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

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round]
2 \node[anchor=east] at (-0.22000,4.13000) {$N=2$};
3 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.00000,3.80000) rectangle ++(0.72000,0.62000);
4 \node[] at (0.36000,4.11000) {$0$};
5 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.72000,3.80000) rectangle ++(0.72000,0.62000);
6 \node[] at (1.08000,4.11000) {$2$};
7 \draw[fill=black!10,draw=black!45,line width=.35pt,] (1.44000,3.80000) rectangle ++(0.72000,0.62000);
8 \node[] at (1.80000,4.11000) {$4$};
9 \draw[fill=black!10,draw=black!45,line width=.35pt,] (2.16000,3.80000) rectangle ++(0.72000,0.62000);
10 \node[] at (2.52000,4.11000) {$6$};
11 \draw[fill=black!10,draw=black!45,line width=.35pt,] (2.88000,3.80000) rectangle ++(0.72000,0.62000);
12 \node[] at (3.24000,4.11000) {$8$};
13 \draw[fill=black!10,draw=black!45,line width=.35pt,] (3.60000,3.80000) rectangle ++(0.72000,0.62000);
14 \node[] at (3.96000,4.11000) {$10$};
15 \draw[fill=black!10,draw=black!45,line width=.35pt,] (4.32000,3.80000) rectangle ++(0.72000,0.62000);
16 \node[] at (4.68000,4.11000) {$12$};
17 \draw[fill=black!10,draw=black!45,line width=.35pt,] (5.04000,3.80000) rectangle ++(0.72000,0.62000);
18 \node[] at (5.40000,4.11000) {$14$};
19 \draw[line width=1.0pt] (0.00000,4.48000) -- (5.76000,4.48000);
20 \node[font=\sffamily\scriptsize] at (2.88000,4.80000) {$r=0$};
21 \draw[fill=black!27,draw=black!45,line width=.35pt,] (5.76000,3.80000) rectangle ++(0.72000,0.62000);
22 \node[] at (6.12000,4.11000) {$1$};
23 \draw[fill=black!27,draw=black!45,line width=.35pt,] (6.48000,3.80000) rectangle ++(0.72000,0.62000);
24 \node[] at (6.84000,4.11000) {$3$};
25 \draw[fill=black!27,draw=black!45,line width=.35pt,] (7.20000,3.80000) rectangle ++(0.72000,0.62000);
26 \node[] at (7.56000,4.11000) {$5$};
27 \draw[fill=black!27,draw=black!45,line width=.35pt,] (7.92000,3.80000) rectangle ++(0.72000,0.62000);
28 \node[] at (8.28000,4.11000) {$7$};
29 \draw[fill=black!27,draw=black!45,line width=.35pt,] (8.64000,3.80000) rectangle ++(0.72000,0.62000);
30 \node[] at (9.00000,4.11000) {$9$};
31 \draw[fill=black!27,draw=black!45,line width=.35pt,] (9.36000,3.80000) rectangle ++(0.72000,0.62000);
32 \node[] at (9.72000,4.11000) {$11$};
33 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.08000,3.80000) rectangle ++(0.72000,0.62000);
34 \node[] at (10.44000,4.11000) {$13$};
35 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.80000,3.80000) rectangle ++(0.72000,0.62000);
36 \node[] at (11.16000,4.11000) {$15$};
37 \draw[line width=1.0pt] (5.76000,4.48000) -- (11.52000,4.48000);
38 \node[font=\sffamily\scriptsize] at (8.64000,4.80000) {$r=1$};
39 \node[anchor=west] at (12.00000,4.12000) {16};
40 \node[anchor=east] at (-0.22000,2.48000) {$N=4$};
41 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.00000,2.15000) rectangle ++(0.72000,0.62000);
42 \node[] at (0.36000,2.46000) {$0$};
43 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.72000,2.15000) rectangle ++(0.72000,0.62000);
44 \node[] at (1.08000,2.46000) {$4$};
45 \draw[fill=black!10,draw=black!45,line width=.35pt,] (1.44000,2.15000) rectangle ++(0.72000,0.62000);
46 \node[] at (1.80000,2.46000) {$8$};
47 \draw[fill=black!10,draw=black!45,line width=.35pt,] (2.16000,2.15000) rectangle ++(0.72000,0.62000);
48 \node[] at (2.52000,2.46000) {$12$};
49 \draw[line width=1.0pt] (0.00000,2.83000) -- (2.88000,2.83000);
50 \node[font=\sffamily\scriptsize] at (1.44000,3.15000) {$r=0$};
51 \draw[fill=black!27,draw=black!45,line width=.35pt,] (2.88000,2.15000) rectangle ++(0.72000,0.62000);
52 \node[] at (3.24000,2.46000) {$2$};
53 \draw[fill=black!27,draw=black!45,line width=.35pt,] (3.60000,2.15000) rectangle ++(0.72000,0.62000);
54 \node[] at (3.96000,2.46000) {$6$};
55 \draw[fill=black!27,draw=black!45,line width=.35pt,] (4.32000,2.15000) rectangle ++(0.72000,0.62000);
56 \node[] at (4.68000,2.46000) {$10$};
57 \draw[fill=black!27,draw=black!45,line width=.35pt,] (5.04000,2.15000) rectangle ++(0.72000,0.62000);
58 \node[] at (5.40000,2.46000) {$14$};
59 \draw[line width=1.0pt] (2.88000,2.83000) -- (5.76000,2.83000);
60 \node[font=\sffamily\scriptsize] at (4.32000,3.15000) {$r=2$};
61 \draw[fill=black!10,draw=black!45,line width=.35pt,] (5.76000,2.15000) rectangle ++(0.72000,0.62000);
62 \node[] at (6.12000,2.46000) {$1$};
63 \draw[fill=black!10,draw=black!45,line width=.35pt,] (6.48000,2.15000) rectangle ++(0.72000,0.62000);
64 \node[] at (6.84000,2.46000) {$5$};
65 \draw[fill=black!10,draw=black!45,line width=.35pt,] (7.20000,2.15000) rectangle ++(0.72000,0.62000);
66 \node[] at (7.56000,2.46000) {$9$};
67 \draw[fill=black!10,draw=black!45,line width=.35pt,] (7.92000,2.15000) rectangle ++(0.72000,0.62000);
68 \node[] at (8.28000,2.46000) {$13$};
69 \draw[line width=1.0pt] (5.76000,2.83000) -- (8.64000,2.83000);
70 \node[font=\sffamily\scriptsize] at (7.20000,3.15000) {$r=1$};
71 \draw[fill=black!27,draw=black!45,line width=.35pt,] (8.64000,2.15000) rectangle ++(0.72000,0.62000);
72 \node[] at (9.00000,2.46000) {$3$};
73 \draw[fill=black!27,draw=black!45,line width=.35pt,] (9.36000,2.15000) rectangle ++(0.72000,0.62000);
74 \node[] at (9.72000,2.46000) {$7$};
75 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.08000,2.15000) rectangle ++(0.72000,0.62000);
76 \node[] at (10.44000,2.46000) {$11$};
77 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.80000,2.15000) rectangle ++(0.72000,0.62000);
78 \node[] at (11.16000,2.46000) {$15$};
79 \draw[line width=1.0pt] (8.64000,2.83000) -- (11.52000,2.83000);
80 \node[font=\sffamily\scriptsize] at (10.08000,3.15000) {$r=3$};
81 \node[anchor=west] at (12.00000,2.47000) {16};
82 \node[anchor=east] at (-0.22000,0.83000) {$N=8$};
83 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.00000,0.50000) rectangle ++(0.72000,0.62000);
84 \node[] at (0.36000,0.81000) {$0$};
85 \draw[fill=black!10,draw=black!45,line width=.35pt,] (0.72000,0.50000) rectangle ++(0.72000,0.62000);
86 \node[] at (1.08000,0.81000) {$8$};
87 \draw[line width=1.0pt] (0.00000,1.18000) -- (1.44000,1.18000);
88 \node[font=\sffamily\scriptsize] at (0.72000,1.50000) {$r=0$};
89 \draw[fill=black!27,draw=black!45,line width=.35pt,] (1.44000,0.50000) rectangle ++(0.72000,0.62000);
90 \node[] at (1.80000,0.81000) {$4$};
91 \draw[fill=black!27,draw=black!45,line width=.35pt,] (2.16000,0.50000) rectangle ++(0.72000,0.62000);
92 \node[] at (2.52000,0.81000) {$12$};
93 \draw[line width=1.0pt] (1.44000,1.18000) -- (2.88000,1.18000);
94 \node[font=\sffamily\scriptsize] at (2.16000,1.50000) {$r=4$};
95 \draw[fill=black!10,draw=black!45,line width=.35pt,] (2.88000,0.50000) rectangle ++(0.72000,0.62000);
96 \node[] at (3.24000,0.81000) {$2$};
97 \draw[fill=black!10,draw=black!45,line width=.35pt,] (3.60000,0.50000) rectangle ++(0.72000,0.62000);
98 \node[] at (3.96000,0.81000) {$10$};
99 \draw[line width=1.0pt] (2.88000,1.18000) -- (4.32000,1.18000);
100 \node[font=\sffamily\scriptsize] at (3.60000,1.50000) {$r=2$};
101 \draw[fill=black!27,draw=black!45,line width=.35pt,] (4.32000,0.50000) rectangle ++(0.72000,0.62000);
102 \node[] at (4.68000,0.81000) {$6$};
103 \draw[fill=black!27,draw=black!45,line width=.35pt,] (5.04000,0.50000) rectangle ++(0.72000,0.62000);
104 \node[] at (5.40000,0.81000) {$14$};
105 \draw[line width=1.0pt] (4.32000,1.18000) -- (5.76000,1.18000);
106 \node[font=\sffamily\scriptsize] at (5.04000,1.50000) {$r=6$};
107 \draw[fill=black!10,draw=black!45,line width=.35pt,] (5.76000,0.50000) rectangle ++(0.72000,0.62000);
108 \node[] at (6.12000,0.81000) {$1$};
109 \draw[fill=black!10,draw=black!45,line width=.35pt,] (6.48000,0.50000) rectangle ++(0.72000,0.62000);
110 \node[] at (6.84000,0.81000) {$9$};
111 \draw[line width=1.0pt] (5.76000,1.18000) -- (7.20000,1.18000);
112 \node[font=\sffamily\scriptsize] at (6.48000,1.50000) {$r=1$};
113 \draw[fill=black!27,draw=black!45,line width=.35pt,] (7.20000,0.50000) rectangle ++(0.72000,0.62000);
114 \node[] at (7.56000,0.81000) {$5$};
115 \draw[fill=black!27,draw=black!45,line width=.35pt,] (7.92000,0.50000) rectangle ++(0.72000,0.62000);
116 \node[] at (8.28000,0.81000) {$13$};
117 \draw[line width=1.0pt] (7.20000,1.18000) -- (8.64000,1.18000);
118 \node[font=\sffamily\scriptsize] at (7.92000,1.50000) {$r=5$};
119 \draw[fill=black!10,draw=black!45,line width=.35pt,] (8.64000,0.50000) rectangle ++(0.72000,0.62000);
120 \node[] at (9.00000,0.81000) {$3$};
121 \draw[fill=black!10,draw=black!45,line width=.35pt,] (9.36000,0.50000) rectangle ++(0.72000,0.62000);
122 \node[] at (9.72000,0.81000) {$11$};
123 \draw[line width=1.0pt] (8.64000,1.18000) -- (10.08000,1.18000);
124 \node[font=\sffamily\scriptsize] at (9.36000,1.50000) {$r=3$};
125 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.08000,0.50000) rectangle ++(0.72000,0.62000);
126 \node[] at (10.44000,0.81000) {$7$};
127 \draw[fill=black!27,draw=black!45,line width=.35pt,] (10.80000,0.50000) rectangle ++(0.72000,0.62000);
128 \node[] at (11.16000,0.81000) {$15$};
129 \draw[line width=1.0pt] (10.08000,1.18000) -- (11.52000,1.18000);
130 \node[font=\sffamily\scriptsize] at (10.80000,1.50000) {$r=7$};
131 \node[anchor=west] at (12.00000,0.82000) {16};
132 \node[] at (5.76000,-0.18000) {Each cell is an address n; its coefficient travels with it.};
133 \end{tikzpicture}

Back to source controls

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