Difference between revisions of "File:Superpower2map.jpg"

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  +
[[Complex map]] of the superpower function
−
Importing image file
  
  +
  +
$F(z)=\exp(2^z)$
  +
  +
is shown with lines
  +
  +
$u\!+\mathrm i v=F(x\!+\mathrm i y)$
  +
  +
== Superpower ==
  +
  +
Superpower function $F$ is [[superfunction]] for the quadratic transfer function $T(z)\!=\!z^2$,
  +
which is in its turn, interpreted as [[power function]] to base 2.
  +
  +
Superpower function $F$ is solution of the transfer equation
  +
  +
$F(z\!+!1)=T(F(z))$
  +
  +
==[[C++]] generator of map==
  +
Files [[ado.cin]] and [[conto.cin]] should be loaded in order to compile the code below.
  +
<poem><nomathjax><nowiki>
  +
#include <math.h>
  +
#include <stdio.h>
  +
#include <stdlib.h>
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=501,M1=M+1;
  +
int N=501,N1=N+1;
  +
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
  +
char v[M1*N1]; // v is working array
  +
FILE *o;o=fopen("superpower2ma.eps","w");ado(o,1020,1020);
  +
fprintf(o,"510 510 translate\n 100 100 scale\n");
  +
DO(m,M1) X[m]=-5.+.02*(m-.5);
  +
DO(n,N1) Y[n]=-5.+.02*(n-.5);
  +
for(m=-5;m<6;m++){if(m==0){M(m,-5.1)L(m,5.1)} else{M(m,-5)L(m,5)}}
  +
for(n=-5;n<6;n++){ M( -5,n)L(5,n)}
  +
fprintf(o,".01 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=1.e15; f[m*N1+n]=1.e15;}
  +
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
c=exp(pow(2.,z));
  +
p=Re(c);
  +
q=Im(c);
  +
if(p>-2.e15 && p<1.e15
  +
&& q>-1.e15 && q<1.e15
  +
)
  +
{g[m*N1+n]=p;
  +
f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=10;q=.4;
  +
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".013 W 0 .6 0 RGB S\n");
  +
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".011 W .9 0 0 RGB S\n");
  +
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".011 W 0 0 .9 RGB S\n");
  +
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".028 W .9 0 0 RGB S\n");
  +
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".028 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-1.e14,1.e14); fprintf(o,".028 W .6 0 .6 RGB S\n");
  +
for(m=-10;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-1.e14,1.e14); fprintf(o,".024 W 0 0 0 RGB S\n");
  +
for(m=1;m<11;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf superpower2ma.eps");
  +
system( "open superpower2ma.pdf");
  +
getchar(); system("killall Preview");//for mac
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphicx}
  +
\usepackage{geometry}
  +
%\usepackage{rotate}
  +
%\usepackage{rotation}
  +
\usepackage{rotating}
  +
\paperwidth 1044px
  +
\paperheight 1036px
  +
\topmargin -98px
  +
\oddsidemargin -90px
  +
\textwidth 2000px
  +
\textheight 2000px
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
\begin{picture}(1030,1026) \normalsize
  +
\put(30,20){\ing{"superpower2ma"}}
  +
\put(16,1016){\sx{3}{$y$}}
  +
\put(16,918){\sx{3}{$4$}}
  +
\put(16,818){\sx{3}{$3$}}
  +
\put(16,718){\sx{3}{$2$}}
  +
\put(16,618){\sx{3}{$1$}}
  +
\put(16,518){\sx{3}{$0$}}
  +
\put(-10,418){\sx{3}{$-1$}}
  +
\put(-10,318){\sx{3}{$-2$}}
  +
\put(-10,218){\sx{3}{$-3$}}
  +
\put(-10,118){\sx{3}{$-4$}}
  +
\put(-10,18){\sx{3}{$-5$}}
  +
\put(10,-1){\sx{3}{$-5$}}
  +
\put(110,-1){\sx{3}{$-4$}}
  +
\put(210,-1){\sx{3}{$-3$}}
  +
\put(310,-1){\sx{3}{$-2$}}
  +
\put(410,-1){\sx{3}{$-1$}}
  +
\put(534,-1){\sx{3}{$0$}}
  +
\put(634,-1){\sx{3}{$1$}}
  +
\put(734,-1){\sx{3}{$2$}}
  +
\put(834,-1){\sx{3}{$3$}}
  +
\put(934,-1){\sx{3}{$4$}}
  +
\put(1028,-1){\sx{3.1}{$x$}}
  +
\put(150,978){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero}
  +
\put(150,744){\rot{-1}{ \sx{3}{$v\!=\!1$}} \ero}
  +
\put(150,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
  +
\put(150,302){\rot{1}{ \sx{3}{$v\!=\!1$}} \ero}
  +
\put(150,68){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero}
  +
%
  +
\put(304,484){\rot{90}{ \sx{3}{$u\!=\!1.2$}} \ero}
  +
\put(400,484){\rot{90}{ \sx{3}{$u\!=\!1.4$}} \ero}
  +
\put(444,484){\rot{90}{ \sx{3}{$u\!=\!1.6$}} \ero}
  +
\put(474,484){\rot{90}{ \sx{3}{$u\!=\!1.8$}} \ero}
  +
\put(500,489){\rot{90}{ \sx{3}{$u\!=\!2$}} \ero}
  +
%
  +
\put(780,564){\rot{-11}{ \sx{3}{$u\!=\!0$}} \ero}
  +
\put(790,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
  +
\put(782,482){\rot{10}{ \sx{3}{$u\!=\!0$}} \ero}
  +
%
  +
\put(336,926){\rot{90}{ \sx{3}{$u\!=\!0.8$}} \ero}
  +
\put(454,926){\rot{90}{ \sx{3}{$u\!=\!0.6$}} \ero}
  +
\put(538,926){\rot{90}{ \sx{3}{$u\!=\!0.4$}} \ero}
  +
\put(618,926){\rot{90}{ \sx{3}{$u\!=\!0.2$}} \ero}
  +
%
  +
\put(710,906){\rot{20}{ \sx{3}{$u\!=\!0$}} \ero} %
  +
\put(760,870){\rot{30}{ \sx{3}{$v\!=\!0$}} \ero}
  +
\put(800,852){\rot{36}{ \sx{3}{$u\!=\!0$}} \ero}
  +
%
  +
\put(336,26){\rot{90}{ \sx{3}{$u\!=\!0.8$}} \ero}
  +
\put(454,26){\rot{90}{ \sx{3}{$u\!=\!0.6$}} \ero}
  +
\put(538,26){\rot{90}{ \sx{3}{$u\!=\!0.4$}} \ero}
  +
\put(618,26){\rot{90}{ \sx{3}{$u\!=\!0.2$}} \ero}
  +
%
  +
\put(658,172){\rot{-31}{ \sx{3}{$u\!=\!0$}} \ero}
  +
%
  +
\put(480,896){\rot{2}{ \sx{3}{$v\!=\!0.2$}} \ero}%%
  +
\put(482,832){\rot{8}{ \sx{3}{$v\!=\!0.4$}} \ero}%%
  +
\put(500,776){\rot{16}{ \sx{3}{$v\!=\!0.6$}} \ero}
  +
\put(528,718){\rot{28}{ \sx{3}{$v\!=\!1$}} \ero}
  +
%
  +
\put(508,348){\rot{-33}{ \sx{3}{$v\!=\!-1$}} \ero}
  +
\put(480,268){\rot{-12}{ \sx{3}{$v\!=\!-0.6$}} \ero}
  +
\put(480,210){\rot{-4}{ \sx{3}{$v\!=\!-0.4$}} \ero}
  +
\put(480,150){\rot{0}{ \sx{3}{$v\!=\!-0.2$}} \ero}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:Complex map]]
  +
[[Category:Power function]]
  +
[[Category:Superfunction]]
  +
[[Category:Superpower]]

Latest revision as of 08:53, 1 December 2018

Complex map of the superpower function

$F(z)=\exp(2^z)$

is shown with lines

$u\!+\mathrm i v=F(x\!+\mathrm i y)$

Superpower

Superpower function $F$ is superfunction for the quadratic transfer function $T(z)\!=\!z^2$, which is in its turn, interpreted as power function to base 2.

Superpower function $F$ is solution of the transfer equation

$F(z\!+!1)=T(F(z))$

C++ generator of map

Files ado.cin and conto.cin should be loaded in order to compile the code below.


#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=501,M1=M+1;
int N=501,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
FILE *o;o=fopen("superpower2ma.eps","w");ado(o,1020,1020);
fprintf(o,"510 510 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-5.+.02*(m-.5);
DO(n,N1) Y[n]=-5.+.02*(n-.5);
for(m=-5;m<6;m++){if(m==0){M(m,-5.1)L(m,5.1)} else{M(m,-5)L(m,5)}}
for(n=-5;n<6;n++){ M( -5,n)L(5,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=1.e15; f[m*N1+n]=1.e15;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
       c=exp(pow(2.,z));
      p=Re(c);
      q=Im(c);
      if(p>-2.e15 && p<1.e15
       && q>-1.e15 && q<1.e15
      )
 {g[m*N1+n]=p;
  f[m*N1+n]=q;
 }
                      }}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=10;q=.4;
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".013 W 0 .6 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".011 W .9 0 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".011 W 0 0 .9 RGB S\n");
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".028 W .9 0 0 RGB S\n");
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".028 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-1.e14,1.e14); fprintf(o,".028 W .6 0 .6 RGB S\n");
for(m=-10;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
             m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-1.e14,1.e14); fprintf(o,".024 W 0 0 0 RGB S\n");
for(m=1;m<11;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf superpower2ma.eps");
      system( "open superpower2ma.pdf");
      getchar(); system("killall Preview");//for mac
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{geometry}
%\usepackage{rotate}
%\usepackage{rotation}
\usepackage{rotating}
\paperwidth 1044px
\paperheight 1036px
\topmargin -98px
\oddsidemargin -90px
\textwidth 2000px
\textheight 2000px
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(1030,1026) \normalsize
\put(30,20){\ing{"superpower2ma"}}
\put(16,1016){\sx{3}{$y$}}
\put(16,918){\sx{3}{$4$}}
\put(16,818){\sx{3}{$3$}}
\put(16,718){\sx{3}{$2$}}
\put(16,618){\sx{3}{$1$}}
\put(16,518){\sx{3}{$0$}}
\put(-10,418){\sx{3}{$-1$}}
\put(-10,318){\sx{3}{$-2$}}
\put(-10,218){\sx{3}{$-3$}}
\put(-10,118){\sx{3}{$-4$}}
\put(-10,18){\sx{3}{$-5$}}
\put(10,-1){\sx{3}{$-5$}}
\put(110,-1){\sx{3}{$-4$}}
\put(210,-1){\sx{3}{$-3$}}
\put(310,-1){\sx{3}{$-2$}}
\put(410,-1){\sx{3}{$-1$}}
\put(534,-1){\sx{3}{$0$}}
\put(634,-1){\sx{3}{$1$}}
\put(734,-1){\sx{3}{$2$}}
\put(834,-1){\sx{3}{$3$}}
\put(934,-1){\sx{3}{$4$}}
\put(1028,-1){\sx{3.1}{$x$}}
\put(150,978){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero}
\put(150,744){\rot{-1}{ \sx{3}{$v\!=\!1$}} \ero}
\put(150,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
\put(150,302){\rot{1}{ \sx{3}{$v\!=\!1$}} \ero}
\put(150,68){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero}
%
\put(304,484){\rot{90}{ \sx{3}{$u\!=\!1.2$}} \ero}
\put(400,484){\rot{90}{ \sx{3}{$u\!=\!1.4$}} \ero}
\put(444,484){\rot{90}{ \sx{3}{$u\!=\!1.6$}} \ero}
\put(474,484){\rot{90}{ \sx{3}{$u\!=\!1.8$}} \ero}
\put(500,489){\rot{90}{ \sx{3}{$u\!=\!2$}} \ero}
%
\put(780,564){\rot{-11}{ \sx{3}{$u\!=\!0$}} \ero}
\put(790,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
\put(782,482){\rot{10}{ \sx{3}{$u\!=\!0$}} \ero}
%
\put(336,926){\rot{90}{ \sx{3}{$u\!=\!0.8$}} \ero}
\put(454,926){\rot{90}{ \sx{3}{$u\!=\!0.6$}} \ero}
\put(538,926){\rot{90}{ \sx{3}{$u\!=\!0.4$}} \ero}
\put(618,926){\rot{90}{ \sx{3}{$u\!=\!0.2$}} \ero}
%
\put(710,906){\rot{20}{ \sx{3}{$u\!=\!0$}} \ero} %
\put(760,870){\rot{30}{ \sx{3}{$v\!=\!0$}} \ero}
\put(800,852){\rot{36}{ \sx{3}{$u\!=\!0$}} \ero}
%
\put(336,26){\rot{90}{ \sx{3}{$u\!=\!0.8$}} \ero}
\put(454,26){\rot{90}{ \sx{3}{$u\!=\!0.6$}} \ero}
\put(538,26){\rot{90}{ \sx{3}{$u\!=\!0.4$}} \ero}
\put(618,26){\rot{90}{ \sx{3}{$u\!=\!0.2$}} \ero}
%
\put(658,172){\rot{-31}{ \sx{3}{$u\!=\!0$}} \ero}
%
\put(480,896){\rot{2}{ \sx{3}{$v\!=\!0.2$}} \ero}%%
\put(482,832){\rot{8}{ \sx{3}{$v\!=\!0.4$}} \ero}%%
\put(500,776){\rot{16}{ \sx{3}{$v\!=\!0.6$}} \ero}
\put(528,718){\rot{28}{ \sx{3}{$v\!=\!1$}} \ero}
%
\put(508,348){\rot{-33}{ \sx{3}{$v\!=\!-1$}} \ero}
\put(480,268){\rot{-12}{ \sx{3}{$v\!=\!-0.6$}} \ero}
\put(480,210){\rot{-4}{ \sx{3}{$v\!=\!-0.4$}} \ero}
\put(480,150){\rot{0}{ \sx{3}{$v\!=\!-0.2$}} \ero}
\end{picture}
\end{document}

References

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