Difference between revisions of "File:Ack4bFragment.jpg"

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Importing image file
 +
Fragment of image
  +
http://mizugadro.mydns.jp/t/index.php/File:Ack4b600.jpg
  +
  +
[[Complex map]] of [[tetration]] to base $b\!=\!2$
  +
  +
$u\!+\!\mathrm i v=\mathrm{tet}_b(x\!+\!\mathrm i y)$
  +
  +
==Usage==
  +
This is fragment of image fig.3b (with improved resolution) of publication "Evaluation of holomorphic ackermanns", 2014.
  +
<ref>
  +
http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf <br>
  +
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf
  +
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.
  +
</ref>
  +
==[[C++]] Generator of map]==
  +
Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[fsexp.cin]]
  +
should be loaded to the working directory 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>
  +
#define z_type std::complex<double>
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
  +
//#include "tet2f4c.cin"
  +
#include "conto.cin"
  +
  +
#include "filog.cin"
  +
  +
//z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
  +
z_type b=M_E;
  +
/*
  +
z_type a=log(b);
  +
z_type Zo=Filog(a);
  +
z_type Zc=conj(Filog(conj(a)));
  +
DB A=32.;
  +
*/
  +
//#include "tet2f4c.cin"
  +
#include "fsexp.cin"
  +
  +
int main(){ int j,k,m,m1,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=641,M1=M+1;
  +
int N=402,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("tet2m2.eps","w");ado(o,1604,804);
  +
// FILE *o;o=fopen("tettenm2.eps","w");ado(o,1604,804);
  +
// FILE *o;o=fopen("amsfig4dFragmen.eps","w");ado(o,1604,804);
  +
// FILE *o;o=fopen("amsfig4aFragmen.eps","w");ado(o,1604,804);
  +
FILE *o;o=fopen("amsfig4bFragmen.eps","w");ado(o,1604,804);
  +
fprintf(o,"802 402 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
  +
DO(m,M1)X[m]=-8.+.05*(m-.3);
  +
DO(n,200)Y[n]=-4.+.02*n;
  +
Y[200]=-.01;
  +
Y[201]= .01;
  +
for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
  +
for(m=-8;m<9;m++){M(m,-4)L(m,4)}
  +
for(n=-4;n<5;n++){ M( -8,n)L(8,n)}
  +
fprintf(o,".008 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=990999; f[m*N1+n]=909999;}
  +
  +
DO(n,N1){y=Y[n];
  +
for(m=150;m<170;m++)
  +
{x=X[m]; //printf("%5.2f\n",x);
  +
z=z_type(x,y);
  +
// c=tetb(z);
  +
// c=F4TEN(z);
  +
// c=F4(z);
  +
c=FSEXP(z);
  +
p=Re(c);q=Im(c);
  +
if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m+k*20; if(m1>M) break;
  +
// d=exp(a*d);
  +
// d=exp(d*log(2.));
  +
d=exp(d);
  +
p=Re(d);q=Im(d);
  +
if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m-k*20; if(m1<0) break;
  +
d=log(d);
  +
// d=log(d)/a;
  +
// d=log(d)/log(2.);
  +
p=Re(d);q=Im(d);
  +
if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=20;q=1;
  +
for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .6 0 RGB S\n");
  +
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .9 0 0 RGB S\n");
  +
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .9 RGB S\n");
  +
for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n");
  +
for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .6 0 .6 RGB S\n");
  +
for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");
  +
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 1 .5 0 RGB S\n");
  +
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 0 .5 1 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
//system( "ggv fig3.eps");
  +
system("epstopdf amsfig4bFragmen.eps");
  +
system( "open amsfig4bFragmen.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] Generator of labels]==
  +
<poem><nomathjax><nowiki>
  +
\documentclass{amsproc}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {{\rm e}} %%
  +
%\newcommand \rme {{e}} %%
  +
\newcommand \rmi {{\rm i}} %%imaginary unity \newcommand \ds {\displaystyle}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing \includegraphics
  +
\usepackage{geometry}
  +
\topmargin -94pt
  +
\oddsidemargin -70pt
  +
\paperwidth 1666pt
  +
\paperheight 856pt
  +
\textwidth 1900px
  +
\textheight 900px
  +
\begin{document}
  +
\parindent 0pt
  +
  +
\newcommand \mapax {
  +
\put(18,820){\sx{5}{$y$}}
  +
\put(18,730){\sx{5}{$3$}}
  +
\put(18,630){\sx{5}{$2$}}
  +
\put(18,530){\sx{5}{$1$}}
  +
\put(18,430){\sx{5}{$0$}}
  +
\put(-14,329){\sx{5}{$-1$}}
  +
\put(-14,229){\sx{5}{$-2$}}
  +
\put(-14,129){\sx{5}{$-3$}}
  +
\put(-14, 29){\sx{5}{$-4$}}
  +
\put(14, 0){\sx{5}{$-8$}}
  +
\put(114, 0){\sx{5}{$-7$}}
  +
\put(214, 0){\sx{5}{$-6$}}
  +
\put(314, 0){\sx{5}{$-5$}}
  +
\put(414, 0){\sx{5}{$-4$}}
  +
\put(514, 0){\sx{5}{$-3$}}
  +
\put(614, 0){\sx{5}{$-2$}}
  +
\put(714, 0){\sx{5}{$-1$}}
  +
\put(844, 0){\sx{5}{$0$}}
  +
\put(944, 0){\sx{5}{$1$}}
  +
\put(1044, 0){\sx{5}{$2$}}
  +
\put(1144, 0){\sx{5}{$3$}}
  +
\put(1244, 0){\sx{5}{$4$}}
  +
\put(1344, 0){\sx{5}{$5$}}
  +
\put(1444, 0){\sx{5}{$6$}}
  +
\put(1544, 0){\sx{5}{$7$}}
  +
\put(1634, 0){\sx{5}{$x$}}
  +
}
  +
%\flushright{$b=\mathrm e \approx 2.71$}
  +
  +
{\begin{picture}(1620,850) %%%
  +
\put(50,40){\ing{amsfig4bFragmen}} \mapax
  +
\put(114,660){\sx{8}{$b=\mathrm e$}}
  +
\put(76,798){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764\!+\! 1.33723570143069 \,\mathrm i$}}
  +
\put(80,90){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764 \!-\! 1.33723570143069 \,\mathrm i$}}
  +
\put(60,434){\sx{4}{\bf cut}}
  +
\put(760,434){\sx{4}{$v\!=\!0$}}
  +
\multiput(46,550)(448,105){3}{\sx{4}{$v\!=\!1.4$}}
  +
\multiput(268,584)(448,105){3}{\sx{4}{$u\!=\!0.4$}}
  +
\multiput(46,316)(448,-105){4}{\sx{4}{$v\!=\!-1.4$}}
  +
\multiput(298,464)(448,105){4}{\sx{4}{$v\!=\!1$}}
  +
\multiput(290,404)(448,-105){4}{\sx{4}{$v\!=\!-1$}}
  +
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==Refrences==
  +
<references/>
  +
  +
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.mydns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. [[Mathematics of Computation]], 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.
  +
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
  +
D.Kouznetsov. Superexponential as special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45.
  +
  +
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. (In Russian)
  +
  +
http://mizugadro.mydns.jp/BOOK/437.pdf
  +
D.Kouznetsov. Suparfuncctions. Mizugadro, 2015. (In English)
  +
  +
http://myweb.astate.edu/wpaulsen/tetration.html
  +
William Paulsen. Tetration is repeated exponentiation. (2016). We can define $^0b = 1, ^1b = b, ^2b = b^b$, &3b = b^{b^b}$, etc. ..
  +
  +
  +
[[Category:AMS]]
  +
[[Category:Applied and Computational Mathematics]]
  +
[[Category:Book]]
  +
[[Category:Bookmap]]
  +
[[Category:Complex map]]
  +
[[Category:C++]]
  +
[[Category:Fragment]]
  +
[[Category:Latex]]
  +
[[Category:Natural tetration]]
  +
[[Category:Tetration]]

Latest revision as of 08:28, 1 December 2018

Fragment of image http://mizugadro.mydns.jp/t/index.php/File:Ack4b600.jpg

Complex map of tetration to base $b\!=\!2$

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

Usage

This is fragment of image fig.3b (with improved resolution) of publication "Evaluation of holomorphic ackermanns", 2014. [1]

C++ Generator of map]

Files ado.cin, conto.cin, fsexp.cin should be loaded to the working directory 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>
#define z_type std::complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

//#include "tet2f4c.cin"
#include "conto.cin"

#include "filog.cin"

//z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type b=M_E;
/*
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;
*/
//#include "tet2f4c.cin"
#include "fsexp.cin"

int main(){ int j,k,m,m1,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=641,M1=M+1;
 int N=402,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("tet2m2.eps","w");ado(o,1604,804);
// FILE *o;o=fopen("tettenm2.eps","w");ado(o,1604,804);
// FILE *o;o=fopen("amsfig4dFragmen.eps","w");ado(o,1604,804);
// FILE *o;o=fopen("amsfig4aFragmen.eps","w");ado(o,1604,804);
 FILE *o;o=fopen("amsfig4bFragmen.eps","w");ado(o,1604,804);
 fprintf(o,"802 402 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
 DO(m,M1)X[m]=-8.+.05*(m-.3);
 DO(n,200)Y[n]=-4.+.02*n;
         Y[200]=-.01;
         Y[201]= .01;
 for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
 for(m=-8;m<9;m++){M(m,-4)L(m,4)}
 for(n=-4;n<5;n++){ M( -8,n)L(8,n)}
 fprintf(o,".008 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){g[m*N1+n]=990999; f[m*N1+n]=909999;}

 DO(n,N1){y=Y[n];
          for(m=150;m<170;m++)
          {x=X[m]; //printf("%5.2f\n",x);
           z=z_type(x,y);
// c=tetb(z);
// c=F4TEN(z);
// c=F4(z);
           c=FSEXP(z);
           p=Re(c);q=Im(c);
           if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
           d=c;
           for(k=1;k<31;k++)
                { m1=m+k*20; if(m1>M) break;
// d=exp(a*d);
// d=exp(d*log(2.));
                d=exp(d);
                p=Re(d);q=Im(d);
                if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
           d=c;
           for(k=1;k<31;k++)
                { m1=m-k*20; if(m1<0) break;
              d=log(d);
// d=log(d)/a;
// d=log(d)/log(2.);
                p=Re(d);q=Im(d);
                if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
        }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=20;q=1;
 for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .6 0 RGB S\n");
 for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .9 0 0 RGB S\n");
 for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .9 RGB S\n");
 for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n");
 for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n");
                    conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .6 0 .6 RGB S\n");
 for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 1 .5 0 RGB S\n");
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 0 .5 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
//system( "ggv fig3.eps");
system("epstopdf amsfig4bFragmen.eps");
system( "open amsfig4bFragmen.pdf");
getchar(); system("killall Preview");
}

Latex Generator of labels]


\documentclass{amsproc}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{hyperref}
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%
%\newcommand \rme {{e}} %%
\newcommand \rmi {{\rm i}} %%imaginary unity \newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\usepackage{geometry}
\topmargin -94pt
\oddsidemargin -70pt
\paperwidth 1666pt
\paperheight 856pt
\textwidth 1900px
\textheight 900px
\begin{document}
\parindent 0pt

\newcommand \mapax {
\put(18,820){\sx{5}{$y$}}
\put(18,730){\sx{5}{$3$}}
\put(18,630){\sx{5}{$2$}}
\put(18,530){\sx{5}{$1$}}
\put(18,430){\sx{5}{$0$}}
\put(-14,329){\sx{5}{$-1$}}
\put(-14,229){\sx{5}{$-2$}}
\put(-14,129){\sx{5}{$-3$}}
\put(-14, 29){\sx{5}{$-4$}}
\put(14, 0){\sx{5}{$-8$}}
\put(114, 0){\sx{5}{$-7$}}
\put(214, 0){\sx{5}{$-6$}}
\put(314, 0){\sx{5}{$-5$}}
\put(414, 0){\sx{5}{$-4$}}
\put(514, 0){\sx{5}{$-3$}}
\put(614, 0){\sx{5}{$-2$}}
\put(714, 0){\sx{5}{$-1$}}
\put(844, 0){\sx{5}{$0$}}
\put(944, 0){\sx{5}{$1$}}
\put(1044, 0){\sx{5}{$2$}}
\put(1144, 0){\sx{5}{$3$}}
\put(1244, 0){\sx{5}{$4$}}
\put(1344, 0){\sx{5}{$5$}}
\put(1444, 0){\sx{5}{$6$}}
\put(1544, 0){\sx{5}{$7$}}
\put(1634, 0){\sx{5}{$x$}}
}
%\flushright{$b=\mathrm e \approx 2.71$}

{\begin{picture}(1620,850) %%%
\put(50,40){\ing{amsfig4bFragmen}} \mapax
\put(114,660){\sx{8}{$b=\mathrm e$}}
\put(76,798){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764\!+\! 1.33723570143069 \,\mathrm i$}}
\put(80,90){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764 \!-\! 1.33723570143069 \,\mathrm i$}}
\put(60,434){\sx{4}{\bf cut}}
\put(760,434){\sx{4}{$v\!=\!0$}}
\multiput(46,550)(448,105){3}{\sx{4}{$v\!=\!1.4$}}
\multiput(268,584)(448,105){3}{\sx{4}{$u\!=\!0.4$}}
\multiput(46,316)(448,-105){4}{\sx{4}{$v\!=\!-1.4$}}
\multiput(298,464)(448,105){4}{\sx{4}{$v\!=\!1$}}
\multiput(290,404)(448,-105){4}{\sx{4}{$v\!=\!-1$}}

\end{picture}}
\end{document}

Refrences

  1. ↑ http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.14.pdf
    http://mizugadro.mydns.jp/PAPERS/2014acker.pdf D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

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.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

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. (In Russian)

http://mizugadro.mydns.jp/BOOK/437.pdf D.Kouznetsov. Suparfuncctions. Mizugadro, 2015. (In English)

http://myweb.astate.edu/wpaulsen/tetration.html William Paulsen. Tetration is repeated exponentiation. (2016). We can define $^0b = 1, ^1b = b, ^2b = b^b$, &3b = b^{b^b}$, etc. ..

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