imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Explicit plot]] of 5 first ackermanns to base e: addition of e, multiplication by e, exponnt, tetration and pentation.
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| <math>y=A_{\mathrm e,m}(x)</math> is plottec versus <math>x</math> for
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| <math>m=1</math>,
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| <math>m=2</math>,
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| <math>m=3</math>,
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| <math>m=4</math> and
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| <math>m=5</math>.
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2014
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| |pub-country = Japan, Germany
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| |notes = This is figure 19.6 from the Russian book [[Суперфункции]]<ref>
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| https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
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| http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
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| http://mizugadro.mydns.jp/BOOK/202.pdf
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| Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014, in Russian; page 273, figure 19.6.</ref>.
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| |versions = http://mizugadro.mydns.jp/t/index.php/File:Ackerplot400.jpg
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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| ==Description==
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| <math>A_{\mathrm e,1}= z\mapsto b+z</math>
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| <math>A_{\mathrm e,2}= z\mapsto b\,z</math>
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| <math>A_{\mathrm e,3}= \exp</math>
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| <math>A_{\mathrm e,4}= \mathrm{tet}</math>
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| <math>A_{\mathrm e,5}= \mathrm{pen}</math>
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| The ackermanns <math>A</math> satisfy equations
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| <math>A_{b,1}(z)=b+z</math>
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| <math>A_{b,m}(1)=b</math> for <math> m>1</math>
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| <math>A_{b,m+1}(z\!+\!1)=A_{m}(A(m\!+\!1,z))</math> for <math> m>1</math>
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| The additional requirements are applied for the uniqueness.
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| In particular, for the real <math>b</math>, the real holomorphism is assumed, <math>A_{b,1}(z^*)=A_{b,1}(z^*)</math>.
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| ==[[C++]] generator of curves==
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| Files [[fsexp.cin]] should be loaded in order to compile the code below.
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| <nowiki>
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| #include <math.h>
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| #include <stdio.h>
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| #include <stdlib.h>
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| #define DB double
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| #define DO(x,y) for(x=0;x<y;x++)
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| #include <complex>
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| typedef std::complex<double> z_type;
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| #define Re(x) x.real()
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| #define Im(x) x.imag()
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| #define I z_type(0.,1.)
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| void ado(FILE *O, int X, int Y)
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| { fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
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| fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
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| fprintf(O,"/M {moveto} bind def\n");
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| fprintf(O,"/L {lineto} bind def\n");
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| fprintf(O,"/S {stroke} bind def\n");
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| fprintf(O,"/s {show newpath} bind def\n");
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| fprintf(O,"/C {closepath} bind def\n");
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| fprintf(O,"/F {fill} bind def\n");
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| fprintf(O,"/o {.01 0 360 arc C F} bind def\n");
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| fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
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| fprintf(O,"/W {setlinewidth} bind def\n");
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| fprintf(O,"/RGB {setrgbcolor} bind def\n");}
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| /* end of routine */
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| #include "fsexp.cin"
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| z_type pen0(z_type z){ DB Lp=-1.8503545290271812; DB k,a,b;
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| k=1.86573322821; a=-.6263241; b=0.4827;
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| z_type e=exp(k*z); return Lp + e*(1.+e*(a+b*e)); }
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| z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.));
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| DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;}
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| return z; }
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| z_type pen(z_type z){ DB x; int m,n;
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| x=Re(z); if(x<= -4.) return pen0(z);
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| m=int(x+5.);
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| z-=DB(m);
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| z=pen0(z);
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| DO(n,m) z=FSEXP(z);
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| return z;
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| }
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| int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
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| FILE *o;o=fopen("ackerplo.eps","w"); ado(o,608,808);
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| fprintf(o,"304 204 translate\n 100 100 scale\n");
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| #define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
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| #define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
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| #define o(x,y) fprintf(o,"%8.4f %8.4f o\n",0.+x,0.+y);
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| for(m=-3;m<4;m++) {M(m,-2)L(m,6)}
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| for(n=-2;n<9;n++) {M( -3,n)L(3,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");
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| M(-3.02,-3.02+M_E)L(3.02,3.02+M_E) fprintf(o,".007 W .3 0 .3 RGB S\n");
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| M(-1., -M_E)L(3.02,3.02*M_E) fprintf(o,".007 W 0 .5 0 RGB S\n");
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| fprintf(o,"1 0 0 RGB\n");
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| DO(n,306){x=-3.02+.02*(n-.5);y=exp(x); o(x,y); if(y>6.)break;} fprintf(o,".02 W 0 .8 0 RGB S\n");
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| DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y));
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| if(n/2*2==n) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");
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| DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");
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| DB L=-1.8503545290271812;
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| DB K=1.86573322821;
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| DB a=-.6263241;
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| DB b=0.4827;
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| M(-3,L)L(0,L)
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| M(0,M_E) L(1,M_E) fprintf(o,".002 W 0 0 0 RGB S\n");
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| DB t2=M_PI/1.86573322821;
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| DB tx=-2.32;
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| printf("pen7(-1)=%18.14f\n", Re(pen7(-1.)));
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| printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821);
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| system("epstopdf ackerplo.eps");
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| system( "open ackerplo.pdf");
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| }
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| </nowiki>
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| ==[[Latex]] generator of curves==
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| <nowiki>
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| \documentclass[12pt]{article}
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| \paperwidth 604px
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| \paperheight 806px
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| \textwidth 1394px
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| \textheight 1300px
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| \topmargin -104px
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| \oddsidemargin -92px
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| \usepackage{graphics}
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| \usepackage{rotating}
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| \newcommand \sx {\scalebox}
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| \newcommand \rot {\begin{rotate}}
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| \newcommand \ero {\end{rotate}}
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| \newcommand \ing {\includegraphics}
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| \newcommand \rmi {\mathrm{i}}
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| \begin{document}
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| {\begin{picture}(608,806) %\put(12,0){\ing{penma}}
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| \put(0,0){\ing{ackerplo}}
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| \put(277,788){\sx{3.}{$y$}}
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| \put(277,695){\sx{3.}{$5$}}
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| \put(277,594){\sx{3.}{$4$}}
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| \put(277,494){\sx{3.}{$3$}}
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| \put(278,468){\sx{3.}{$\mathrm e$}}
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| \put(277,394){\sx{3.}{$2$}}
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| \put(277,294){\sx{3.}{$1$}}
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| \put(277,194){\sx{3.}{$0$}}
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| \put(258, 93){\sx{3.}{$-1$}}
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| \put( 80,174){\sx{3.}{$-2$}}
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| \put(180,174){\sx{3.}{$-1$}}
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| \put(296,174){\sx{3.}{$0$}}
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| \put(396,174){\sx{3.}{$1$}}
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| \put(496,174){\sx{3.}{$2$}}
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| \put(586,174){\sx{3.}{$x$}}
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| \put(438,714){\sx{1.8}{\rot{85}$y\!=\!\mathrm{pen}(x)$\ero}}
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| \put(460,716){\sx{1.8}{\rot{82}$y\!=\!\mathrm{tet}(x)$\ero}}
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| \put(478,712){\sx{1.8}{\rot{77}$y\!=\!\mathrm{exp}(x)$\ero}}
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| \put(504,712){\sx{1.8}{\rot{70}$y\!=\!\mathrm{e}x$\ero}}
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| \put(538,718){\sx{1.96}{\rot{44}$y\!=\!\mathrm{e}\!+\!x$\ero}}
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| \put(86,222){\sx{1.9}{\rot{11}$y\!=\!\mathrm{exp}(x)$\ero}}
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| \put(20,30){\sx{1.9}{\rot{30}$y\!=\!\mathrm{pen}(x)$\ero}}
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| \put(138,22){\sx{1.9}{\rot{74}$y\!=\!\mathrm{tet}(x)$\ero}}
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| \put(252,22){\sx{1.9}{\rot{70}$y\!=\!\mathrm{e} x$\ero}}
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| \put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}}
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| \end{picture}
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| \end{document}
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| </nowiki>
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| ==References==
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| <references/>
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| D.Kouznetsov. Holomorphic ackermanns. 2014-2015, in preparation.
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| [[Category:Ackermann]]
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| [[Category:Tetration]]
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| [[Category:Pentation]]
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| [[Category:Explicit plot]]
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| [[Category:C++]]
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| [[Category:Latex]]
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| [[Category:TORI]]
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