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Diffstat (limited to 'vorlesungen/slides/8/laplace.tex')
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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 + + |