summability.tex

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

summability.tex

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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) -- (0.8,3.85);
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) -- (2.925,3.85);
7 \draw[] (2.925,0.59) -- (2.925,0.71);
8 \node[font=\small,align=center,font=\scriptsize] at (2.925,0.39) {4};
9 \draw[black!13,line width=.25pt] (5.05,0.65) -- (5.05,3.85);
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11 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.39) {8};
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21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,1.564286) {1};
22 \draw[black!13,line width=.25pt] (0.8,2.478571) -- (9.3,2.478571);
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24 \draw[black!13,line width=.25pt] (0.8,3.392857) -- (9.3,3.392857);
25 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.392857) {3};
26 \draw[->,line width=.6pt] (0.8,0.65) -- (9.42,0.65);
27 \draw[->,line width=.6pt] (0.8,0.65) -- (0.8,3.97);
28 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.1) {partial-sum cutoff $N$};
29 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,4.25) {sum of defect norms};
30 \draw[line width=1.1pt] plot coordinates {(1.33125,1.564286) (1.8625,2.021429) (2.39375,2.32619) (2.925,2.554762) (3.45625,2.737619) (3.9875,2.89) (4.51875,3.020612) (5.05,3.134898) (5.58125,3.236485) (6.1125,3.327914) (6.64375,3.411031) (7.175,3.487221) (7.70625,3.557551) (8.2375,3.622857) (8.76875,3.683809) (9.3,3.740952)};
31 \draw[dashed,line width=.9pt] plot coordinates {(1.33125,1.564286) (1.8625,1.792857) (2.39375,1.894444) (2.925,1.951587) (3.45625,1.988159) (3.9875,2.013556) (4.51875,2.032214) (5.05,2.0465) (5.58125,2.057788) (6.1125,2.06693) (6.64375,2.074487) (7.175,2.080836) (7.70625,2.086246) (8.2375,2.09091) (8.76875,2.094974) (9.3,2.098545)};
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33 \node[font=\small,align=center,font=\small] at (11.7,3.95) {solid: $\sum_{n\le N}n^{-1}$};
34 \node[font=\small,align=center,font=\small] at (11.7,2.95) {dashed: $\sum_{n\le N}n^{-2}$};
35 \node[font=\small,align=center,font=\small] at (11.7,1.95) {dotted: $\frac12\sum_{n\le N}n^{-2}$};
36 \node[font=\small,align=center,font=\scriptsize] at (11.7,0.55) {The infinite-series conclusion\\uses the exact formulas.};
37 \end{tikzpicture}

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