Difference between revisions of "File:Arqnem00zt.jpg"

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[[Complex map]] of function [[ArqNem]] with parameter $q\!=\!0$:
−
Importing image file
  
  +
  +
$u\!+\!\mathrm i v = \mathrm{ArqNem}_1(x\!+\!\mathrm i y)$.
  +
  +
Additional grid lines mark the branch point of function $\mathrm{arqNem}_q~~$;
  +
  +
$x_0 +\mathrm i y_0 = \mathrm{Nem}_0(z_0)$
  +
  +
where $z_0$ is solution of equation $\mathrm{Nem}_1^{~\prime} (z_0)=0$
  +
  +
Technically, the algorithm for positive $q\!=\!10^{-6}$ is used to generate the map
  +
  +
==References==
  +
<references/>
  +
  +
==[[C++]] generator of map==
  +
  +
[[ado.cin]], [[conto.cin]], [[nembran.cin]], [[arqnem.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++)
  +
//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 "conto.cin"
  +
  +
DB Q=1.e-6;
  +
  +
z_type nem(z_type z){ return z*(1.+z*z*(1.+z*Q)); }
  +
z_type nem1(z_type z){ return 1.+z*z*(3.+z*(4.*Q)); } // WARNING: Q is global!
  +
  +
#include"nembran.cin"
  +
z_type NemZo=nembra(Q);
  +
z_type ANemZo=nem(NemZo);
  +
DB tr=Re(ANemZo);
  +
DB ti=Im(ANemZo);
  +
  +
#include "arqnem.cin"
  +
  +
int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
// DB rr,ti;
  +
int M=2001,M1=M+1;
  +
int N=1001,N1=N+1;
  +
DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array.
  +
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
char v[M1*N1]; // v is working array
  +
//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("arqnem00z.eps","w");ado(o,2020,2020);
  +
fprintf(o,"1010 1010 translate\n 1000 1000 scale 2 setlinecap\n");
  +
DO(m,M1) X[m]=-1.+.001*(m-.5);
  +
DO(n,N1) Y[n]=-1.+.002*(n-.5);
  +
for(m=-10;m<11;m+=5){ M(.1*m,-1)L(.1*m,1)}
  +
for(n=-10;n<11;n+=5){ M(-1,.1*n)L(1,.1*n)}
  +
fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");
  +
  +
M(tr, -2) L(tr, 2)
  +
M(-2, ti) L(2, ti)
  +
M(-2,-ti) L(2,-ti)
  +
fprintf(o,".002 W 0 0 0 RGB S\n");
  +
  +
M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
  +
fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");
  +
printf("ti, tr= %9.5lf %9.5lf\n",tr,ti);
  +
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999999; f[m*N1+n]=9999999;}
  +
DO(m,M1){x=X[m]; if(m/10*10==m) printf("run at x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
//c=sunem(z);
  +
//c=aun(z);
  +
c=arqnem(z);
  +
//c=arnemD(z);
  +
//c=nem(c);
  +
//p=abs(c-z)/abs(c+z); p=-log(p)/log(10.);
  +
p=Re(c); q=Im(c);
  +
//if(p>-85 && p<85) g[m*N1+n]=p;
  +
if(p>-1001 && p<1001 &&
  +
q >-1001 && q<1001 ) { g[m*N1+n]=p; f[m*N1+n]=q; }
  +
}}
  +
  +
M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
  +
fprintf(o,".002 W 1 1 0 RGB 0 setlinecap S\n");
  +
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n");
  +
p=20.;q=.5;
  +
//#include"plofu.cin"
  +
//p=2;q=1;
  +
  +
for(m=-5;m<5;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
  +
fprintf(o,".006 W 0 .6 0 RGB S\n");
  +
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
  +
fprintf(o,".006 W .9 0 0 RGB S\n");
  +
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
  +
fprintf(o,".006 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,".01 W .8 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,".01 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".01 W .5 0 .5 RGB S\n");
  +
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");
  +
  +
/*
  +
// conto(o,g,w,v,X,Y,M,N,15.5,-p,p);fprintf(o,".02 W .3 0 .3 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".004 W 0 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".002 W 0 1 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".002 W 1 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".004 W 0 0 .7 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,9.,-p,p);fprintf(o,".004 W 0 .6 .8 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,8.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,7.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,6.,-p,p);fprintf(o,".004 W 0 .6 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,5.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,4.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,3.,-p,p);fprintf(o,".004 W 1 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,2.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,1.,-p,p);fprintf(o,".005 W .5 0 0 RGB S\n");
  +
*/
  +
  +
M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
  +
fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");
  +
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf arqnem00z.eps");
  +
system( "open arqnem00z.pdf"); //mac
  +
  +
//getchar(); system("killall Preview");// mac
  +
  +
return 0;
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphics}
  +
\paperwidth 2020pt
  +
\paperheight 2020pt
  +
\usepackage{geometry}
  +
\usepackage{rotating}
  +
\textwidth 2260pt
  +
\textheight 2260pt
  +
\topmargin -97pt
  +
\oddsidemargin -84pt
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
\begin{picture}(2020,2020)
  +
%\put(24,20){\ing{35z}}
  +
\put(10,10){\ing{arqnem00z}}
  +
\put(226,1000){\sx{8}{\rot{0.} \bf cut \ero}}
  +
\put(1044,1044){\sx{8}{\rot{90} \bf cut \ero}}
  +
  +
\put(940,1980){\sx{8}{$y$}}
  +
\put(900,1500){\sx{8}{$0.5$}}
  +
\put(900,1390){\sx{8}{$y_0$}}
  +
\put(930,1000){\sx{8}{$0$}}
  +
\put(860,620){\sx{8}{$-y_0$}}
  +
\put(830, 492){\sx{8}{$-0.5$}}
  +
\put(400, 942){\sx{8}{$-0.5$}}
  +
\put(1000, 942){\sx{8}{$0$}}
  +
%\put(1064, 942){\sx{8}{$x_0$}}
  +
\put(1464, 942){\sx{8}{$0.5$}}
  +
\put(1980, 942){\sx{8}{$x$}}
  +
%
  +
\put(668,1830){\sx{8}{\rot{-79} $v\!=\!0.8$\ero}}
  +
\put(464,1670){\sx{8}{\rot{-62} $v\!=\!0.9$\ero}}
  +
\put(280,1460){\sx{8}{\rot{-52} $v\!=\!1$\ero}}
  +
\put(40,1270){\sx{8}{\rot{-46} $v\!=\!1.1$\ero}}
  +
%
  +
\put(234,400){\sx{8}{\rot{58} $v\!=\!-1$\ero}}
  +
%
  +
\put(920,1644){\sx{8}{\rot{87} $v\!=\!0.7$\ero}}
  +
\put(1066,1670){\sx{8}{\rot{70} $v\!=\!0.6$\ero}}
  +
\put(1224,1660){\sx{8}{\rot{56} $v\!=\!0.5$\ero}}
  +
\put(1384,1604){\sx{8}{\rot{43} $v\!=\!0.4$\ero}}
  +
\put(1550,1520){\sx{8}{\rot{30} $v\!=\!0.3$\ero}}
  +
\put(1660,1368){\sx{8}{\rot{19} $v\!=\!0.2$\ero}}
  +
\put(1720,1190){\sx{8}{\rot{9} $v\!=\!0.1$\ero}}
  +
\put(1720,1000){\sx{8}{\rot{0} $v\!=\!0$\ero}}
  +
\put(1690, 814){\sx{8}{\rot{-10} $v\!=\!-0.1$\ero}}
  +
\put(1690, 620){\sx{8}{\rot{-22} $v\!=\!-0.2$\ero}}
  +
%
  +
\put(1128, 740){\sx{8}{\rot{84} $u\!=\!0.1$\ero}}
  +
\put(1210, 720){\sx{8}{\rot{77} $u\!=\!0.2$\ero}}
  +
\put(1302, 660){\sx{8}{\rot{71} $u\!=\!0.3$\ero}}
  +
\put(1410, 594){\sx{8}{\rot{67} $u\!=\!0.4$\ero}}
  +
\put(1490, 442){\sx{8}{\rot{57} $u\!=\!0.5$\ero}}
  +
\put(1624, 298){\sx{8}{\rot{56} $u\!=\!0.6$\ero}}
  +
\put(1820, 172){\sx{8}{\rot{56} $u\!=\!0.7$\ero}}
  +
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
  +
[[Category:Abel function]]
  +
[[Category:ArqNem]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Latex]]
  +
[[Category:Nemtsov function]]
  +
[[Category:Superfunction]]

Latest revision as of 08:30, 1 December 2018

Complex map of function ArqNem with parameter $q\!=\!0$:

$u\!+\!\mathrm i v = \mathrm{ArqNem}_1(x\!+\!\mathrm i y)$.

Additional grid lines mark the branch point of function $\mathrm{arqNem}_q~~$;

$x_0 +\mathrm i y_0 = \mathrm{Nem}_0(z_0)$

where $z_0$ is solution of equation $\mathrm{Nem}_1^{~\prime} (z_0)=0$

Technically, the algorithm for positive $q\!=\!10^{-6}$ is used to generate the map

References


C++ generator of map

ado.cin, conto.cin, nembran.cin, arqnem.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 "conto.cin"

DB Q=1.e-6;

z_type nem(z_type z){ return z*(1.+z*z*(1.+z*Q)); }
z_type nem1(z_type z){ return 1.+z*z*(3.+z*(4.*Q)); } // WARNING: Q is global!

#include"nembran.cin"
z_type NemZo=nembra(Q);
z_type ANemZo=nem(NemZo);
DB tr=Re(ANemZo);
DB ti=Im(ANemZo);

#include "arqnem.cin"

int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
// DB rr,ti;
 int M=2001,M1=M+1;
 int N=1001,N1=N+1;
DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
//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("arqnem00z.eps","w");ado(o,2020,2020);
fprintf(o,"1010 1010 translate\n 1000 1000 scale 2 setlinecap\n");
DO(m,M1) X[m]=-1.+.001*(m-.5);
DO(n,N1) Y[n]=-1.+.002*(n-.5);
for(m=-10;m<11;m+=5){ M(.1*m,-1)L(.1*m,1)}
for(n=-10;n<11;n+=5){ M(-1,.1*n)L(1,.1*n)}
fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

M(tr, -2) L(tr, 2)
M(-2, ti) L(2, ti)
M(-2,-ti) L(2,-ti)
fprintf(o,".002 W 0 0 0 RGB S\n");

M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");
printf("ti, tr= %9.5lf %9.5lf\n",tr,ti);

DO(m,M1)DO(n,N1){g[m*N1+n]=9999999; f[m*N1+n]=9999999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
//c=sunem(z);
//c=aun(z);
c=arqnem(z);
//c=arnemD(z);
//c=nem(c);
//p=abs(c-z)/abs(c+z); p=-log(p)/log(10.);
        p=Re(c); q=Im(c);
//if(p>-85 && p<85) g[m*N1+n]=p;
        if(p>-1001 && p<1001 &&
        q >-1001 && q<1001 ) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
fprintf(o,".002 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=20.;q=.5;
//#include"plofu.cin"
//p=2;q=1;

for(m=-5;m<5;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".006 W 0 .6 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".006 W .9 0 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".006 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,".01 W .8 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,".01 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".01 W .5 0 .5 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".01 W 0 0 0 RGB S\n");

/*
// conto(o,g,w,v,X,Y,M,N,15.5,-p,p);fprintf(o,".02 W .3 0 .3 RGB S\n");
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".004 W 0 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".002 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".002 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".004 W 0 0 .7 RGB S\n");
conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,9.,-p,p);fprintf(o,".004 W 0 .6 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N,8.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,7.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,6.,-p,p);fprintf(o,".004 W 0 .6 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,5.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,4.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,3.,-p,p);fprintf(o,".004 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,2.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,1.,-p,p);fprintf(o,".005 W .5 0 0 RGB S\n");
*/

M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf arqnem00z.eps");
        system( "open arqnem00z.pdf"); //mac

//getchar(); system("killall Preview");// mac

return 0;
}

Latex generator of labels


 \documentclass[12pt]{article}
 \usepackage{graphics}
 \paperwidth 2020pt
 \paperheight 2020pt
 \usepackage{geometry}
 \usepackage{rotating}
 \textwidth 2260pt
 \textheight 2260pt
 \topmargin -97pt
 \oddsidemargin -84pt
 \parindent 0pt
 \pagestyle{empty}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(2020,2020)
%\put(24,20){\ing{35z}}
\put(10,10){\ing{arqnem00z}}
\put(226,1000){\sx{8}{\rot{0.} \bf cut \ero}}
\put(1044,1044){\sx{8}{\rot{90} \bf cut \ero}}

\put(940,1980){\sx{8}{$y$}}
\put(900,1500){\sx{8}{$0.5$}}
\put(900,1390){\sx{8}{$y_0$}}
\put(930,1000){\sx{8}{$0$}}
\put(860,620){\sx{8}{$-y_0$}}
\put(830, 492){\sx{8}{$-0.5$}}
\put(400, 942){\sx{8}{$-0.5$}}
\put(1000, 942){\sx{8}{$0$}}
%\put(1064, 942){\sx{8}{$x_0$}}
\put(1464, 942){\sx{8}{$0.5$}}
\put(1980, 942){\sx{8}{$x$}}
%
\put(668,1830){\sx{8}{\rot{-79} $v\!=\!0.8$\ero}}
\put(464,1670){\sx{8}{\rot{-62} $v\!=\!0.9$\ero}}
\put(280,1460){\sx{8}{\rot{-52} $v\!=\!1$\ero}}
\put(40,1270){\sx{8}{\rot{-46} $v\!=\!1.1$\ero}}
%
\put(234,400){\sx{8}{\rot{58} $v\!=\!-1$\ero}}
%
\put(920,1644){\sx{8}{\rot{87} $v\!=\!0.7$\ero}}
\put(1066,1670){\sx{8}{\rot{70} $v\!=\!0.6$\ero}}
\put(1224,1660){\sx{8}{\rot{56} $v\!=\!0.5$\ero}}
\put(1384,1604){\sx{8}{\rot{43} $v\!=\!0.4$\ero}}
\put(1550,1520){\sx{8}{\rot{30} $v\!=\!0.3$\ero}}
\put(1660,1368){\sx{8}{\rot{19} $v\!=\!0.2$\ero}}
\put(1720,1190){\sx{8}{\rot{9} $v\!=\!0.1$\ero}}
\put(1720,1000){\sx{8}{\rot{0} $v\!=\!0$\ero}}
\put(1690, 814){\sx{8}{\rot{-10} $v\!=\!-0.1$\ero}}
\put(1690, 620){\sx{8}{\rot{-22} $v\!=\!-0.2$\ero}}
%
\put(1128, 740){\sx{8}{\rot{84} $u\!=\!0.1$\ero}}
\put(1210, 720){\sx{8}{\rot{77} $u\!=\!0.2$\ero}}
\put(1302, 660){\sx{8}{\rot{71} $u\!=\!0.3$\ero}}
\put(1410, 594){\sx{8}{\rot{67} $u\!=\!0.4$\ero}}
\put(1490, 442){\sx{8}{\rot{57} $u\!=\!0.5$\ero}}
\put(1624, 298){\sx{8}{\rot{56} $u\!=\!0.6$\ero}}
\put(1820, 172){\sx{8}{\rot{56} $u\!=\!0.7$\ero}}

\end{picture}
\end{document}

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