Merge pull request #1 from AndreasFMueller/master

update

1 files changed, 213 insertions, 0 deletions

diff --git a/vorlesungen/slides/8/laplace.tex b/vorlesungen/slides/8/laplace.tex
new file mode 100644
index 0000000..a1c364d
--- /dev/null
+++ b/vorlesungen/slides/8/laplace.tex
@@ -0,0 +1,213 @@
+%
+% laplace.tex
+%
+% (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{Laplace-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D);
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A);
+
+\uncover<2-4>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<3-7>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<4-13>{
+\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<5->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<6-8>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<7-10>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<8->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<9->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<10-11>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<11->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<12->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<13-14>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\uncover<14->{
+\draw[<->,color=darkgreen,line width=8pt,shorten <= 0cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<15->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\fill[color=red] (A) circle[radius=0.3];
+\fill[color=red!50] (B) circle[radius=0.3];
+\fill[color=white] (C) circle[radius=0.3];
+\fill[color=blue!50] (D) circle[radius=0.3];
+\fill[color=blue] (E) circle[radius=0.3];
+
+\draw (A) circle[radius=0.3];
+\draw (B) circle[radius=0.3];
+\draw (C) circle[radius=0.3];
+\draw (D) circle[radius=0.3];
+\draw (E) circle[radius=0.3];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+
+\end{tikzpicture}
+\end{center}
+\uncover<16->{%
+\begin{block}{Definition}
+Laplace-Matrix
+\[
+L(G) = D(G) - A(G)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{align*}
+f
+&=
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\frac{df}{dt}
+&=
+-\kappa
+\begin{pmatrix*}[r]
+\only<1>{\phantom{-0}}
+	\only<2>{\phantom{-}1}
+	\only<3>{\phantom{-}2}
+	\only<4->{\phantom{-}3}
+	&\only<1>{\phantom{-0}}\only<2->{{-1}}%
+	&\only<-2>{\phantom{-0}}\only<3->{{-1}}%
+	&\uncover<16->{ 0}
+	&\only<-3>{\phantom{-0}}\only<4->{{-1}}\\
+\only<-4>{\phantom{-0}}\only<5->{{-1}}
+	&\only<-4>{\phantom{-0}}
+	\only<5>{\phantom{-}1}
+	\only<6>{\phantom{-}2}
+	\only<7->{\phantom{-}3}
+	&\only<-5>{\phantom{-0}}\only<6->{{-1}}
+	&\only<-6>{\phantom{-0}}\only<7->{{-1}}
+	&\uncover<16->{ 0}\\
+\only<-7>{\phantom{-0}}\only<8->{{-1}}
+	&\only<-8>{\phantom{-0}}\only<9->{{-1}}
+	&\only<-7>{\phantom{-0}}
+	\only<8>{\phantom{-}1}
+	\only<9>{\phantom{-}2}
+	\only<10->{\phantom{-}3}
+	&\only<-9>{\phantom{-0}}\only<10->{{-1}}
+	&\uncover<16->{ 0}\\
+\uncover<16->{ 0}
+	&\only<-10>{\phantom{-0}}\only<11->{{-1}}
+	&\only<-11>{\phantom{-0}}\only<12->{{-1}}
+	&\only<-10>{\phantom{-0}}
+	\only<11>{\phantom{-}1}
+	\only<12>{\phantom{-}2}
+	\only<13->{\phantom{-}3}
+	&\only<-12>{\phantom{-0}}\only<13->{{-1}}\\
+\only<-13>{\phantom{-0}}\only<14->{{-1}}
+	&\uncover<16->{ 0}
+	&\uncover<16->{ 0}
+	&\only<-14>{\phantom{-0}}\only<15->{{-1}}
+	&\only<-13>{\phantom{-0}}
+	\only<14>{\phantom{-}1}
+	\only<15->{\phantom{-}2}
+\end{pmatrix*}
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\uncover<17->{
+&=
+-\kappa L f}
+\end{align*}
+\vspace{-20pt}
+\uncover<18->{%
+\begin{block}{Rekonstruktion}
+Der Graph lässt sich aus $L$ rekonstruieren
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+
+\egroup
+
+