dev/null +++ b/vorlesungen/slides/8/chrwilf.tex @@ -0,0 +1,115 @@ +% +% chrwilf.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\def\kante#1#2{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2); +} +\def\knoten#1#2{ + \uncover<8->{ + \fill[color=#2!30] (#1) circle[radius=0.2]; + \draw[color=#2] (#1) circle[radius=0.2]; + } + \only<-7>{ + \draw (#1) circle[radius=0.2]; + } +} +\def\R{1.5} +\definecolor{rot}{rgb}{1,0,0} +\definecolor{gruen}{rgb}{0,0.6,0} +\definecolor{blau}{rgb}{0,0,1} +\begin{frame}[t] +\frametitle{Schranke für die chromatische Zahl} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Wilf)} +$\uncover<2->{\operatorname{chr}(X) \le 1+}\alpha_{\text{max}} \le\uncover<2->{ 1 + }d$ +\end{block} +\uncover<3->{% +\begin{block}{Beispiel} +\begin{align*} +\uncover<4->{d&= 4} +&&\uncover<5->{\Rightarrow& \operatorname{chr}(G) &\le 5}\\ +\uncover<6->{\alpha_{\text{max}} &= +2.9565} +&&\uncover<7->{\Rightarrow& \operatorname{chr}(G) &\le 3}\\ +\uncover<4->{\overline{d} &= \frac{24}{9}=\rlap{$2.6666$}} +\end{align*} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (A) at (0:\R); +\coordinate (B) at (40:\R); +\coordinate (C) at (80:\R); +\coordinate (D) at (120:\R); +\coordinate (E) at (160:\R); +\coordinate (F) at (200:\R); +\coordinate (G) at (240:\R); +\coordinate (H) at (280:\R); +\coordinate (I) at (320:\R); + +\knoten{A}{rot} +\knoten{B}{blau} +\knoten{C}{gruen} +\knoten{D}{blau} +\knoten{E}{rot} +\knoten{F}{blau} +\knoten{G}{rot} +\knoten{H}{gruen} +\knoten{I}{blau} + +\kante{A}{B} +\kante{B}{C} +\kante{C}{D} +\kante{D}{E} +\kante{E}{F} +\kante{F}{G} +\kante{G}{H} +\kante{H}{I} +\kante{I}{A} + +\kante{A}{C} +\kante{A}{D} +\kante{D}{G} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<9->{% +\begin{proof}[Beweis] +Induktion nach der Grösse $n$ des Graphen. +\begin{itemize} +\item<10-> +Entferne $v\in X$ mit minimalem Grad: $X'=X\setminus \{v\}$ +\item<11-> +Induktionsannahme: +\[ +\operatorname{chr}(X') +\le +1+ +\alpha_{\text{max}}' +\] +\item<12-> +$X'$ kann mit höhcstens $1+\alpha_{\text{max}}'\le 1+\alpha_{\text{max}}$ +Farben eingefärbt werden. +\item<13-> +Wegen +\( +\deg(v) \le \overline{d} \le \alpha_{\text{max}} +\) +hat $v$ höchstens $\alpha_{\text{max}}$ Nachbarn, um $v$ zu färben, +braucht man also höchstens $1+\alpha_{\text{max}}$ Farben. +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex index 952c85b..10f88cd 100644 --- a/vorlesungen/slides/8/inzidenz.tex +++ b/vorlesungen/slides/8/inzidenz.tex @@ -5,6 +5,8 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \begin{frame}[t] \frametitle{Inzidenz- und Adjazenzmatrix} \vspace{-20pt} @@ -67,7 +69,7 @@ \vspace{-10pt} \uncover<5->{% \begin{block}{Definition} -\vspace{-15pt} +%\vspace{-15pt} \begin{align*} B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\ A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$} diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex index 5f2f51a..43e5330 100644 --- a/vorlesungen/slides/8/inzidenzd.tex +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -5,6 +5,8 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \begin{frame}[t] \frametitle{Inzidenz- und Adjazenz-Matrix} \vspace{-20pt} @@ -67,7 +69,7 @@ \vspace{-15pt} \uncover<5->{% \begin{block}{Definition} -\vspace{-20pt} +%\vspace{-20pt} \begin{align*} B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\ B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\ diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex index 1d8b725..93333bc 100644 --- a/vorlesungen/slides/8/produkt.tex +++ b/vorlesungen/slides/8/produkt.tex @@ -56,7 +56,7 @@ \end{center} \vspace{-15pt} \begin{block}{Berechne} -\vspace{-20pt} +%\vspace{-20pt} \begin{align*} \uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t \end{align*} diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex index 425fe1c..62180d9 100644 --- a/vorlesungen/slides/8/spanningtree.tex +++ b/vorlesungen/slides/8/spanningtree.tex @@ -3,6 +3,7 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % +\bgroup \begin{frame} \frametitle{Spannbäume} @@ -121,7 +122,7 @@ Wieviele Spannbäume gibt es? \begin{column}{0.56\hsize} \uncover<5->{% \begin{block}{Laplace-Matrix} -\vspace{-15pt} +%\vspace{-15pt} \[ L= \tiny @@ -162,3 +163,4 @@ L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{arr \end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/8/subgraph.tex b/vorlesungen/slides/8/subgraph.tex new file mode 100644 index 0000000..f3005f9 --- /dev/null +++ b/vorlesungen/slides/8/subgraph.tex @@ -0,0 +1,60 @@ +% +% subgraph.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\alpha_{\text{max}}$ eines Untergraphen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$X'$ ein echter Untergraph von $X$ mit Adjazenzmatrix $A'$ und grösstem +Eigenwert $\alpha_{\text{max}}'$ +\[ +\alpha_{\text{max}}' \le \alpha_{\text{max}} +\] +\end{block} +\uncover<2->{$V'$ die Knoten von $X'$} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> +$f'$ der positive Eigenvektor von $A'$ +\item<5-> +Definiere +\[ +g(v) += +\begin{cases} +f'(v) &\qquad v\in V'\\ +0 &\qquad \text{sonst} +\end{cases} +\] +\item<6-> Skalarprodukte: +\begin{align*} +\uncover<7->{\langle f',f'\rangle &= \langle g,g\rangle} +\\ +\uncover<8->{\langle A'f',f'\rangle &\le \langle Ag,g\rangle} +\end{align*} +\item<9-> Vergleich +\[ +\alpha_{\text{max}}' += +\frac{\langle A'f',f'\rangle}{\langle f',f'\rangle} +\uncover<10->{\le +\frac{\langle Ag,g\rangle}{\langle g,g\rangle}} +\uncover<11->{\le +\alpha_{\text{max}}} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/Makefile b/vorlesungen/slides/8/wavelets/Makefile new file mode 100644 index 0000000..3b4a5ce --- /dev/null +++ b/vorlesungen/slides/8/wavelets/Makefile @@ -0,0 +1,8 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +vektoren.tex: ev.m + octave ev.m diff --git a/vorlesungen/slides/8/wavelets/beispiel.tex b/vorlesungen/slides/8/wavelets/beispiel.tex new file mode 100644 index 0000000..dcc33d4 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/beispiel.tex @@ -0,0 +1,44 @@ +% +% beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{ +\node at (0,0) [rotate=-90] +{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets auf einer Kugel} +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\only<1>{ \bild{0.6}{wavelets-phi-sphere-334.pdf} } + +\only<2>{ \bild{0.6}{wavelets-psi-5-sphere-334.pdf} } +\only<3>{ \bild{0.6}{wavelets-psi-4-sphere-334.pdf} } +\only<4>{ \bild{0.6}{wavelets-psi-3-sphere-334.pdf} } +\only<5>{ \bild{0.6}{wavelets-psi-2-sphere-334.pdf} } +\only<6>{ \bild{0.6}{wavelets-psi-1-sphere-334.pdf} } + +\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; } +\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; } +\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; } +\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; } +\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; } +\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; } + +\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; } +\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; } +\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; } +\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; } +\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; } +\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; } + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/dilatation.tex b/vorlesungen/slides/8/wavelets/dilatation.tex new file mode 100644 index 0000000..881f760 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/dilatation.tex @@ -0,0 +1,62 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Dilatation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Dilatation in $\mathbb{R}$} +$f\colon \mathbb{R}\to\mathbb{R}$ +Definition im Ortsraum: +\[ +(D_af)(x) += +\frac{1}{\sqrt{|a|}} +f\biggl(\frac{x}{a}\biggr) +\] +\uncover<2->{% +Dilatation im Frequenzraum: +\[ +\widehat{D_af}(\omega) += +D_{1/a}\hat{f}(\omega) +\]} +\uncover<3->{% +Spektrum wird mit $1/a$ skaliert!} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{``Dilatation'' auf einem Graphen} +\begin{itemize} +\item<5-> Dilatation auf dem Graphen gibt es nicht +\item<6-> Dilatation im Spektrum $\{\lambda_1,\dots,\lambda_n\}$ gibt es nicht +\item<7-> ``Spektrale Dilatation'' verwenden +\begin{enumerate} +\item<8-> Start: $e_k$ +\item<9-> Fourier-Transformation: $\chi^te_k$ +\item<10-> Spektrum skalieren: mit +$D_{1/a}g$ filtern +\item<11-> Rücktransformation +\[ +D_{g,a}e_k += +\chi +\uncover<12->{\operatorname{diag}(\tilde{D}_{1/a}g(\lambda_*)) +\chi^t e_k} +\] +\end{enumerate} +\end{itemize} + + +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/dilbei.tex b/vorlesungen/slides/8/wavelets/dilbei.tex new file mode 100644 index 0000000..fc66a0a --- /dev/null +++ b/vorlesungen/slides/8/wavelets/dilbei.tex @@ -0,0 +1,46 @@ +% +% beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{ +\node at (0,0) [rotate=-90] +{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets einer Strecke} +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\only<1>{ \bild{0.6}{wavelets-psi-line-5-10.pdf} } +\only<2>{ \bild{0.6}{wavelets-psi-line-4-10.pdf} } +\only<3>{ \bild{0.6}{wavelets-psi-line-3-10.pdf} } +\only<4>{ \bild{0.6}{wavelets-psi-line-2-10.pdf} } +\only<5>{ \bild{0.6}{wavelets-psi-line-1-10.pdf} } + +\only<6>{ \bild{0.6}{wavelets-phi-line-10.pdf} } + +\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; } +\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; } +\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; } +\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; } +\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; } +\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; } + + +\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; } +\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; } +\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; } +\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; } +\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; } + +\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; } + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/ev.m b/vorlesungen/slides/8/wavelets/ev.m new file mode 100644 index 0000000..7f4dd55 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/ev.m @@ -0,0 +1,97 @@ +# +# ev.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +L = [ + 2, -1, 0, -1, 0; + -1, 4, -1, -1, -1; + 0, -1, 2, 0, -1; + -1, -1, 0, 3, -1; + 0, -1, -1, -1, 3 +]; + +[v, lambda] = eig(L); + +function knoten(fn, wert, punkt) + if (wert > 0) + farbe = sprintf("red!%02d", round(100 * wert)); + else + farbe = sprintf("blue!%02d", round(-100 * wert)); + end + fprintf(fn, "\t\\fill[color=%s] %s circle[radius=0.25];\n", + farbe, punkt); + fprintf(fn, "\t\\draw %s circle[radius=0.25];\n", punkt); +endfunction + +function vektor(fn, v, name, lambda) + fprintf(fn, "\\def\\%s{\n", name); + fprintf(fn, "\t\\coordinate (A) at ({0*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (B) at ({1*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (C) at ({2*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (D) at ({0.5*\\a},{-\\b});\n"); + fprintf(fn, "\t\\coordinate (E) at ({1.5*\\a},{-\\b});\n"); + fprintf(fn, "\t\\draw (A) -- (B);\n"); + fprintf(fn, "\t\\draw (A) -- (D);\n"); + fprintf(fn, "\t\\draw (B) -- (C);\n"); + fprintf(fn, "\t\\draw (B) -- (D);\n"); + fprintf(fn, "\t\\draw (B) -- (E);\n"); + fprintf(fn, "\t\\draw (C) -- (E);\n"); + fprintf(fn, "\t\\draw (D) -- (E);\n"); + fprintf(fn, "\t\\node at (-2.8,{-0.5*\\b}) [right] {$\\lambda=%.4f$};\n", + round(1000 * abs(lambda)) / 10000); + w = v / max(abs(v)); + knoten(fn, w(1,1), "(A)"); + knoten(fn, w(2,1), "(B)"); + knoten(fn, w(3,1), "(C)"); + knoten(fn, w(4,1), "(D)"); + knoten(fn, w(5,1), "(E)"); + fprintf(fn, "}\n"); +endfunction + +function punkt(fn, x, wert) + fprintf(fn, "({%.4f*\\c},{%.4f*\\d})", x, wert); +endfunction + +function funktion(fn, v, name, lambda) + fprintf(fn, "\\def\\%s{\n", name); + fprintf(fn, "\t\\draw[color=red,line width=1.4pt]\n\t\t"); + punkt(fn, -2, v(1,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, -1, v(4,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 0, v(2,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 1, v(5,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 2, v(3,1)); + fprintf(fn, ";\n"); + fprintf(fn, "\t\\draw[->] ({-2.1*\\c},0) -- ({2.1*\\c},0);\n"); + fprintf(fn, "\t\\draw[->] (0,{-1.1*\\d}) -- (0,{1.1*\\d});\n"); + for x = (-2:2) + fprintf(fn, "\t\\fill ({%d*\\c},0) circle[radius=0.05];\n", x); + endfor + fprintf(fn, "}\n"); +endfunction + +fn = fopen("vektoren.tex", "w"); + +vektor(fn, v(:,1), "vnull", lambda(1,1)); +funktion(fn, v(:,1), "fnull", lambda(1,1)); + +vektor(fn, v(:,2), "vone", lambda(2,2)); +funktion(fn, v(:,2), "fone", lambda(2,2)); + +vektor(fn, v(:,3), "vtwo", lambda(3,3)); +funktion(fn, v(:,3), "ftwo", lambda(3,3)); + +vektor(fn, v(:,4), "vthree", lambda(4,4)); +funktion(fn, v(:,4), "fthree", lambda(4,4)); + +vektor(fn, v(:,5), "vfour", lambda(5,5)); +funktion(fn, v(:,5), "ffour", lambda(5,5)); + +fclose(fn); + + diff --git a/vorlesungen/slides/8/wavelets/fourier.tex b/vorlesungen/slides/8/wavelets/fourier.tex new file mode 100644 index 0000000..3195ec8 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/fourier.tex @@ -0,0 +1,86 @@ +% +% fourier.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Fourier-Transformation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Gegeben: Funktion $f$ auf dem Graphen +\\ +\uncover<2->{% +Gesucht: Koeffizienten $\hat{f}$ der Darstellung in der Laplace-Basis} +\end{block} +\uncover<3->{% +\begin{block}{Definition $\chi$-Matrix} +Eigenwerte $0=\lambda_1<\lambda_2\le \dots \le \lambda_n$ von $L$ +\vspace{-10pt} +\begin{center} +\begin{tikzpicture} +\node at (-1.9,0) [left] {$\chi=\mathstrut$}; +\node at (0,0) {$\left(\raisebox{0pt}[1.7cm][1.7cm]{\hspace{3.5cm}}\right)$}; + +\fill[color=blue!20] (-1.7,-1.7) rectangle (-1.1,1.7); +\draw[color=blue] (-1.7,-1.7) rectangle (-1.1,1.7); +\node at (-1.4,0) [rotate=90] {$v_1=\mathstrut$EV zum EW $\lambda_1$\strut}; + +\fill[color=blue!20] (-1.0,-1.7) rectangle (-0.4,1.7); +\draw[color=blue] (-1.0,-1.7) rectangle (-0.4,1.7); +\node at (-0.7,0) [rotate=90] {$v_2=\mathstrut$EV zum EW $\lambda_2$\strut}; + +\fill[color=blue!20] (1.1,-1.7) rectangle (1.7,1.7); +\draw[color=blue] (1.1,-1.7) rectangle (1.7,1.7); +\node at (1.4,0) [rotate=90] {$v_n=\mathstrut$EV zum EW $\lambda_n$\strut}; + +\node at (0.4,0) {$\dots$}; + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Transformation} +$L$ symmetrisch +\\ +\uncover<5->{$\Rightarrow$ +Die Eigenvektoren von $L$ können orthonormiert gewählt werden} +\\ +\uncover<6->{$\Rightarrow$ +Koeffizienten können durch Skalarprodukte ermittelt werden:} +\uncover<7->{% +\[ +\hat{f}(k) += +\hat{f}(\lambda_k) +\uncover<8->{= +\langle v_k, f\rangle +\quad\Rightarrow\quad +\hat{f}} +\uncover<9->{= +\chi^tf} +\]} +\uncover<10->{% +$\chi$ ist die {\em Fourier-Transformation}} +\end{block}} +\uncover<11->{% +\begin{block}{Rücktransformation} +Eigenvektoren orthonormiert +\\ +\uncover<12->{$\Rightarrow$ +$\chi$ orthogonal} +\uncover<13->{ +\[ +\chi\chi^t = I +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/frame.tex b/vorlesungen/slides/8/wavelets/frame.tex new file mode 100644 index 0000000..4d0c7d1 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/frame.tex @@ -0,0 +1,66 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Graph Wavelet Frame} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frame-Vektoren} +Zu Dilatationsfaktoren $A=\{a_i\,|\,i=1,\dots,N\}$ +konstruiere das Frame +\begin{align*} +F= +\{&D_he_1,\dots,D_he_n,\\ + &Dg_1e_1,\dots,Dg_1e_n,\\ + &Dg_2e_1,\dots,Dg_2e_n,\\ + &\dots\\ + &Dg_Ne_1,\dots,Dg_Ne_n\} +\end{align*} +\uncover<2->{Notation: +\begin{align*} +v_{0,k} +&= +D_he_k +\\ +v_{i,k} +&= +Dg_ie_k +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Frameoperator} +\begin{align*} +\mathcal{T}\colon \mathbb{R}^n\to\mathbb{R}^{nN} +: +v +&\mapsto +\begin{pmatrix} +\uncover<4->{\langle D_he_1,v\rangle}\\ +\uncover<4->{\vdots}\\ +\uncover<4->{\langle D_he_n,v\rangle}\\ +\hline +\uncover<5->{\langle D_{g_1}e_1,v\rangle}\\ +\uncover<5->{\vdots}\\ +\uncover<5->{\langle D_{g_1}e_n,v\rangle}\\ +\hline +\uncover<6->{\vdots}\\ +\uncover<6->{\vdots}\\ +\hline +\uncover<7->{\langle D_{g_N}e_1,v\rangle}\\ +\uncover<7->{\vdots}\\ +\uncover<7->{\langle D_{g_N}e_n,v\rangle} +\end{pmatrix} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/framekonstanten.tex b/vorlesungen/slides/8/wavelets/framekonstanten.tex new file mode 100644 index 0000000..a436536 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/framekonstanten.tex @@ -0,0 +1,71 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +%\setlength{\abovedisplayskip}{5pt} +%\setlength{\belowdisplayskip}{5pt} +\frametitle{Framekonstanten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Menge $\mathcal{F}$ von Vektoren heisst ein Frame, +falls es Konstanten $A$ und $B$ gibt derart, dass +\[ +A\|v\|^2 +\le +\|\mathcal{T}v\|^2 +\sum_{b\in\mathcal{F}} |\langle b,v\rangle|^2 +\le +B\|v\|^2 +\] +\uncover<2->{$A>0$ garantiert Invertierbarkeit} +\end{block} +\uncover<3->{% +\begin{block}{$\|\mathcal{T}v\|$ für Graph-Wavelets} +\begin{align*} +\|\mathcal{T}v\|^2 +&= +\sum_k |\langle D_he_k,v\rangle|^2 ++ +\sum_{i,k} |\langle D_{g_i}e_k, v\rangle|^2 +\\ +&\uncover<4->{= +\sum_k |h(\lambda_k) \hat{v}(k)|^2 ++ +\sum_{k,i} |g_i(\lambda_k) \hat{v}(k)|^2} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{$A$ und $B$} +Frame-Norm-Funktion +\begin{align*} +f(\lambda) +&= +h(\lambda) ++ +\sum_i g_i(\lambda) +\\ +&\uncover<6->{= +h(\lambda) ++ +\sum_i g(a_i\lambda)} +\end{align*} +\uncover<7->{Abschätzung für Frame-Konstanten +\begin{align*} +A&\uncover<8->{= +\min_{i} f(\lambda_i)} +\\ +B&\uncover<9->{= +\max_{i} f(\lambda_i)} +\end{align*}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex new file mode 100644 index 0000000..c78e6dd --- /dev/null +++ b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex @@ -0,0 +1,78 @@ +% +% frequenzlokalisierung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\def\kurve#1#2{ + \draw[color=#2,line width=1.4pt] + plot[domain=0:6.3,samples=400] + ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1}); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lokalisierung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Bandpass} +Gegeben durch $g(\lambda)\ge 0$: +\begin{align*} +g(0) &= 0\\ +\lim_{\lambda\to\infty}g(\lambda)&= 0 +\end{align*} +\vspace{-10pt} +\begin{enumerate} +\item<3-> Fourier-transformieren +\item<4-> Amplituden mit $g(\lambda)$ multiplizieren +\item<5-> Rücktransformieren +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Tiefpass} +Gegeben durch $h(\lambda)\ge0$: +\begin{align*} +h(0) &= 1\\ +\lim_{\lambda\to\infty}h(\lambda)&= 0 +\end{align*} +\vspace{-10pt} +\begin{enumerate} +\item<8-> Fourier-Transformation +\item<9-> Amplituden mit $h(\lambda)$ multiplizieren +\item<10-> Rücktransformation +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.8] + +\uncover<2->{ +\begin{scope}[xshift=-4.5cm] +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; +\kurve{3}{red} +\draw[->] (0,-0.1) -- (0,3.3); +\end{scope} +} + +\uncover<7->{ +\begin{scope}[xshift=4.5cm] +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; +\draw[color=darkgreen,line width=1.4pt] + plot[domain=0:6.3,samples=100] + ({\x},{3*exp(-(\x/0.5)*(\x/0.5)}); + +\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}]; +\end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/funktionen.tex b/vorlesungen/slides/8/wavelets/funktionen.tex new file mode 100644 index 0000000..2e3ae9b --- /dev/null +++ b/vorlesungen/slides/8/wavelets/funktionen.tex @@ -0,0 +1,78 @@ +% +% funktionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\knoten#1#2{ + \draw #1 circle[radius=0.25]; + \node at #1 {$#2$}; +} +\def\kante#1#2{ + \draw[shorten >= 0.25cm,shorten <= 0.25cm] #1 -- #2; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionen auf einem Graphen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein Graph $G=(V,E)$, eine Funktion auf dem Graphen ist +\[ +f\colon V \to \mathbb{R} : v\mapsto f(v) +\] +Knoten: $V=\{1,\dots,n\}$ +\\ +\uncover<2->{% +Vektorschreibweise +\[ +f = \begin{pmatrix} +f(1)\\f(2)\\\vdots\\f(n) +\end{pmatrix} +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Matrizen} +Adjazenz-, Grad- und Laplace-Matrix operieren auf Funktionen auf Graphen: +\[ +L += +\begin{pmatrix*}[r] + 2&-1& 0&-1& 0\\ +-1& 4&-1&-1&-1\\ + 0&-1& 2& 0&-1\\ +-1&-1& 0& 3&-1\\ + 0&-1&-1&-1& 3\\ +\end{pmatrix*} +\] +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{2} +\coordinate (A) at (0,0); +\coordinate (B) at (\a,0); +\coordinate (C) at ({2*\a},0); +\coordinate (D) at ({0.5*\a},{-0.5*sqrt(3)*\a}); +\coordinate (E) at ({1.5*\a},{-0.5*sqrt(3)*\a}); +\knoten{(A)}{1} +\knoten{(B)}{2} +\knoten{(C)}{3} +\knoten{(D)}{4} +\knoten{(E)}{5} +\kante{(A)}{(B)} +\kante{(B)}{(C)} +\kante{(A)}{(D)} +\kante{(B)}{(D)} +\kante{(B)}{(E)} +\kante{(C)}{(E)} +\kante{(D)}{(E)} +\end{tikzpicture} +\end{center}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/gundh.tex b/vorlesungen/slides/8/wavelets/gundh.tex new file mode 100644 index 0000000..2d6c677 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/gundh.tex @@ -0,0 +1,85 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\kurve#1#2{ + \draw[color=#2,line width=1.4pt] + plot[domain=0:6.3,samples=400] + ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1}); +} + +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Mutterwavelets + Dilatation} +Eine Menge von Dilatationsfaktoren +\[ +A= \{a_1,a_2,\dots,a_N\} +\] +wählen\uncover<2->{, und mit Funktionen +\[ +{\color{blue}g_i} = \tilde{D}_{1/a_i}{\color{red}g} +\] +die Standardbasisvektoren filtern} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{ +\begin{block}{Vaterwavelets} +Tiefpass mit Funktion ${\color{darkgreen}h(\lambda)}$, +Standardbasisvektoren mit ${\color{darkgreen}h}$ filtern: +\[ +D_{\color{darkgreen}h}e_k +\] +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope} + +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; + +\kurve{1}{red} +\uncover<4->{ +\foreach \k in {0,...,4}{ + \pgfmathparse{0.30*exp(ln(2)*\k)} + \xdef\l{\pgfmathresult} + \kurve{\l}{blue} +} +} + +\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$}; +\uncover<4->{ +\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$}; +} + +\draw[->] (0,-0.1) -- (0,3.3); +\end{scope} + +\begin{scope}[xshift=7cm] + +\uncover<6->{ +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; + +\draw[color=darkgreen,line width=1.4pt] + plot[domain=0:6.3,samples=100] + ({\x},{3*exp(-(\x/0.5)*(\x/0.5)}); + +\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}]; +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/laplacebasis.tex b/vorlesungen/slides/8/wavelets/laplacebasis.tex new file mode 100644 index 0000000..ced4c09 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/laplacebasis.tex @@ -0,0 +1,62 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\a{2} +\def\b{0.8} +\def\c{1} +\def\d{0.6} +\input{../slides/8/wavelets/vektoren.tex} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Laplace-Basis} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\begin{scope}[yshift=-0.4cm,xshift=-5.5cm] +\fnull +\end{scope} + +\begin{scope}[yshift=-1.8cm,xshift=-5.5cm] +\fone +\end{scope} + +\begin{scope}[yshift=-3.2cm,xshift=-5.5cm] +\ftwo +\end{scope} + +\begin{scope}[yshift=-4.6cm,xshift=-5.5cm] +\fthree +\end{scope} + +\begin{scope}[yshift=-6.0cm,xshift=-5.5cm] +\ffour +\end{scope} + +\begin{scope}[yshift=0cm] +\vnull +\end{scope} + +\begin{scope}[yshift=-1.4cm] +\vone +\end{scope} + +\begin{scope}[yshift=-2.8cm] +\vtwo +\end{scope} + +\begin{scope}[yshift=-4.2cm] +\vthree +\end{scope} + +\begin{scope}[yshift=-5.6cm] +\vfour +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex new file mode 100644 index 0000000..d6575d0 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex @@ -0,0 +1,46 @@ +% +% lokalisierungsvergleich.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lokalisierung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ortsraum} +Ortsraum$\mathstrut=V$ +\begin{itemize} +\item<3-> Standardbasis +\item<5-> lokalisiert in den Knoten +\item<7-> die meisten $\hat{f}(k)$ gross +\item<9-> vollständig delokalisiert im Frequenzraum +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Frequenzraum} +\uncover<2->{Frequenzraum $\mathstrut=\{\lambda_1,\lambda_2,\dots,\lambda_n\}$} +\begin{itemize} +\item<4-> Laplace-Basis +\item<6-> lokalisiert in den Eigenwerten +\item<8-> die meisten Komponenten gross +\item<10-> vollständig delokalisiert im Ortsraum +\end{itemize} +\end{block} +\end{column} +\end{columns} +\uncover<11->{% +\begin{block}{Plan} +Gesucht sind Funktionen auf dem Graphen derart, die +\begin{enumerate} +\item<12-> in der Nähe einzelner Knoten konzentriert/lokalisiert sind und +\item<13-> deren Fourier-Transformation in der Nähe einzelner Eigenwerte +konzentriert/lokalisiert ist +\end{enumerate} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/matrixdilatation.tex b/vorlesungen/slides/8/wavelets/matrixdilatation.tex new file mode 100644 index 0000000..3536736 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/matrixdilatation.tex @@ -0,0 +1,39 @@ +% +% matrixdilatation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Dilatation in Matrixform} +Dilatationsfaktor $a$, skaliertes Wavelet beim Knoten $k$ mit Spektrum +$\tilde{D}_{1/a}g$ +\begin{align*} +D_{g,a}e_k +&= +\chi +\begin{pmatrix} +g(a\lambda_1)& 0 & \dots & 0 \\ + 0 &g(a\lambda_2)& \dots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \dots &g(a\lambda_n) +\end{pmatrix} +\chi^t +e_k +\intertext{\uncover<2->{``verschmierter'' Standardbasisvektor am Knoten $k$}} +\uncover<2->{D_he_k +&= +\chi +\begin{pmatrix} +h(\lambda_1)& 0 & \dots & 0 \\ + 0 &h(\lambda_2)& \dots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \dots &h(\lambda_n) +\end{pmatrix} +\chi^t +e_k} +\end{align*} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/vektoren.tex b/vorlesungen/slides/8/wavelets/vektoren.tex new file mode 100644 index 0000000..2315d53 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/vektoren.tex @@ -0,0 +1,200 @@ +\def\vnull{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.0000$}; + \fill[color=red!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=red!100] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=red!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fnull{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.4472*\d}) -- + ({-1.0000*\c},{0.4472*\d}) -- + ({0.0000*\c},{0.4472*\d}) -- + ({1.0000*\c},{0.4472*\d}) -- + ({2.0000*\c},{0.4472*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vone{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.1586$}; + \fill[color=blue!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!41] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!41] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fone{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{-0.6533*\d}) -- + ({-1.0000*\c},{-0.2706*\d}) -- + ({0.0000*\c},{-0.0000*\d}) -- + ({1.0000*\c},{0.2706*\d}) -- + ({2.0000*\c},{0.6533*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vtwo{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.3000$}; + \fill[color=red!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=blue!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\ftwo{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.5000*\d}) -- + ({-1.0000*\c},{-0.5000*\d}) -- + ({0.0000*\c},{-0.0000*\d}) -- + ({1.0000*\c},{-0.5000*\d}) -- + ({2.0000*\c},{0.5000*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vthree{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.4414$}; + \fill[color=red!41] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=red!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=blue!41] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fthree{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.2706*\d}) -- + ({-1.0000*\c},{-0.6533*\d}) -- + ({0.0000*\c},{0.0000*\d}) -- + ({1.0000*\c},{0.6533*\d}) -- + ({2.0000*\c},{-0.2706*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vfour{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.5000$}; + \fill[color=red!25] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!100] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!25] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=red!25] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!25] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\ffour{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.2236*\d}) -- + ({-1.0000*\c},{0.2236*\d}) -- + ({0.0000*\c},{-0.8944*\d}) -- + ({1.0000*\c},{0.2236*\d}) -- + ({2.0000*\c},{0.2236*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} diff --git a/vorlesungen/slides/8/weitere.tex b/vorlesungen/slides/8/weitere.tex new file mode 100644 index 0000000..46a3da0 --- /dev/null +++ b/vorlesungen/slides/8/weitere.tex @@ -0,0 +1,43 @@ +% +% weitere.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Weitere Resultate der spektralen Graphentheorie} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Hoffmann)} +\[ +\operatorname{chr} X \ge 1 + \frac{\alpha_{\text{max}}}{-\alpha_{\text{min}}} +\] +\end{block} +\uncover<2->{% +\begin{block}{Satz (Hoffmann)} +\[ +\operatorname{ind} X \le n \biggl(1-\frac{d_{\text{min}}}{\lambda_{\text{max}}}\biggr) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Korollar} +Für einen regulären Graphen mit $n$ Knoten gilt +\begin{align*} +\operatorname{ind} X +&\le +\frac{n}{\displaystyle 1-\frac{d}{\alpha_{\text{min}}}} +\\ +\operatorname{chr} X +&\ge +1-\frac{d}{\alpha_{\text{min}}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wilf.m b/vorlesungen/slides/8/wilf.m new file mode 100644 index 0000000..49dc161 --- /dev/null +++ b/vorlesungen/slides/8/wilf.m @@ -0,0 +1,22 @@ +# +# wilf.m -- chromatische Zahl für einen Graphen +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +N = 9; +A = zeros(N,N); + +for i = (1:N) + j = 1 + rem(i, N) + A(i,j) = 1; +endfor +for i = (1:3:N-3) + j = 1 + rem(i + 2, N) + A(i,j) = 1; +endfor + +A(1,3) = 1; + +A = A + A' + +eig(A) diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc index fa6c29b..2257810 100644 --- a/vorlesungen/slides/9/Makefile.inc +++ b/vorlesungen/slides/9/Makefile.inc @@ -10,5 +10,20 @@ chapter9 = \ ../slides/9/irreduzibel.tex \ ../slides/9/stationaer.tex \ ../slides/9/pf.tex \ + ../slides/9/potenz.tex \ + ../slides/9/pf/positiv.tex \ + ../slides/9/pf/primitiv.tex \ + ../slides/9/pf/trennung.tex \ + ../slides/9/pf/vergleich.tex \ + ../slides/9/pf/vergleich3d.tex \ + ../slides/9/pf/dreieck.tex \ + ../slides/9/pf/folgerungen.tex \ + ../slides/9/parrondo/uebersicht.tex \ + ../slides/9/parrondo/erwartung.tex \ + ../slides/9/parrondo/spiela.tex \ + ../slides/9/parrondo/spielb.tex \ + ../slides/9/parrondo/spielbmod.tex \ + ../slides/9/parrondo/kombiniert.tex \ + ../slides/9/parrondo/deformation.tex \ ../slides/9/chapter.tex diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex index 9e26587..cbab0f0 100644 --- a/vorlesungen/slides/9/chapter.tex +++ b/vorlesungen/slides/9/chapter.tex @@ -10,5 +10,21 @@ \folie{9/stationaer.tex} \folie{9/irreduzibel.tex} \folie{9/pf.tex} +\folie{9/potenz.tex} +\folie{9/pf/positiv.tex} +\folie{9/pf/primitiv.tex} +\folie{9/pf/trennung.tex} +\folie{9/pf/vergleich.tex} +\folie{9/pf/vergleich3d.tex} +\folie{9/pf/dreieck.tex} +\folie{9/pf/folgerungen.tex} + +\folie{9/parrondo/uebersicht.tex} +\folie{9/parrondo/erwartung.tex} +\folie{9/parrondo/spiela.tex} +\folie{9/parrondo/spielb.tex} +\folie{9/parrondo/spielbmod.tex} +\folie{9/parrondo/kombiniert.tex} +\folie{9/parrondo/deformation.tex} diff --git a/vorlesungen/slides/9/parrondo/deformation.tex b/vorlesungen/slides/9/parrondo/deformation.tex new file mode 100644 index 0000000..40d2eb9 --- /dev/null +++ b/vorlesungen/slides/9/parrondo/deformation.tex @@ -0,0 +1,45 @@ +% +% deformation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Deformation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Verlustspiele} +Durch Deformation (Parameter $e$ und $\varepsilon$) kann man +aus $A_e$ und $B_\varepsilon$ Spiele mit negativer Gewinnerwartung machen +\uncover<2->{% +\begin{align*} +E(X)&=0&&\rightarrow&E(X_e)&<0\\ +E(Y)&=0&&\rightarrow&E(Y_\varepsilon)&<0\\ +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Kombiniertes Spiel} +\uncover<3->{% +Die Deformation für das Spiel $C$ startet mit Erwartungswert $\frac{18}{709}$}% +\begin{align*} +\uncover<4->{E(Z)&=\frac{18}{709}>0} +&&\uncover<5->{\rightarrow& +E(Z_*)&>0} +\end{align*} +\uncover<6->{Wegen Stetigkeit!} +\\ +\uncover<5->{Die Deformation ist immer noch ein Gewinnspiel (für Parameter klein genug)} +\end{block} +\uncover<7->{% +\begin{block}{Parrondo-Paradoxon} +Zufällig zwischen zwei Verlustspielen auswählen kann trotzdem ein +Gewinnspiel ergeben +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/erwartung.tex b/vorlesungen/slides/9/parrondo/erwartung.tex new file mode 100644 index 0000000..b58c37f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/erwartung.tex @@ -0,0 +1,81 @@ +% +% erwartung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Erwartung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zufallsvariable} +\begin{center} +\[ +\begin{array}{c|c} +\text{Werte $X$}&\text{Wahrscheinlichkeit $p$}\\ +\hline +x_1&p_1=P(X=x_1)\\ +x_2&p_2=P(X=x_2)\\ +\vdots&\vdots\\ +x_n&p_n=P(X=x_n) +\end{array} +\] +\end{center} +\end{block} +\uncover<4->{% +\begin{block}{Einervektoren/-matrizen} +\[ +U=\begin{pmatrix} +1&1&\dots&1\\ +1&1&\dots&1\\ +\vdots&\vdots&\ddots&\vdots\\ +1&1&\dots&1 +\end{pmatrix} +\in +M_{n\times m}(\Bbbk) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Erwartungswerte} +\begin{align*} +E(X) +&= +\sum_i x_ip_i += +x^tp +\uncover<5->{= +U^t x\odot p} +\hspace*{3cm} +\\ +\uncover<2->{E(X^2) +&= +\sum_i x_i^2p_i} +\ifthenelse{\boolean{presentation}}{ +\only<6>{= +(x\odot x)^tp}}{} +\uncover<7->{= +U^t (x\odot x) \odot p} +\\ +\uncover<3->{E(X^k) +&= +\sum_i x_i^kp_i} +\uncover<8->{= +U^t x^{\odot k}\odot p} +\end{align*} +\uncover<9->{% +Substitution: +\begin{align*} +\uncover<10->{\sum_i &\to U^t}\\ +\uncover<11->{x_i^k &\to x^{\odot k}} +\end{align*}}% +\uncover<12->{Kann für Übergangsmatrizen von Markov-Ketten verallgemeinert werden} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/kombiniert.tex b/vorlesungen/slides/9/parrondo/kombiniert.tex new file mode 100644 index 0000000..5012d06 --- /dev/null +++ b/vorlesungen/slides/9/parrondo/kombiniert.tex @@ -0,0 +1,73 @@ +% +% kombiniert.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kombiniertes Spiel $C$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein fairer Münzwurf entscheidet, ob +Spiel $A$ oder Spiel $B$ gespielt wird +\end{block} +\uncover<2->{% +\begin{block}{Übergangsmatrix} +Münzwurf $X$ +\begin{align*} +C +&= +P(X=\text{Kopf})\cdot A ++ +P(X=\text{Zahl})\cdot B +\\ +&\uncover<3->{= +\begin{pmatrix} + 0&\frac{3}{8}&\frac{5}{8}\\ +\frac{3}{10}& 0&\frac{3}{8}\\ +\frac{7}{10}&\frac{5}{8}& 0 +\end{pmatrix}} +\end{align*} +\end{block}} +\vspace{-8pt} +\uncover<4->{% +\begin{block}{Gewinnerwartung im Einzelspiel} +\[ +p=\frac13U +\Rightarrow +U^t(G\odot C)p +\uncover<5->{= +-\frac{1}{30}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Iteriertes Spiel} +\[ +\overline{p}=C\overline{p} +\quad +\uncover<7->{\Rightarrow +\quad +\overline{p}=\frac{1}{709}\begin{pmatrix}245\\180\\284\end{pmatrix}} +\] +\end{block}} +\uncover<8->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +E(Z) +&= +U^t (G\odot C) \overline{p} +\uncover<9->{= +\frac{18}{709}} +\end{align*} +\uncover<10->{$C$ ist ein Gewinnspiel!} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spiela.tex b/vorlesungen/slides/9/parrondo/spiela.tex new file mode 100644 index 0000000..629586f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spiela.tex @@ -0,0 +1,52 @@ +% +% spiela.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spiel $A$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn = Zufallsvariable $X$ mit Werten $\pm 1$ +\begin{align*} +P(X=\phantom{+}1) +&= +\frac12\uncover<2->{+e} +\\ +P(X= - 1) +&= +\frac12\uncover<2->{-e} +\end{align*} +Bernoulli-Experiment mit $p=\frac12\uncover<2->{+e}$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{ +\begin{block}{Gewinnerwartung} +\begin{align*} +E(X) +&=\uncover<4->{ +P(X=1)\cdot (1)} +\\ +&\qquad +\uncover<4->{+ +P(X=-1)\cdot (-1)} +\\ +&\uncover<5->{= +\biggl(\frac12+e\biggr)\cdot 1 ++ +\biggl(\frac12-e\biggr)\cdot (-1)} +\\ +&\uncover<6->{=2e} +\end{align*} +\uncover<7->{$\Rightarrow$ {\usebeamercolor[fg]{title}Verlustspiel für $e<0$}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spielb.tex b/vorlesungen/slides/9/parrondo/spielb.tex new file mode 100644 index 0000000..f65564f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spielb.tex @@ -0,0 +1,100 @@ +% +% spielb.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spiel $B$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des +aktuellen Kapitals $K$: +\begin{center} +\uncover<2->{% +\begin{tikzpicture}[>=latex,thick] +\coordinate (A0) at (90:2); +\coordinate (A1) at (210:2); +\coordinate (A2) at (330:2); + +\node at (A0) {$0$}; +\node at (A1) {$1$}; +\node at (A2) {$2$}; + +\draw (A0) circle[radius=0.4]; +\draw (A1) circle[radius=0.4]; +\draw (A2) circle[radius=0.4]; + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2); + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1); + +\def\R{1.9} +\def\r{0.7} + +\node at (30:\r) {$\frac{9}{10}$}; +\node at (150:\r) {$\frac1{10}$}; +\node at (270:\r) {$\frac34$}; + +\node at (30:\R) {$\frac{3}{4}$}; +\node at (150:\R) {$\frac1{4}$}; +\node at (270:\R) {$\frac14$}; + +\end{tikzpicture}} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Markov-Kette $Y$} +Übergangsmatrix +\[ +B=\begin{pmatrix} +0&\frac14&\frac34\\ +\frac{1}{10}&0&\frac14\\ +\frac{9}{10}&\frac34&0 +\end{pmatrix} +\] +\vspace{-10pt} + +\uncover<4->{% +Gewinnmatrix: +\vspace{-2pt} +\[ +G=\begin{pmatrix*}[r] +0&-1&1\\ +1&0&-1\\ +-1&1&0 +\end{pmatrix*} +\]} +\end{block}} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +&&&& +E(Y) +&= +U^t(G\odot B)p +\\ +p&={\textstyle\frac13}U +&&\Rightarrow& +E(Y)&={\textstyle\frac1{15}} +\\ +\overline{p}&={\tiny\frac{1}{13}\begin{pmatrix}5\\2\\6\end{pmatrix}} +&&\Rightarrow& +E(Y)&=0 +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex new file mode 100644 index 0000000..66d39bc --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spielbmod.tex @@ -0,0 +1,103 @@ +% +% spielb.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Modifiziertes Spiel $\tilde{B}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des +aktuellen Kapitals $K$: +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A0) at (90:2); +\coordinate (A1) at (210:2); +\coordinate (A2) at (330:2); + +\node at (A0) {$0$}; +\node at (A1) {$1$}; +\node at (A2) {$2$}; + +\draw (A0) circle[radius=0.4]; +\draw (A1) circle[radius=0.4]; +\draw (A2) circle[radius=0.4]; + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2); + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1); + +\def\R{1.9} +\def\r{0.7} + +\node at (30:{0.9*\r}) {\tiny $\frac{9}{10}\uncover<2->{+\varepsilon}$}; +\node at (150:{0.9*\r}) {\tiny $\frac1{10}\uncover<2->{-\varepsilon}$}; +\node at (270:\r) {$\frac34\uncover<2->{-\varepsilon}$}; + +\node at (30:{1.1*\R}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$}; +\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$}; +\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Markov-Kette $\tilde{Y}$} +Übergangsmatrix +\[ +\tilde{B}= +B\uncover<2->{+\varepsilon F} +\uncover<3->{= +B+\varepsilon\begin{pmatrix*}[r] +0&1&-1\\ +-1&0&1\\ +1&-1&0 +\end{pmatrix*}} +\] +\vspace{-12pt} + +\uncover<4->{% +Gewinnmatrix: +\[ +G=\begin{pmatrix*}[r] +0&-1&1\\ +1&0&-1\\ +-1&1&0 +\end{pmatrix*} +\]} +\end{block} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +\uncover<6->{E(\tilde{Y}) +&= +U^t(G\odot \tilde{B})p} +\\ +&\uncover<7->{= +E(Y) + \varepsilon U^t(G\odot F)p} +\uncover<8->{= +{\textstyle\frac1{15}}+2\varepsilon} +\\ +\uncover<9->{ +\text{rep.} +&= +-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2) +\quad\text{Verlustspiel} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/uebersicht.tex b/vorlesungen/slides/9/parrondo/uebersicht.tex new file mode 100644 index 0000000..2f3597a --- /dev/null +++ b/vorlesungen/slides/9/parrondo/uebersicht.tex @@ -0,0 +1,17 @@ +% +% uebersicht.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Parrondo-Paradoxon} +\begin{center} +\Large +Zufällige +Wahl zwischen zwei Verlustspielen = Gewinnspiel? +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/dreieck.tex b/vorlesungen/slides/9/pf/dreieck.tex new file mode 100644 index 0000000..0a572f3 --- /dev/null +++ b/vorlesungen/slides/9/pf/dreieck.tex @@ -0,0 +1,44 @@ +% +% dreieck.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Verallgemeinerte Dreiecksungleichung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.32\textwidth} +\begin{block}{Satz} +\[ +|u+v|\le |u|+|v| +\] +Gleichheit wenn lin.~abh. +\end{block} +\begin{block}{Satz} +\[ +\biggl|\sum_i u_i\biggr| +\le +\sum_i |u_i| +\] +Gleichheit wenn $u_i = \lambda_i u$ +\end{block} +\begin{block}{Satz} +\[ +\biggl|\sum_i z_i\biggr| +\le +\sum_i |z_i| +\] +Gleichheit, wenn $z_i=|z_i|c$, $c\in\mathbb{C}$ +\end{block} +\end{column} +\begin{column}{0.68\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/folgerungen.tex b/vorlesungen/slides/9/pf/folgerungen.tex new file mode 100644 index 0000000..5042c78 --- /dev/null +++ b/vorlesungen/slides/9/pf/folgerungen.tex @@ -0,0 +1,203 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Folgerungen für $A>0$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u\ge 0$ ein EV zum EW $ \lambda\ne 0$, +dann ist $u>0$ und $\lambda >0$ +\end{block} +\uncover<6->{% +\begin{block}{Satz} +$v$ ein EV zum EW $\lambda$ mit $|\lambda| = \varrho(A)$, +dann ist $u=|v|$ mit $u_i=|v_i|$ ein EV mit EW $\varrho(A)$ +\end{block}} +\uncover<29->{% +\begin{block}{Satz} +$v$ ein EV zum EW $\lambda$ mit $|\lambda|=\varrho(A)$, +dann ist $\lambda=\varrho(A)$ +\end{block}} +\uncover<46->{% +\begin{block}{Satz} +Der \only<57->{verallgemeinerte }Eigenraum zu EW $\varrho(A)$ +ist eindimensional +\end{block} +} +\end{column} +\ifthenelse{\boolean{presentation}}{ +\only<-6>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<3-> +Vergleich: $Au>0$ +\item<4-> +$Au=\lambda u > 0$ +\item<5-> +$\lambda >0$ und $u>0$ +\end{itemize} +\end{proof} +\end{column}} +\only<7-20>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{align*} +(Au)_i +&\only<-8>{= +\sum_j a_{ij}u_j} +\only<8-9>{= +\sum_j |a_{ij}v_j|} +\only<9->{\ge} +\only<9-10>{ +\biggl|\sum_j a_{ij}v_j\biggr|} +\only<10>{=} +\only<10-11>{ +|(Av)_i|} +\only<11>{=} +\only<11-12>{ +|\lambda v_i|} +\only<12>{=} +\only<12-13>{ +\varrho(A) |v_i|} +\only<13>{=} +\uncover<13->{ +\varrho(A) u_i} +\hspace*{5cm} +\\ +\uncover<14->{Au&\ge \varrho(A)u} +\intertext{\uncover<15->{Vergleich}} +\uncover<16->{A^2u&> \varrho(A)Au} +\intertext{\uncover<17->{Trennung: $\exists \vartheta >1$ mit}} +\uncover<18->{A^2u&\ge \vartheta \varrho(A) Au }\\ +\uncover<19->{A^3u&\ge (\vartheta \varrho(A))^2 Au }\\ +\uncover<20->{A^ku&\ge (\vartheta \varrho(A))^{k-1} Au }\\ +\end{align*} +\end{proof} +\end{column}} +\only<21-29>{% +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis, Fortsetzung] +Abschätzung der Operatornorm: +\begin{align*} +\|A^k\|\, |Au| +\ge +\|A^{k+1}u\| +\uncover<22->{ +\ge +(\vartheta\varrho(A))^k |Au|} +\end{align*} +\uncover<23->{Abschätzung des Spektralradius} +\begin{align*} +\uncover<24->{\|A^k\| &\ge (\vartheta\varrho(A))^k} +\\ +\uncover<25->{\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)} +\\ +\uncover<26->{\lim_{k\to\infty}\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)} +\\ +\uncover<27->{\varrho(A) &\ge \underbrace{\vartheta}_{>1} \varrho(A)} +\end{align*} +\uncover<28->{Widerspruch: $u=v$} +\end{proof} +\end{column}} +\only<30-46>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +$u$ ist EV mit EW $\varrho(A)$: +\[ +Au=\varrho(A)u +\uncover<31->{\Rightarrow +\sum_j a_{ij}|v_j| = {\color<38->{red}\varrho(A) |v_i|}} +\] +\uncover<33->{Andererseits: $Av=\lambda v$} +\[ +\uncover<34->{\sum_{j}a_{ij}v_j=\lambda v_i} +\] +\uncover<35->{Betrag} +\begin{align*} +\uncover<36->{\biggl|\sum_j a_{ij}v_j\biggr| +&= +|\lambda v_i|} +\uncover<37->{= +{\color<38->{red}\varrho(A) |v_i|}} +\uncover<39->{= +\sum_j a_{ij}|v_j|} +\end{align*} +\uncover<40->{Dreiecksungleichung: $v_j=|v_j|c, c\in\mathbb{C}$} +\[ +\uncover<41->{\lambda v = Av} +\uncover<42->{= Acu} +\uncover<43->{= c\varrho(A) u} +\uncover<44->{= \varrho(A)v} +\] +\uncover<45->{$\Rightarrow +\lambda=\varrho(A) +$} +\end{proof} +\end{column}} +\only<47-57>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<48-> $u>0$ ein EV zum EW $\varrho(A)$ +\item<49-> $v$ ein weiterer EV, man darf $v\in\mathbb{R}^n$ annehmen +\item<50-> Da $u>0$ gibt es $c>0$ mit $u\ge cv$ aber $u\not > cv$ +\item<51-> $u-cv\ge 0$ aber $u-cv\not > 0$ +\item<52-> $A$ anwenden: +\[ +\begin{array}{ccc} +\uncover<53->{A(u-cv)}&\uncover<54->{>&0} +\\ +\uncover<53->{\|}&& +\\ +\uncover<53->{\varrho(A)(u-cv)}&\uncover<55->{\not>&0} +\end{array} +\] +\uncover<56->{Widerspruch: $v$ existiert nicht} +\end{itemize} +\end{proof} +\end{column}} +\only<58->{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<59-> $Au=\varrho(A)u$ und $A^tp^t=\varrho(A)p^t$ +\item<60-> $u>0$ und $p>0$ $\Rightarrow$ $up>0$ +\item<61-> $px=0$, dann ist +\[ +\uncover<62->{pAx} +\only<62-63>{= +(A^tp^t)^t x} +\only<63-64>{= +\varrho(A) (p^t)^t x} +\uncover<64->{= +\varrho(A) px} +\uncover<65->{= 0} +\] +\uncover<66->{also ist $\{x\in\mathbb{R}^n\;|\; px=0\}$ +invariant} +\item<67-> Annahme: $v\in \mathcal{E}_{\varrho(A)}$ +\item<68-> Dann muss es einen EV zum EW $\varrho(A)$ in +$\mathcal{E}_{\varrho(A)}$ geben +\item<69-> Widerspruch: der Eigenraum ist eindimensional +\end{itemize} +\end{proof} +\end{column}} +}{ +\begin{column}{0.48\textwidth} +\begin{block}{} +\usebeamercolor[fg]{title} +Beweise: Buch Abschnitt 9.3 +\end{block} +\end{column} +} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/positiv.tex b/vorlesungen/slides/9/pf/positiv.tex new file mode 100644 index 0000000..d7e833d --- /dev/null +++ b/vorlesungen/slides/9/pf/positiv.tex @@ -0,0 +1,64 @@ +% +% positiv.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Positive und nichtnegative Matrizen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positive Matrix\strut} +Eine Matrix $A$ heisst positiv, wenn +\[ +a_{ij} > 0\quad\forall i,j +\] +Man schreibt $A>0\mathstrut$ +\end{block} +\uncover<2->{% +\begin{block}{Relation $>\mathstrut$} +Man schreibt $A>B$ wenn $A-B > 0\mathstrut$ +\end{block}} +\uncover<5->{% +\begin{block}{Wahrscheinlichkeitsmatrix} +\[ +W=\begin{pmatrix} +0.7&0.2&0.1\\ +0.2&0.6&0.1\\ +0.1&0.2&0.8 +\end{pmatrix} +\] +Spaltensumme$\mathstrut=1$, Zeilensumme$\mathstrut=?$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Nichtnegative Matrix\strut} +Eine Matrix $A$ heisst nichtnegativ, wenn +\[ +a_{ij} \ge 0\quad\forall i,j +\] +Man schreibt $A\ge 0\mathstrut$ +\end{block}} +\uncover<4->{% +\begin{block}{Relation $\ge\mathstrut$} +Man schreibt $A\ge B$ wenn $A-B \ge 0\mathstrut$ +\end{block}} +\uncover<6->{% +\begin{block}{Permutationsmatrix} +\[ +P=\begin{pmatrix} +0&0&1\\ +1&0&0\\ +0&1&0 +\end{pmatrix} +\] +Genau eine $1$ in jeder Zeile/Spalte +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/primitiv.tex b/vorlesungen/slides/9/pf/primitiv.tex new file mode 100644 index 0000000..961b1d5 --- /dev/null +++ b/vorlesungen/slides/9/pf/primitiv.tex @@ -0,0 +1,84 @@ +% +% primitiv.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Primitive Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +$A\ge 0$ heisst primitiv, wenn es ein $n>0$ gibt mit $A^n>0$ +\end{block} +\uncover<9->{% +\begin{block}{Intuition} +\begin{itemize} +\item<10-> +Markov-Ketten: $a_{ij} > 0$ bedeutet, $i$ von $j$ aus erreichbar. +\item<11-> +Band: {\em alle} Verbindung mit allen Nachbarn +\item<12-> +$n$-te Potenz: Pfade der Länge $n$ +\item<13-> +Durchmesser: wenn $n>\text{Durchmesser des Zustandsdiagramms}$, +dann ist $A^n>0$ +\end{itemize} +\end{block} +} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiel: Reduzible W'keitsmatrix} +\vspace{-5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\fill[color=gray!40] (-1,0) rectangle (0,1); +\fill[color=gray!40] (0,-1) rectangle (1,0); +\draw[line width=0.3pt] (0,-1) -- (0,1); +\draw[line width=0.3pt] (-1,0) -- (1,0); +%\draw (-1,-1) rectangle (1,1); +\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$}; +\node at (-1.3,0) [left] {$\mathstrut W=$}; +\node at (0.5,0.5) {$0$}; +\node at (-0.5,-0.5) {$0$}; +\end{tikzpicture} +\end{center} +\vspace{-10pt} + +$\Rightarrow$ $W$ ist nicht primitiv +\end{block}} +\uncover<3->{% +\begin{block}{Beispiel: Bandmatrix} +\centering +\begin{tikzpicture}[>=latex,thick] +\begin{scope} +\clip (-1,-1) rectangle (1,1); +\foreach \n in {3,...,8}{ + \pgfmathparse{0.3*(\n-2)} + \xdef\x{\pgfmathresult} + \only<\n>{ + \fill[color=gray!40] + ({-1.2-\x},1) -- (1,{-1.2-\x}) -- (1,{-0.8+\x}) + -- ({-0.8+\x},1) -- cycle; + } +} +\fill[color=gray] (-1.2,1) -- (1,-1.2) -- (1,-0.8) -- (-0.8,1) -- cycle; +\end{scope} +\foreach \n in {2,...,8}{ + \uncover<\n>{ + \pgfmathparse{int(\n-2)} + \xdef\k{\pgfmathresult} + \node at (-1.3,0) [left] {$\mathstrut B^{\k}=$}; + } +} +\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$}; +\end{tikzpicture} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/trennung.tex b/vorlesungen/slides/9/pf/trennung.tex new file mode 100644 index 0000000..9c85849 --- /dev/null +++ b/vorlesungen/slides/9/pf/trennung.tex @@ -0,0 +1,99 @@ +% +% trennung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Trennung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (u) at (3.5,4.5); +\coordinate (v) at (2.5,2); +\coordinate (va) at ({(3.5/2.5)*2.5},{(3.5/2.5)*2}); + +\uncover<3->{ +\fill[color=darkgreen!20] (0,0) rectangle (5.3,5.3); +\node[color=darkgreen] at (1.5,4.9) {$u\not\ge w$}; +\node[color=darkgreen] at (4.4,0.6) {$u\not\ge w$}; +} + +\uncover<5->{ +\begin{scope} +\clip (0,0) rectangle (5.3,5.3); +\draw[color=darkgreen] (0,0) -- ($3*(v)$); +\end{scope} + +\node[color=darkgreen] at ($1.2*(va)$) + [below,rotate={atan(2/2.5)}] {$(1+\mu)v$}; +} + +\uncover<2->{ + \fill[color=red!20] (0,0) rectangle (u); +} + +\fill[color=red] (u) circle[radius=0.08]; +\node[color=red] at (u) [above right] {$u$}; + +\uncover<4->{ + \fill[color=blue!40,opacity=0.5] (0,0) rectangle (v); +} + +\uncover<2->{ + \fill[color=blue] (v) circle[radius=0.08]; + \node[color=blue] at (v) [above] {$v$}; +} + +\uncover<4->{ + \draw[color=blue] (0,0) -- (va); + + \fill[color=blue] (va) circle[radius=0.08]; + \node[color=blue] at (va) [above left] {$(1+\varepsilon)v$}; +} + +\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}]; +\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}]; + +\uncover<2->{ + \draw[->,color=red] (3.0,-0.2) -- (3.0,1.5); + \node[color=red] at (3.0,-0.2) [below] + {$\{w\in\mathbb{R}^n\;|\; w<u\}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u>v\ge 0$\uncover<4->{, dann gibt es $\varepsilon>0$ mit +\[ +u\ge (1+\varepsilon)v +\]}% +\uncover<5->{und für $\mu>\varepsilon$ ist +\[ +u \not\ge (1+\mu)v +\]} +\uncover<6->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<7-> +$u>v$ $\Rightarrow$ $u_i/v_i>1$ falls $v_i>0$ +\item<8-> +\[ +\vartheta = \min_{v_i\ne 0} \frac{u_i}{v_i} > 1 +\] +\uncover<9->{$\varepsilon = \vartheta - 1$} +\end{itemize} +\end{proof}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/vergleich.tex b/vorlesungen/slides/9/pf/vergleich.tex new file mode 100644 index 0000000..c1a1f7a --- /dev/null +++ b/vorlesungen/slides/9/pf/vergleich.tex @@ -0,0 +1,113 @@ +% +% vergleich.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Vergleich} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\a{1.2} \def\b{0.35} +\def\c{0.5} \def\d{1.25} +\def\r{4} + +\coordinate (u) at (3.5,0); +\coordinate (v) at (2.5,0); + +\coordinate (Au) at ({3.5*\a},{3.5*\c}); +\coordinate (Av) at ({2.5*\a},{2.5*\c}); + +\uncover<2->{ + \begin{scope} + \clip (0,0) rectangle (5,5); + \fill[color=red!20] (0,0) circle[radius=4]; + \end{scope} + \node[color=red] at (0,4) [below right] {$\mathbb{R}^n$}; + + \fill[color=blue!40,opacity=0.5] (0,0) -- ({\a*\r},{\c*\r}) + -- plot[domain=0:90,samples=100] + ({\r*(\a*cos(\x)+\b*sin(\x))},{\r*(\c*cos(\x)+\d*sin(\x))}) + -- ({\b*\r},{\d*\r}) -- cycle; + \node[color=blue] at ({\r*\b},{\r*\d}) [below right] {$A\mathbb{R}^n$}; +} + +\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}]; +\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}]; + +\uncover<3->{ + \fill[color=darkgreen!30,opacity=0.5] + (0,0) rectangle ({3.5*\a},{3.5*\c}); + \draw[color=white,line width=0.7pt] + ({3.5*\a},0) -- ({3.5*\a},{3.5*\c}) -- (0,{3.5*\c}); +} + +\uncover<2->{ + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\a},{\r*\c}); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\b},{\r*\d}); + + \draw[->,color=red,line width=1.4pt] (0,0) -- (4,0); + \draw[->,color=red,line width=1.4pt] (0,0) -- (0,4); +} + +\draw[color=darkgreen,line width=2pt] (u) -- (v); +\fill[color=darkgreen] (u) circle[radius=0.08]; +\fill[color=darkgreen] (v) circle[radius=0.08]; + +\node[color=darkgreen] at (u) [below right] {$u$}; +\node[color=darkgreen] at (v) [below left] {$v$}; +\node[color=darkgreen] at ($0.5*(u)+0.5*(v)$) [above] {$v\le u$}; + +\uncover<3->{ + \draw[color=darkgreen,line width=2pt] (Au) -- (Av); + \fill[color=darkgreen] (Au) circle[radius=0.08]; + \fill[color=darkgreen] (Av) circle[radius=0.08]; + + \node[color=darkgreen] at (Au) [above left] {$Au$}; + \node[color=darkgreen] at (Av) [above left] {$Av$}; + + \node[color=darkgreen] at ($0.5*(Au)+0.5*(Av)$) + [below,rotate={atan(\c/\a)}] {$Av<Au$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u\ge v\ge 0$ \uncover<2->{und $A > 0$}\uncover<3->{ $\Rightarrow$ $Au>Av$} +\end{block} +\uncover<4->{% +\begin{block}{intuitiv} +$A>0$ befördert $\ge$ zu $>$ +\end{block}} +\uncover<5->{% +\begin{proof}[Beweis] +$d=u-v\ge 0$ +\begin{align*} +(Ad)_i +\uncover<6->{= +\sum_{j} +\underbrace{a_{ij}}_{>0}d_j} +\uncover<7->{> +0} +\uncover<8->{\quad\Rightarrow\quad +Au > Av} +\end{align*} +\uncover<7->{da mindestens ein $d_j>0$ ist} +\end{proof}} +\uncover<9->{% +\begin{block}{Korollar} +$A>0$ und $d\ge 0$ $\Rightarrow$ $Ad > 0$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/vergleich3d.tex b/vorlesungen/slides/9/pf/vergleich3d.tex new file mode 100644 index 0000000..1c019a6 --- /dev/null +++ b/vorlesungen/slides/9/pf/vergleich3d.tex @@ -0,0 +1,26 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Vergleich} + +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.57\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf} +\end{center} +\end{column} +\begin{column}{0.38\textwidth} +\begin{block}{Satz} +$u\ge v\ge 0$ $\Rightarrow$ $Au>Av$ +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/potenz.tex b/vorlesungen/slides/9/potenz.tex new file mode 100644 index 0000000..2c3afa3 --- /dev/null +++ b/vorlesungen/slides/9/potenz.tex @@ -0,0 +1,15 @@ +% +% potenz.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Potenzmethode} +\begin{center} +\includegraphics[width=0.9\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/positiv.pdf} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 4bf9348..a9d72be 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -9,9 +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) $(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 e4b9ad7..4289c44 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -1,17 +1,28 @@ % % test.tex collection of all slides % -% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{5/verzerrung.tex} - -% XXX Visualisierung Cayley-Hamilton-Produkte -% XXX \folie{5/chvisual.tex} +%\folie{9/google.tex} +%\folie{9/markov.tex} +%\folie{9/stationaer.tex} +%\folie{9/irreduzibel.tex} +%\folie{9/pf.tex} -% XXX stone weierstrass incomplete -%\folie{5/stoneweierstrass.tex} +%\folie{9/pf/positiv.tex} +%\folie{9/pf/primitiv.tex} +%\folie{9/pf/trennung.tex} +%\folie{9/pf/vergleich.tex} +%\folie{9/pf/vergleich3d.tex} +%\folie{9/pf/dreieck.tex} +%\folie{9/pf/folgerungen.tex} +%\folie{9/potenz.tex} -% XXX polynome auf dem spektrum -% XXX Motiviation für *-Operation -%\folie{5/normal.tex} +\folie{9/parrondo/erwartung.tex} +%\folie{9/parrondo/uebersicht.tex} +\folie{9/parrondo/spiela.tex} +\folie{9/parrondo/spielb.tex} +\folie{9/parrondo/spielbmod.tex} +\folie{9/parrondo/kombiniert.tex} +\folie{9/parrondo/deformation.tex} |
