23_mean_cancellation.tex

Paper X: Universal Residual Seeds and Conditional Occupied-System Completion

23_mean_cancellation.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!55,line width=.45pt] (3.35,2.45) -- (3.013173,3.263173);
4 \path[draw,fill=white] (3.35,2.45) circle (0.1);
5 \node[font=\small,align=center,font=\scriptsize] at (3.84,2.45) {$(0,1)$};
6 \draw[->,black!55,line width=.45pt] (3.013173,3.263173) -- (2.2,3.6);
7 \path[draw,fill=black] (3.013173,3.263173) circle (0.1);
8 \node[font=\small,align=center,font=\scriptsize] at (3.359655,3.609655) {$(4,0)$};
9 \draw[->,black!55,line width=.45pt] (2.2,3.6) -- (1.386827,3.263173);
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17 \node[font=\small,align=center,font=\scriptsize] at (0.56,2.45) {$(0,4)$};
18 \draw[->,black!55,line width=.45pt] (1.386827,1.636827) -- (2.2,1.3);
19 \path[draw,fill=black] (1.386827,1.636827) circle (0.1);
20 \node[font=\small,align=center,font=\scriptsize] at (1.040345,1.290345) {$(1,0)$};
21 \draw[->,black!55,line width=.45pt] (2.2,1.3) -- (3.013173,1.636827);
22 \path[draw,fill=white] (2.2,1.3) circle (0.1);
23 \node[font=\small,align=center,font=\scriptsize] at (2.2,0.81) {$(0,2)$};
24 \draw[->,black!55,line width=.45pt] (3.013173,1.636827) -- (3.35,2.45);
25 \path[draw,fill=black] (3.013173,1.636827) circle (0.1);
26 \node[font=\small,align=center,font=\scriptsize] at (3.359655,1.290345) {$(3,0)$};
27 \node[font=\small,align=center,font=\scriptsize] at (2.2,2.45) {sum $=0$};
28 \draw[->,black!55,line width=.45pt] (8,2.45) -- (7.663173,3.263173);
29 \path[draw,fill=black] (8,2.45) circle (0.1);
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31 \draw[->,black!55,line width=.45pt] (7.663173,3.263173) -- (6.85,3.6);
32 \path[draw,fill=white] (7.663173,3.263173) circle (0.1);
33 \node[font=\small,align=center,font=\scriptsize] at (8.009655,3.609655) {$(4,2)$};
34 \draw[->,black!55,line width=.45pt] (6.85,3.6) -- (6.036827,3.263173);
35 \path[draw,fill=black] (6.85,3.6) circle (0.1);
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37 \draw[->,black!55,line width=.45pt] (6.036827,3.263173) -- (5.7,2.45);
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40 \draw[->,black!55,line width=.45pt] (5.7,2.45) -- (6.036827,1.636827);
41 \path[draw,fill=black] (5.7,2.45) circle (0.1);
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43 \draw[->,black!55,line width=.45pt] (6.036827,1.636827) -- (6.85,1.3);
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46 \draw[->,black!55,line width=.45pt] (6.85,1.3) -- (7.663173,1.636827);
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51 \node[font=\small,align=center,font=\scriptsize] at (8.009655,1.290345) {$(3,4)$};
52 \node[font=\small,align=center,font=\scriptsize] at (6.85,2.45) {sum $=0$};
53 \draw[->,black!55,line width=.45pt] (12.65,2.45) -- (12.313173,3.263173);
54 \path[draw,fill=white] (12.65,2.45) circle (0.1);
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56 \draw[->,black!55,line width=.45pt] (12.313173,3.263173) -- (11.5,3.6);
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61 \node[font=\small,align=center,font=\scriptsize] at (11.5,4.09) {$(3,1)$};
62 \draw[->,black!55,line width=.45pt] (10.686827,3.263173) -- (10.35,2.45);
63 \path[draw,fill=black] (10.686827,3.263173) circle (0.1);
64 \node[font=\small,align=center,font=\scriptsize] at (10.340345,3.609655) {$(4,1)$};
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67 \node[font=\small,align=center,font=\scriptsize] at (9.86,2.45) {$(4,3)$};
68 \draw[->,black!55,line width=.45pt] (10.686827,1.636827) -- (11.5,1.3);
69 \path[draw,fill=black] (10.686827,1.636827) circle (0.1);
70 \node[font=\small,align=center,font=\scriptsize] at (10.340345,1.290345) {$(2,3)$};
71 \draw[->,black!55,line width=.45pt] (11.5,1.3) -- (12.313173,1.636827);
72 \path[draw,fill=white] (11.5,1.3) circle (0.1);
73 \node[font=\small,align=center,font=\scriptsize] at (11.5,0.81) {$(2,4)$};
74 \draw[->,black!55,line width=.45pt] (12.313173,1.636827) -- (12.65,2.45);
75 \path[draw,fill=black] (12.313173,1.636827) circle (0.1);
76 \node[font=\small,align=center,font=\scriptsize] at (12.659655,1.290345) {$(1,4)$};
77 \node[font=\small,align=center,font=\scriptsize] at (11.5,2.45) {sum $=0$};
78 \node[font=\small,align=center,font=\footnotesize] at (7,0.1) {Multiply by $v=2u$. Its norm is $2$, so $\chi(N(vz))=-\chi(N(z))$.};
79 \node[font=\small,align=center,font=\scriptsize] at (7,-0.4) {Filled node: $+1$. Open node: $-1$. The omitted zero element has weight $0$.};
80 \end{tikzpicture}

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