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version 1.1, 2001/09/20 09:27:56 version 1.4, 2001/09/25 02:28:27
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 %% $OpenXM$  %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.3 2001/09/25 01:17:08 takayama Exp $
 \documentclass{slides}  \documentclass{slides}
 %%\documentclass[12pt]{article}  %%\documentclass[12pt]{article}
 \usepackage{color}  \usepackage{color}
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 {\color{green} Design and Implementation of OpenXM-RFC 100 and 101}  {\color{green} Design and Implementation of OpenXM-RFC 100 and 101}
   
 \noindent  \noindent
 M.Maekawa, M.Noro, K.Ohara, N.Takayama, Y.Tamura \\  M.Maekawa ($BA0(B $B@n(B $B!!(B $B>-(B $B=((B), \\ M.Noro ($BLn(B $BO$(B $B!!(B $B@5(B $B9T(B), \\
 \htmladdnormallink{http://www.openxm.org}{http://www.openxm.org}  K.Ohara ($B>.(B $B86(B $B!!(B $B8y(B $BG$(B), \\ N.Takayama ($B9b(B $B;3(B $B!!(B $B?.(B $B5#(B), \\
   Y.Tamura ($BED(B $BB<(B $B!!(B $B63(B $B;N(B)\\
   \htmladdnormallink{{\color{red}http://www.openxm.org}}{{\color{red}http://www.openxm.org}}
   
   
 \newpage  \newpage
 \noindent  \noindent
 {\color{red} 1. Architecture} \\  {\color{red} 1. Architecture} \\
Line 21  Two main applications of the project \\
Line 24  Two main applications of the project \\
 \begin{enumerate}  \begin{enumerate}
 \item Providing an environment for interactive distributed computation.  \item Providing an environment for interactive distributed computation.
 {\color{blue} Risa/Asir}  {\color{blue} Risa/Asir}
   (computer algebra system for general purpose,
    open source (c) Fujitsu, \\
    http://www.openxm.org, \\
    http://risa.cs.ehime-u.ac.jp, \\
    http://www.math.kobe-u.ac.jp/Asir/asir.html)
 \item e-Bateman project  \item e-Bateman project
 (Electronic version of higher transcendental functions of the 21st century)\\  (Electronic version of higher transcendental functions of the 21st century)\\
 1st step: Generate and verify hypergeometric function identities.  1st step: Generate and verify hypergeometric function identities.
Line 131  ox_reset(P);
Line 139  ox_reset(P);
 \noindent{\color{red} 4. Easy to try and evaluate distributed algorithms} \\  \noindent{\color{red} 4. Easy to try and evaluate distributed algorithms} \\
   
 \noindent  \noindent
   {\color{green} Example 1} \\
 Theorem (Cantor-Zassenhaus) \\  Theorem (Cantor-Zassenhaus) \\
 Let $f_1$ and $f_2$ be degree $d$ polynomials in $F_q[x]$.  Let $f_1$ and $f_2$ be degree $d$ irreducible polynomials in $F_q[x]$.
 For a random degree $2d-1$ polynomial $g \in F_q[x]$,  For a random degree $2d-1$ polynomial $g \in F_q[x]$,
 the chance of  the chance of
 $$ GCD(g^{(q^d-1)/2}-1,f_1 f_2) = f_1 \,\mbox{or}\, f_2 $$  $$ GCD(g^{(q^d-1)/2}-1,f_1 f_2) = f_1 \,\mbox{or}\, f_2 $$
Line 195  def c_z(F,E,Level)
Line 204  def c_z(F,E,Level)
 }  }
 \end{verbatim}  \end{verbatim}
 \newpage  \newpage
   
   \epsfxsize=17cm
   \epsffile{cz.ps}
   
 \noindent  \noindent
 {\color{red} 5. e-Bateman project} \\  {\color{blue} Performance of parallel CZ algorithm} \\
   $d=1$, $k=200$ : product of $200$ linear forms. \\
   $d=2$, $k=50$ : product of $50$ irreducible degree $2$ polynomials. \\
   
   \newpage
   {\color{green} Example 2} \\
   Shoup's algorithm to multiply polynomials.  \\
   {\color{green} Example 3} \\
   Competitive Gr\"obner basis computation. \\
   \newpage
   
   \noindent
   {\color{red} 5. e-Bateman project} (Electronic mathematical formula book)\\
 First Step: \\  First Step: \\
 Gauss Hypergeometric function:  Gauss Hypergeometric function:
 $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty  $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty
   \frac{(a)_n (b)_n}{(1)_n}{(c)_n} x^n    \frac{(a)_n (b)_n}{(1)_n (c)_n} x^n
 $$  $$
 where  where
 $$ (a)_n = a(a+1) \cdots (a+n-1). $$  $$ (a)_n = a(a+1) \cdots (a+n-1). $$
 $$ F(?,?,?;x) = \log (1+x). $$  {\color{green}
   $$ \log (1+x) = x F(1,1,2;-x) $$
   $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$
   }
   
 \noindent  \noindent
 Appell's $F_1$:  Appell's $F_1$:
Line 218  Erdelyi: {\color{green} Higher Transcendental Function
Line 246  Erdelyi: {\color{green} Higher Transcendental Function
 The solution space of the ordinary differential equation  The solution space of the ordinary differential equation
 $$ x(1-x) \frac{d^2f}{dx^2} -\left( c-(a+b+1)x \right) \frac{df}{dx} - a b f = 0$$  $$ x(1-x) \frac{d^2f}{dx^2} -\left( c-(a+b+1)x \right) \frac{df}{dx} - a b f = 0$$
 is spanned by  is spanned by
 $$ F(a,b,c;x) , \  x^{1-c} F(a,b,c;x) $$  $$ F(a,b,c;x) = {\color{red}1} + O(x), \
      x^{1-c} F(a,b,c;x) = {\color{red}x^{1-c}}+O(x^{2-c}))$$
   
 when $c \not\in {\bf Z}$. \\  when $c \not\in {\bf Z}$. \\
 {\color{blue} Formula (type B)}\\  {\color{blue} Formula (type B)}\\
 \begin{eqnarray*}  \begin{eqnarray*}
Line 247  Implementation & partially done & NO \\ \hline
Line 277  Implementation & partially done & NO \\ \hline
 \noindent  \noindent
 Our ox servers  Our ox servers
 {\tt ox\_asir}, {\tt ox\_sm1}, {\tt ox\_tigers}, {\tt ox\_gnuplot},  {\tt ox\_asir}, {\tt ox\_sm1}, {\tt ox\_tigers}, {\tt ox\_gnuplot},
 {\tt ox\_mathematica}, {\tt OMproxy} {\tt ox\_m2}  {\tt ox\_mathematica}, {\tt OMproxy} (JavaClasses), {\tt ox\_m2}
 are used to generate, verify and present formulas of type A  are used to generate, verify and present formulas of type A
 for GKZ hypergeometric systems.  for GKZ hypergeometric systems.
   
   \newpage
   \noindent
   {\color{green} Example 3. Competitive Gr\"obner Basis Computation}
   \begin{verbatim}
   extern Proc1,Proc2$
   Proc1 = -1$ Proc2 = -1$
   /* G:set of polys; V:list of variables */
   /* Mod: the Ground field GF(Mod); O:type of order */
   def dgr(G,V,Mod,O)
   {
     /* invoke servers if necessary */
     if ( Proc1 == -1 ) Proc1 = ox_launch();
     if ( Proc2 == -1 ) Proc2 = ox_launch();
     P = [Proc1,Proc2];
     map(ox_reset,P); /* reset servers */
     /* P0 executes Buchberger algorithm over GF(Mod) */
     ox_cmo_rpc(P[0],"dp_gr_mod_main",G,V,0,Mod,O);
     /* P1 executes F4 algorithm over GF(Mod) */
     ox_cmo_rpc(P[1],"dp_f4_mod_main",G,V,Mod,O);
     map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
     F = ox_select(P); /* wait for data */
     /* F[0] is a server's id which is ready */
     R = ox_get(F[0]);
     if ( F[0] == P[0] ) { Win = "Buchberger"; Lose = P[1]; }
     else { Win = "F4"; Lose = P[0]; }
     ox_reset(Lose); /* reset the loser */
     return [Win,R];
   }
   \end{verbatim}
   \newpage
   
 \end{document}  \end{document}
   
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