===================================================================
RCS file: /home/cvs/OpenXM/doc/issac2000/heterotic-network.tex,v
retrieving revision 1.6
retrieving revision 1.12
diff -u -p -r1.6 -r1.12
--- OpenXM/doc/issac2000/heterotic-network.tex	2000/01/15 03:47:59	1.6
+++ OpenXM/doc/issac2000/heterotic-network.tex	2000/01/17 08:06:15	1.12
@@ -1,4 +1,5 @@
-% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.5 2000/01/15 03:23:59 takayama Exp $
+% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.11 2000/01/17 07:06:53 noro Exp $
+\section{Applications}
 
 \subsection{Heterogeneous Servers}
 
@@ -6,14 +7,14 @@
 
 By using OpenXM, we can treat OpenXM servers essentially 
 like a subroutine.
-Since OpenXM provides a universal stackmachine which does not
+Since OpenXM provides a universal stack machine which does not
 depend each servers, 
 it is relatively easy to install new servers.
 We can build a new computer math system by assembling
 different OpenXM servers.
 It is similar to building a toy house by LEGO blocks.
 
-We will see two examples of custom made systems
+We will see two examples of custom-made systems
 built by OpenXM servers.
 
 \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
@@ -32,13 +33,13 @@ equations for the function $f^{-1}$.
 An algorithm to determine generators of the annihilating ideal
 was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
 His algorithm reduces the problem to computations of Gr\"obner bases
-in $D$ and to find the maximal integral root of a polynomial.
+in $D$ and to find the minimal integral root of a polynomial.
 This algorithm (the function {\tt annfs}) is implemented by
 kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
-ox\_asir, to factorize polynomials to find the integral
+{\tt ox\_asir}, to factorize polynomials to find the integral
 roots.
-These two OpenXM complient systems are integrated by
-OpenXM protocol.
+These two OpenXM compliant systems are integrated by
+the OpenXM protocol.
 
 For example, the following is a sm1 session to find the annihilating
 ideal for $f = x^3 - y^2 z^2$.
@@ -62,39 +63,45 @@ an algorithm to stratify singularity 
 \cite{oaku-advance}
 is implemented by
 kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
-ox\_asir, for primary ideal decompositions.
+{\tt ox\_asir}, for primary ideal decompositions.
 
 \subsubsection{A Course on Solving Algebraic Equations}
 
-Risa/asir \cite{asir} is a general computer algebra system
-which is good at Gr\"obner basis computations for zero dimensional ideal
+Risa/Asir \cite{asir} is a general computer algebra system
+which can be used for Gr\"obner basis computations for zero dimensional ideal
 with ${\bf Q}$ coefficients.
 However, it is not good at graphical presentations and
 numerical methods.
-We integrated Risa/asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
+We integrated Risa/Asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
 for the polyhedral homotopy method) and
 ox\_gnuplot (GNUPLOT) servers
 to teach a course on solving algebraic equations.
-This course was presented with the text book \cite{CLO} which discusses
+This course was presented with the text book \cite{CLO},
+which discusses 
 on the Gr\"obner basis method and the polyhedral homotopy method
 to solve systems of algebraic equations.
-We could teach a course
+We taught the course
 with a unified environment
-controlled by asir user language, which is similar to C.
-The following is an asir session to solve algebraic equations by calling
-the PHC pack.
+controlled by Asir user language, which is similar to C.
+The following is an Asir session to solve algebraic equations by calling
+the PHC pack (Figure \ref{katsura} is the output of {\tt [292]}):
 \begin{verbatim}
-[257] phc([x^2+y^2-4,x*y-1]);
+[287] phc(katsura(7));
 The detailed output is in the file tmp.output.*
 The answer is in the variable Phc.
 0
-[260] Phc ;
-[[[-0.517638,0],[-1.93185,0]],
-[[1.93185,0],[0.517638,0]],
-[[-1.93185,0],[-0.517638,0]],
-[[0.517638,0],[1.93185,0]]]
-[261] 
+[290] B=map(first,Phc)$
+[291] gnuplot_plotDots([],0)$
+[292] gnuplot_plotDots(B,0)$
 \end{verbatim}
+
+\begin{figure}[htbp]
+\epsfxsize=8.5cm
+\epsffile{katsura7.ps}
+\caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
+\label{katsura}
+\end{figure}
+