===================================================================
RCS file: /home/cvs/OpenXM/doc/ascm2001p/ohp.tex,v
retrieving revision 1.2
retrieving revision 1.4
diff -u -p -r1.2 -r1.4
--- OpenXM/doc/ascm2001p/ohp.tex	2001/09/23 08:31:18	1.2
+++ OpenXM/doc/ascm2001p/ohp.tex	2001/09/25 02:28:27	1.4
@@ -1,4 +1,4 @@
-%% $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[12pt]{article}
 \usepackage{color}
@@ -9,9 +9,12 @@
 {\color{green} Design and Implementation of OpenXM-RFC 100 and 101}
 
 \noindent
-M.Maekawa, M.Noro, K.Ohara, N.Takayama, Y.Tamura \\
-\htmladdnormallink{http://www.openxm.org}{http://www.openxm.org}
+M.Maekawa (前 川 　 将 秀), \\ M.Noro (野 呂 　 正 行), \\
+K.Ohara (小 原 　 功 任), \\ N.Takayama (高 山 　 信 毅), \\
+Y.Tamura (田 村 　 恭 士)\\
+\htmladdnormallink{{\color{red}http://www.openxm.org}}{{\color{red}http://www.openxm.org}}
 
+
 \newpage
 \noindent
 {\color{red} 1. Architecture} \\
@@ -21,6 +24,11 @@ Two main applications of the project \\
 \begin{enumerate}
 \item Providing an environment for interactive distributed computation.
 {\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 
 (Electronic version of higher transcendental functions of the 21st century)\\
 1st step: Generate and verify hypergeometric function identities.
@@ -131,8 +139,9 @@ ox_reset(P);
 \noindent{\color{red} 4. Easy to try and evaluate distributed algorithms} \\
 
 \noindent
+{\color{green} Example 1} \\
 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 $g \in F_q[x]$,
 the chance of
 $$ GCD(g^{(q^d-1)/2}-1,f_1 f_2) = f_1 \,\mbox{or}\, f_2 $$
@@ -195,17 +204,35 @@ def c_z(F,E,Level)
 }
 \end{verbatim}
 \newpage
+
+\epsfxsize=17cm
+\epsffile{cz.ps}
+
 \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: \\
 Gauss Hypergeometric function:
 $$ {\color{blue} F(a,b,c;x)} = \sum_{n=1}^\infty
-  \frac{(a)_n (b)_n}{(1)_n}{(c)_n} x^n
+  \frac{(a)_n (b)_n}{(1)_n (c)_n} x^n
 $$
 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
 Appell's $F_1$:
@@ -219,7 +246,9 @@ Erdelyi: {\color{green} Higher Transcendental Function
 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
-$$ F(a,b,c;x) , \  x^{1-c} F(a,b,c;x) $$
+$$ F(a,b,c;x) = {\color{red}1} + O(x), \  
+   x^{1-c} F(a,b,c;x) = {\color{red}x^{1-c}}+O(x^{2-c}))$$
+
 when $c \not\in {\bf Z}$. \\
 {\color{blue} Formula (type B)}\\
 \begin{eqnarray*}
@@ -254,7 +283,7 @@ for GKZ hypergeometric systems.
 
 \newpage
 \noindent
-{\color{green} Competitive Gr\"obner Basis Computation}
+{\color{green} Example 3. Competitive Gr\"obner Basis Computation}
 \begin{verbatim}
 extern Proc1,Proc2$
 Proc1 = -1$ Proc2 = -1$