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diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex new file mode 100644 index 0000000..974f238 --- /dev/null +++ b/vorlesungen/slides/5/bloecke.tex @@ -0,0 +1,141 @@ +% +% bloecke.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\sx{1} +\def\sy{0.1} +\def\block#1#2{ + \fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2}); +} +\def\kreuz#1{ + \draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1}); + \draw[color=white,line width=0.1pt] (#1,0)--(#1,-60); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (A) at ({1*\sx},{20*\sy}); +\coordinate (B) at ({2*\sx},{(20+15)*\sy}); +\coordinate (C) at ({3*\sx},{(20+15+10)*\sy}); +\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy}); +\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy}); +\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy}); +\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F) + -- ({6*\sx},0) -- cycle; + +\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle; +\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle; +\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle; +\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle; +\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle; +\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle; + +\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F); + +\draw[color=gray] (A) -- (0,{20*\sy}); +\draw[color=gray] (B) -- (0,{(20+15)*\sy}); +\draw[color=gray] (C) -- (0,{(20+15+10)*\sy}); +\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy}); +\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy}); +\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy}); + +\node at ({0.5*\sx},{0.5*20*\sy}) + [right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$}; +\node at ({1.5*\sx},{0.5*(20+20+15)*\sy}) + [right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$}; +\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$}; +\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$}; +\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$}; +\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$}; + +\fill (A) circle[radius=0.08]; +\fill (B) circle[radius=0.08]; +\fill (C) circle[radius=0.08]; +\fill (D) circle[radius=0.08]; +\fill (E) circle[radius=0.08]; +\fill (F) circle[radius=0.08]; + +\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}]; +\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}]; + +\foreach \x in {0,1,...,6}{ + \draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05); + \node at ({\sx*\x},{-0.1}) [below] {$\x$}; +} + +\node at (0,{60*\sy}) [left] {\llap{$n$}}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.43\textwidth} +\vspace{-10pt} +\begin{center} +\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}} +k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\ +\hline +0& 0& &\\ +1& 20& d_1-d_2&5\\ +2& 15& d_2-d_3&5\\ +3& 10& d_3-d_4&2\\ +4& 8& d_4-d_5&3\\ +5& 5& d_5-d_6&3\\ +6& 2& d_6 &2\\ +\end{tabular} +\end{center} +\vspace{-13pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.05] +\fill[color=gray!40] (0,0) rectangle (60,-60); +\node[color=white] at (30,-30) [scale=6] {$A$}; +\kreuz{5} +\kreuz{15} +\kreuz{21} +\kreuz{33} +\kreuz{48} +\node at (0,-2.5) [left] {$k=1$}; +\node at (60,-2.5) [right] {$5$ Blöcke}; +\node at (0,-10) [left] {$k=2$}; +\node at (60,-10) [right] {$5$ Blöcke}; +\node at (0,-18) [left] {$k=3$}; +\node at (60,-18) [right] {$2$ Blöcke}; +\node at (0,-27) [left] {$k=4$}; +\node at (60,-27) [right] {$3$ Blöcke}; +\node at (0,-40.5) [left] {$k=5$}; +\node at (60,-40.5) [right] {$3$ Blöcke}; +\node at (0,-54) [left] {$k=6$}; +\node at (60,-54) [right] {$2$ Blöcke}; +\block{0}{1} +\block{1}{1} +\block{2}{1} +\block{3}{1} +\block{4}{1} +\block{5}{2} +\block{7}{2} +\block{9}{2} +\block{11}{2} +\block{13}{2} +\block{15}{3} +\block{18}{3} +\block{21}{4} +\block{25}{4} +\block{29}{4} +\block{33}{5} +\block{38}{5} +\block{43}{5} +\block{48}{6} +\block{54}{6} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup |