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version 1.1, 2001/06/19 07:32:58 version 1.5, 2001/06/21 00:15:34
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 % $OpenXM$  % $OpenXM: OpenXM/doc/ascm2001p/heterotic-network.tex,v 1.4 2001/06/20 05:42:47 takayama Exp $
 \section{Applications}  \section{Applications}
   
 \subsection{Heterogeneous Servers}  \subsection{Heterogeneous Servers}
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 By using OpenXM, we can treat OpenXM servers essentially  By using OpenXM, we can treat OpenXM servers essentially
 like a subroutine.  like a subroutine.
 Since OpenXM provides a universal stack machine which does not  Since OpenXM provides a universal stack machine which does not
 depend each servers,  depend on each server,
 it is relatively easy to install new servers.  it is relatively easy to install new servers.
 We can build a new computer math system by assembling  We can build a new mathematical software system by assembling
 different OpenXM servers.  different OpenXM servers.
 It is similar to building a toy house by LEGO blocks.  OpenXM servers currently provide 1077 functions
   \cite{openxm-1077}.
   We can use these as building blocks for a new system.
   
 We will see two examples of custom-made systems  %We present an example of a custom-made system
 built by OpenXM servers.  %built by OpenXM servers.
   
 \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}  %\subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
   %
   %Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
   %be the ring of differential operators.
   %For a given polynomial
   %$ f \in {\bf Q}[x_1, \ldots, x_n] $,
   %the annihilating ideal of $f^{-1}$ is defined as
   %$$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
   %  \ell \bullet f^{-1} = 0 \}.
   %$$
   %Here, $\bullet$ denotes the action of $D$ to functions.
   %The annihilating ideal can be regarded as the maximal differential
   %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 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
   %{\tt ox\_asir}, to factorize polynomials to find the integral
   %roots.
   %These two OpenXM compliant systems are integrated by
   %the OpenXM protocol.
   
 Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$  
 be the ring of differential operators.  
 For a given polynomial  
 $ f \in {\bf Q}[x_1, \ldots, x_n] $,  
 the annihilating ideal of $f^{-1}$ is defined as  
 $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,  
   \ell \bullet f^{-1} = 0 \}.  
 $$  
 Here, $\bullet$ denotes the action of $D$ to functions.  
 The annihilating ideal can be regarded as the maximal differential  
 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 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  
 {\tt ox\_asir}, to factorize polynomials to find the integral  
 roots.  
 These two OpenXM compliant systems are integrated by  
 the OpenXM protocol.  
   
 %For example, the following is a sm1 session to find the annihilating  %For example, the following is a sm1 session to find the annihilating
 %ideal for $f = x^3 - y^2 z^2$.  %ideal for $f = x^3 - y^2 z^2$.
 %\begin{verbatim}  %\begin{verbatim}
Line 102  the OpenXM protocol.
Line 104  the OpenXM protocol.
 %\label{katsura}  %\label{katsura}
 %\end{figure}  %\end{figure}
   
 \subsubsection{Asir-contrib-HG package to solve GKZ hypergeometric systems}  Asir-contrib/Hypergeometric package is an
   example of a custom-made system built by OpenXM servers.
 GKZ hypergeometric system is a system of linear partial differential  GKZ hypergeometric system is a system of linear partial differential
 equations associated to $A=(a_{ij})$  equations associated to $A=(a_{ij})$
 (an integer $d\times n$-matrix of rank $d$)  (an integer $d\times n$-matrix of rank $d$)
Line 116  these algorithms.
Line 118  these algorithms.
 What we need for the implementation are mainly  What we need for the implementation are mainly
 (1) Gr\"obner basis computation both in the ring of polynomials  (1) Gr\"obner basis computation both in the ring of polynomials
 and in the ring of differential operators,  and in the ring of differential operators,
   (2) enumeration of all the Gr\"obner bases of toric ideals,
   and
   (3) primary ideal decomposition.
   Asir and kan/sm1 provide functions for (1),
   {\tt TiGERS} provides a function for (2),
 and  and
 (2) enumeration of all the Gr\"obner bases of toric ideals.  Asir provides a function for (3).
 Asir and kan/sm1 provide functions for (1) and  These software systems communicate with each other
 {\tt TiGERS} provides a function for (2).  by the OpenXM-RFC 100 protocol
 These components communicate each other by OpenXM-RFC 100 protocol.  and form a unified package for GKZ hypergeometric systems.
   See the chapter of {\tt dsolv} of Asir Contrib User's manual \cite{openxm-web}
   for details.
   
 %Let us see an example how to construct series solution of a GKZ hypergeometric  %Let us see an example how to construct series solution of a GKZ hypergeometric
 %system.  %system.
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