resolvent.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) -- (0.8,3.9);
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] (2.925,0.65) -- (2.925,3.9);
7 \draw[] (2.925,0.59) -- (2.925,0.71);
8 \node[font=\small,align=center,font=\scriptsize] at (2.925,0.39) {1};
9 \draw[black!13,line width=.25pt] (5.05,0.65) -- (5.05,3.9);
10 \draw[] (5.05,0.59) -- (5.05,0.71);
11 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.39) {2};
12 \draw[black!13,line width=.25pt] (7.175,0.65) -- (7.175,3.9);
13 \draw[] (7.175,0.59) -- (7.175,0.71);
14 \node[font=\small,align=center,font=\scriptsize] at (7.175,0.39) {3};
15 \draw[black!13,line width=.25pt] (9.3,0.65) -- (9.3,3.9);
16 \draw[] (9.3,0.59) -- (9.3,0.71);
17 \node[font=\small,align=center,font=\scriptsize] at (9.3,0.39) {4};
18 \draw[black!13,line width=.25pt] (0.8,0.65) -- (9.3,0.65);
19 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,0.65) {0};
20 \draw[black!13,line width=.25pt] (0.8,1.733333) -- (9.3,1.733333);
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,1.733333) {1};
22 \draw[black!13,line width=.25pt] (0.8,2.816667) -- (9.3,2.816667);
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.816667) {2};
24 \draw[black!13,line width=.25pt] (0.8,3.9) -- (9.3,3.9);
25 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.9) {3};
26 \draw[->,line width=.6pt] (0.8,0.65) -- (9.42,0.65);
27 \draw[->,line width=.6pt] (0.8,0.65) -- (0.8,4.02);
28 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.1) {$z>0$};
29 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,4.3) {resolvent coefficient};
30 \draw[line width=1.1pt] plot coordinates {(1.225,3.604545) (1.30575,3.167942) (1.3865,2.850551) (1.46725,2.609135) (1.548,2.419126) (1.62875,2.265528) (1.7095,2.138668) (1.79025,2.032029) (1.871,1.941056) (1.95175,1.862472) (2.0325,1.793856) (2.11325,1.733384) (2.194,1.679652) (2.27475,1.631564) (2.3555,1.588249) (2.43625,1.549011) (2.517,1.513281) (2.59775,1.480594) (2.6785,1.450563) (2.75925,1.422866) (2.84,1.397232) (2.92075,1.373428) (3.0015,1.351259) (3.08225,1.330554) (3.163,1.311168) (3.24375,1.292972) (3.3245,1.275856) (3.40525,1.259723) (3.486,1.244486) (3.56675,1.230069) (3.6475,1.216405) (3.72825,1.203433) (3.809,1.1911) (3.88975,1.179359) (3.9705,1.168164) (4.05125,1.157477) (4.132,1.147263) (4.21275,1.137489) (4.2935,1.128127) (4.37425,1.119149) (4.455,1.110532) (4.53575,1.102252) (4.6165,1.09429) (4.69725,1.086627) (4.778,1.079245) (4.85875,1.072129) (4.9395,1.065263) (5.02025,1.058635) (5.101,1.052231) (5.18175,1.04604) (5.2625,1.04005) (5.34325,1.034253) (5.424,1.028637) (5.50475,1.023195) (5.5855,1.017918) (5.66625,1.012798) (5.747,1.007829) (5.82775,1.003002) (5.9085,0.998313) (5.98925,0.993755) (6.07,0.989322) (6.15075,0.985009) (6.2315,0.980811) (6.31225,0.976723) (6.393,0.972741) (6.47375,0.96886) (6.5545,0.965077) (6.63525,0.961388) (6.716,0.957789) (6.79675,0.954277) (6.8775,0.950848) (6.95825,0.9475) (7.039,0.944229) (7.11975,0.941034) (7.2005,0.93791) (7.28125,0.934856) (7.362,0.93187) (7.44275,0.928948) (7.5235,0.92609) (7.60425,0.923292) (7.685,0.920553) (7.76575,0.91787) (7.8465,0.915243) (7.92725,0.912669) (8.008,0.910147) (8.08875,0.907675) (8.1695,0.905251) (8.25025,0.902875) (8.331,0.900544) (8.41175,0.898257) (8.4925,0.896014) (8.57325,0.893812) (8.654,0.891651) (8.73475,0.889529) (8.8155,0.887446) (8.89625,0.8854) (8.977,0.88339) (9.05775,0.881416) (9.1385,0.879475) (9.21925,0.877569) (9.3,0.875694)};
31 \draw[dashed,line width=.8pt] plot coordinates {(1.225,1.552778) (1.30575,1.525067) (1.3865,1.499007) (1.46725,1.474455) (1.548,1.451282) (1.62875,1.429376) (1.7095,1.408637) (1.79025,1.388972) (1.871,1.370301) (1.95175,1.352551) (2.0325,1.335654) (2.11325,1.319551) (2.194,1.304187) (2.27475,1.289512) (2.3555,1.275481) (2.43625,1.262053) (2.517,1.249189) (2.59775,1.236854) (2.6785,1.225018) (2.75925,1.213649) (2.84,1.202721) (2.92075,1.192209) (3.0015,1.182089) (3.08225,1.17234) (3.163,1.162942) (3.24375,1.153876) (3.3245,1.145125) (3.40525,1.136673) (3.486,1.128504) (3.56675,1.120605) (3.6475,1.112963) (3.72825,1.105565) (3.809,1.0984) (3.88975,1.091456) (3.9705,1.084724) (4.05125,1.078195) (4.132,1.071859) (4.21275,1.065707) (4.2935,1.059733) (4.37425,1.053927) (4.455,1.048284) (4.53575,1.042797) (4.6165,1.037458) (4.69725,1.032263) (4.778,1.027205) (4.85875,1.022279) (4.9395,1.017481) (5.02025,1.012804) (5.101,1.008245) (5.18175,1.003799) (5.2625,0.999462) (5.34325,0.995231) (5.424,0.9911) (5.50475,0.987067) (5.5855,0.983128) (5.66625,0.979281) (5.747,0.975521) (5.82775,0.971846) (5.9085,0.968253) (5.98925,0.964739) (6.07,0.961303) (6.15075,0.95794) (6.2315,0.954649) (6.31225,0.951428) (6.393,0.948275) (6.47375,0.945186) (6.5545,0.942161) (6.63525,0.939197) (6.716,0.936293) (6.79675,0.933447) (6.8775,0.930656) (6.95825,0.92792) (7.039,0.925237) (7.11975,0.922605) (7.2005,0.920023) (7.28125,0.91749) (7.362,0.915003) (7.44275,0.912563) (7.5235,0.910167) (7.60425,0.907814) (7.685,0.905503) (7.76575,0.903234) (7.8465,0.901004) (7.92725,0.898813) (8.008,0.896661) (8.08875,0.894545) (8.1695,0.892465) (8.25025,0.89042) (8.331,0.88841) (8.41175,0.886432) (8.4925,0.884488) (8.57325,0.882575) (8.654,0.880693) (8.73475,0.878841) (8.8155,0.877019) (8.89625,0.875225) (8.977,0.87346) (9.05775,0.871722) (9.1385,0.870011) (9.21925,0.868326) (9.3,0.866667)};
32 \node[font=\small,align=center] at (11.7,3.7) {exact: $\dfrac{z+1}{z(z+2)}$};
33 \node[font=\small,align=center] at (11.7,2.5) {omit hidden state: $\dfrac1{z+1}$};
34 \node[font=\small,align=center,font=\small] at (11.7,1) {The pole at zero survives\\through hidden feedback.};
35 \end{tikzpicture}

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