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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/9/pf/dreieck.tex | 19 | ||||
| -rw-r--r-- | vorlesungen/slides/9/pf/trennung.tex | 99 | ||||
| -rw-r--r-- | vorlesungen/slides/9/pf/vergleich.tex | 113 | ||||
| -rw-r--r-- | vorlesungen/slides/9/pf/vergleich3d.tex | 26 |
4 files changed, 257 insertions, 0 deletions
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 |