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1.1     ! noro        1: % $OpenXM$
        !             2: \section{Applications}
        !             3:
        !             4: \subsection{Heterogeneous Servers}
        !             5:
        !             6: \def\pd#1{ \partial_{#1} }
        !             7:
        !             8: By using OpenXM, we can treat OpenXM servers essentially
        !             9: like a subroutine.
        !            10: Since OpenXM provides a universal stack machine which does not
        !            11: depend each servers,
        !            12: it is relatively easy to install new servers.
        !            13: We can build a new computer math system by assembling
        !            14: different OpenXM servers.
        !            15: It is similar to building a toy house by LEGO blocks.
        !            16:
        !            17: We will see two examples of custom-made systems
        !            18: built by OpenXM servers.
        !            19:
        !            20: \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
        !            21:
        !            22: Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
        !            23: be the ring of differential operators.
        !            24: For a given polynomial
        !            25: $ f \in {\bf Q}[x_1, \ldots, x_n] $,
        !            26: the annihilating ideal of $f^{-1}$ is defined as
        !            27: $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
        !            28:   \ell \bullet f^{-1} = 0 \}.
        !            29: $$
        !            30: Here, $\bullet$ denotes the action of $D$ to functions.
        !            31: The annihilating ideal can be regarded as the maximal differential
        !            32: equations for the function $f^{-1}$.
        !            33: An algorithm to determine generators of the annihilating ideal
        !            34: was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
        !            35: His algorithm reduces the problem to computations of Gr\"obner bases
        !            36: in $D$ and to find the minimal integral root of a polynomial.
        !            37: This algorithm (the function {\tt annfs}) is implemented by
        !            38: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
        !            39: {\tt ox\_asir}, to factorize polynomials to find the integral
        !            40: roots.
        !            41: These two OpenXM compliant systems are integrated by
        !            42: the OpenXM protocol.
        !            43:
        !            44: For example, the following is a sm1 session to find the annihilating
        !            45: ideal for $f = x^3 - y^2 z^2$.
        !            46: \begin{verbatim}
        !            47: sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
        !            48: Starting ox_asir server.
        !            49: Byte order for control process is network byte order.
        !            50: Byte order for engine process is network byte order.
        !            51: [[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
        !            52: -2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx],
        !            53:  [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
        !            54:      -7898796*s^3-5220720*s^2-1900500*s-294000]]
        !            55: \end{verbatim}
        !            56: The last polynomial is factored as
        !            57: $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
        !            58: and the minimal integral root is $-1$
        !            59: as shown in the output.
        !            60:
        !            61: Similarly,
        !            62: an algorithm to stratify singularity
        !            63: \cite{oaku-advance}
        !            64: is implemented by
        !            65: kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
        !            66: {\tt ox\_asir}, for primary ideal decompositions.
        !            67:
        !            68: \subsubsection{A Course on Solving Algebraic Equations}
        !            69:
        !            70: Risa/Asir \cite{asir} is a general computer algebra system
        !            71: which can be used for Gr\"obner basis computations for zero dimensional ideal
        !            72: with ${\bf Q}$ coefficients.
        !            73: However, it is not good at graphical presentations and
        !            74: numerical methods.
        !            75: We integrated Risa/Asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
        !            76: for the polyhedral homotopy method) and
        !            77: ox\_gnuplot (GNUPLOT) servers
        !            78: to teach a course on solving algebraic equations.
        !            79: This course was presented with the text book \cite{CLO},
        !            80: which discusses
        !            81: on the Gr\"obner basis method and the polyhedral homotopy method
        !            82: to solve systems of algebraic equations.
        !            83: We taught the course
        !            84: with a unified environment
        !            85: controlled by the Asir user language, which is similar to C.
        !            86: The following is an Asir session to solve algebraic equations by calling
        !            87: the PHC pack (Figure \ref{katsura} is the output of {\tt [292]}):
        !            88: \begin{verbatim}
        !            89: [287] phc(katsura(7));
        !            90: The detailed output is in the file tmp.output.*
        !            91: The answer is in the variable Phc.
        !            92: 0
        !            93: [290] B=map(first,Phc)$
        !            94: [291] gnuplot_plotDots([],0)$
        !            95: [292] gnuplot_plotDots(B,0)$
        !            96: \end{verbatim}
        !            97:
        !            98: \begin{figure}[htbp]
        !            99: \epsfxsize=8.5cm
        !           100: \epsffile{katsura7.ps}
        !           101: \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
        !           102: \label{katsura}
        !           103: \end{figure}
        !           104:
        !           105:
        !           106:
        !           107: