jordan.tex

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 \draw[black!13,line width=.25pt] (0.8,0.65) -- (0.8,3.8);
4 \draw[] (0.8,0.59) -- (0.8,0.71);
5 \node[font=\small,align=center,font=\scriptsize] at (0.8,0.39) {0};
6 \draw[black!13,line width=.25pt] (2.216667,0.65) -- (2.216667,3.8);
7 \draw[] (2.216667,0.59) -- (2.216667,0.71);
8 \node[font=\small,align=center,font=\scriptsize] at (2.216667,0.39) {0.5};
9 \draw[black!13,line width=.25pt] (3.633333,0.65) -- (3.633333,3.8);
10 \draw[] (3.633333,0.59) -- (3.633333,0.71);
11 \node[font=\small,align=center,font=\scriptsize] at (3.633333,0.39) {1};
12 \draw[black!13,line width=.25pt] (5.05,0.65) -- (5.05,3.8);
13 \draw[] (5.05,0.59) -- (5.05,0.71);
14 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.39) {1.5};
15 \draw[black!13,line width=.25pt] (6.466667,0.65) -- (6.466667,3.8);
16 \draw[] (6.466667,0.59) -- (6.466667,0.71);
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18 \draw[black!13,line width=.25pt] (7.883333,0.65) -- (7.883333,3.8);
19 \draw[] (7.883333,0.59) -- (7.883333,0.71);
20 \node[font=\small,align=center,font=\scriptsize] at (7.883333,0.39) {2.5};
21 \draw[black!13,line width=.25pt] (9.3,0.65) -- (9.3,3.8);
22 \draw[] (9.3,0.59) -- (9.3,0.71);
23 \node[font=\small,align=center,font=\scriptsize] at (9.3,0.39) {3};
24 \draw[black!13,line width=.25pt] (0.8,0.65) -- (9.3,0.65);
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26 \draw[black!13,line width=.25pt] (0.8,2.15) -- (9.3,2.15);
27 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.15) {0.1};
28 \draw[black!13,line width=.25pt] (0.8,3.65) -- (9.3,3.65);
29 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.65) {0.2};
30 \draw[->,line width=.6pt] (0.8,0.65) -- (9.42,0.65);
31 \draw[->,line width=.6pt] (0.8,0.65) -- (0.8,3.92);
32 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.1) {time $t$};
33 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,4.2) {memory coefficient $t e^{-2t}$};
34 \draw[line width=1.1pt] plot coordinates {(0.8,0.65) (0.885,1.073794) (0.97,1.448228) (1.055,1.777615) (1.14,2.06593) (1.225,2.316841) (1.31,2.533726) (1.395,2.719697) (1.48,2.87762) (1.565,3.01013) (1.65,3.119652) (1.735,3.208414) (1.82,3.278462) (1.905,3.331675) (1.99,3.369776) (2.075,3.394345) (2.16,3.406829) (2.245,3.408551) (2.33,3.400724) (2.415,3.384453) (2.5,3.360748) (2.585,3.330531) (2.67,3.294639) (2.755,3.253838) (2.84,3.20882) (2.925,3.160214) (3.01,3.108592) (3.095,3.054469) (3.18,2.998312) (3.265,2.940541) (3.35,2.881535) (3.435,2.821633) (3.52,2.76114) (3.605,2.700328) (3.69,2.639439) (3.775,2.578689) (3.86,2.518267) (3.945,2.458342) (4.03,2.39906) (4.115,2.34055) (4.2,2.282923) (4.285,2.226275) (4.37,2.170687) (4.455,2.116227) (4.54,2.062953) (4.625,2.010912) (4.71,1.96014) (4.795,1.910666) (4.88,1.862511) (4.965,1.815689) (5.05,1.770209) (5.135,1.726073) (5.22,1.683278) (5.305,1.641818) (5.39,1.601683) (5.475,1.562858) (5.56,1.525329) (5.645,1.489074) (5.73,1.454073) (5.815,1.420304) (5.9,1.387741) (5.985,1.356357) (6.07,1.326128) (6.155,1.297023) (6.24,1.269016) (6.325,1.242076) (6.41,1.216174) (6.495,1.191282) (6.58,1.167368) (6.665,1.144405) (6.75,1.122361) (6.835,1.101208) (6.92,1.080916) (7.005,1.061458) (7.09,1.042805) (7.175,1.024929) (7.26,1.007802) (7.345,0.991399) (7.43,0.975693) (7.515,0.960659) (7.6,0.946271) (7.685,0.932505) (7.77,0.919338) (7.855,0.906746) (7.94,0.894708) (8.025,0.883201) (8.11,0.872204) (8.195,0.861697) (8.28,0.85166) (8.365,0.842075) (8.45,0.832922) (8.535,0.824183) (8.62,0.815842) (8.705,0.807882) (8.79,0.800286) (8.875,0.79304) (8.96,0.786128) (9.045,0.779536) (9.13,0.77325) (9.215,0.767257) (9.3,0.761544)};
35 \path[fill=black] (2.216667,3.409096) circle (0.065);
36 \node[font=\small,align=center] at (11.7,3.9) {$J=-2I+N,\quad N^2=0$};
37 \node[font=\small,align=center] at (11.7,2.8) {$e^{tJ}=e^{-2t}(I+tN)$};
38 \node[font=\small,align=center] at (11.7,1.35) {$\widehat K(z)=\dfrac1{(z+2)^2}$};
39 \node[font=\small,align=center,font=\scriptsize] at (11.7,0.4) {A repeated pole carries a jet.};
40 \end{tikzpicture}

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