Difference between revisions of "File:E1egi4.jpg"

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(Importing image file)
 
($ -> \( ; description ; refs ; pre ; keywords)
 
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  +
{{oq|File:E1egi4.jpg|Original file ‎(2,236 × 1,660 pixels, file size: 828 KB, MIME type: image/jpeg) }}
−
Importing image file
  
  +
  +
[[Complex map]] of [[Abelfunction]] of the exponential to base \(\eta=\exp(1/\mathrm e)\):
  +
  +
\[
  +
u+\mathrm i y=\mathrm{AuExp}_{\eta, 3}(x\!+\!\mathrm i y)
  +
\]
  +
  +
The superexponential \(\ \mathrm{SuExp}_{\eta, 3}\ \) of real argument remains bigger than
  +
\(\mathrm e\) ;
  +
so, the inverse function \(\ \mathrm{AuExp}_{\eta, 3}\!=\!\mathrm{SuExp}_{\eta, 3}^{~ -1} \ \)
  +
has the inevitable cut from \(-\infty\) to \(\mathrm e\) ; it is shown in the map with dashed line.
  +
  +
==Descriptions==
  +
The growing [[Abelfunction]] \(\mathrm{AuExp}_{\eta, 3}\)
  +
to "[[base e1e]]", est, to base \(\eta=\exp(1/\mathrm e)\)
  +
is described in [[Mathematics of Computation]], 2012
  +
<ref name="e1e">
  +
https://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/home.html <br>
  +
https://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), p. 2207-2227. ISSN 1088-6842(e) ISSN 0025-5718(p)
  +
</ref>. The similar map appears as top part of Fig.2 at page 2216.
  +
  +
The similar map
  +
is used also as bottom part of Fig.11.6 at page 141 of book «[[Суперфункции]]», (Russian version), 2014 <ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0<br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  +
</ref>. Both maps of Fig.11.6 of the Russian version are loaded as
  +
http://mizugadro.mydns.jp/t/index.php/File:E1eAuMap600.jpg
  +
  +
In the English version, book «[[Superfunctions]]», year 2020
  +
<ref>
  +
http://mizugadro.mydns.jp/BOOK/448.pdf
  +
D,Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref> the closest analogy is Fig.10.7 at page 130,
  +
but the map for abelfunction \(AuExp_{\eta,3}\) appears at the top
  +
of the figure as it is described first in the text (the map of the [[arctetration]] to base \(\eta\)
  +
http://mizugadro.mydns.jp/t/index.php/File:E1eti4.jpg appears at the bottom of that figure).
  +
  +
==[[C++]] generator of map==
  +
//Files [[ado.cin]] , [[conto.cin]] , [[e1egf.cin]] , [[e1egi.cin]] should be loaded in order to compile the code below.
  +
//<pre>
  +
#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 "e1egi.cin"
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=401,M1=M+1;
  +
int N=301,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 Growing Funciton
  +
FILE *o;o=fopen("e1egi.eps","w");ado(o,402,302);
  +
fprintf(o,"101 151 translate\n 10 10 scale\n");
  +
DO(m,M1) X[m]=-10.+.1*(m-.5);
  +
DO(n,N1) Y[n]=-15.+.1*(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,".006 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=E1EGI(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;
  +
//#include"plofu.cin"
  +
//p=2;q=1;
  +
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,".02 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,".02 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,".02 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,".07 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,".07 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".07 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,".07 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"0 setlinecap\n");
  +
M(M_E,0)L(-10,0) fprintf(o,".08 W 1 1 1 RGB S\n");
  +
DO(m,36){M(M_E-.4*(m),0)L(M_E-.4*(m+.5),0)} fprintf(o,".09 W 1 0 1 RGB S\n");
  +
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf e1egi.eps");
  +
system( "open e1egi.pdf"); //mac
  +
// system( "xpdf e1egi.pdf"); // linux
  +
getchar(); system("killall Preview");// mac
  +
}
  +
</pre>
  +
  +
==[[Latex]] generator of labels==
  +
<pre>
  +
\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
  +
\put(0,0){\ing{e1egi} }
  +
\normalsize
  +
\figaxe
  +
\put( 4,220){\sx{1.2}{$u\!=\!19$}}
  +
%\put(166,216){\sx{1.4}{$p\!=\!18$}}
  +
\put(352,276){\sx{1.2}{$v\!=\!0.2$}}
  +
\put(350, 26){\sx{1.2}{$v\!=\!-0.2$}}
  +
\put(350,186){\sx{1.2}{$u\!=\!18.4$}}
  +
\put(270,176){\sx{1.2}{$u\!=\!18.2$}}
  +
\put(233,158){\sx{1.2}{$u\!=\!18$}}
  +
\put( 4, 74){\sx{1.2}{$u\!=\!19$}}
  +
% \put(166, 80){\sx{1.4}{$p\!=\!18$}}
  +
\put(20,186){\sx{1.2}{$v\!=\!1$}}
  +
\put(20,110){\sx{1.2}{$v\!=\!-1$}}
  +
\put(20,148.4){\sx{1.2}{\bf cut }}
  +
\put(377,148.4){\sx{1.1}{$v\!=\!0$}}
  +
\end{picture}}
  +
\end{document}
  +
</pre>
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
«[[Abel function]]»,
  +
«[[Abelfunction]]»,
  +
«[[Base e1e]]»,
  +
«[[Exotic iteration]]»,
  +
«[[Fixed point]]»,
  +
«[[Mahtematics of Computation]]»,
  +
«[[Superfunctions]]»,
  +
«[[Superexponential]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:Abel function]]
  +
[[Category:Abelfunction]]
  +
[[Category:AuExp]]
  +
[[Category:Base e1e]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:e1e]]
  +
[[Category:Exotic iteration]]
  +
[[Category:Latex]]
  +
[[Category:Mahtematics of Computation]]
  +
[[Category:Superexponential]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 07:35, 24 August 2025


Complex map of Abelfunction of the exponential to base \(\eta=\exp(1/\mathrm e)\):

\[ u+\mathrm i y=\mathrm{AuExp}_{\eta, 3}(x\!+\!\mathrm i y) \]

The superexponential \(\ \mathrm{SuExp}_{\eta, 3}\ \) of real argument remains bigger than \(\mathrm e\) ; so, the inverse function \(\ \mathrm{AuExp}_{\eta, 3}\!=\!\mathrm{SuExp}_{\eta, 3}^{~ -1} \ \) has the inevitable cut from \(-\infty\) to \(\mathrm e\) ; it is shown in the map with dashed line.

Descriptions

The growing Abelfunction \(\mathrm{AuExp}_{\eta, 3}\) to "base e1e", est, to base \(\eta=\exp(1/\mathrm e)\) is described in Mathematics of Computation, 2012 [1]. The similar map appears as top part of Fig.2 at page 2216.

The similar map is used also as bottom part of Fig.11.6 at page 141 of book «Суперфункции», (Russian version), 2014 [2]. Both maps of Fig.11.6 of the Russian version are loaded as http://mizugadro.mydns.jp/t/index.php/File:E1eAuMap600.jpg

In the English version, book «Superfunctions», year 2020 [3] the closest analogy is Fig.10.7 at page 130, but the map for abelfunction \(AuExp_{\eta,3}\) appears at the top of the figure as it is described first in the text (the map of the arctetration to base \(\eta\) http://mizugadro.mydns.jp/t/index.php/File:E1eti4.jpg appears at the bottom of that figure).

C++ generator of map

//Files ado.cin , conto.cin , e1egf.cin , e1egi.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++)
//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 "e1egi.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=401,M1=M+1;
 int N=301,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 Growing Funciton
FILE *o;o=fopen("e1egi.eps","w");ado(o,402,302);
fprintf(o,"101 151 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.1*(m-.5);
DO(n,N1) Y[n]=-15.+.1*(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,".006 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=E1EGI(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;
//#include"plofu.cin"
//p=2;q=1;
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,".02 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,".02 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,".02 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,".07 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,".07 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0.  ),-p,p);fprintf(o,".07 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,".07 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinecap\n");
M(M_E,0)L(-10,0) fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,36){M(M_E-.4*(m),0)L(M_E-.4*(m+.5),0)} fprintf(o,".09 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1egi.eps");
        system(    "open e1egi.pdf"); //mac
//      system(    "xpdf e1egi.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
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  \put(0,0){\ing{e1egi} } 
  \normalsize
 \figaxe
 \put(  4,220){\sx{1.2}{$u\!=\!19$}} 
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 \put(350, 26){\sx{1.2}{$v\!=\!-0.2$}}
 \put(350,186){\sx{1.2}{$u\!=\!18.4$}}
 \put(270,176){\sx{1.2}{$u\!=\!18.2$}}
 \put(233,158){\sx{1.2}{$u\!=\!18$}}
 \put(  4, 74){\sx{1.2}{$u\!=\!19$}} 
% \put(166, 80){\sx{1.4}{$p\!=\!18$}}
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 \put(20,110){\sx{1.2}{$v\!=\!-1$}}
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\end{document}

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