(九州大学)

場所: 理学部３号館３３１１室 (変更です!). Room No.3311 (3F), Third Building of Faculty of Sciences

All talks will be presented in English.

Koji Nakagawa (九州大学) [中川 康二]

Converting Printed Mathematical Document into a Content Base Format [印刷された数学文書からその内容をとりだすことについて] , 10:00--10:45

OCR (Optical Character Recognition) technology makes it possible to digitize printed mathematical documents into presentation-base XML languages. In order to use the data stored in the XML languages usefully (e.g. search), the data should be converted into content base formats that show logical structures. In this research we convert a presentation-base XML language KML, obtained from an integrated OCR environment Infty, to a content-base XML language OMDoc.

Tadashi Takahashi (神戸大学) [高橋正]

Groebner Bases and Invariants [グレブナ基底と不変量] , 11:00--11:45

Invariants of rational singularities are well known. They obtained by using actions of quotient ring. We can get the invariants of singularities by using their Groebner Basis. We will show them in this presentation.

Bruno Buchberger (京都大学、RISC-Linz)

Algorithmic Algorithm Synthesis: Case Study Groebner Bases [アルゴリズム合成のアルゴリズム: グレブナ基底の場合] , 13:30--15:00

Recently, we introduced a new method, called "Lazy Thinking", for the algorithmic invention of algorithms. The method starts from a given formal specifications of a problem in predicate logic and tries out various algorithm schemes from a library of possible algorithm schemes. An algorithm scheme is a recursive definition of an algorithm in terms of unspecified auxiliary functions. For each algorithm scheme in the library, the method tries an (automated) correctness proof w.r.t. to the given problem specification. Normally, this proof will fail because nothing is known about the unspecificied subalgorithms. From the failing proof, the method automatically generates specifications for the unknown subalgorithms that will guarantee that the correctness proof can be completed. Now, there are two possibilities: Either we know algorithms that meet the specifications generated for the subalgorithms. Then we are done, i.e. we have synthesized an algorithm for the original problem. Or we apply the lazy thinking method recursively to the specifications of the unknown subalgorithms until we arrive at problem specifications that can be met by known algorithms. We implemented the method in the Theorema system, which is a software system for mathematical theory exploration. At the beginning, for consolidating the method, we successfully tried out the lazy thinking algorithm synthesis method for synthesizing various sorting algorithms. Recently, we were able to show that the method is powerful enough to synthesize the author's Groebner bases algorithm, which is deemed to be a nontrivial algorithm in the area of computer algebra and, hence, may well serve as a benchmark for the logical power of algorithm synthesis methods. In the talk, we will explain the method and, in particular, the synthesis of the Groebner bases algorithm in detail. We will finally draw some conclusions for the methodology of algorithmic mathematics, for the future of mathematical software systems, and for the future education of mathematics and computer science students. I will show how this method can be successfully applied to the automated invention of my Groebner bases algorithm starting from a formal specification of the Groebner bases problem. This talk is dedicated to the memory of Mikhail Mescsheryakov, director of LCTA at JINR, where I was a visiting researcher in 1970 / 71.

Mitsushi Fujimoto (福岡教育大学) [藤本光史]

On polynomial curves in the affine plane [アフィン平面の多項式曲線について] , 15:15--16:00

A curve that can be parametrized by polynomials is called a polynomial curve. It is well-known that a polynomial curve has only one place at infinity. In 1977, Sathaye indicated the Abhyankar's question for curves with one place at infinity. Let $C$ be a curve with one place at infinity, and $\Omega$ be the semigroup generated by pole orders of $C$ at infinity. Is there a polynomial curve associated with $\Omega$? In 1990, Sathaye-Stenerson suggested a candidate of counter example for this question. However, they could not give the answer to the question since the root computation for a huge polynomial system was required. We found a counter example for the Abhyankar's question using Groebner basis computation. In this talk, we give the details.

Seiichi Toyoda (九州大学) [豊田晴一] joint work with Masakazu Suzuki, Mitsushi Fujimoto

An Implementation of Symbolic Integration on Risa/Asir [Risa/Asirへの記号積分法の実装] , 16:15--17:00

We implemented symbolic integration on Risa/Asir. Algorithms which we adopted are heuristic. We think that heuristic algorithms are useful in education. At present, our package works for a part of elementary functions. In our talk, we would like to present our algorithms, implementation, educational use and future works.

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The computer algebra seminar is co-organized by

M.Noro, noro@math.kobe-u.ac.jp

N.Takayama, takayama@math.kobe-u.ac.jp

K.Yokoyama, yokoyama@math.kyushu-u.ac.jp

For more info see http://www.math.kobe-u.ac.jp/seminar.html/compalg.html

http://www.math.kobe-u.ac.jp/seminars.html