diff options
| -rw-r--r-- | vorlesungen/slides/7/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/7/drehung.tex | 4 | ||||
| -rw-r--r-- | vorlesungen/slides/7/parameter.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/7/zusammenhang.tex | 99 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 18 |
6 files changed, 105 insertions, 20 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 7afeea1..3e99415 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -18,5 +18,6 @@ chapter5 = \ ../slides/7/liealgebra.tex \ ../slides/7/kommutator.tex \ ../slides/7/dg.tex \ + ../slides/7/zusammenhang.tex \ ../slides/7/chapter.tex diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 079cf16..7f9f72a 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -17,3 +17,4 @@ \folie{7/liealgebra.tex} \folie{7/kommutator.tex} \folie{7/dg.tex} +\folie{7/zusammenhang.tex} diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 2d7b317..02201d4 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -58,7 +58,7 @@ D_{60^\circ} \begin{column}{0.58\textwidth} \uncover<4->{% \begin{block}{Ansatz} -\vspace{-12pt} +%\vspace{-12pt} \begin{align*} DST &= @@ -101,7 +101,7 @@ c^{-1}&0\\ \vspace{-10pt} \uncover<7->{% \begin{block}{Koeffizientenvergleich} -\vspace{-15pt} +%\vspace{-15pt} \begin{align*} \uncover<8->{ {\color{red} c} diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index 52c8e4a..f3579a3 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -14,7 +14,7 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.4\textwidth} \begin{block}{Drehung um Achsen} -\vspace{-12pt} +%\vspace{-12pt} \begin{align*} \uncover<2->{ D_{x,\alpha} 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/test.tex b/vorlesungen/slides/test.tex index 43da30a..0c13de0 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,20 +4,4 @@ % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{a/dc/prinzip.tex} -%\folie{a/dc/effizient.tex} -%\folie{a/dc/beispiel.tex} - -%\folie{a/ecc/gruppendh.tex} -%\folie{a/ecc/kurve.tex} -%\folie{a/ecc/inverse.tex} -%\folie{a/ecc/operation.tex} -%\folie{a/ecc/quadrieren.tex} -%\folie{a/ecc/oakley.tex} - -%\folie{a/aes/bytes.tex} -%\folie{a/aes/sinverse.tex} -%\folie{a/aes/blocks.tex} -\folie{a/aes/keys.tex} -%\folie{a/aes/runden.tex} - +\folie{7/zusammenhang.tex} |