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diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex
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-%
-% irreduzibel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Irreduzible Markovkette}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{2}
-\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)});
-\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)});
-\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)});
-\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)});
-\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)});
-\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)});
-
-\uncover<-2>{
-\draw (A) -- (B);
-\draw (A) -- (C);
-\draw (A) -- (D);
-\draw (A) -- (E);
-\draw (A) -- (F);
-
-\draw (B) -- (A);
-\draw (B) -- (C);
-\draw (B) -- (D);
-\draw (B) -- (E);
-\draw (B) -- (F);
-
-\draw (C) -- (A);
-\draw (C) -- (B);
-\draw (C) -- (D);
-\draw (C) -- (E);
-\draw (C) -- (F);
-
-\draw (D) -- (A);
-\draw (D) -- (B);
-\draw (D) -- (C);
-\draw (D) -- (E);
-\draw (D) -- (F);
-
-\draw (E) -- (A);
-\draw (E) -- (B);
-\draw (E) -- (C);
-\draw (E) -- (D);
-\draw (E) -- (F);
-
-\draw (F) -- (A);
-\draw (F) -- (B);
-\draw (F) -- (C);
-\draw (F) -- (D);
-\draw (F) -- (E);
-}
-
-\uncover<3->{
-
-\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B);
-\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C);
-\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C);
-\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B);
-\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A);
-\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B);
-
-\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
-	(D) to[out=-90,in=150] (E);
-\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt]
-	(E) -- (D);
-\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt]
-	(E) to[out=-30,in=-150] (F);
-\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
-	(F) -- (E);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
-	(F) to[out=120,in=0] (D);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
-	(D) -- (F);
-}
-
-\fill[color=white] (A) circle[radius=0.2];
-\fill[color=white] (B) circle[radius=0.2];
-\fill[color=white] (C) circle[radius=0.2];
-\fill[color=white] (D) circle[radius=0.2];
-\fill[color=white] (E) circle[radius=0.2];
-\fill[color=white] (F) circle[radius=0.2];
-
-\draw (A) circle[radius=0.2];
-\draw (B) circle[radius=0.2];
-\draw (C) circle[radius=0.2];
-\draw (D) circle[radius=0.2];
-\draw (E) circle[radius=0.2];
-\draw (F) circle[radius=0.2];
-
-\node at (A) {$1$};
-\node at (B) {$2$};
-\node at (C) {$3$};
-\node at (D) {$4$};
-\node at (E) {$5$};
-\node at (F) {$6$};
-
-\end{tikzpicture}
-\end{center}
-\uncover<2->{%
-\begin{block}{Irreduzibel}
-Graph zusammenhängend $\Rightarrow$
-Keine Zerlegung in Teilgraphen möglich
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Reduzibel}
-Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$
-und es gibt keine Übergängen zwischen den Mengen:
-\uncover<4->{%
-\begin{align*}
-P
-&=
-\begin{pmatrix*}[l]
-0  &0.2&0.5&   &   &   \\
-0.3&0.2&0.5&   &   &   \\
-0.7&0.6&0  &   &   &   \\
-   &   &   &0  &0.3&0.4\\
-   &   &   &0.4&0  &0.6\\
-   &   &   &0.6&0.7&0  
-\end{pmatrix*}
-\end{align*}}%
-\uncover<5->{%
-$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}