6 files changed, 265 insertions, 0 deletions
diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc
index a64a09a..e23cf09 100644
--- a/vorlesungen/slides/9/Makefile.inc
+++ b/vorlesungen/slides/9/Makefile.inc
@@ -12,5 +12,9 @@ chapter9 =								\
 	../slides/9/pf.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/chapter.tex
 
diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex
index ded0176..595dec9 100644
--- a/vorlesungen/slides/9/chapter.tex
+++ b/vorlesungen/slides/9/chapter.tex
@@ -13,5 +13,9 @@
 
 \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}
 
 
diff --git a/vorlesungen/slides/9/pf/dreieck.tex b/vorlesungen/slides/9/pf/dreieck.tex
new file mode 100644
index 0000000..ba26e9e
--- /dev/null
+++ b/vorlesungen/slides/9/pf/dreieck.tex
@@ -0,0 +1,19 @@
+%
+% 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}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\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