Difference between revisions of "File:QQFacMT.jpg"

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  +
Something strange with names..
−
Importing image file
  
  +
  +
[[Complex map]] of function $H(z)=h(h(z))$,
  +
which is second iterate of the half iterate of factorial.
  +
  +
The half iterate is evaluatef through functions [[SuFac]] and [[AuFac]]:
  +
  +
$\displaystyle
  +
h(z)=\mathrm{Factorial}^{0.5}(z)=\mathrm{SuFac}\left( \frac{1}{2}+\mathrm{AuFac}(z)\right)
  +
$
  +
  +
The map is show with lines of constant $u$ and lines of constant $v$;
  +
  +
$u+\mathrm i v=H(x+\mathrm i u)$
  +
  +
In the right hand side of the map, it coincides with map of [[Factorial]].
  +
The image represents figure 8.10 from the Russian version of the Book [[Суперфункции]]
  +
<ref>
  +
Д.Кузнецов. [[Суперфункции]]. Lambert Academic Publishing, 2014.
  +
</ref>
  +
  +
==[[C++]] generator of map==
  +
Files
  +
[[fac.cin]]
  +
[[facp.cin]]
  +
[[afacc.cin]]
  +
[[superfactorial.cin]]
  +
[[abelfac.cin]]
  +
[[conto.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 "sinc.cin"
  +
#include "facp.cin"
  +
#include "afacc.cin"
  +
#include "superfactorial.cin"
  +
#include "abelfac.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("QQFacMap.eps","w");ado(o,402,402);
  +
fprintf(o,"201 201 translate\n 20 20 scale\n");
  +
DO(m,M1)X[m]=-8.04+.04*(m+.5);
  +
// DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);}
  +
DO(n,N1)Y[n]=-8.04+.04*(n+.5);
  +
// DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);}
  +
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
  +
for(n=-8;n<9;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]=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=afacc(z);
  +
// c=fac(z);
  +
c=abelfac(z); c=superfac(.5+c);
  +
c=abelfac(c); c=superfac(.5+c);
  +
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.;
  +
p=Re(c);q=Im(c);
  +
if(p>-60 && p<60 &&
  +
// (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
  +
q>-60 && q<60 && fabs(q)> 1.e-16
  +
)
  +
{g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
//fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
  +
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,".01 W 0 .5 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,".01 W .8 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,".01 W 0 0 .8 RGB S\n");
  +
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n");
  +
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n");
  +
for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
//#include"plofu.cin"
  +
// x=0.8856031944;
  +
// conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
  +
M(1,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W 1 1 1 RGB S\n");
  +
DO(m,23){ M(1-.4*(m+.2),0)L(1-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n");
  +
DO(m,23){ M(1-.4*(m+.7),0)L(1-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n");
  +
//M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n");
  +
// May it be, that, some printers do not interpret well the dashing ?
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf QQFacMap.eps");
  +
system( "open QQFacMap.pdf"); //for LINUX
  +
// getchar(); system("killall Preview");//for mac
  +
}
  +
//</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
%<poem><nomathjax><nowiki>
  +
  +
\documentclass[12pt,a4paper,oneside]{book}
  +
%\newcommand \EN[1] {{#1}} % make the English version
  +
\newcommand \EN[1] {{}} % suppress the English version
  +
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
  +
%\newcommand \RU[1] {{}} % suppress the Russian version
  +
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
  +
\newcommand \JP[1] {{}} % ореsuppress some text
  +
%\usepackage[space]{cite }% If exist.
  +
\usepackage[utf8]{inputenc}
  +
\usepackage[T2A]{fontenc}
  +
\usepackage[russian]{babel}
  +
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
  +
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\usepackage{wrapfig}
  +
\usepackage{geometry}
  +
%\renewcommand\baselinestretch{.94}
  +
%\renewcommand\baselinestretch{.94}
  +
%\renewcommand\baselinestretch{1.15}
  +
%\renewcommand\baselinestretch{1.2}
  +
%\renewcommand\baselinestretch{1.2}
  +
\paperwidth150mm
  +
%\paperheight 257mm
  +
%\paperheight 216mm
  +
\paperheight 147mm
  +
%\textwidth 175mm
  +
\textwidth 150mm
  +
%\oddsidemargin 5mm
  +
\oddsidemargin -72pt
  +
%\topmargin -17mm
  +
\topmargin -103pt
  +
\textheight 257mm
  +
\pagestyle{empty}
  +
\large
  +
  +
\usepackage{color}
  +
\definecolor{rose}{rgb}{1,.9,1}
  +
\renewcommand\thesection{\arabic{section}}
  +
\renewcommand\figurename{Рис.}
  +
%\renewcommand\refname{References}
  +
%\renewcommand\figurename{Fig.}
  +
\newcommand \be {\begin{eqnarray}}
  +
\newcommand \ee {\end{eqnarray} }
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
  +
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
  +
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \bN {\mathbb{N}}
  +
\newcommand \bC {\mathbb{C}}
  +
\newcommand \bR {\mathbb{R}}
  +
\newcommand \cO {\mathcal{O}}
  +
\newcommand \cF {\mathcal{F}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
  +
\newcommand \pS {{~}~{~}}
  +
\newcommand \fac {\mathrm{Factorial}}
  +
\newcommand {\rf}[1] {(\ref{#1})}
  +
\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
  +
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
  +
\newcommand \eL[1] {\iL{#1}\ee}
  +
\newcommand \ing \includegraphics
  +
\newcommand \tet {\mathrm{tet}}
  +
\parindent 0pt
  +
\begin{document}
  +
\hskip -28pt
  +
%{\sx{.72}{\begin{picture}(362,330)
  +
{\sx{1.25}{\begin{picture}(362,363)
  +
\put(0,0){\ing{QQFacMap}}
  +
\put(30,357){\sx{1.3}{$y$}}
  +
\put(30,317){\sx{1.3}{$6$}}
  +
\put(30,277){\sx{1.3}{$4$}}
  +
\put(30,237){\sx{1.3}{$2$}}
  +
\put(29,196){\sx{1.3}{$0$}}
  +
\put(20,156){\sx{1.3}{$-2$}}
  +
\put(20,116){\sx{1.3}{$-4$}}
  +
\put(20,76){\sx{1.3}{$-6$}}
  +
\put(20,36){\sx{1.2}{$-8$}}
  +
\put(70,30){\sx{1.1}{$-6$}}
  +
\put(111,30){\sx{1.1}{$-4$}}
  +
\put(151,30){\sx{1.1}{$-2$}}
  +
\put(198.6,30){\sx{1.1}{$0$}}
  +
\put(238.6,30){\sx{1.1}{$2$}}
  +
\put(278.6,30){\sx{1.1}{$4$}}
  +
\put(318.6,30){\sx{1.1}{$6$}}
  +
\put(355,30){\sx{1.1}{$x$}}
  +
\put(70,198){\sx{1.3}{\bf cut}} %central
  +
%\put(182,263){\rot{42}\sx{1.6}{\bf cut}\ero}
  +
\put(189,260){\rot{42}\sx{1.6}{\bf cut}\ero}
  +
\put(193,209){\rot{-24}\sx{1.6}{\bf cut}\ero}
  +
\put(221,290){\rot{-35}\sx{1.6}{$u\!=\!0$}\ero}
  +
\put(212,277){\rot{-40}\sx{1.6}{$v\!=\!0$}\ero}
  +
\put(199,266){\rot{-47}\sx{1.6}{$u\!=\!0$}\ero}
  +
\put(299,196.6){\sx{1.6}{$v\!=\!0$}}
  +
\end{picture}}}
  +
\end{document}
  +
  +
  +
%JP{
  +
\color{red}
  +
\begin{picture}(425,729.88)
  +
\put(0,0){\rule{1pt}{729.88pt}}
  +
\put(425,0){\rule{1pt}{729.88pt}}
  +
\put(0,0){\rule{426pt}{1pt}}
  +
\put(0,728.88){\rule{426pt}{1pt}}
  +
\end{picture}
  +
%\vskip -740pt
  +
\vskip -729pt
  +
\color{black}
  +
%}
  +
\end{document}
  +
  +
%</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Factorial]]
  +
[[Category:SuFac]]
  +
[[Category:AuFac]]
  +
[[Category:Superfunction]]
  +
[[Category:Abelfunction]]
  +
[[Category:Square root of factorial]]
  +
[[Category:Halfiterate of factorial]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:48, 1 December 2018

Something strange with names..

Complex map of function $H(z)=h(h(z))$, which is second iterate of the half iterate of factorial.

The half iterate is evaluatef through functions SuFac and AuFac:

$\displaystyle h(z)=\mathrm{Factorial}^{0.5}(z)=\mathrm{SuFac}\left( \frac{1}{2}+\mathrm{AuFac}(z)\right) $

The map is show with lines of constant $u$ and lines of constant $v$;

$u+\mathrm i v=H(x+\mathrm i u)$

In the right hand side of the map, it coincides with map of Factorial. The image represents figure 8.10 from the Russian version of the Book Суперфункции [1]

C++ generator of map

Files fac.cin facp.cin afacc.cin superfactorial.cin abelfac.cin conto.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 "sinc.cin"
 #include "facp.cin"
 #include "afacc.cin"
 #include "superfactorial.cin"
 #include "abelfac.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("QQFacMap.eps","w");ado(o,402,402);
 fprintf(o,"201 201 translate\n 20 20 scale\n");
 DO(m,M1)X[m]=-8.04+.04*(m+.5);
 // DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);}
 DO(n,N1)Y[n]=-8.04+.04*(n+.5);
 // DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);}
 for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
 for(n=-8;n<9;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]=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=afacc(z);
 // c=fac(z);
        c=abelfac(z); c=superfac(.5+c);
        c=abelfac(c); c=superfac(.5+c);
 // p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);
        if(p>-60 && p<60 &&
 // (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
           q>-60 && q<60 && fabs(q)> 1.e-16
        )
        {g[m*N1+n]=p;f[m*N1+n]=q;}
                        }}
 //fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
 fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
 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,".01 W 0 .5 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,".01 W .8 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,".01 W 0 0 .8 RGB S\n");
 for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n");
 for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n");
 for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
           m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n");
 for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
 //#include"plofu.cin"
 // x=0.8856031944;
 // conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
 M(1,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W 1 1 1 RGB S\n");
 DO(m,23){ M(1-.4*(m+.2),0)L(1-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n");
 DO(m,23){ M(1-.4*(m+.7),0)L(1-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n");
 //M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n");
 // May it be, that, some printers do not interpret well the dashing ?
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf QQFacMap.eps");
        system( "open QQFacMap.pdf"); //for LINUX
 // getchar(); system("killall Preview");//for mac
 }
//

Latex generator of labels

%



\documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}} % make the English version
\newcommand \EN[1] {{}} % suppress the English version
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}} % suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}} % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{geometry}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{1.15}
%\renewcommand\baselinestretch{1.2}
%\renewcommand\baselinestretch{1.2}
\paperwidth150mm
%\paperheight 257mm
%\paperheight 216mm
\paperheight 147mm
%\textwidth 175mm
\textwidth 150mm
%\oddsidemargin 5mm
\oddsidemargin -72pt
%\topmargin -17mm
\topmargin -103pt
\textheight 257mm
\pagestyle{empty}
\large

\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\renewcommand\thesection{\arabic{section}}
\renewcommand\figurename{Рис.}
%\renewcommand\refname{References}
%\renewcommand\figurename{Fig.}
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font
%\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}} %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \bN {\mathbb{N}}
\newcommand \bC {\mathbb{C}}
\newcommand \bR {\mathbb{R}}
\newcommand \cO {\mathcal{O}}
\newcommand \cF {\mathcal{F}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\newcommand \pS {{~}~{~}}
\newcommand \fac {\mathrm{Factorial}}
\newcommand {\rf}[1] {(\ref{#1})}
\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
\newcommand \eL[1] {\iL{#1}\ee}
\newcommand \ing \includegraphics
\newcommand \tet {\mathrm{tet}}
\parindent 0pt
\begin{document}
\hskip -28pt
%{\sx{.72}{\begin{picture}(362,330)
{\sx{1.25}{\begin{picture}(362,363)
\put(0,0){\ing{QQFacMap}}
\put(30,357){\sx{1.3}{$y$}}
\put(30,317){\sx{1.3}{$6$}}
\put(30,277){\sx{1.3}{$4$}}
\put(30,237){\sx{1.3}{$2$}}
\put(29,196){\sx{1.3}{$0$}}
\put(20,156){\sx{1.3}{$-2$}}
\put(20,116){\sx{1.3}{$-4$}}
\put(20,76){\sx{1.3}{$-6$}}
\put(20,36){\sx{1.2}{$-8$}}
\put(70,30){\sx{1.1}{$-6$}}
\put(111,30){\sx{1.1}{$-4$}}
\put(151,30){\sx{1.1}{$-2$}}
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References

  1. ↑ Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.

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