Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round]
2 \node[] at (1.05000,3.75000) {$\sum_k4^{jk}2^{rk}\pmod7$};
3 \draw[fill=black!12,draw=black!45,line width=.35pt,] (0.00000,2.50000) rectangle ++(0.70000,0.70000);
4 \node[] at (0.35000,2.85000) {$3$};
5 \draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,2.50000) rectangle ++(0.70000,0.70000);
6 \node[] at (1.05000,2.85000) {$0$};
7 \draw[fill=black!0,draw=black!45,line width=.35pt,] (1.40000,2.50000) rectangle ++(0.70000,0.70000);
8 \node[] at (1.75000,2.85000) {$0$};
9 \draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.70000,0.70000);
10 \node[] at (0.35000,2.15000) {$0$};
11 \draw[fill=black!12,draw=black!45,line width=.35pt,] (0.70000,1.80000) rectangle ++(0.70000,0.70000);
12 \node[] at (1.05000,2.15000) {$3$};
13 \draw[fill=black!0,draw=black!45,line width=.35pt,] (1.40000,1.80000) rectangle ++(0.70000,0.70000);
14 \node[] at (1.75000,2.15000) {$0$};
15 \draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.10000) rectangle ++(0.70000,0.70000);
16 \node[] at (0.35000,1.45000) {$0$};
17 \draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000);
18 \node[] at (1.05000,1.45000) {$0$};
19 \draw[fill=black!12,draw=black!45,line width=.35pt,] (1.40000,1.10000) rectangle ++(0.70000,0.70000);
20 \node[] at (1.75000,1.45000) {$3$};
21 \node[] at (3.70000,2.05000) {$\times\,5$};
22 \node[] at (6.05000,3.75000) {$P_j(z^r):\ 5=3^{-1}\pmod7$};
23 \draw[fill=black!12,draw=black!45,line width=.35pt,] (5.00000,2.50000) rectangle ++(0.70000,0.70000);
24 \node[] at (5.35000,2.85000) {$1$};
25 \draw[fill=black!0,draw=black!45,line width=.35pt,] (5.70000,2.50000) rectangle ++(0.70000,0.70000);
26 \node[] at (6.05000,2.85000) {$0$};
27 \draw[fill=black!0,draw=black!45,line width=.35pt,] (6.40000,2.50000) rectangle ++(0.70000,0.70000);
28 \node[] at (6.75000,2.85000) {$0$};
29 \draw[fill=black!0,draw=black!45,line width=.35pt,] (5.00000,1.80000) rectangle ++(0.70000,0.70000);
30 \node[] at (5.35000,2.15000) {$0$};
31 \draw[fill=black!12,draw=black!45,line width=.35pt,] (5.70000,1.80000) rectangle ++(0.70000,0.70000);
32 \node[] at (6.05000,2.15000) {$1$};
33 \draw[fill=black!0,draw=black!45,line width=.35pt,] (6.40000,1.80000) rectangle ++(0.70000,0.70000);
34 \node[] at (6.75000,2.15000) {$0$};
35 \draw[fill=black!0,draw=black!45,line width=.35pt,] (5.00000,1.10000) rectangle ++(0.70000,0.70000);
36 \node[] at (5.35000,1.45000) {$0$};
37 \draw[fill=black!0,draw=black!45,line width=.35pt,] (5.70000,1.10000) rectangle ++(0.70000,0.70000);
38 \node[] at (6.05000,1.45000) {$0$};
39 \draw[fill=black!12,draw=black!45,line width=.35pt,] (6.40000,1.10000) rectangle ++(0.70000,0.70000);
40 \node[] at (6.75000,1.45000) {$1$};
41 \node[] at (10.15000,3.05000) {$\omega=2,\quad\omega^3=1$};
42 \node[] at (10.15000,2.15000) {$\omega^{-1}=4,\quad\eta=5$};
43 \node[] at (10.15000,1.25000) {$j,r\in\{0,1,2\}$};
44 \draw[black!35] (0.00000,0.25000) -- (12.20000,0.25000);
45 \node[] at (6.00000,-0.40000) {In $\mathbb F_3\times\mathbb F_3$: order two does not imply orthogonality.};
46 \node[] at (6.00000,-1.15000) {$\omega=(-1,1),\quad\eta=(2,2),\quad \eta(1+\omega)=(0,1)\ne(0,0)$};
47 \end{tikzpicture}

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