Difference between revisions of "File:Vladi08.jpg"

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Agreement of approximations of tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.
−
Importing image file
  
  +
  +
The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.
  +
  +
Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.
  +
  +
$\displaystyle
  +
{\rm fse}(z)=\left\{ \begin{array}{ccrcccr}
  +
{\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\
  +
{\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\
  +
{\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\
  +
{\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\
  +
{\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5
  +
\end{array}
  +
\right.$
  +
  +
Mnemonics for the name of the approximation: Fast Super Exponent.
  +
  +
The agreement plotted is
  +
  +
$\displaystyle
  +
D=D_{8}(z)=-\lg\left( \frac
  +
{|\mathrm{fse}(z)-F_{4}(z)|}
  +
{|\mathrm{fse}(z)|+|F_{4}(z)|}
  +
\right)
  +
$
  +
  +
where $F_4$ denotes the 4th [[ackermann]], id set, natural [[tetration]] evaluated through the Cauchi integral
  +
<ref>
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf <br>
  +
http://mizugadro.mudns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  +
</ref>.
  +
  +
Usage: this is figure 14.11 of the book [[Суперфункции]] (2014, 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 Publishing]], 2014.
  +
</ref>; the English version is in preparation in 2015.
  +
  +
First time published in the [[Vladikavkaz Matehmatical Journal]]
  +
<ref>
  +
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
  +
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
Figure 8.
  +
</ref>.
  +
  +
==Refereces==
  +
<references/>
  +
  +
==[[C++]] generator of the first picture==
  +
[[Fsexp.cin]],
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[plodi.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++)
  +
#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 "superex.cin"
  +
#include "fsexp.cin"
  +
#include "f4natu.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
  +
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#include "superlo.cin"
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
  +
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
  +
  +
int M=100,M1=M+1;
  +
int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1) X[m]=-4.+.08*(m-.5);
  +
DO(n,N1) Y[n]=-1.+.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n=0;n<7;n++) { M( -3,n)L(3,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=fsexp(z);
  +
d=F4natu(z);
  +
p=abs(c-d);///(abs(c)+abs(d));
  +
p=-log(p)/log(10.);
  +
// p=Re(c); q=Im(c);
  +
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
  +
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
  +
}}
  +
#include"plodi.cin"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
// system( "gv fig08a.eps");
  +
system("epstopdf vladi08a.eps");
  +
system( "open vladi08a.pdf"); // for linux
  +
// getchar(); system("killall Preview");// for macintosh
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of the second picture==
  +
  +
[[vladif5c.cin]] also 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 "superex.cin"
  +
#include "fsexp.cin"
  +
//#include "f4natu.cin"
  +
#include "vladif5c.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
  +
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#include "superlo.cin"
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
  +
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
  +
  +
int M=100,M1=M+1;
  +
int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1)X[m]=-4.+.08*(m-.5);
  +
DO(n,N1)Y[n]=-1.+.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n= 0;n<7;n++) { M( -3,n)L(3,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=fsexp(z);
  +
d=F5(z);
  +
p=abs(c-d);///(abs(c)+abs(d));
  +
p=-log(p)/log(10.);
  +
// p=Re(c); q=Im(c);
  +
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
  +
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
  +
}}
  +
  +
#include"plodi.cin"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf vladi08b.eps");
  +
system( "open vladi08b.pdf");//linux
  +
//getchar(); system("killall Preview");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] combiner==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\usepackage{geometry}
  +
\paperwidth 378px
  +
%\paperheight 134px
  +
\paperheight 180px
  +
\topmargin -104pt
  +
\oddsidemargin -94pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
  +
\sx{2.2}{\begin{picture}(90,80)
  +
%\put(0,0){\includegraphics{figsexpF4}}
  +
%\put(0,0){\includegraphics{figcf4small}}
  +
\put(0,0){\includegraphics{vladi08a}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put(70 , 6){\sx{.45}{$\Re(z)$}}
  +
\put(60 , 6){\sx{.5}{$2$}}
  +
\put(40 , 6){\sx{.5}{$0$}}
  +
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(25,62){\sx{.5}{$D\!>\!14$}}
  +
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
  +
\end{picture}}
  +
\sx{2.2}{\begin{picture}(80,80)
  +
%\put(0,0){\includegraphics{figsexpF5}}
  +
%\put(0,0){\includegraphics{figcf5small}}
  +
\put(0,0){\includegraphics{vladi08b}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put(70 , 6){\sx{.45}{$\Re(z)$}}
  +
\put(60 , 6){\sx{.5}{$2$}}
  +
\put(40 , 6){\sx{.5}{$0$}}
  +
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(31,56){\sx{.5}{$D\!>\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
  +
\end{picture}}
  +
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
  +
[[Category:Agreement]]
  +
[[Category:Book]]
  +
[[Category:Agreement]]
  +
[[Category:BookMap]]
  +
[[Category:Tetration]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:56, 1 December 2018

Agreement of approximations of tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.

The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.

Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.

$\displaystyle {\rm fse}(z)=\left\{ \begin{array}{ccrcccr} {\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\ {\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\ {\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\ {\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\ {\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5 \end{array} \right.$

Mnemonics for the name of the approximation: Fast Super Exponent.

The agreement plotted is

$\displaystyle D=D_{8}(z)=-\lg\left( \frac

\right) $ where $F_4$ denotes the 4th ackermann, id set, natural tetration evaluated through the Cauchi integral [1]. Usage: this is figure 14.11 of the book Суперфункции (2014, In Russian) [2]; the English version is in preparation in 2015. First time published in the Vladikavkaz Matehmatical Journal[3].

Refereces

  1. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
    http://mizugadro.mudns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  2. ↑ 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 Publishing, 2014.
  3. ↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. Figure 8.

C++ generator of the first picture

Fsexp.cin, ado.cin, conto.cin, plodi.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++)
#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 "superex.cin"
#include "fsexp.cin"
#include "f4natu.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

  int M=100,M1=M+1;
  int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1) Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n=0;n<7;n++) { M( -3,n)L(3,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=fsexp(z);
        d=F4natu(z);
        p=abs(c-d);///(abs(c)+abs(d));
        p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}
#include"plodi.cin"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
// system( "gv fig08a.eps");
        system("epstopdf vladi08a.eps");
        system( "open vladi08a.pdf"); // for linux
// getchar(); system("killall Preview");// for macintosh
}

C++ generator of the second picture

vladif5c.cin also 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 "superex.cin"
#include "fsexp.cin"
//#include "f4natu.cin"
#include "vladif5c.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

  int M=100,M1=M+1;
  int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1)X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n= 0;n<7;n++) { M( -3,n)L(3,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=fsexp(z);
        d=F5(z);
        p=abs(c-d);///(abs(c)+abs(d));
        p=-log(p)/log(10.);
// p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
                        }}

#include"plodi.cin"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi08b.eps");
        system( "open vladi08b.pdf");//linux
//getchar(); system("killall Preview");
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 378px
%\paperheight 134px
\paperheight 180px
\topmargin -104pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\sx{2.2}{\begin{picture}(90,80)
%\put(0,0){\includegraphics{figsexpF4}}
%\put(0,0){\includegraphics{figcf4small}}
\put(0,0){\includegraphics{vladi08a}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(25,62){\sx{.5}{$D\!>\!14$}}
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\sx{2.2}{\begin{picture}(80,80)
%\put(0,0){\includegraphics{figsexpF5}}
%\put(0,0){\includegraphics{figcf5small}}
\put(0,0){\includegraphics{vladi08b}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(31,56){\sx{.5}{$D\!>\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}

\end{document}

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Date/TimeThumbnailDimensionsUserComment
current06:14, 1 December 2018Thumbnail for version as of 06:14, 1 December 2018787 × 375 (117 KB)Maintenance script (talk | contribs)Importing image file

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