diff options
Diffstat (limited to 'vorlesungen')
| -rw-r--r-- | vorlesungen/slides/4/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/qundr.tex | 138 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 1 |
4 files changed, 141 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 88ae3bb..6616f56 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -20,6 +20,7 @@ chapter4 = \ ../slides/4/charakteristik.tex \ ../slides/4/char2.tex \ ../slides/4/frobenius.tex \ + ../slides/4/qundr.tex \ ../slides/4/alpha.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 8c9a30b..6872018 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -19,3 +19,4 @@ \folie{4/charakteristik.tex} \folie{4/char2.tex} \folie{4/frobenius.tex} +\folie{4/qundr.tex} 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/test.tex b/vorlesungen/slides/test.tex index 466256b..99f9cda 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -17,3 +17,4 @@ \folie{4/char2.tex} \folie{4/frobenius.tex} +\folie{4/qundr.tex} |