Difference between revisions of "File:Apow2ma4.jpg"

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Two complex maps of the real-holomorphic [[abelpower]] functions,
−
Importing image file
  
  +
  +
[[AuPow]]$_2(z)\!=\!\log_2(\ln(z))~$ and $~$
  +
[[AdPow]]$_2(z)\!=\!\log_2(\ln(1/z))$
  +
  +
Left: $u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)$
  +
  +
Right: $u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)$
  +
  +
==Description==
  +
  +
The two real–holomorphic solutions $G$ of the [[Abel equation]]
  +
  +
$G(T(z))=T(z)\!+\!1$
  +
  +
for $T(z)\!=\!z^2$
  +
are shown.
  +
  +
Lines of the constant real part are the same for both maps.
  +
  +
The imaginary parts differ for a constant, but this constant has jumps at the cutlines.
  +
All these cutlines are at the real axis.
  +
  +
Both function have cut for negative values of argument.
  +
Each of them has also one additional cut:
  +
  +
[[AuPow]] has cut segment between zero and unity.
  +
  +
[[AdPow]] has cut from unity to infinity.
  +
  +
Many other abelfunctions for the same transfer function can be obtained adding to the function some periodic function with period unity. Such functions show fast (at least exponential) growth in the imaginary direction; the abelfunctions shown seem to be simplest possible.
  +
  +
==[[C++]] generator of left map==
  +
// [[ado.cin]] and [[conto.cin]] should be loaded for compilation of the codes 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=402,M1=M+1;
  +
int N=402,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("aupow2ma4.eps","w");ado(o,420,420);
  +
fprintf(o,"210 210 translate\n 100 100 scale\n");
  +
DO(m,M1) X[m]=-2.+.01*(m-.5);
  +
for(n=0;n<250;n++) Y[n]=-2.+.01*(n);
  +
Y[200]=-.006;
  +
Y[201]= .006;
  +
for(n=202;n<N1;n++)Y[n]=-2.+.01*(n-1);
  +
//DO(m,M1) X[m]=Y[m];
  +
  +
for(m=-2;m<3;m++){if(m==0){M(m,-2.1)L(m,2.1)} else{M(m,-2)L(m,2)}}
  +
for(n=-2;n<3;n++){ M( -2,n)L(2,n)}
  +
fprintf(o,".01 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=999999; f[m*N1+n]=999999;}
  +
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));
  +
// c=log(log(1./z))/log(2.);
  +
c=log(log(z))/log(2.);
  +
p=Re(c);
  +
q=Im(c);
  +
if(p>-99. && p<99.
  +
&& q>-99. && q<99.
  +
)
  +
{g[m*N1+n]=p;
  +
f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1;q=.5;
  +
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf aupow2ma4.eps");
  +
system( "open aupow2ma4.pdf");
  +
getchar(); system("killall Preview");//for mac
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of right map==
  +
[[ado.cin]] and [[conto.cin]] should be loaded
  +
<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=402,M1=M+1;
  +
int N=402,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("adpow2ma4.eps","w");ado(o,420,420);
  +
fprintf(o,"210 210 translate\n 100 100 scale\n");
  +
DO(m,M1) X[m]=-2.+.01*(m-.5);
  +
for(n=0;n<250;n++) Y[n]=-2.+.01*(n);
  +
Y[200]=-.006;
  +
Y[201]= .006;
  +
for(n=202;n<N1;n++)Y[n]=-2.+.01*(n-1);
  +
//DO(m,M1) X[m]=Y[m];
  +
  +
for(m=-2;m<3;m++){if(m==0){M(m,-2.1)L(m,2.1)} else{M(m,-2)L(m,2)}}
  +
for(n=-2;n<3;n++){ M( -2,n)L(2,n)}
  +
fprintf(o,".01 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=999999; f[m*N1+n]=999999;}
  +
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));
  +
c=log(log(1./z))/log(2.);
  +
// c=log(log(z))/log(2.);
  +
p=Re(c);
  +
q=Im(c);
  +
if(p>-99. && p<99.
  +
&& q>-99. && q<99.
  +
)
  +
{g[m*N1+n]=p;
  +
f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1;q=.5;
  +
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf adpow2ma4.eps");
  +
system( "open adpow2ma4.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{rotating}
  +
%\paperwidth 432px
  +
\paperwidth 854px
  +
\paperheight 422px
  +
\topmargin -98px
  +
\oddsidemargin -90px
  +
\textwidth 2000px
  +
\textheight 2000px
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \axey {
  +
\put(12,412){\sx{2}{$y$}}
  +
\put(12,312){\sx{2}{$1$}}
  +
\put(12,212){\sx{2}{$0$}}
  +
\put(-3,112){\sx{2}{$-1$}}
  +
\put(-3,11){\sx{2}{$-2$}}}
  +
\newcommand \axexb {
  +
\put( 13, 1){\sx{2}{$-2$}}
  +
\put(112, 1){\sx{2}{$-1$}}
  +
\put(225, 1){\sx{2}{$0$}}
  +
\put(325, 1){\sx{2}{$1$}}
  +
\put(419, 1){\sx{2.1}{$x$}}
  +
\put(32, 252){\rot{-18}{ \sx{1.8}{$u\!=\!1.6$}} \ero}
  +
\put( 68,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero}
  +
\put(32, 178){\rot{18}{ \sx{1.8}{$u\!=\!1.6$}} \ero}
  +
\put(36, 64){\rot{51}{ \sx{1.8}{$u\!=\!1.4$}} \ero}
  +
\put(109, 23){\rot{71}{ \sx{1.8}{$u\!=\!1.2$}} \ero}
  +
\put(174, 23){\rot{86}{ \sx{1.9}{$u\!=\!1$}} \ero}
  +
\put(348,104){\rot{11}{ \sx{1.8}{$u\!=\!-0.2$}} \ero}
  +
\put(354,83){\rot{0}{ \sx{1.8}{$u\!=\!0$}} \ero}
  +
\put(348,55){\rot{-5}{ \sx{1.8}{$u\!=\!0.2$}} \ero}
  +
}
  +
\begin{document}
  +
\begin{picture}(420,413) \normalsize \put(20,10){\ing{"aupow2ma4"}} \axey \axexb
  +
\put(240,351){\rot{-27}{ \sx{1.8}{$v\!=\!2$}} \ero}
  +
\put(240,266){\rot{-6}{ \sx{1.8}{$v\!=\!3$}} \ero}
  +
\put(242,229){\rot{0}{ \sx{1.8}{$v\!=\!4$}} \ero}
  +
\put(390,290){\rot{69}{ \sx{1.8}{$v\!=\!1$}} \ero}
  +
\put(380,215.3){\rot{0}{ \sx{1.78}{$v\!=\!0$}} \ero}
  +
\put(358,187){\rot{-63}{ \sx{1.78}{$v\!=\!-1$}} \ero}
  +
\put(210, 76){\rot{16}{ \sx{1.8}{$v\!=\!-2$}} \ero}
  +
\end{picture}
  +
\begin{picture}(420,413) \normalsize \put(20,10){\ing{"adpow2ma4"}} \axexb
  +
\put( 396,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero}
  +
\put(241,215){\rot{0}{ \sx{1.8}{$v\!=\!0$}} \ero}
  +
\put(236,189){\rot{-2}{ \sx{1.8}{$v\!=\!1$}} \ero}
  +
\put(216,142){\rot{-5}{ \sx{1.8}{$v\!=\!2$}} \ero}
  +
\put(300, 28){\rot{51}{ \sx{1.8}{$v\!=\!3$}} \ero}
  +
  +
\put(383,240){\rot{35}{ \sx{1.8}{$v\!=\!-4$}} \ero}
  +
\put(384,190){\rot{-35}{ \sx{1.8}{$v\!=\!4$}} \ero}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Abelfunction]]
  +
[[Category:Abelpower]]
  +
[[Category:AdPow]]
  +
[[Category:AuPow]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:Complex map]]
  +
[[Category:Superfunctions]]

Latest revision as of 08:29, 1 December 2018

Two complex maps of the real-holomorphic abelpower functions,

AuPow$_2(z)\!=\!\log_2(\ln(z))~$ and $~$ AdPow$_2(z)\!=\!\log_2(\ln(1/z))$

Left: $u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)$

Right: $u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)$

Description

The two real–holomorphic solutions $G$ of the Abel equation

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

for $T(z)\!=\!z^2$ are shown.

Lines of the constant real part are the same for both maps.

The imaginary parts differ for a constant, but this constant has jumps at the cutlines. All these cutlines are at the real axis.

Both function have cut for negative values of argument. Each of them has also one additional cut:

AuPow has cut segment between zero and unity.

AdPow has cut from unity to infinity.

Many other abelfunctions for the same transfer function can be obtained adding to the function some periodic function with period unity. Such functions show fast (at least exponential) growth in the imaginary direction; the abelfunctions shown seem to be simplest possible.

C++ generator of left map

// ado.cin and conto.cin should be loaded for compilation of the codes 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=402,M1=M+1;
int N=402,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("aupow2ma4.eps","w");ado(o,420,420);
fprintf(o,"210 210 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-2.+.01*(m-.5);
for(n=0;n<250;n++) Y[n]=-2.+.01*(n);
Y[200]=-.006;
Y[201]= .006;
for(n=202;n<N1;n++)Y[n]=-2.+.01*(n-1);
//DO(m,M1) X[m]=Y[m];

for(m=-2;m<3;m++){if(m==0){M(m,-2.1)L(m,2.1)} else{M(m,-2)L(m,2)}}
for(n=-2;n<3;n++){ M( -2,n)L(2,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=999999; f[m*N1+n]=999999;}
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));
// c=log(log(1./z))/log(2.);
       c=log(log(z))/log(2.);
      p=Re(c);
      q=Im(c);
      if(p>-99. && p<99.
       && q>-99. && q<99.
      )
 {g[m*N1+n]=p;
  f[m*N1+n]=q;
 }
                      }}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1;q=.5;
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
             m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf aupow2ma4.eps");
      system( "open aupow2ma4.pdf");
      getchar(); system("killall Preview");//for mac
}

C++ generator of right map

ado.cin and conto.cin should be loaded


#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=402,M1=M+1;
int N=402,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("adpow2ma4.eps","w");ado(o,420,420);
fprintf(o,"210 210 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-2.+.01*(m-.5);
for(n=0;n<250;n++) Y[n]=-2.+.01*(n);
Y[200]=-.006;
Y[201]= .006;
for(n=202;n<N1;n++)Y[n]=-2.+.01*(n-1);
//DO(m,M1) X[m]=Y[m];

for(m=-2;m<3;m++){if(m==0){M(m,-2.1)L(m,2.1)} else{M(m,-2)L(m,2)}}
for(n=-2;n<3;n++){ M( -2,n)L(2,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=999999; f[m*N1+n]=999999;}
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));
       c=log(log(1./z))/log(2.);
// c=log(log(z))/log(2.);
      p=Re(c);
      q=Im(c);
      if(p>-99. && p<99.
       && q>-99. && q<99.
      )
 {g[m*N1+n]=p;
  f[m*N1+n]=q;
 }
                      }}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1;q=.5;
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");
             m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf adpow2ma4.eps");
      system( "open adpow2ma4.pdf");
      getchar(); system("killall Preview");//for mac
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{rotating}
%\paperwidth 432px
\paperwidth 854px
\paperheight 422px
\topmargin -98px
\oddsidemargin -90px
\textwidth 2000px
\textheight 2000px
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \axey {
\put(12,412){\sx{2}{$y$}}
\put(12,312){\sx{2}{$1$}}
\put(12,212){\sx{2}{$0$}}
\put(-3,112){\sx{2}{$-1$}}
\put(-3,11){\sx{2}{$-2$}}}
\newcommand \axexb {
\put( 13, 1){\sx{2}{$-2$}}
\put(112, 1){\sx{2}{$-1$}}
\put(225, 1){\sx{2}{$0$}}
\put(325, 1){\sx{2}{$1$}}
\put(419, 1){\sx{2.1}{$x$}}
\put(32, 252){\rot{-18}{ \sx{1.8}{$u\!=\!1.6$}} \ero}
\put( 68,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero}
\put(32, 178){\rot{18}{ \sx{1.8}{$u\!=\!1.6$}} \ero}
\put(36, 64){\rot{51}{ \sx{1.8}{$u\!=\!1.4$}} \ero}
\put(109, 23){\rot{71}{ \sx{1.8}{$u\!=\!1.2$}} \ero}
\put(174, 23){\rot{86}{ \sx{1.9}{$u\!=\!1$}} \ero}
\put(348,104){\rot{11}{ \sx{1.8}{$u\!=\!-0.2$}} \ero}
\put(354,83){\rot{0}{ \sx{1.8}{$u\!=\!0$}} \ero}
\put(348,55){\rot{-5}{ \sx{1.8}{$u\!=\!0.2$}} \ero}
}
\begin{document}
\begin{picture}(420,413) \normalsize \put(20,10){\ing{"aupow2ma4"}} \axey \axexb
\put(240,351){\rot{-27}{ \sx{1.8}{$v\!=\!2$}} \ero}
\put(240,266){\rot{-6}{ \sx{1.8}{$v\!=\!3$}} \ero}
\put(242,229){\rot{0}{ \sx{1.8}{$v\!=\!4$}} \ero}
\put(390,290){\rot{69}{ \sx{1.8}{$v\!=\!1$}} \ero}
\put(380,215.3){\rot{0}{ \sx{1.78}{$v\!=\!0$}} \ero}
\put(358,187){\rot{-63}{ \sx{1.78}{$v\!=\!-1$}} \ero}
\put(210, 76){\rot{16}{ \sx{1.8}{$v\!=\!-2$}} \ero}
\end{picture}
\begin{picture}(420,413) \normalsize \put(20,10){\ing{"adpow2ma4"}} \axexb
\put( 396,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero}
\put(241,215){\rot{0}{ \sx{1.8}{$v\!=\!0$}} \ero}
\put(236,189){\rot{-2}{ \sx{1.8}{$v\!=\!1$}} \ero}
\put(216,142){\rot{-5}{ \sx{1.8}{$v\!=\!2$}} \ero}
\put(300, 28){\rot{51}{ \sx{1.8}{$v\!=\!3$}} \ero}

\put(383,240){\rot{35}{ \sx{1.8}{$v\!=\!-4$}} \ero}
\put(384,190){\rot{-35}{ \sx{1.8}{$v\!=\!4$}} \ero}
\end{picture}
\end{document}

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