From 1fb743df08b0734932d510c6b11405d0a2dbbe47 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 19:57:13 +0200 Subject: new slides --- vorlesungen/slides/7/semi.tex | 109 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 109 insertions(+) create mode 100644 vorlesungen/slides/7/semi.tex (limited to 'vorlesungen/slides/7/semi.tex') diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex new file mode 100644 index 0000000..46f6d03 --- /dev/null +++ b/vorlesungen/slides/7/semi.tex @@ -0,0 +1,109 @@ +% +% 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} +\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} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(e^{s_1},t_1)(e^{s_2},t_2)x +&= +(e^{s_1},t_1)(e^{s_2}x+t_2) +\\ +&= +e^{s_1+s_2}x + e^{s_1}t_2+t_1 +\\ +(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} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +\vec{x} +&= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +\\ +&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +\\ +(\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} +\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} +\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 -- cgit v1.2.1 From 15881729aa3f1293d546a1692a02094ed3f24e2b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 10:30:05 +0200 Subject: phases --- vorlesungen/slides/7/semi.tex | 34 +++++++++++++++++++++------------- 1 file changed, 21 insertions(+), 13 deletions(-) (limited to 'vorlesungen/slides/7/semi.tex') diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 46f6d03..66b8d27 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -21,6 +21,7 @@ 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= @@ -32,51 +33,57 @@ Wirkung auf $\mathbb{R}^2$: \[ \vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} \] -\end{block} +\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 -&= -(e^{s_1},t_1)(e^{s_2}x+t_2) +&\uncover<4->{= +(e^{s_1},t_1)(e^{s_2}x+t_2)} \\ -&= -e^{s_1+s_2}x + e^{s_1}t_2+t_1 +&\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) +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} \end{align*} -\end{block} +\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} -&= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +&\uncover<8->{= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} \\ -&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +&\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{block}} \end{column} \end{columns} \vspace{-10pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<11->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -88,9 +95,10 @@ e^s&t\\ \quad\text{auf}\quad \begin{pmatrix}x\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<12->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -102,7 +110,7 @@ D_{\alpha}&\vec{t}\\ \quad\text{auf}\quad \begin{pmatrix}\vec{x}\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} -- cgit v1.2.1 From 9247dde37fc0283a4600ad73586ba155565ffe3f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 29 Apr 2021 17:42:24 +0200 Subject: some improvements --- vorlesungen/slides/7/semi.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) (limited to 'vorlesungen/slides/7/semi.tex') diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 66b8d27..cd974c9 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -41,7 +41,7 @@ Wirkung auf $\mathbb{R}^2$: \begin{column}{0.48\textwidth} \uncover<3->{% \begin{block}{Verknüpfung} -\vspace{-15pt} +%\vspace{-15pt} \begin{align*} (e^{s_1},t_1)(e^{s_2},t_2)x &\uncover<4->{= @@ -60,7 +60,7 @@ e^{s_1+s_2}x + e^{s_1}t_2+t_1} \begin{column}{0.48\textwidth} \uncover<7->{% \begin{block}{Verknüpfung} -\vspace{-15pt} +%\vspace{-15pt} \begin{align*} (\alpha_1,\vec{t}_1) (\alpha_2,\vec{t}_2) @@ -85,7 +85,7 @@ e^{s_1+s_2}x + e^{s_1}t_2+t_1} \begin{column}{0.48\textwidth} \uncover<11->{% \begin{block}{Matrixschreibweise} -\vspace{-12pt} +%\vspace{-12pt} \[ g=(e^s,t) = \begin{pmatrix} @@ -100,7 +100,7 @@ e^s&t\\ \begin{column}{0.48\textwidth} \uncover<12->{% \begin{block}{Matrixschreibweise} -\vspace{-12pt} +%\vspace{-12pt} \[ g=(\alpha,\vec{t}) = \begin{pmatrix} -- cgit v1.2.1