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version 1.3, 2001/09/25 01:17:08 version 1.5, 2001/09/29 08:33:41
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 %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.2 2001/09/23 08:31:18 takayama Exp $  %% $OpenXM: OpenXM/doc/ascm2001p/ohp.tex,v 1.4 2001/09/25 02:28:27 takayama Exp $
 \documentclass{slides}  \documentclass{slides}
 %%\documentclass[12pt]{article}  %%\documentclass[12pt]{article}
 \usepackage{color}  \usepackage{color}
Line 44  OpenXM-RFC 100 \\
Line 44  OpenXM-RFC 100 \\
 OpenXM {\color{red} stackmachine}.  OpenXM {\color{red} stackmachine}.
 \item execute\_string  \item execute\_string
 \begin{verbatim}  \begin{verbatim}
   P = ox_launch(0,"ox_asir");    Pid = ox_launch(0,"ox_asir");
   ox_execute_string(Pid," poly_factor(x^10-1);");    ox_execute_string(Pid," poly_factor(x^10-1);");
 \end{verbatim}  \end{verbatim}
 \end{enumerate}  \end{enumerate}
Line 136  ox_reset(P);
Line 136  ox_reset(P);
 \end{picture}  \end{picture}
 \newpage  \newpage
   
 \noindent{\color{red} 4. Easy to try and evaluate distributed algorithms} \\  \noindent
   {\color{red} 4. e-Bateman project} (Electronic mathematical formula book)\\
   First Step: \\
   Gauss Hypergeometric function:
   $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty
     \frac{(a)_n (b)_n}{(1)_n (c)_n} x^n
   $$
   where
   $$ (a)_n = a(a+1) \cdots (a+n-1). $$
   {\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 $F_1$:
   $$ {\color{blue} F_1(a,b,b',c;x,y)} = \sum_{m,n=1}^\infty
     \frac{(a)_{m+n} (b)_m (b')_n}{(c)_{m+n}(1)_m (1)_n} x^m y^n.
   $$
   \newpage
   Mathematical formula book, e.g.,
   Erdelyi: {\color{green} Higher Transcendental Functions} \\
   {\color{blue} Formula (type A)}\\
   The solution space of the ordinary differential equation
   $$ x(1-x) \frac{d^2f}{dx^2} -\left( c-(a+b+1)x \right) \frac{df}{dx} - a b f = 0$$
   is spanned by
   $$ F(a,b,c;x) = {\color{red}1} + O(x), \
      x^{1-c} F(a,b,c;x) = {\color{red}x^{1-c}}+O(x^{2-c}))$$
   
   when $c \not\in {\bf Z}$. \\
   {\color{blue} Formula (type B)}\\
   \begin{eqnarray*}
   &\ & F(a_1, a_2, b_2;z) \, F(-a_1,-a_2,2-b_2;z)  \\
   &+& \frac{z}{e_2}\, F'(a_1, a_2, b_2;z) \, F(-a_1,-a_2,2-b_2;z)  \\
   &-& \frac{z}{e_2}\, F(a_1, a_2, b_2;z) \, F'(-a_1,-a_2,2-b_2;z)  \\
   &-& \frac{a_1+a_2-e_2}{a_1 a_2 e_2}z^2\,
     F'(a_1, a_2, b_2;z)\,F'(-a_1,-a_2,2-b_2;z) \\
   &=& 1
   \end{eqnarray*}
   where $e_2 = b_2-1$ and $a_1, a_2, e_2, e_2-a_2 \not\in {\bf Z}$.  \\
   (generalization of $\sin^2 x + \cos^2 x =1$.)
   
   \noindent
   Project in progress: \\
   We are trying to generate or verify type A formulas and type B formulas
   for {\color{blue} GKZ hypergeometric systems}.
   
   \begin{tabular}{|c|c|c|}
   \hline
     & type A & type B \\ \hline
   Algorithm &  {\color{red} OK} (SST book) &  in progress \\ \hline
   Implementation & partially done & NO \\ \hline
   \end{tabular}
   
   \noindent
   Our ox servers
   {\tt ox\_asir}, {\tt ox\_sm1}, {\tt ox\_tigers}, {\tt ox\_gnuplot},
   {\tt ox\_mathematica}, {\tt OpenMathproxy} (JavaClasses), {\tt ox\_m2}
   are used to generate, verify and present formulas of type A
   for GKZ hypergeometric systems.
   
   \newpage
   
   \noindent{\color{red} 5. Easy to try and evaluate distributed algorithms} \\
   
   \noindent
 {\color{green} Example 1} \\  {\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 151  $$ \frac{1}{2}-\frac{1}{(2q)^d}. $$
Line 214  $$ \frac{1}{2}-\frac{1}{(2q)^d}. $$
 \begin{picture}(20,14)(0,0)  \begin{picture}(20,14)(0,0)
 \put(7,12){\framebox(4,1.5){client}}  \put(7,12){\framebox(4,1.5){client}}
 \put(2,6){\framebox(4,1.5){server}}  \put(2,6){\framebox(4,1.5){server}}
 \put(7,6){\framebox(4,1.5){server}}  %%\put(7,6){\framebox(4,1.5){server}}
 \put(12,6){\framebox(4,1.5){server}}  \put(12,6){\framebox(4,1.5){server}}
 \put(0,0){\framebox(4,1.5){server}}  \put(0,0){\framebox(4,1.5){server}}
 \put(5,0){\framebox(4,1.5){server}}  \put(5,0){\framebox(4,1.5){server}}
 \put(13.5,0){\framebox(4,1.5){server}}  \put(13.5,0){\framebox(4,1.5){server}}
   
 \put(9,12){\vector(-1,-1){4.3}}  \put(9,12){\vector(-1,-1){4.3}}
 \put(9,12){\vector(0,-1){4.3}}  %%\put(9,12){\vector(0,-1){4.3}}
 \put(9,12){\vector(1,-1){4.3}}  \put(9,12){\vector(1,-1){4.3}}
 \put(4,6){\vector(-1,-2){2.2}}  \put(4,6){\vector(-1,-2){2.2}}
 \put(4,6){\vector(1,-2){2.2}}  \put(4,6){\vector(1,-2){2.2}}
Line 205  def c_z(F,E,Level)
Line 268  def c_z(F,E,Level)
 \end{verbatim}  \end{verbatim}
 \newpage  \newpage
   
 {\color{green} Example 2} \\  \epsfxsize=17cm
 Shoup's algorithm to multyply polynomials.  \epsffile{cz.ps}
 \newpage  
   
 \noindent  \noindent
 {\color{red} 5. e-Bateman project} \\  {\color{blue} Performance of parallel CZ algorithm} \\
 First Step: \\  $d=1$, $k=200$ : product of $200$ linear forms. \\
 Gauss Hypergeometric function:  $d=2$, $k=50$ : product of $50$ irreducible degree $2$ polynomials. \\
 $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty  
   \frac{(a)_n (b)_n}{(1)_n}{(c)_n} x^n  
 $$  
 where  
 $$ (a)_n = a(a+1) \cdots (a+n-1). $$  
 $$ \log (1+x) = x F(1,1,2;-x) $$  
 $$ \arcsin x = x F(1/2,1/2,3/2;x^2) $$  
   
 \noindent  
 Appell's $F_1$:  
 $$ {\color{blue} F_1(a,b,b',c;x,y)} = \sum_{m,n=1}^\infty  
   \frac{(a)_{m+n} (b)_m (b')_n}{(c)_{m+n}(1)_m (1)_n} x^m y^n.  
 $$  
 \newpage  \newpage
 Mathematical formula book, e.g.,  {\color{green} Example 2} \\
 Erdelyi: {\color{green} Higher Transcendental Functions} \\  Shoup's algorithm to multiply polynomials.  \\
 {\color{blue} Formula (type A)}\\  {\color{green} Example 3} \\
 The solution space of the ordinary differential equation  Competitive Gr\"obner basis computation. \\
 $$ x(1-x) \frac{d^2f}{dx^2} -\left( c-(a+b+1)x \right) \frac{df}{dx} - a b f = 0$$  \newpage
 is spanned by  
 $$ F(a,b,c;x) = {\color{red}1} + O(x), \  
    x^{1-c} F(a,b,c;x) = {\color{red}x^{1-c}}+O(x^{2-c}))$$  
   
 when $c \not\in {\bf Z}$. \\  
 {\color{blue} Formula (type B)}\\  
 \begin{eqnarray*}  
 &\ & F(a_1, a_2, b_2;z) \, F(-a_1,-a_2,2-b_2;z)  \\  
 &+& \frac{z}{e_2}\, F'(a_1, a_2, b_2;z) \, F(-a_1,-a_2,2-b_2;z)  \\  
 &-& \frac{z}{e_2}\, F(a_1, a_2, b_2;z) \, F'(-a_1,-a_2,2-b_2;z)  \\  
 &-& \frac{a_1+a_2-e_2}{a_1 a_2 e_2}z^2\,  
   F'(a_1, a_2, b_2;z)\,F'(-a_1,-a_2,2-b_2;z) \\  
 &=& 1  
 \end{eqnarray*}  
 where $e_2 = b_2-1$ and $a_1, a_2, e_2, e_2-a_2 \not\in {\bf Z}$.  \\  
 (generalization of $\sin^2 x + \cos^2 x =1$.)  
   
 \noindent  \noindent
 Project in progress: \\  {\color{green} Example 3. Competitive Gr\"obner Basis Computation}
 We are trying to generate or verify type A formulas and type B formulas  
 for {\color{blue} GKZ hypergeometric systems}.  
   
 \begin{tabular}{|c|c|c|}  
 \hline  
   & type A & type B \\ \hline  
 Algorithm &  {\color{red} OK} (SST book) &  in progress \\ \hline  
 Implementation & partially done & NO \\ \hline  
 \end{tabular}  
   
 \noindent  
 Our ox servers  
 {\tt ox\_asir}, {\tt ox\_sm1}, {\tt ox\_tigers}, {\tt ox\_gnuplot},  
 {\tt ox\_mathematica}, {\tt OMproxy} (JavaClasses), {\tt ox\_m2}  
 are used to generate, verify and present formulas of type A  
 for GKZ hypergeometric systems.  
   
 \newpage  
 \noindent  
 {\color{green} 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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