Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \node[font=\small,align=center,font=\small] at (2.18,4.3) {$A_f$};
4 \path[draw=black!30,fill=black!8] (0.8,2.68) rectangle (1.72,3.6);
5 \node[font=\small,align=center,font=\small] at (1.26,3.14) {$2$};
6 \path[draw=black!30,fill=black!8] (1.72,2.68) rectangle (2.64,3.6);
7 \node[font=\small,align=center,font=\small] at (2.18,3.14) {$1$};
8 \path[draw=black!30,fill=white] (2.64,2.68) rectangle (3.56,3.6);
9 \node[font=\small,align=center,font=\small] at (3.1,3.14) {$0$};
10 \path[draw=black!30,fill=white] (0.8,1.76) rectangle (1.72,2.68);
11 \node[font=\small,align=center,font=\small] at (1.26,2.22) {$0$};
12 \path[draw=black!30,fill=black!8] (1.72,1.76) rectangle (2.64,2.68);
13 \node[font=\small,align=center,font=\small] at (2.18,2.22) {$2$};
14 \path[draw=black!30,fill=white] (2.64,1.76) rectangle (3.56,2.68);
15 \node[font=\small,align=center,font=\small] at (3.1,2.22) {$0$};
16 \path[draw=black!30,fill=white] (0.8,0.84) rectangle (1.72,1.76);
17 \node[font=\small,align=center,font=\small] at (1.26,1.3) {$0$};
18 \path[draw=black!30,fill=white] (1.72,0.84) rectangle (2.64,1.76);
19 \node[font=\small,align=center,font=\small] at (2.18,1.3) {$0$};
20 \path[draw=black!30,fill=black!8] (2.64,0.84) rectangle (3.56,1.76);
21 \node[font=\small,align=center,font=\small] at (3.1,1.3) {$9$};
22 \node[font=\small,align=center,font=\small] at (6.78,4.3) {$F(A_f)$ for $F(z)=(1+z)^{-1}$};
23 \path[draw=black!30,fill=black!8] (5.4,2.68) rectangle (6.32,3.6);
24 \node[font=\small,align=center,font=\small] at (5.86,3.14) {$\frac{1}{3}$};
25 \path[draw=black!30,fill=black!8] (6.32,2.68) rectangle (7.24,3.6);
26 \node[font=\small,align=center,font=\small] at (6.78,3.14) {$- \frac{1}{9}$};
27 \path[draw=black!30,fill=white] (7.24,2.68) rectangle (8.16,3.6);
28 \node[font=\small,align=center,font=\small] at (7.7,3.14) {$0$};
29 \path[draw=black!30,fill=white] (5.4,1.76) rectangle (6.32,2.68);
30 \node[font=\small,align=center,font=\small] at (5.86,2.22) {$0$};
31 \path[draw=black!30,fill=black!8] (6.32,1.76) rectangle (7.24,2.68);
32 \node[font=\small,align=center,font=\small] at (6.78,2.22) {$\frac{1}{3}$};
33 \path[draw=black!30,fill=white] (7.24,1.76) rectangle (8.16,2.68);
34 \node[font=\small,align=center,font=\small] at (7.7,2.22) {$0$};
35 \path[draw=black!30,fill=white] (5.4,0.84) rectangle (6.32,1.76);
36 \node[font=\small,align=center,font=\small] at (5.86,1.3) {$0$};
37 \path[draw=black!30,fill=white] (6.32,0.84) rectangle (7.24,1.76);
38 \node[font=\small,align=center,font=\small] at (6.78,1.3) {$0$};
39 \path[draw=black!30,fill=black!8] (7.24,0.84) rectangle (8.16,1.76);
40 \node[font=\small,align=center,font=\small] at (7.7,1.3) {$\frac{1}{10}$};
41 \path[draw=black,dashed,line width=1pt] (5.38,1.73) rectangle (7.26,3.63);
42 \node[font=\small,align=center] at (11.7,3.6) {$F(2)=1/3$};
43 \node[font=\small,align=center] at (11.7,2.65) {$F'(2)=-1/9$};
44 \node[font=\small,align=center,font=\small] at (11.7,1.25) {The dashed block is\\$F(A_c)$, not just $F(2)I$.};
45 \end{tikzpicture}

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