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author | LordMcFungus <mceagle117@gmail.com> | 2021-03-22 18:05:11 +0100 |
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committer | GitHub <noreply@github.com> | 2021-03-22 18:05:11 +0100 |
commit | 76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7 (patch) | |
tree | 11b2d41955ee4bfa0ae5873307c143f6b4d55d26 /vorlesungen/slides/8/pfade/langepfade.tex | |
parent | more chapter structure (diff) | |
parent | add title image (diff) | |
download | SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.tar.gz SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.zip |
Merge pull request #1 from AndreasFMueller/master
update
Diffstat (limited to 'vorlesungen/slides/8/pfade/langepfade.tex')
-rw-r--r-- | vorlesungen/slides/8/pfade/langepfade.tex | 59 |
1 files changed, 59 insertions, 0 deletions
diff --git a/vorlesungen/slides/8/pfade/langepfade.tex b/vorlesungen/slides/8/pfade/langepfade.tex new file mode 100644 index 0000000..8c0dd0d --- /dev/null +++ b/vorlesungen/slides/8/pfade/langepfade.tex @@ -0,0 +1,59 @@ +% +% langepfade.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\definecolor{darkred}{rgb}{0.5,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame} +\frametitle{Wieviele Pfade der Länge $k$?} +\begin{definition} +Anzahl Pfade der Länge $k$ zwischen zwei Knoten +\[ +a_{{\color{darkred}i}{\color{blue}j}}^{(k)} += +\#\{\text{Pfade der Länge $k$ von $\color{blue}j$ nach $\color{darkred}i$}\}, +\qquad +A^{(k)} += +\left( +a_{{\color{darkred}i}{\color{blue}j}}^{(k)} +\right) +\] +\end{definition} +\uncover<2->{ +{\usebeamercolor[fg]{title}Spezialfall:} $A^{(1)}=A$. +} + +\uncover<3->{ +\begin{block}{Rekursionsformel} +\vspace{-25pt} +\begin{align*} +a_{{\color{darkred}i}{\color{blue}{\color{blue}j}}}^{(k)} +&\uncover<4->{= +\sum_{{\color{darkgreen}l}=1}^n +\#\{\text{Pfade der Länge $1$ von $\color{darkgreen}l$ nach $\color{darkred}i$}\}} +\\[-11pt] +&\uncover<4->{\qquad\qquad\times +\#\{\text{Pfade der Länge $k-1$ von $\color{blue}j$ nach $\color{darkgreen}l$}\}} +\\ +&\uncover<5->{= +\sum_{{\color{darkgreen}l}=1}^n +a_{{\color{darkred}i}{\color{darkgreen}l}}^{(1)} +\cdot +a_{{\color{darkgreen}l}{\color{blue}j}}^{(k-1)}} +\\ +\uncover<6->{ +\Rightarrow\qquad +A^{(k)}} +&\uncover<6->{= +A\;A^{(k-1)}} +\uncover<7->{ +\qquad\Rightarrow\qquad +A^{(k)} = A^k} +\end{align*} +\end{block} } + +\end{frame} +\egroup |