Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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,line join=round]
2 \node[] at (2.50000,4.40000) {Four weighted votes for $z^m$};
3 \draw[->,line width=1pt] (2.50000,2.00000) -- (3.95000,2.00000);
4 \node[] at (4.40000,2.00000) {$1$};
5 \draw[->,line width=1pt] (2.50000,2.00000) -- (2.50000,3.45000);
6 \node[] at (2.50000,3.90000) {$i$};
7 \draw[->,line width=1pt] (2.50000,2.00000) -- (1.05000,2.00000);
8 \node[] at (0.60000,2.00000) {$-1$};
9 \draw[->,line width=1pt] (2.50000,2.00000) -- (2.50000,0.55000);
10 \node[] at (2.50000,0.10000) {$-i$};
11 \node[] at (2.50000,-0.30000) {$\frac14\sum_{\ell=0}^3 i^{(m-j)\ell}$};
12 \node[anchor=east] at (6.10000,3.70000) {$j=0$};
13 \draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,3.40000) rectangle ++(1.70000,0.60000);
14 \node[] at (7.30000,3.70000) {$z^{0}$};
15 \draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,3.40000) rectangle ++(1.70000,0.60000);
16 \node[] at (9.00000,3.70000) {$z^{4}$};
17 \draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,3.40000) rectangle ++(1.70000,0.60000);
18 \node[] at (10.70000,3.70000) {$z^{8}$};
19 \node[anchor=east] at (6.10000,2.85000) {$j=1$};
20 \draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,2.55000) rectangle ++(1.70000,0.60000);
21 \node[] at (7.30000,2.85000) {$z^{1}$};
22 \draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,2.55000) rectangle ++(1.70000,0.60000);
23 \node[] at (9.00000,2.85000) {$z^{5}$};
24 \draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,2.55000) rectangle ++(1.70000,0.60000);
25 \node[] at (10.70000,2.85000) {$z^{9}$};
26 \node[anchor=east] at (6.10000,2.00000) {$j=2$};
27 \draw[fill=black!18,draw=black!40,line width=.35pt] (6.45000,1.70000) rectangle ++(1.70000,0.60000);
28 \node[] at (7.30000,2.00000) {$z^{2}$};
29 \draw[fill=black!18,draw=black!40,line width=.35pt] (8.15000,1.70000) rectangle ++(1.70000,0.60000);
30 \node[] at (9.00000,2.00000) {$z^{6}$};
31 \draw[fill=black!18,draw=black!40,line width=.35pt] (9.85000,1.70000) rectangle ++(1.70000,0.60000);
32 \node[] at (10.70000,2.00000) {$z^{10}$};
33 \node[anchor=east] at (6.10000,1.15000) {$j=3$};
34 \draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,0.85000) rectangle ++(1.70000,0.60000);
35 \node[] at (7.30000,1.15000) {$z^{3}$};
36 \draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,0.85000) rectangle ++(1.70000,0.60000);
37 \node[] at (9.00000,1.15000) {$z^{7}$};
38 \draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,0.85000) rectangle ++(1.70000,0.60000);
39 \node[] at (10.70000,1.15000) {$z^{11}$};
40 \node[] at (9.10000,-0.30000) {only the matching phase survives};
41 \end{tikzpicture}

Back to source controls

Back to Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data