Difference between revisions of "File:E1esuma8.jpg"

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  +
[[Complex map]] of the growing [[superfunction]] of exponential to [[base e1e]]
−
Importing image file
  
  +
  +
$u\!+\!\mathrm i v = \mathrm{SuExp}_{\eta,3}(x\!+\!\mathrm i y)$
  +
  +
where $\eta=\exp(1/\mathrm e)$ is the [[Henryk base]]; and $F=\mathrm{SuExp}_{\eta,3}$ is real–holomorphic solution of the [[transfer equation]]
  +
  +
$\exp_\eta(F(z))=F(z+1)$,
  +
  +
with specific value at zero, $F(0)=3$ and specific asymptotic behaviour,
  +
  +
==Use of this image==
  +
Similar map is used in the book [[Суперфункции]] (in Russian) <ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0<br>
  +
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf<br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014.
  +
</ref>
  +
  +
and in the article <ref>
  +
http://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/home.html<br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf<br>
  +
http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf
  +
H.Trappmann, D.Kouznetsov. COMPUTATION OF THE TWO REGULAR SUPER-EXPONENTIALS TO BASE EXP(1/E).
  +
Mathematics of Computation, v.81 (2012), 2207-2227.
  +
</ref>
  +
  +
==References==
  +
  +
<references/>
  +
  +
==[[C++]] generator of curves==
  +
<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++)
  +
//using namespace std;
  +
#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 "e1egf.cin"
  +
//#include "e1eti.cin"
  +
#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=601,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
  +
//E1E Growung Function
  +
//FILE *o;o=fopen("e1egf.eps","w");ado(o,402,302);
  +
//FILE *o;o=fopen("e1esuexpmap.eps","w");ado(o,402,302);
  +
FILE *o;o=fopen("e1esuma.eps","w");ado(o,402,302);
  +
fprintf(o,"101 151 translate\n 10 10 scale\n");
  +
// DO(m,M1) X[m]=-10.+.05*(m-.5);
  +
DO(m,M1) X[m]=-10.+.08*(m-.5);
  +
DO(n,N1) Y[n]=-15.+.05*(n-.5);
  +
for(m=-10;m<31;m++){ //if(m==0){ M(m,-10.2)L(m,10.2)} else
  +
{ M(m,-15)L(m,15) }}
  +
for(n=-15;n<16;n++){ M(-10,n)L(30,n)}
  +
fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
// c=E1ETF(z);
  +
c=E1EGF(z);
  +
// d=z;
  +
// p=abs(c-d)/abs(c+d); p=-log(p)/log(10.);
  +
p=Re(c); q=Im(c);
  +
// if(p>-85 && p<85) g[m*N1+n]=p;
  +
if(p>-33 && p<33 && fabs(p)> 1.e-9 && fabs(p-1.)>1.e-9 &&
  +
q >-33 && q<33 && fabs(q)> 1.e-9) { g[m*N1+n]=p; f[m*N1+n]=q; }
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n");
  +
p=1.;q=.5;
  +
for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
  +
fprintf(o,".03 W 0 .6 0 RGB S\n");
  +
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
  +
fprintf(o,".03 W .9 0 0 RGB S\n");
  +
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
  +
fprintf(o,".03 W 0 0 .9 RGB S\n");
  +
  +
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".09 W .8 0 0 RGB S\n");
  +
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".09 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".09 W .5 0 .5 RGB S\n");
  +
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".09 W 0 0 0 RGB S\n");
  +
/*
  +
fprintf(o,"0 setlinecap\n");
  +
M(-2,0)L(-8,0) fprintf(o,".08 W 1 1 1 RGB S\n");
  +
DO(m,16){M(-2-.4*(m),0)L(-2-.4*(m+.5),0)} fprintf(o,".09 W 0 0 0 RGB S\n");
  +
*/
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf e1esuma.eps");
  +
system( "open e1esuma.pdf"); //mac
  +
// system( "xpdf e1egf.pdf"); // linux
  +
getchar(); system("killall Preview");// mac
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt,a4paper,oneside]{book}
  +
%\newcommand \EN[1] {{#1}} % make the English version
  +
\newcommand \EN[1] {{}} % suppress the English version
  +
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
  +
%\newcommand \RU[1] {{}} % suppress the Russian version
  +
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
  +
\newcommand \JP[1] {{}} % ореsuppress some text
  +
%\usepackage[space]{cite }% If exist.
  +
\usepackage[utf8]{inputenc}
  +
\usepackage[T2A]{fontenc}
  +
\usepackage[russian]{babel}
  +
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
  +
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\usepackage{wrapfig}
  +
\usepackage{geometry}
  +
\paperwidth 404pt
  +
\paperheight 300pt
  +
\oddsidemargin -56pt
  +
\topmargin -110pt
  +
\pagestyle{empty}
  +
\large
  +
\usepackage{color}
  +
\definecolor{rose}{rgb}{1,.9,1}
  +
\renewcommand\thesection{\arabic{section}}
  +
\renewcommand\figurename{Рис.}
  +
\newcommand \be {\begin{eqnarray}}
  +
\newcommand \ee {\end{eqnarray} }
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
  +
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
  +
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing \includegraphics
  +
\newcommand \tet {\mathrm{tet}}
  +
\begin{document}
  +
\parindent 0pt
  +
\newcommand \figaxe {
  +
\put( -6,298){\sx{1}{$y$}}
  +
%\put( -11,289){\sx{1}{$14$}}
  +
\put( -11,268){\sx{.9}{$12$}}
  +
\put( -11,248){\sx{.9}{$10$}}
  +
\put( -6,228){\sx{.9}{$8$}}
  +
\put( -6,208){\sx{.9}{$6$}}
  +
\put( -6,188){\sx{.9}{$4$}}
  +
\put( -6,168){\sx{.9}{$2$}}
  +
\put( -6, 148){\sx{.9}{$0$}}
  +
\put(-14,128){\sx{.9}{$-2$}}
  +
\put(-14,108){\sx{.9}{$-4$}}
  +
\put(-14, 88){\sx{.9}{$-6$}}
  +
\put(-14, 68){\sx{.9}{$-8$}}
  +
\put(-18.4, 48){\sx{.9}{$-10$}}
  +
\put(-18.4, 28){\sx{.9}{$-12$}}
  +
\put(-18.4, 8){\sx{.9}{$-14$}}
  +
%\put(-13, 79){\sx{1}{$-6$}}
  +
\put(397,-7.5){\sx{1}{$x$}}
  +
\put(377,-8){\sx{.9}{$28$}}
  +
\put(357,-8){\sx{.9}{$26$}}
  +
\put(337,-8){\sx{.9}{$24$}}
  +
\put(317,-8){\sx{.9}{$22$}}
  +
\put(297,-8){\sx{.9}{$20$}}
  +
\put(277,-8){\sx{.9}{$18$}}
  +
\put(257,-8){\sx{.9}{$16$}}
  +
\put(237,-8){\sx{.9}{$14$}}
  +
\put(217,-8){\sx{.9}{$12$}}
  +
\put(197,-8){\sx{.9}{$10$}}
  +
\put(179,-8){\sx{.9}{$8$}}
  +
\put(159,-8){\sx{.9}{$6$}}
  +
\put(139,-8){\sx{.9}{$4$}}
  +
\put(119,-8){\sx{.9}{$2$}}
  +
\put( 99,-8){\sx{.9}{$0$}}
  +
\put( 73,-8){\sx{.9}{$-2$}}
  +
\put( 53,-8){\sx{.9}{$-4$}}
  +
\put( 33,-8){\sx{.9}{$-6$}}
  +
\put( 13,-8){\sx{.9}{$-8$}}
  +
}
  +
%\hskip -28pt
  +
%\vskip 18pt
  +
\sx{.96}{\begin{picture}(410,310)
  +
%\put(0,0){\ing{e1egf} } % suppress this string to boost
  +
\put(0,0){\ing{e1esuma} } % suppress this string to boost
  +
\figaxe
  +
\put(100,194){\sx{1.2}{$u\!=\!3$}}
  +
\put(150,280){\sx{1.1}{$v\!=\!0.2$}}
  +
\put(150, 30){\sx{1.1}{$v\!=\!-0.2$}}
  +
\put(215,224){\sx{1.1}{$v\!=\!0.4$}}
  +
\put(259,268){\sx{1.1}{$u\!=\!2.8$}}%
  +
\put(258, 38){\sx{1.1}{$u\!=\!2.8$}}
  +
\put(306,260){\sx{1.1}{$u\!=\!2.6$}}
  +
\put(308, 39){\sx{1.1}{$u\!=\!2.6$}}
  +
\put(346,252){\sx{1.1}{$v\!=\!0.4$}}
  +
% \put(390,239){\sx{1.2}{$u\!=\!2.4$}}
  +
\put(365,241){\sx{1.1}{$u\!=\!2.4$}}
  +
% \put(370, 59){\sx{1.2}{$u\!=\!2.4$}}
  +
% \put(390,205){\sx{1.2}{$v\!=\!0.2$}}
  +
% \put(380, 88){\sx{1.2}{$v\!=\!-0.2$}}
  +
\put(352, 73){\sx{1.1}{$v\!=\!-0.4$}}
  +
% \put(390,172){\sx{1.2}{$u\!=\!2.2$}}
  +
% \put(390,122){\sx{1.2}{$u\!=\!2.2$}}
  +
\put(403,158){\sx{1.1}{$v\!=\!0$}}
  +
\put(403,143){\sx{1.1}{$v\!=\!0$}}
  +
\put(362,129){\sx{1.2}{$u\!=\!2$}}
  +
\put(174,165){\sx{1.1}{$u\!=\!3.2$}}
  +
\put( 60,148){\sx{1.2}{$v\!=\!0$}}
  +
\put(334,149){\sx{.9}{\bf huge values}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:BookE]]
  +
[[Category:BookMapE]]
  +
[[Category:Superfunction]]
  +
[[Category:Exotic iteration]]
  +
[[Category:Base e1e]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:34, 1 December 2018

Complex map of the growing superfunction of exponential to base e1e

$u\!+\!\mathrm i v = \mathrm{SuExp}_{\eta,3}(x\!+\!\mathrm i y)$

where $\eta=\exp(1/\mathrm e)$ is the Henryk base; and $F=\mathrm{SuExp}_{\eta,3}$ is real–holomorphic solution of the transfer equation

$\exp_\eta(F(z))=F(z+1)$,

with specific value at zero, $F(0)=3$ and specific asymptotic behaviour,

Use of this image

Similar map is used in the book Суперфункции (in Russian) [1]

and in the article [2]

References

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014.
  2. ↑ http://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/home.html
    http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf
    http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. COMPUTATION OF THE TWO REGULAR SUPER-EXPONENTIALS TO BASE EXP(1/E). Mathematics of Computation, v.81 (2012), 2207-2227.

C++ generator of curves


#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
//using namespace std;
#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 "e1egf.cin"
//#include "e1eti.cin"
#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=601,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
//E1E Growung Function
//FILE *o;o=fopen("e1egf.eps","w");ado(o,402,302);
//FILE *o;o=fopen("e1esuexpmap.eps","w");ado(o,402,302);
FILE *o;o=fopen("e1esuma.eps","w");ado(o,402,302);
fprintf(o,"101 151 translate\n 10 10 scale\n");
// DO(m,M1) X[m]=-10.+.05*(m-.5);
        DO(m,M1) X[m]=-10.+.08*(m-.5);
        DO(n,N1) Y[n]=-15.+.05*(n-.5);
for(m=-10;m<31;m++){ //if(m==0){ M(m,-10.2)L(m,10.2)} else
                                { M(m,-15)L(m,15) }}
for(n=-15;n<16;n++){ M(-10,n)L(30,n)}
fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=E1ETF(z);
        c=E1EGF(z);
// d=z;
// p=abs(c-d)/abs(c+d); p=-log(p)/log(10.);
        p=Re(c); q=Im(c);
// if(p>-85 && p<85) g[m*N1+n]=p;
        if(p>-33 && p<33 && fabs(p)> 1.e-9 && fabs(p-1.)>1.e-9 &&
          q >-33 && q<33 && fabs(q)> 1.e-9) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=1.;q=.5;
for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".03 W 0 .6 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".03 W .9 0 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".03 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".09 W .8 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".09 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".09 W .5 0 .5 RGB S\n");
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".09 W 0 0 0 RGB S\n");
/*
fprintf(o,"0 setlinecap\n");
M(-2,0)L(-8,0) fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,16){M(-2-.4*(m),0)L(-2-.4*(m+.5),0)} fprintf(o,".09 W 0 0 0 RGB S\n");
*/
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1esuma.eps");
        system( "open e1esuma.pdf"); //mac
// system( "xpdf e1egf.pdf"); // linux
getchar(); system("killall Preview");// mac
}

Latex generator of labels


\documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}} % make the English version
\newcommand \EN[1] {{}} % suppress the English version
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}} % suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}} % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{geometry}
\paperwidth 404pt
\paperheight 300pt
\oddsidemargin -56pt
\topmargin -110pt
\pagestyle{empty}
\large
\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\renewcommand\thesection{\arabic{section}}
\renewcommand\figurename{Рис.}
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\newcommand \tet {\mathrm{tet}}
\begin{document}
\parindent 0pt
\newcommand \figaxe {
\put( -6,298){\sx{1}{$y$}}
%\put( -11,289){\sx{1}{$14$}}
\put( -11,268){\sx{.9}{$12$}}
\put( -11,248){\sx{.9}{$10$}}
\put( -6,228){\sx{.9}{$8$}}
\put( -6,208){\sx{.9}{$6$}}
\put( -6,188){\sx{.9}{$4$}}
\put( -6,168){\sx{.9}{$2$}}
\put( -6, 148){\sx{.9}{$0$}}
\put(-14,128){\sx{.9}{$-2$}}
\put(-14,108){\sx{.9}{$-4$}}
\put(-14, 88){\sx{.9}{$-6$}}
\put(-14, 68){\sx{.9}{$-8$}}
\put(-18.4, 48){\sx{.9}{$-10$}}
\put(-18.4, 28){\sx{.9}{$-12$}}
\put(-18.4, 8){\sx{.9}{$-14$}}
%\put(-13, 79){\sx{1}{$-6$}}
\put(397,-7.5){\sx{1}{$x$}}
\put(377,-8){\sx{.9}{$28$}}
\put(357,-8){\sx{.9}{$26$}}
\put(337,-8){\sx{.9}{$24$}}
\put(317,-8){\sx{.9}{$22$}}
\put(297,-8){\sx{.9}{$20$}}
\put(277,-8){\sx{.9}{$18$}}
\put(257,-8){\sx{.9}{$16$}}
\put(237,-8){\sx{.9}{$14$}}
\put(217,-8){\sx{.9}{$12$}}
\put(197,-8){\sx{.9}{$10$}}
\put(179,-8){\sx{.9}{$8$}}
\put(159,-8){\sx{.9}{$6$}}
\put(139,-8){\sx{.9}{$4$}}
\put(119,-8){\sx{.9}{$2$}}
\put( 99,-8){\sx{.9}{$0$}}
\put( 73,-8){\sx{.9}{$-2$}}
\put( 53,-8){\sx{.9}{$-4$}}
\put( 33,-8){\sx{.9}{$-6$}}
\put( 13,-8){\sx{.9}{$-8$}}
}
%\hskip -28pt
%\vskip 18pt
 \sx{.96}{\begin{picture}(410,310)
 %\put(0,0){\ing{e1egf} } % suppress this string to boost
 \put(0,0){\ing{e1esuma} } % suppress this string to boost
 \figaxe
 \put(100,194){\sx{1.2}{$u\!=\!3$}}
 \put(150,280){\sx{1.1}{$v\!=\!0.2$}}
 \put(150, 30){\sx{1.1}{$v\!=\!-0.2$}}
 \put(215,224){\sx{1.1}{$v\!=\!0.4$}}
 \put(259,268){\sx{1.1}{$u\!=\!2.8$}}%
 \put(258, 38){\sx{1.1}{$u\!=\!2.8$}}
 \put(306,260){\sx{1.1}{$u\!=\!2.6$}}
 \put(308, 39){\sx{1.1}{$u\!=\!2.6$}}
 \put(346,252){\sx{1.1}{$v\!=\!0.4$}}
% \put(390,239){\sx{1.2}{$u\!=\!2.4$}}
 \put(365,241){\sx{1.1}{$u\!=\!2.4$}}
% \put(370, 59){\sx{1.2}{$u\!=\!2.4$}}
% \put(390,205){\sx{1.2}{$v\!=\!0.2$}}
% \put(380, 88){\sx{1.2}{$v\!=\!-0.2$}}
 \put(352, 73){\sx{1.1}{$v\!=\!-0.4$}}
% \put(390,172){\sx{1.2}{$u\!=\!2.2$}}
% \put(390,122){\sx{1.2}{$u\!=\!2.2$}}
 \put(403,158){\sx{1.1}{$v\!=\!0$}}
 \put(403,143){\sx{1.1}{$v\!=\!0$}}
 \put(362,129){\sx{1.2}{$u\!=\!2$}}
 \put(174,165){\sx{1.1}{$u\!=\!3.2$}}
 \put( 60,148){\sx{1.2}{$v\!=\!0$}}
 \put(334,149){\sx{.9}{\bf huge values}}
 \end{picture}}
\end{document}

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