From e11b789a18229366389b70dd90bfe2dda4829696 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Mon, 13 Jan 2014 21:33:49 +0100 Subject: [PATCH] zu lange Aufgaben entfernt --- documents/GeoTopo/GeoTopo.pdf | Bin 4720515 -> 4717954 bytes documents/GeoTopo/Kapitel3-UB.tex | 25 +++++++------ documents/GeoTopo/Kapitel4.tex | 4 +-- documents/GeoTopo/Loesungen.tex | 33 ++++++++++++++---- .../topology-triangle-in-text/Makefile | 31 ++++++++++++++++ .../topology-triangle-in-text/Readme.md | 3 ++ .../topology-oriented-triangle.tex | 19 ++++++++++ 7 files changed, 96 insertions(+), 19 deletions(-) create mode 100644 documents/GeoTopo/figures/topology-triangle-in-text/Makefile create mode 100644 documents/GeoTopo/figures/topology-triangle-in-text/Readme.md create mode 100644 documents/GeoTopo/figures/topology-triangle-in-text/topology-oriented-triangle.tex diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 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zBm{L|iYJ;LbxUKHVLUL~=lZ?CvqOsn+zYMtKhqup2H}R~bfU>&M%L|`$r6U=`w=7# zz|}qd#%o!W((vj;yUmd2Ue8<&1MtUAt0d`hAlj)|WIxM|xqSqAz^P6fg8|ST5|FGa zXBp}~o;Z=Op?Z-p@$9LwIW$kFYGo;qJgferol4_0sPCLb<20;4lGp4gjbYMWD5+K5 zcozoHIE`#iaO0lD_HDSUB)c13R>>wm8D8GQISnNhbT>E_>m=QkI-2n%t$Q{m@~dUO z$I>7UU1hyD(~uZFk`YW7=&7l>H~JG-Uo8b+JQS6^%+43ktKoTxCiffNP7!OR#mpVx zIGV{T>sT(3ylO;(sd*_}pxL(uK7lJuXv6@XVpm^>g1-^OeSjQq#_5Wgf8qZ`{sWG) zBoD9URmFb(&uAw6X|F*7m;iU~z>(drIz9sJ&jIeKk;H_%Ffg|jiZ?RUY+wRWJ~0o1 zZT6BcIhpIEZy$IY`4*Y}hL$#jH~%^291v_qKH|xLmVJ*AeuoP<{|N4!J8*2D1D)?{ z?08_XgFZh7{VWH4H3$7Y$$bRK{q)Iw<;nf+i+uhs6T2Kp^&G_I|BqjFRIEUHtYFxb z017tb89MNq7(e!uU;t9}pEzqe*i|UfH>S)7V25v%-Dq1T^+AF3Eg_Wm{|RRv20h+?x`Gq5n!oZ|B3zxp p8Ce@TINBTOS^Y&e`sQ#Bj(YaV)~}+l@XV}q^l&63!ZIRo{|8o3XORE^ diff --git a/documents/GeoTopo/Kapitel3-UB.tex b/documents/GeoTopo/Kapitel3-UB.tex index 8c5baa4..c8980bd 100644 --- a/documents/GeoTopo/Kapitel3-UB.tex +++ b/documents/GeoTopo/Kapitel3-UB.tex @@ -1,13 +1,16 @@ -\clearpage -\section*{Übungsaufgaben} -\addcontentsline{toc}{section}{Übungsaufgaben} +%\clearpage +%\section*{Übungsaufgaben} +%\addcontentsline{toc}{section}{Übungsaufgaben} -\begin{aufgabe}\label{ub7:aufg1} - Berechnen Sie die Homologiegruppen von $S^1$ und $S^2$, indem Sie - zu $S^1$ bzw. $S^2$ homöomorphe Simplizialkomplexe betrachten. -\end{aufgabe} +%Die Lösung ist zu lang (vgl. Loesungen.tex) +%\begin{aufgabe}\label{ub7:aufg1} +% Berechnen Sie die Homotogiegruppen von $S^1$ und $S^2$, indem Sie +% zu $S^1$ bzw. $S^2$ homöomorphe Simplizialkomplexe betrachten. +%\end{aufgabe} -\begin{aufgabe}\label{ub7:aufg3} - Es sei $G$ eine topologische Gruppe und $e$ ihr neutrales - Element. Man beweise, dass $\pi_1(G,e)$ abelsch ist. -\end{aufgabe} + +% Auch diese Aufgabe ist zu lang... +%\begin{aufgabe}\label{ub7:aufg3} +% Es sei $G$ eine topologische Gruppe und $e$ ihr neutrales +% Element. Man beweise, dass $\pi_1(G,e)$ abelsch ist. +%\end{aufgabe} diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 225c781..368c559 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -34,8 +34,8 @@ aufgestellt. \item Parallelenaxiom: Euklid:\\ Wird eine Gerade so von zwei Geraden geschnitten, dass die Summe der Innenwinkel zwei Rechte ist, dann schneiden sich - diese Geraden auf der Seite dieser Winkel. - \todo[inline]{Bild} + diese Geraden auf der Seite dieser Winkel.\\ + \\ Man mache sich klar, dass das nur dann nicht der Fall ist, wenn beide Geraden parallel sind und senkrecht auf die erste stehen. \end{itemize} diff --git a/documents/GeoTopo/Loesungen.tex b/documents/GeoTopo/Loesungen.tex index c16f102..dbd3839 100644 --- a/documents/GeoTopo/Loesungen.tex +++ b/documents/GeoTopo/Loesungen.tex @@ -184,10 +184,31 @@ \end{enumerate} \end{solution} -\begin{solution}[\ref{ub7:aufg1}] - \todo[inline]{Kommt noch.} -\end{solution} +%Das scheint mir etwas zu lang zu sein... +%\begin{solution}[\ref{ub7:aufg1}] +% \textbf{Beh.:} $H_k = \begin{cases}\mdr &\text{für } k\in \Set{0,1}\\ +% 0 &\text{für } k \geq 2$ +% \newcommand{\triangleSimplizialkomplex}{\mathord{\includegraphics[height=5ex]{figures/triangleSimplizialkomplex.pdf}}} +% \textbf{Bew.:} $S^1$ ist homöomorph zum Simplizialkomplex +% $X = \triangleSimplizialkomplex$, d.~h. dem Rand +% von $\Delta^2$. Es gilt: +% \[X = \Set{\underbrace{v_0, v_1, v_2}_{A_0(X)}, \underbrace{\Delta (v_1, v_2)}_{=: a_0}, \underbrace{\underbrace{\Delta (v_0, v_2)}_{=: a_1}, \underbrace{\Delta(v_0, v_1)}_{=: a_2}}_{A_1(X)}}\] +% Damit folgt: +% \begin{enumerate} +% \item Für $k \geq 2$ ist $C_k(X) \cong 0$, da es in diesen +% Dimensionen keine Simplizes gibt, d.~h. $A_k(X) = \emptyset$ gilt.\\ +% Also: $H_k(X) \cong 0 \; \forall k \geq 2$ +% \item $C_0(X) = \Set{\sum_{i=0}^2 c_i v_i | c_i \in \mdr}$, da +% $A_0(x)$ Basis von $C_0(X)$ ist;\\ +% $C_1(X) = \Set{\sum_{i=0}^2 c_i a_i | c_i \in \mdr}$, da +% $A_1(X)$ Basis von $C_1(X)$ ist. +% \item Für die Randabbildungen $d_i: C_i(X) \rightarrow C_{i-1}(X)$ gilt: +% $d_0 \equiv 0$, $d_1: C_1(X) \rightarrow C_0(X)$ ist definiert durch +% $d_1(a_k) = \sum_{i=0}^1 (-1)^i \partial_i(a_k) = \partial_0 (a_k) - \partial_1(a_k) \; \forall k \in \Set{0,1,2}$ +% \end{enumerate} +%\end{solution} -\begin{solution}[\ref{ub7:aufg3}] - \todo[inline]{Kommt noch.} -\end{solution} +%Auch diese Aufgabe ist zu lang +%\begin{solution}[\ref{ub7:aufg3}] +% +%\end{solution} diff --git a/documents/GeoTopo/figures/topology-triangle-in-text/Makefile b/documents/GeoTopo/figures/topology-triangle-in-text/Makefile new file mode 100644 index 0000000..0c1b58a --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-in-text/Makefile @@ -0,0 +1,31 @@ +SOURCE = topology-oriented-triangle +DELAY = 80 +DENSITY = 300 +WIDTH = 512 + +make: + pdflatex $(SOURCE).tex -output-format=pdf + make clean + +clean: + rm -rf $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot + +gif: + pdfcrop $(SOURCE).pdf + convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif + make clean + +png: + make + make svg + inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png + +transparentGif: + convert $(SOURCE).pdf -transparent white result.gif + make clean + +svg: + #inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg + pdf2svg $(SOURCE).pdf $(SOURCE).svg + # Necessary, as pdf2svg does not always create valid svgs: + inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg diff --git a/documents/GeoTopo/figures/topology-triangle-in-text/Readme.md b/documents/GeoTopo/figures/topology-triangle-in-text/Readme.md new file mode 100644 index 0000000..1de15f1 --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-in-text/Readme.md @@ -0,0 +1,3 @@ +Compiled example +---------------- +![Example](topology-oriented-triangle.png) diff --git a/documents/GeoTopo/figures/topology-triangle-in-text/topology-oriented-triangle.tex b/documents/GeoTopo/figures/topology-triangle-in-text/topology-oriented-triangle.tex new file mode 100644 index 0000000..1ef8c3d --- /dev/null +++ b/documents/GeoTopo/figures/topology-triangle-in-text/topology-oriented-triangle.tex @@ -0,0 +1,19 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{tikz} +\usetikzlibrary{calc,shadings} +\usepackage{pgfplots} + +\begin{document} +\begin{tikzpicture}[scale=0.5] + \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt] + \node (a)[point,label={[label distance=0cm]180:$v_0$}] at (210:1cm) {}; + \node (b)[point,label={[label distance=0cm]0:$v_2$}] at (330:1cm) {}; + \node (c)[point,label={[label distance=-0.1cm]90:$v_1$}] at (90:1cm) {}; + + \draw[very thick] (a) edge node {} (b); + \draw[very thick] (b) edge node {} (c); + \draw[very thick] (c) edge node {} (a); +\end{tikzpicture} +%\newcommand{\triangleSimplizialkomplex}{\mathord{\includegraphics[height=5ex]{triangle.pdf}}} +%Es gilt $x=\triangleSimplizialkomplex$ und +\end{document}