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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-14 07:26:10 +0200 |
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committer | GitHub <noreply@github.com> | 2021-06-14 07:26:10 +0200 |
commit | 114633b43a0f1ebedbc5dfd85f75ede9841f26fd (patch) | |
tree | 18e61c7d69883a1c9b69098b7d36856abaed5c1e /vorlesungen/slides/8/subgraph.tex | |
parent | Delete buch.pdf (diff) | |
parent | Fix references.bib (diff) | |
download | SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.tar.gz SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.zip |
Merge branch 'master' into master
Diffstat (limited to 'vorlesungen/slides/8/subgraph.tex')
-rw-r--r-- | vorlesungen/slides/8/subgraph.tex | 60 |
1 files changed, 60 insertions, 0 deletions
diff --git a/vorlesungen/slides/8/subgraph.tex b/vorlesungen/slides/8/subgraph.tex new file mode 100644 index 0000000..f3005f9 --- /dev/null +++ b/vorlesungen/slides/8/subgraph.tex @@ -0,0 +1,60 @@ +% +% subgraph.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\alpha_{\text{max}}$ eines Untergraphen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$X'$ ein echter Untergraph von $X$ mit Adjazenzmatrix $A'$ und grösstem +Eigenwert $\alpha_{\text{max}}'$ +\[ +\alpha_{\text{max}}' \le \alpha_{\text{max}} +\] +\end{block} +\uncover<2->{$V'$ die Knoten von $X'$} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> +$f'$ der positive Eigenvektor von $A'$ +\item<5-> +Definiere +\[ +g(v) += +\begin{cases} +f'(v) &\qquad v\in V'\\ +0 &\qquad \text{sonst} +\end{cases} +\] +\item<6-> Skalarprodukte: +\begin{align*} +\uncover<7->{\langle f',f'\rangle &= \langle g,g\rangle} +\\ +\uncover<8->{\langle A'f',f'\rangle &\le \langle Ag,g\rangle} +\end{align*} +\item<9-> Vergleich +\[ +\alpha_{\text{max}}' += +\frac{\langle A'f',f'\rangle}{\langle f',f'\rangle} +\uncover<10->{\le +\frac{\langle Ag,g\rangle}{\langle g,g\rangle}} +\uncover<11->{\le +\alpha_{\text{max}}} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup |