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version 1.2, 2001/09/23 08:31:18 version 1.4, 2001/09/25 02:28:27
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 %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.1 2001/09/20 09:27:56 takayama Exp $  %% $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
 $$ (a)_n = a(a+1) \cdots (a+n-1). $$  $$ (a)_n = a(a+1) \cdots (a+n-1). $$
   {\color{green}
 $$ \log (1+x) = x F(1,1,2;-x) $$  $$ \log (1+x) = x F(1,1,2;-x) $$
 $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$  $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$
   }
   
 \noindent  \noindent
 Appell's $F_1$:  Appell's $F_1$:
Line 219  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 254  for GKZ hypergeometric systems.
Line 283  for GKZ hypergeometric systems.
   
 \newpage  \newpage
 \noindent  \noindent
 {\color{green} Competitive Gr\"obner Basis Computation}  {\color{green} Example 3. Competitive Gr\"obner Basis Computation}
 \begin{verbatim}  \begin{verbatim}
 extern Proc1,Proc2$  extern Proc1,Proc2$
 Proc1 = -1$ Proc2 = -1$  Proc1 = -1$ Proc2 = -1$
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