Difference between revisions of "File:Nembraplot.jpg"

From TORI
Jump to navigation Jump to search
(Importing image file)
 
 
Line 1: Line 1:
  +
[[Parametric plot]] of position of the [[branch point]] of the [[ArcNem]] function:
−
Importing image file
  
  +
  +
$x\!+\!\mathrm i y = \mathrm{NemBra}(q)$
  +
  +
for positive $q$. Function [[NemBra]] is solution $z=\mathrm{NemBra}(q)$
  +
of equation
  +
  +
$\mathrm{Nem}_q'(z)=0$
  +
  +
where the [[Nemtsov function]] Nem is special polynomial
  +
  +
$\mathrm{Nem}_q(z)=z+z^3+qz^4$
  +
  +
and, correspondently,
  +
  +
$\mathrm{Nem}_q'(z)=1+3z^2+4qz^3$
  +
  +
Function [[NemBra]] is necessary for implementation of function [[ArqNem]], that, in its turn,
  +
happen to be necessary for implementation of the non-integer iterates of the [[Nemtsov function]],
  +
suggested as example of [[exotic iterate]]s for the book [[Superfunctions]].
  +
  +
==References==
  +
<references/>
  +
  +
==[[Mathematica]] generator of the solution==
  +
  +
<poem><nomathjax><nowiki>
  +
T[z_] = z + z^3 + q z^4
  +
s123 = Solve[T'[z] == 0, z]
  +
s1[q_] = ReplaceAll[z, Extract[s123, 1]]
  +
Simplify[T'[ 1/4 (-(1/q) + 1/( q (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)) + (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)/q)]]
  +
Print[N[Normal[Simplify[Series[s1[q], {q, 0, 7}]]], 16]]
  +
Simplify[Series[s1[q], {q, Infinity, 4}]]
  +
</nowiki></nomathjax></poem>
  +
  +
  +
==[[C++]] generator of curve==
  +
  +
<poem><nomathjax><nowiki>
  +
#include <stdio.h>
  +
#include <math.h>
  +
#include <stdlib.h>
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
  +
void ado(FILE *O, int X, int Y)
  +
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
  +
fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
  +
fprintf(O,"/M {moveto} bind def\n");
  +
fprintf(O,"/L {lineto} bind def\n");
  +
fprintf(O,"/S {stroke} bind def\n");
  +
fprintf(O,"/s {show newpath} bind def\n");
  +
fprintf(O,"/C {closepath} bind def\n");
  +
fprintf(O,"/F {fill} bind def\n");
  +
fprintf(O,"/o {.002 0 360 arc C S} bind def\n");
  +
fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
  +
fprintf(O,"/W {setlinewidth} bind def\n");
  +
fprintf(O,"/RGB {setrgbcolor} bind def\n");}
  +
//#include "ado.cin"
  +
  +
z_type nem(DB q,z_type z){ return z*(1.+z*z*(1.+z*q)); }
  +
z_type nem1(DB q,z_type z){ return 1.+z*z*(3.+z*(4.*q)); }
  +
z_type nembra0(DB q){
  +
return -0.5773502691896258*I+
  +
q*(2./9+
  +
q*(0.2138334330331947*I+
  +
q*(-0.2633744855967078 +
  +
q*(-0.3658927631901332*I+
  +
q*(0.5462581923487273 +
  +
q*(0.8556857213229570*I+
  +
q*(-1.387322393266609
  +
))))))) ;}
  +
  +
z_type nembrao(DB q){ z_type x,y,z,s; x=conj(pow(z_type(-.25/q,0.),1./3.)); y=x*x; z=y*y;
  +
s=1.+y*(1.+y*(1.+y*(2./3.+z*(-2./3.+y*(-7./9.+z*(11./9.+y*(130./91.)
  +
) ) ) ) ) );
  +
return s*x;}
  +
  +
z_type nembra(DB q){
  +
if(fabs(q)<.021) return nembra0(q);
  +
if(fabs(q) >20.) return nembrao(q);
  +
z_type Q,v,V; Q=q*q; v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q); V=pow(v,1./3.); return (.25/q)*(-1.+1./V+V); }
  +
  +
int main(){ int m,n; z_type z,c,d,e; DB x,y,q,u,v;
  +
  +
DO(n,41){ q=-.04+.01*n; c=nembra0(q); d=nem(q,c); e=nem1(q,c);
  +
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
  +
printf("\n");
  +
DO(n,41){ q=.1*n; c=nembra(q); d=nem(q,c); e=nem1(q,c);
  +
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
  +
printf("\n");
  +
DO(n,41){ q=1.+1.*n; c=nembrao(q); d=nem(q,c); e=nem1(q,c);
  +
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
  +
  +
//FILE *o; o=fopen("08.eps","w"); ado(o,154,604);
  +
FILE *o; o=fopen("nembraplo.eps","w"); ado(o,154,604);
  +
fprintf(o,"2 2 translate 1000 1000 scale 2 setlinecap\n");
  +
#define M(x,y) fprintf(o,"%8.6f %8.6f M\n",(0.+x),(0.+y));
  +
#define L(x,y) fprintf(o,"%8.6f %8.6f L\n",(0.+x),(0.+y));
  +
#define o(x,y) fprintf(o,"%8.6f %8.6f o\n",(0.+x),(0.+y));
  +
  +
for(n=0;n<61;n+=5){ M(0,.01*n) L(.15,.01*n) }
  +
for(n=0;n<31;n+=5){ M(.01*n,0) L(.01*n,.6) }
  +
fprintf(o,".0006 W S 1 setlinejoin\n");
  +
fprintf(o,"1 0 0 RGB .001 W\n");
  +
  +
for(n= 1;n<10;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
for(n=11;n<20;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
for(n=21;n<30;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
fprintf(o,"0 .5 0 RGB\n");
  +
for(n= 1;n<20;n++){ q=1.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
fprintf(o,"0 0 1 RGB\n");
  +
for(n= 1;n<10;n++){ q=10.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
fprintf(o,".5 0 .8 RGB\n");
  +
for(n= 1;n<10;n++){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
fprintf(o,"0 0 0 RGB\n");
  +
for(n= 0;n<11;n+=10){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
  +
q=10000.; c=nembra(q); x=Re(c); y=-Im(c); o(x,y);
  +
  +
for(n= 0;n<400;n++){ q=-.006+.04*(n+.01*n*n+.001*n*n*n); c=nembra(q); x=Re(c); y=-Im(c); if(n==0) M(x,y) else L(x,y);
  +
printf("%9.3lf %16.6lf %16.6lf\n",q,x,y);
  +
}
  +
L(0,0)
  +
fprintf(o,"0 0 0 RGB .001 W 0 setlinecap S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer\n",'%','%');
  +
fclose(o);
  +
  +
system("epstopdf nembraplo.eps");
  +
system("open nembraplo.pdf");
  +
getchar();
  +
return 0;}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\paperwidth 174pt
  +
\paperheight 620pt
  +
\topmargin -100pt
  +
\oddsidemargin -72pt
  +
\textheight 800px
  +
\parindent 0pt
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\begin{document}
  +
\begin{picture}(170,610)
  +
\put(20,14){\ing{nembraplo}}
  +
\put(8,612){\sx{1.4}{$y$}}
  +
\put(0,512){\sx{1.4}{$0.5$}}
  +
\put(0,412){\sx{1.4}{$0.4$}}
  +
\put(0,312){\sx{1.4}{$0.3$}}
  +
\put(0,212){\sx{1.4}{$0.2$}}
  +
\put(0,112){\sx{1.4}{$0.1$}}
  +
\put(8,12){\sx{1.4}{$0$}}
  +
\put(20,1){\sx{1.4}{$0$}}
  +
\put(60,1){\sx{1.4}{$0.05$}}
  +
\put(112,1){\sx{1.4}{$0.1$}}
  +
\put(165,2){\sx{1.4}{$x$}}
  +
%\put(18,601){\sx{1.1}{$q\!=\!0$}}
  +
\put(0,583){\sx{1.1}{$q\!=\!0$}}
  +
%\put(63,593){\sx{1.1}{$q\!=\!0.2$}}
  +
\put(30,576){\sx{1.1}{$q\!=\!0.2$}}
  +
%\put(97,574){\sx{1.1}{$q\!=\!0.4$}}
  +
\put(62,558){\sx{1.1}{$q\!=\!0.4$}}
  +
%\put(122,550){\sx{1.1}{$q\!=\!0.6$}}
  +
\put(85,535){\sx{1.1}{$q\!=\!0.6$}}
  +
%\put(137,528){\sx{1.1}{$q\!=\!0.8$}}
  +
\put(100,514){\sx{1.1}{$q\!=\!0.8$}}
  +
%\put(148,506){\sx{1.1}{$q\!=\!1$}}
  +
\put(121,494){\sx{1.1}{$q\!=\!1$}}
  +
\put(137,424){\sx{1.1}{$q\!=\!2$}}
  +
\put(138,380){\sx{1.1}{$q\!=\!3$}}
  +
\put(136,350){\sx{1.1}{$q\!=\!4$}}
  +
\put(133,330){\sx{1.1}{$q\!=\!5$}}
  +
\put(129,310){\sx{1.1}{$q\!=\!6$}}
  +
\put(122,286){\sx{1.1}{$q\!=\!8$}}
  +
\put(110,268){\sx{1.1}{$q\!=\!10$}}
  +
\put( 90,214){\sx{1.1}{$q\!=\!20$}}
  +
\put( 82,192){\sx{1.1}{$q\!=\!30$}}
  +
\put( 88,127){\sx{1.1}{$q\!=\!100$}}
  +
\put( 77,104){\sx{1.1}{$q\!=\!200$}}
  +
\put( 54,63){\sx{1.1}{$q\!=\!1000$}}
  +
\put( 39,36){\sx{1.1}{$q\!=\!10000$}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:Branch point]]
  +
[[Category:Nemtsov function]]
  +
[[Category:Parametric plot]]

Latest revision as of 08:44, 1 December 2018

Parametric plot of position of the branch point of the ArcNem function:

$x\!+\!\mathrm i y = \mathrm{NemBra}(q)$

for positive $q$. Function NemBra is solution $z=\mathrm{NemBra}(q)$ of equation

$\mathrm{Nem}_q'(z)=0$

where the Nemtsov function Nem is special polynomial

$\mathrm{Nem}_q(z)=z+z^3+qz^4$

and, correspondently,

$\mathrm{Nem}_q'(z)=1+3z^2+4qz^3$

Function NemBra is necessary for implementation of function ArqNem, that, in its turn, happen to be necessary for implementation of the non-integer iterates of the Nemtsov function, suggested as example of exotic iterates for the book Superfunctions.

References


Mathematica generator of the solution


T[z_] = z + z^3 + q z^4
s123 = Solve[T'[z] == 0, z]
s1[q_] = ReplaceAll[z, Extract[s123, 1]]
Simplify[T'[ 1/4 (-(1/q) + 1/( q (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)) + (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)/q)]]
Print[N[Normal[Simplify[Series[s1[q], {q, 0, 7}]]], 16]]
Simplify[Series[s1[q], {q, Infinity, 4}]]


C++ generator of curve


#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#define DB double
#define DO(x,y) for(x=0;x<y;x++)

void ado(FILE *O, int X, int Y)
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.002 0 360 arc C S} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}
//#include "ado.cin"

z_type nem(DB q,z_type z){ return z*(1.+z*z*(1.+z*q)); }
z_type nem1(DB q,z_type z){ return 1.+z*z*(3.+z*(4.*q)); }
z_type nembra0(DB q){
return -0.5773502691896258*I+
        q*(2./9+
        q*(0.2138334330331947*I+
        q*(-0.2633744855967078 +
        q*(-0.3658927631901332*I+
        q*(0.5462581923487273 +
        q*(0.8556857213229570*I+
        q*(-1.387322393266609
        ))))))) ;}

z_type nembrao(DB q){ z_type x,y,z,s; x=conj(pow(z_type(-.25/q,0.),1./3.)); y=x*x; z=y*y;
s=1.+y*(1.+y*(1.+y*(2./3.+z*(-2./3.+y*(-7./9.+z*(11./9.+y*(130./91.)
       ) ) ) ) ) );
return s*x;}

z_type nembra(DB q){
if(fabs(q)<.021) return nembra0(q);
if(fabs(q) >20.) return nembrao(q);
z_type Q,v,V; Q=q*q; v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q); V=pow(v,1./3.); return (.25/q)*(-1.+1./V+V); }

int main(){ int m,n; z_type z,c,d,e; DB x,y,q,u,v;

DO(n,41){ q=-.04+.01*n; c=nembra0(q); d=nem(q,c); e=nem1(q,c);
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
printf("\n");
DO(n,41){ q=.1*n; c=nembra(q); d=nem(q,c); e=nem1(q,c);
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
printf("\n");
DO(n,41){ q=1.+1.*n; c=nembrao(q); d=nem(q,c); e=nem1(q,c);
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}

//FILE *o; o=fopen("08.eps","w"); ado(o,154,604);
FILE *o; o=fopen("nembraplo.eps","w"); ado(o,154,604);
fprintf(o,"2 2 translate 1000 1000 scale 2 setlinecap\n");
#define M(x,y) fprintf(o,"%8.6f %8.6f M\n",(0.+x),(0.+y));
#define L(x,y) fprintf(o,"%8.6f %8.6f L\n",(0.+x),(0.+y));
#define o(x,y) fprintf(o,"%8.6f %8.6f o\n",(0.+x),(0.+y));

for(n=0;n<61;n+=5){ M(0,.01*n) L(.15,.01*n) }
for(n=0;n<31;n+=5){ M(.01*n,0) L(.01*n,.6) }
fprintf(o,".0006 W S 1 setlinejoin\n");
fprintf(o,"1 0 0 RGB .001 W\n");

for(n= 1;n<10;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
for(n=11;n<20;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
for(n=21;n<30;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 .5 0 RGB\n");
for(n= 1;n<20;n++){ q=1.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 0 1 RGB\n");
for(n= 1;n<10;n++){ q=10.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,".5 0 .8 RGB\n");
for(n= 1;n<10;n++){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 0 0 RGB\n");
for(n= 0;n<11;n+=10){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); }
                      q=10000.; c=nembra(q); x=Re(c); y=-Im(c); o(x,y);

for(n= 0;n<400;n++){ q=-.006+.04*(n+.01*n*n+.001*n*n*n); c=nembra(q); x=Re(c); y=-Im(c); if(n==0) M(x,y) else L(x,y);
printf("%9.3lf %16.6lf %16.6lf\n",q,x,y);
}
L(0,0)
fprintf(o,"0 0 0 RGB .001 W 0 setlinecap S\n");

fprintf(o,"showpage\n%c%cTrailer\n",'%','%');
fclose(o);

system("epstopdf nembraplo.eps");
system("open nembraplo.pdf");
getchar();
return 0;}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 174pt
\paperheight 620pt
\topmargin -100pt
\oddsidemargin -72pt
\textheight 800px
\parindent 0pt
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(170,610)
\put(20,14){\ing{nembraplo}}
\put(8,612){\sx{1.4}{$y$}}
\put(0,512){\sx{1.4}{$0.5$}}
\put(0,412){\sx{1.4}{$0.4$}}
\put(0,312){\sx{1.4}{$0.3$}}
\put(0,212){\sx{1.4}{$0.2$}}
\put(0,112){\sx{1.4}{$0.1$}}
\put(8,12){\sx{1.4}{$0$}}
\put(20,1){\sx{1.4}{$0$}}
\put(60,1){\sx{1.4}{$0.05$}}
\put(112,1){\sx{1.4}{$0.1$}}
\put(165,2){\sx{1.4}{$x$}}
%\put(18,601){\sx{1.1}{$q\!=\!0$}}
\put(0,583){\sx{1.1}{$q\!=\!0$}}
%\put(63,593){\sx{1.1}{$q\!=\!0.2$}}
\put(30,576){\sx{1.1}{$q\!=\!0.2$}}
%\put(97,574){\sx{1.1}{$q\!=\!0.4$}}
\put(62,558){\sx{1.1}{$q\!=\!0.4$}}
%\put(122,550){\sx{1.1}{$q\!=\!0.6$}}
\put(85,535){\sx{1.1}{$q\!=\!0.6$}}
%\put(137,528){\sx{1.1}{$q\!=\!0.8$}}
\put(100,514){\sx{1.1}{$q\!=\!0.8$}}
%\put(148,506){\sx{1.1}{$q\!=\!1$}}
\put(121,494){\sx{1.1}{$q\!=\!1$}}
\put(137,424){\sx{1.1}{$q\!=\!2$}}
\put(138,380){\sx{1.1}{$q\!=\!3$}}
\put(136,350){\sx{1.1}{$q\!=\!4$}}
\put(133,330){\sx{1.1}{$q\!=\!5$}}
\put(129,310){\sx{1.1}{$q\!=\!6$}}
\put(122,286){\sx{1.1}{$q\!=\!8$}}
\put(110,268){\sx{1.1}{$q\!=\!10$}}
\put( 90,214){\sx{1.1}{$q\!=\!20$}}
\put( 82,192){\sx{1.1}{$q\!=\!30$}}
\put( 88,127){\sx{1.1}{$q\!=\!100$}}
\put( 77,104){\sx{1.1}{$q\!=\!200$}}
\put( 54,63){\sx{1.1}{$q\!=\!1000$}}
\put( 39,36){\sx{1.1}{$q\!=\!10000$}}
\end{picture}
\end{document}

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current06:13, 1 December 2018Thumbnail for version as of 06:13, 1 December 2018361 × 1,286 (77 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following 2 pages use this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:Nembraplot.jpg&oldid=20388"