Difference between revisions of "File:Lofmap.jpg"

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  +
[[Complex map]] of function [[lof]],
−
Importing image file
  
  +
  +
$u\!+\!\mathrm i v=\mathrm{lof}(x\!+\!\mathrm i y)$
  +
  +
in vicinity of the real axis, [[lof]] is just logarithm of [[Factorial]] of its argument;
  +
in particular, for real $x$,
  +
  +
$\mathrm{lof}(x)=\ln\!\big(\mathrm{Factorial}(x)\big)= \ln(x!)$
  +
  +
However, $\mathrm{lof}(z)$ does not have multiple cut lines (except that along $z\!\le -1$, as
  +
$\ln(z!)$ has; in such a way, [[lof]] is holomorphic in the most of the complex plane.
  +
  +
==[[C++]] generator of curves==
  +
  +
// Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[fac.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 "fac.cin"
  +
//#include "facp.cin"
  +
//#include "afacc.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=401,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("lofma.eps","w");ado(o,1008,1008);
  +
fprintf(o,"504 504 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
  +
DO(m,M1) X[m]=-5.+.025*(m-.5);
  +
for(n=0;n<200;n++)Y[n]=-5.+.025*n;
  +
Y[200]=-.01;
  +
Y[201]=.01;
  +
for(n=202;n<N1;n++)Y[n]=-5.+.025*(n-1.);
  +
for(m=-5;m<6;m++){ M(m,-5)L(m,5) }
  +
for(n=-5;n<6;n++){ M( -5,n)L(5,n)}
  +
fprintf(o,".008 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("%5.2f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
// c=fac(z); //
  +
c=lof(z);
  +
p=Re(c);q=Im(c); if(p>-9999 && p<9999 && q>-9999 && q<9999 ) {g[m*N1+n]=p;f[m*N1+n]=q;}
  +
//p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); g[m*N1+n]=p;
  +
}}
  +
  +
p=1.4;q=.8;
  +
for(m=-15;m<15;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".008 W 0 .6 0 RGB S\n");
  +
for(m=0;m<15;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".008 W .9 0 0 RGB S\n");
  +
for(m=0;m<15;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".008 W 0 0 .9 RGB S\n");
  +
for(m=1;m<16;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<16;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. ),-9,9); fprintf(o,".02 W .6 0 .6 RGB S\n");
  +
for(m=-15;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),-9,9); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
for(m=1;m<16;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(-5.1,0) L(-1,0) fprintf(o,".02 W 1 1 1 RGB 0 setlinecap S\n");
  +
/*
  +
conto(o,g,w,v,X,Y,M,N, 1,-p,p); fprintf(o,".04 W .5 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 2,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 3,-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 4,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 5,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 6,-p,p); fprintf(o,".05 W 1 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 7,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 8,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N, 9,-p,p); fprintf(o,".05 W 0 .8 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,10,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,11,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,12,-p,p); fprintf(o,".05 W 0 0 1 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,13,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,14,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
  +
conto(o,g,w,v,X,Y,M,N,15,-p,p); fprintf(o,".05 W 1 0 1 RGB S\n");
  +
*/
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf lofma.eps");
  +
system( "open lofma.pdf");
  +
}
  +
//</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
%<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\paperwidth 1036pt
  +
\paperheight 1032pt
  +
\topmargin -96pt
  +
\oddsidemargin -68pt
  +
\pagestyle{empty}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\parindent 0pt
  +
\textwidth 1800px
  +
\textheight 1900px
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
\begin{picture}(1008,1008)
  +
\put(20,10){\includegraphics{lofma}}
  +
%\put(20,10){\includegraphics{hermiga6ma}}
  +
%\put(20,10){\includegraphics{hermiten6draft}}
  +
\put(4,1005){\sx{2.3}{$y$}}
  +
\put(4,905){\sx{2.2}{$4$}}
  +
\put(4,805){\sx{2.2}{$3$}}
  +
\put(4,705){\sx{2.2}{$2$}}
  +
\put(4,605){\sx{2.2}{$1$}}
  +
\put(4,505){\sx{2.2}{$0$}}
  +
\put(-13,405){\sx{2.2}{$-1$}}
  +
\put(-13,306){\sx{2.2}{$-2$}}
  +
\put(-13,206){\sx{2.2}{$-3$}}
  +
\put(-13,106){\sx{2.2}{$-4$}}
  +
\put(-13,6){\sx{2.2}{$-5$}}
  +
%\put( 0,-8){\sx{2}{$-3$}}
  +
\put( 3,-8){\sx{2.2}{$-5$}}
  +
\put(102,-8){\sx{2.2}{$-4$}}
  +
\put(202,-8){\sx{2.2}{$-3$}}
  +
\put(302,-8){\sx{2.2}{$-2$}}
  +
\put(402,-8){\sx{2.2}{$-1$}}
  +
\put(520,-8){\sx{2.2}{$0$}}
  +
\put(620,-8){\sx{2.2}{$1$}}
  +
\put(720,-8){\sx{2.2}{$2$}}
  +
\put(820,-8){\sx{2.2}{$3$}}
  +
\put(920,-8){\sx{2.2}{$4$}}
  +
\put(1014,-8){\sx{2.3}{$x$}}
  +
%
  +
\put(270,940){\sx{3}{\rot{38}$u\!=\!-9$\ero}}
  +
\put(300,912){\sx{3}{\rot{38}$u\!=\!-8$\ero}}
  +
\put(330,884){\sx{3}{\rot{37}$u\!=\!-7$\ero}}
  +
\put(360,854){\sx{3}{\rot{34}$u\!=\!-6$\ero}}
  +
\put(388,820){\sx{3}{\rot{32}$u\!=\!-5$\ero}}
  +
\put(418,780){\sx{3}{\rot{32}$u\!=\!-4$\ero}}
  +
\put(444,740){\sx{3}{\rot{27}$u\!=\!-3$\ero}}
  +
\put(482,694){\sx{3}{\rot{27}$u\!=\!-2$\ero}}
  +
\put(530,634){\sx{3}{\rot{27}$u\!=\!-1$\ero}}
  +
\put(640,546){\sx{3}{\rot{61}$u\!=\!0$\ero}}
  +
\put(475,445){\sx{3}{\rot{36}$u\!=\!0$\ero}}%
  +
\put(765,516){\sx{3}{\rot{82}$u\!=\!1$\ero}}
  +
\put(852,492){\sx{3}{\rot{87}$u\!=\!2$\ero}}
  +
\put(923,490){\sx{3}{\rot{87}$u\!=\!3$\ero}}
  +
\put(988,490){\sx{3}{\rot{87}$u\!=\!4$\ero}}
  +
%
  +
\put(830,982){\sx{3}{\rot{-26}$v\!=\!7$\ero}}
  +
\put(810,936){\sx{3}{\rot{-27}$v\!=\!6$\ero}}
  +
\put(790,886){\sx{3}{\rot{-27}$v\!=\!5$\ero}}
  +
\put(760,837){\sx{3}{\rot{-28}$v\!=\!4$\ero}}
  +
\put(730,781){\sx{3}{\rot{-28}$v\!=\!3$\ero}}
  +
\put(706,716){\sx{3}{\rot{-28}$v\!=\!2$\ero}}
  +
\put(684,629){\sx{3}{\rot{-22}$v\!=\!1$\ero}}
  +
\put(644,506){\sx{3}{\rot{0}$v\!=\!0$\ero}}
  +
%
  +
\put(146,383){\sx{3}{\rot{67}$v\!=\!9$\ero}}
  +
\put(178,370){\sx{3}{\rot{68}$v\!=\!8$\ero}}
  +
\put(209,357){\sx{3}{\rot{68}$v\!=\!7$\ero}}
  +
\put(243,346){\sx{3}{\rot{69}$v\!=\!6$\ero}}
  +
\put(279,336){\sx{3}{\rot{70}$v\!=\!5$\ero}}
  +
\put(316,324){\sx{3}{\rot{70}$v\!=\!4$\ero}}
  +
\put(356,314){\sx{3}{\rot{70}$v\!=\!3$\ero}}
  +
\put(401,302){\sx{3}{\rot{70}$v\!=\!2$\ero}}
  +
\put(452,292){\sx{3}{\rot{70}$v\!=\!1$\ero}}
  +
\put(504,272){\sx{3}{\rot{63}$v\!=\!0$\ero}}
  +
\put(547,230){\sx{3}{\rot{51}$v\!=\!-1$\ero}}
  +
\put(590,192){\sx{3}{\rot{43}$v\!=\!-2$\ero}}
  +
\put(628,154){\sx{3}{\rot{37}$v\!=\!-3$\ero}}
  +
\put(656,104){\sx{3}{\rot{33}$v\!=\!-4$\ero}}
  +
\put(684,62){\sx{3}{\rot{30}$v\!=\!-5$\ero}}
  +
\put(714,18){\sx{3}{\rot{30}$v\!=\!-6$\ero}}
  +
%
  +
\put(346,510){\sx{2.4}{\bf cut}}
  +
\end{picture}
  +
\end{document}
  +
%</nowiki></nomathjax></poem>
  +
  +
==References==
  +
  +
<references/>
  +
  +
[[Category:Complex map]]
  +
[[Category:Factorial]]
  +
[[Category:lof]]
  +
[[Category:Norm of oscillator function]]
  +
[[Category:Oscillator function]]

Latest revision as of 08:41, 1 December 2018

Complex map of function lof,

$u\!+\!\mathrm i v=\mathrm{lof}(x\!+\!\mathrm i y)$

in vicinity of the real axis, lof is just logarithm of Factorial of its argument; in particular, for real $x$,

$\mathrm{lof}(x)=\ln\!\big(\mathrm{Factorial}(x)\big)= \ln(x!)$

However, $\mathrm{lof}(z)$ does not have multiple cut lines (except that along $z\!\le -1$, as $\ln(z!)$ has; in such a way, lof is holomorphic in the most of the complex plane.

C++ generator of curves

// Files ado.cin, conto.cin, fac.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 "fac.cin"
//#include "facp.cin"
//#include "afacc.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=401,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("lofma.eps","w");ado(o,1008,1008);
fprintf(o,"504 504 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
DO(m,M1) X[m]=-5.+.025*(m-.5);
for(n=0;n<200;n++)Y[n]=-5.+.025*n;
                Y[200]=-.01;
                Y[201]=.01;
for(n=202;n<N1;n++)Y[n]=-5.+.025*(n-1.);
for(m=-5;m<6;m++){ M(m,-5)L(m,5) }
for(n=-5;n<6;n++){ M( -5,n)L(5,n)}
fprintf(o,".008 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("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=fac(z); //
c=lof(z);
      p=Re(c);q=Im(c); if(p>-9999 && p<9999 && q>-9999 && q<9999 ) {g[m*N1+n]=p;f[m*N1+n]=q;}
//p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); g[m*N1+n]=p;
                       }}

p=1.4;q=.8;
for(m=-15;m<15;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".008 W 0 .6 0 RGB S\n");
for(m=0;m<15;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".008 W .9 0 0 RGB S\n");
for(m=0;m<15;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".008 W 0 0 .9 RGB S\n");
for(m=1;m<16;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<16;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. ),-9,9); fprintf(o,".02 W .6 0 .6 RGB S\n");
for(m=-15;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),-9,9); fprintf(o,".02 W 0 0 0 RGB S\n");
for(m=1;m<16;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(-5.1,0) L(-1,0) fprintf(o,".02 W 1 1 1 RGB 0 setlinecap S\n");
/*
 conto(o,g,w,v,X,Y,M,N, 1,-p,p); fprintf(o,".04 W .5 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 2,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 3,-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 4,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 5,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 6,-p,p); fprintf(o,".05 W 1 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 7,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 8,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, 9,-p,p); fprintf(o,".05 W 0 .8 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,10,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,11,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,12,-p,p); fprintf(o,".05 W 0 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,13,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,14,-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,15,-p,p); fprintf(o,".05 W 1 0 1 RGB S\n");
*/

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

Latex generator of labels

%


\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 1036pt
\paperheight 1032pt
\topmargin -96pt
\oddsidemargin -68pt
\pagestyle{empty}
\usepackage{graphicx}
\usepackage{rotating}
\parindent 0pt
\textwidth 1800px
\textheight 1900px
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(1008,1008)
\put(20,10){\includegraphics{lofma}}
%\put(20,10){\includegraphics{hermiga6ma}}
%\put(20,10){\includegraphics{hermiten6draft}}
\put(4,1005){\sx{2.3}{$y$}}
\put(4,905){\sx{2.2}{$4$}}
\put(4,805){\sx{2.2}{$3$}}
\put(4,705){\sx{2.2}{$2$}}
\put(4,605){\sx{2.2}{$1$}}
\put(4,505){\sx{2.2}{$0$}}
\put(-13,405){\sx{2.2}{$-1$}}
\put(-13,306){\sx{2.2}{$-2$}}
\put(-13,206){\sx{2.2}{$-3$}}
\put(-13,106){\sx{2.2}{$-4$}}
\put(-13,6){\sx{2.2}{$-5$}}
%\put( 0,-8){\sx{2}{$-3$}}
\put( 3,-8){\sx{2.2}{$-5$}}
\put(102,-8){\sx{2.2}{$-4$}}
\put(202,-8){\sx{2.2}{$-3$}}
\put(302,-8){\sx{2.2}{$-2$}}
\put(402,-8){\sx{2.2}{$-1$}}
\put(520,-8){\sx{2.2}{$0$}}
\put(620,-8){\sx{2.2}{$1$}}
\put(720,-8){\sx{2.2}{$2$}}
\put(820,-8){\sx{2.2}{$3$}}
\put(920,-8){\sx{2.2}{$4$}}
\put(1014,-8){\sx{2.3}{$x$}}
%
\put(270,940){\sx{3}{\rot{38}$u\!=\!-9$\ero}}
\put(300,912){\sx{3}{\rot{38}$u\!=\!-8$\ero}}
\put(330,884){\sx{3}{\rot{37}$u\!=\!-7$\ero}}
\put(360,854){\sx{3}{\rot{34}$u\!=\!-6$\ero}}
\put(388,820){\sx{3}{\rot{32}$u\!=\!-5$\ero}}
\put(418,780){\sx{3}{\rot{32}$u\!=\!-4$\ero}}
\put(444,740){\sx{3}{\rot{27}$u\!=\!-3$\ero}}
\put(482,694){\sx{3}{\rot{27}$u\!=\!-2$\ero}}
\put(530,634){\sx{3}{\rot{27}$u\!=\!-1$\ero}}
\put(640,546){\sx{3}{\rot{61}$u\!=\!0$\ero}}
\put(475,445){\sx{3}{\rot{36}$u\!=\!0$\ero}}%
\put(765,516){\sx{3}{\rot{82}$u\!=\!1$\ero}}
\put(852,492){\sx{3}{\rot{87}$u\!=\!2$\ero}}
\put(923,490){\sx{3}{\rot{87}$u\!=\!3$\ero}}
\put(988,490){\sx{3}{\rot{87}$u\!=\!4$\ero}}
%
\put(830,982){\sx{3}{\rot{-26}$v\!=\!7$\ero}}
\put(810,936){\sx{3}{\rot{-27}$v\!=\!6$\ero}}
\put(790,886){\sx{3}{\rot{-27}$v\!=\!5$\ero}}
\put(760,837){\sx{3}{\rot{-28}$v\!=\!4$\ero}}
\put(730,781){\sx{3}{\rot{-28}$v\!=\!3$\ero}}
\put(706,716){\sx{3}{\rot{-28}$v\!=\!2$\ero}}
\put(684,629){\sx{3}{\rot{-22}$v\!=\!1$\ero}}
\put(644,506){\sx{3}{\rot{0}$v\!=\!0$\ero}}
%
\put(146,383){\sx{3}{\rot{67}$v\!=\!9$\ero}}
\put(178,370){\sx{3}{\rot{68}$v\!=\!8$\ero}}
\put(209,357){\sx{3}{\rot{68}$v\!=\!7$\ero}}
\put(243,346){\sx{3}{\rot{69}$v\!=\!6$\ero}}
\put(279,336){\sx{3}{\rot{70}$v\!=\!5$\ero}}
\put(316,324){\sx{3}{\rot{70}$v\!=\!4$\ero}}
\put(356,314){\sx{3}{\rot{70}$v\!=\!3$\ero}}
\put(401,302){\sx{3}{\rot{70}$v\!=\!2$\ero}}
\put(452,292){\sx{3}{\rot{70}$v\!=\!1$\ero}}
\put(504,272){\sx{3}{\rot{63}$v\!=\!0$\ero}}
\put(547,230){\sx{3}{\rot{51}$v\!=\!-1$\ero}}
\put(590,192){\sx{3}{\rot{43}$v\!=\!-2$\ero}}
\put(628,154){\sx{3}{\rot{37}$v\!=\!-3$\ero}}
\put(656,104){\sx{3}{\rot{33}$v\!=\!-4$\ero}}
\put(684,62){\sx{3}{\rot{30}$v\!=\!-5$\ero}}
\put(714,18){\sx{3}{\rot{30}$v\!=\!-6$\ero}}
%
\put(346,510){\sx{2.4}{\bf cut}}
\end{picture}
\end{document}
%

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