2001 Fiscal Year Annual Research Report

格子上のトーリックイデアルのグレブナ基底と数理物理への応用

Research Project

Project/Area Number	12640226
Research Institution	Okayama University of science
Principal Investigator	澤江隆一岡山理科大学, 理学部, 助教授 (20226062)
Co-Investigator(Kenkyū-buntansha)	宇佐美広介広島大学, 総合科学部, 助教授 (90192509) 池田岳岡山理科大学, 理学部, 助手 (40309539) 坂田年男九州芸術工科大学, 芸術工学部, 教授 (20117352) 大江貴司岡山理科大学, 総合情報学部, 助教授 (90258210) 梶本ひろし長崎大学, 教育学部, 助教授 (50194741)
Keywords	分割表 / グレブナ基底 / トーリックイデアル / エシェロン形式 / MCMC法 / 量子コンピュータ / Order-finding問題 / 量子ランダムウォーク
Research Abstract	多項式環のイデアルの特別な基底としてのグレブナ基底は、具体的な計算においてその活用の場を広く様々な分野で見いだされつつある。本研究の出発点となった「非負成分の多重表に於いて周辺和を固定した分割表上の酔歩に対するマルコフ基底を代数的なグレブナ基底を計算する事によって求める」を使って実際に具体的に計算する事であった。この目的での研究実績は、3元分割表である3x3x4,3x4x4,4x4x4などに関するグレブナ基底の計算を、トーリックイデアルを用いた専用の計算プログラムを作成して計算機上(ワークステーション)で計算を実行させる事によって結果を得て、対応するマルコフ基底を求めた。これらの結果は、分割表の離散統計解析への応用として、得られたマルコフ基底より分割表上のランダムウォークを実現し、実際に具体的なデータに対して、P-値などを求めて解析を行った。また、これらの研究過程で、グレブナ基底を求める際のアルゴリズムに関して、分割表に起因するトーリックイデアルのエシェロン形式に関しての補題等の証明を行った。4x4x4分割表に関するSturmfelsの予想に関しても、成立するかどうか判定できるグレブナ基底に関するデータを得た。量子コンピュータに関する結果では、量子アルゴリズムの研究・計算量の過程で現れたOrder-finding問題に関しては、シミュレーションの為のプログラムを作成し、計算量の最適化として線形計画法を用いて、最適解を順次に求めて行き、そこに現れる現象に関しての予想を得た。さらに、量子アルゴリズムに関連して、分割表上でのマルコフ連鎖モンテカルロ法への応用として、分割表上の量子ランダムウォークの為のユニタリー行列を定義し、それに基づくシミュレーションの為のプログラム開発、及びその結果について纏めた。

Research Products
(32 results)

All Other

All Publications (32 results)

[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei, et al.: "Quantum random walks on the set of contingency tables"42^<nd> Sanibel Symposium, Abstract. (発表予定). (2002)
[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei, et al.: "Quantum random walks on the set of contingency tables"42^<nd> Sanibel Symposium, Proceeding. (発表予定). (2002)
[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei: "Application of Groebner basis to Quantum random walks on the set of contingency tables"10^<th> JST International Symposium. (予定). (2002)
[Publications] Ryuichi Sawae, Toshio Sakata: "Quantum walks on the contingency tables"「最近の計算機支援型推測の基礎理論とその応用」,非正則推測理論と情報量の概念に関する研究シンポジウム. 1-11 (2001)
[Publications] 鄭実成, 財部健一}, 澤江隆一他: "NMRによる量子コンピュータの実現"「最近の計算機支援型推測の基礎理論とその応用」,非正則推測理論と情報量の概念に関する研究シンポジウム. 62-80 (2001)
[Publications] Toshio Sakata, Rynichi Sawae: "Hypergeometric Systems, Weyl Algebra, Creation Operator, and Sequential Conditional Inference of Contingency Tables"Bulletin of the Institute of the International Statistical Institute, 53rd Session, Contributed papers. Vol.Book1. 387-388 (2001)
[Publications] Ryuichi Sawae, Toshio Sakata: "Conditional Analysis of Three Way Contingency Tables by Metropolis Walk"Bulletin of the Institute of the International Statistical Institute, 53rd Session, Contributed papers. Vol.Book1. 383-384 (2001)
[Publications] 澤江隆一, 財部健一, 坂田年男, 鄭実成他: "量子コンピューティング最小置換回数問題への応用"電子情報通信学会技術研究報告,信学技報. Vol00.No.586. 49-54 (2001)
[Publications] Ryuichi Sawae, Toshio Sakata, Minaru Tei, Kenichi Takarabe, 他: "Quantum Order-Finding by Linear Programming Theory"International Conference on Experimental Implementation of Quantum Computation, Program and Abstracts. WP-25. (2001)
[Publications] Ryuichi Sawae, Toshio Sakata, Minaru Tei, Kenichi Takarabe, 他: "Optimal probabilistic guess of quantum order-finding computation by linear programming theory"第4回量子情報技術研究会資料. 117-117 (2000)
[Publications] Toshio Sakata, Ryuichi Sawae: "Echelon form and conditional test for three way contingency tables"Proceeding of the Tenth Japan and Korea Joint Conference of Statistics. 233-238 (2000)
[Publications] 澤江隆一, 坂田年男: "整数行列のエシェロン形式と3元表上の酔歩"日本計算機統計学会第14回大会論文集. 106-109 (2000)
[Publications] Toshio Sakata, Ryuichi Sawae: "A study of the sequential conditional test for contingency tables"Preceedings of Internatinal Conference On New trends in Computational statistics with Biomedical applications. 377-382 (2001)
[Publications] Toshio Sakata, Ryuichi Sawae, Valeri Kroumov: "Applications of Groebner basis to analisis of contingency tables and integer programming"16-th International conference On product research, Praha, Proceedings. C.D.E-7.3. (2001)
[Publications] 坂田年男, 澤江隆一: "Hypergeometric Systems, Weyl Algebra, Creation Operator and Sequential Conditional Inference of Contingency Tables"シンポジウム・統計的実験における理論的基礎とその応用報告集. 41-51 (2000)
[Publications] Toshio Sakata, Ryuichi Sawae: "A new development of the conditional inference of the contingency tables"The proceedings of 7-th Japan-China Statistical Symposium. 271-274 (2000)
[Publications] S.Nagamachi, E.Bruning: "Hyperfunction quantum field theory : Analytic structure, modular aspects, and local observable algebras"Journal of Mathematical Physics. 42, No.1. 99-129
[Publications] Erwin Bruning, Shigeaki Nagamachi: "Kernel Theorem for Fourier Hyperfunctions"Journal of Mathematics, The University of Tokushima. 35. 57-64 (2001)
[Publications] Masaru Ikehata, Takashi Ohe: "Numerical method for finding the convex hull of polygonal cavities using Enclosure method"Inverse Problems. Vol.18, No.1. 111-124 (2002)
[Publications] Masaru Ikehata, Takashi Ohe: "Numerical implementation for finding the convex hull of inclusions using Enclosure method"The 7^<th> International Workshop on Electromagnetic Nondestructive Evaluation, Abstract. 45-46 (2001)
[Publications] Takashi Ohe, Katsu Yamatani, Kohzaburo Ohnaka: "An application of charge simulation method for the Laplace equation"International Conference on Recent Advances in Computational Mathematics (ICRACM 2001), Abstract of Talks. 95-96 (2001)
[Publications] 池畠優, 大江貴司: "The enclosure method and its numerical implementation"日本応用数理学会2000年度年会講演予稿集. 34-35 (2000)
[Publications] 池畠優, 大江貴司: "囲い込法の数値的構成"第50回理論応用力学講演会講演論文集(NCTAM2001). 441-442 (2001)
[Publications] V.Kroumov, Keishi Shibayama, Ryuichi Sawae, Toshio Sakata: "PERFORMANCE ENHANCEMENT OF A CLASS OF NONLINEAR NONMINIMUM PHASE SYSTEMS BY ADAPTIVE OBSERVER"
[Publications] Masatsugu MIZUKAMI, Manabu NAITO, Hiroyuki USAMI: "Asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations"
[Publications] Ken-ichi Kamo, Hiroyuki Usami: "Asymptotic forms of positive solutions of third-order Emden-Fowler equations"
[Publications] Takeshi Ikeda, Kanehisa Takasaki: "Toroidal Lie Algebras and Bogoyavlensky's (2+1)-Dimensional Equation"International Mathematics Research Notices. No7. (2001)
[Publications] Yuki Naito, Hiroyuki Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46. 629-652 (2001)
[Publications] Ken-ichi KAMO, Hiroyuki USAMI: "Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations with sub-homogeneity"HIROSHIMA MATHEMATICAL JOURNAL. Vol31, No.1. (2001)
[Publications] Minaru Tei, Y.Manmoto, M.Nobori, T.Yoshimoto: "Effects of Cooling Rate on Growth of Bi-2212 Superconducting Whiskers"JJAP suppl.. 39-1. (2001)
[Publications] Ken-ichi Kamo, Hiroyuki Usami: "Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations"Adv. Math. Sci. Appi.. Vol.2, No.2. 673-688 (2000)
[Publications] M.Ishii, Y.Yoshino, K.Takarabe, O.Shinomura: "X-ray induced photoemission of a localized electron and its application to site-selective x-ray absorption fine structure measurement"Journal of Applied Physics. Vol.88. 3962-3967 (2000)

2001 Fiscal Year Annual Research Report

格子上のトーリックイデアルのグレブナ基底と数理物理への応用

Principal Investigator

澤江 隆一 岡山理科大学, 理学部, 助教授 (20226062)

Research Products

[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei, et al.: "Quantum random walks on the set of contingency tables"42^<nd> Sanibel Symposium, Abstract. (発表予定). (2002)

[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei, et al.: "Quantum random walks on the set of contingency tables"42^<nd> Sanibel Symposium, Proceeding. (発表予定). (2002)

[Publications] R.Sawae, T.SAKATA, K.Takarabe, M.Tei: "Application of Groebner basis to Quantum random walks on the set of contingency tables"10^<th> JST International Symposium. (予定). (2002)

[Publications] Ryuichi Sawae, Toshio Sakata: "Quantum walks on the contingency tables"「最近の計算機支援型推測の基礎理論とその応用」,非正則推測理論と情報量の概念に関する研究シンポジウム. 1-11 (2001)

[Publications] 鄭 実成, 財部 健一}, 澤江隆一 他: "NMRによる量子コンピュータの実現"「最近の計算機支援型推測の基礎理論とその応用」,非正則推測理論と情報量の概念に関する研究シンポジウム. 62-80 (2001)

[Publications] Toshio Sakata, Rynichi Sawae: "Hypergeometric Systems, Weyl Algebra, Creation Operator, and Sequential Conditional Inference of Contingency Tables"Bulletin of the Institute of the International Statistical Institute, 53rd Session, Contributed papers. Vol.Book1. 387-388 (2001)

[Publications] Ryuichi Sawae, Toshio Sakata: "Conditional Analysis of Three Way Contingency Tables by Metropolis Walk"Bulletin of the Institute of the International Statistical Institute, 53rd Session, Contributed papers. Vol.Book1. 383-384 (2001)

[Publications] 澤江 隆一, 財部 健一, 坂田 年男, 鄭 実成 他: "量子コンピューティング最小置換回数問題への応用"電子情報通信学会技術研究報告,信学技報. Vol00.No.586. 49-54 (2001)

[Publications] Ryuichi Sawae, Toshio Sakata, Minaru Tei, Kenichi Takarabe, 他: "Quantum Order-Finding by Linear Programming Theory"International Conference on Experimental Implementation of Quantum Computation, Program and Abstracts. WP-25. (2001)

[Publications] Ryuichi Sawae, Toshio Sakata, Minaru Tei, Kenichi Takarabe, 他: "Optimal probabilistic guess of quantum order-finding computation by linear programming theory"第4回量子情報技術研究会資料. 117-117 (2000)

[Publications] Toshio Sakata, Ryuichi Sawae: "Echelon form and conditional test for three way contingency tables"Proceeding of the Tenth Japan and Korea Joint Conference of Statistics. 233-238 (2000)

[Publications] 澤江 隆一, 坂田 年男: "整数行列のエシェロン形式と3元表上の酔歩"日本計算機統計学会第14回大会論文集. 106-109 (2000)

[Publications] Toshio Sakata, Ryuichi Sawae: "A study of the sequential conditional test for contingency tables"Preceedings of Internatinal Conference On New trends in Computational statistics with Biomedical applications. 377-382 (2001)

[Publications] Toshio Sakata, Ryuichi Sawae, Valeri Kroumov: "Applications of Groebner basis to analisis of contingency tables and integer programming"16-th International conference On product research, Praha, Proceedings. C.D.E-7.3. (2001)

[Publications] 坂田 年男, 澤江 隆一: "Hypergeometric Systems, Weyl Algebra, Creation Operator and Sequential Conditional Inference of Contingency Tables"シンポジウム・統計的実験における理論的基礎とその応用 報告集. 41-51 (2000)

[Publications] Toshio Sakata, Ryuichi Sawae: "A new development of the conditional inference of the contingency tables"The proceedings of 7-th Japan-China Statistical Symposium. 271-274 (2000)

[Publications] S.Nagamachi, E.Bruning: "Hyperfunction quantum field theory : Analytic structure, modular aspects, and local observable algebras"Journal of Mathematical Physics. 42, No.1. 99-129

[Publications] Erwin Bruning, Shigeaki Nagamachi: "Kernel Theorem for Fourier Hyperfunctions"Journal of Mathematics, The University of Tokushima. 35. 57-64 (2001)

[Publications] Masaru Ikehata, Takashi Ohe: "Numerical method for finding the convex hull of polygonal cavities using Enclosure method"Inverse Problems. Vol.18, No.1. 111-124 (2002)

[Publications] Masaru Ikehata, Takashi Ohe: "Numerical implementation for finding the convex hull of inclusions using Enclosure method"The 7^<th> International Workshop on Electromagnetic Nondestructive Evaluation, Abstract. 45-46 (2001)

[Publications] Takashi Ohe, Katsu Yamatani, Kohzaburo Ohnaka: "An application of charge simulation method for the Laplace equation"International Conference on Recent Advances in Computational Mathematics (ICRACM 2001), Abstract of Talks. 95-96 (2001)

[Publications] 池畠優, 大江貴司: "The enclosure method and its numerical implementation"日本応用数理学会2000年度年会講演予稿集. 34-35 (2000)

[Publications] 池畠優, 大江貴司: "囲い込法の数値的構成"第50回理論応用力学講演会講演論文集(NCTAM2001). 441-442 (2001)

[Publications] V.Kroumov, Keishi Shibayama, Ryuichi Sawae, Toshio Sakata: "PERFORMANCE ENHANCEMENT OF A CLASS OF NONLINEAR NONMINIMUM PHASE SYSTEMS BY ADAPTIVE OBSERVER"

[Publications] Masatsugu MIZUKAMI, Manabu NAITO, Hiroyuki USAMI: "Asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations"

[Publications] Ken-ichi Kamo, Hiroyuki Usami: "Asymptotic forms of positive solutions of third-order Emden-Fowler equations"

[Publications] Takeshi Ikeda, Kanehisa Takasaki: "Toroidal Lie Algebras and Bogoyavlensky's (2+1)-Dimensional Equation"International Mathematics Research Notices. No7. (2001)

[Publications] Yuki Naito, Hiroyuki Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46. 629-652 (2001)

[Publications] Ken-ichi KAMO, Hiroyuki USAMI: "Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations with sub-homogeneity"HIROSHIMA MATHEMATICAL JOURNAL. Vol31, No.1. (2001)

[Publications] Minaru Tei, Y.Manmoto, M.Nobori, T.Yoshimoto: "Effects of Cooling Rate on Growth of Bi-2212 Superconducting Whiskers"JJAP suppl.. 39-1. (2001)

[Publications] Ken-ichi Kamo, Hiroyuki Usami: "Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations"Adv. Math. Sci. Appi.. Vol.2, No.2. 673-688 (2000)

[Publications] M.Ishii, Y.Yoshino, K.Takarabe, O.Shinomura: "X-ray induced photoemission of a localized electron and its application to site-selective x-ray absorption fine structure measurement"Journal of Applied Physics. Vol.88. 3962-3967 (2000)

澤江隆一岡山理科大学, 理学部, 助教授 (20226062)

[Publications] 鄭実成, 財部健一}, 澤江隆一他: "NMRによる量子コンピュータの実現"「最近の計算機支援型推測の基礎理論とその応用」,非正則推測理論と情報量の概念に関する研究シンポジウム. 62-80 (2001)

[Publications] 澤江隆一, 財部健一, 坂田年男, 鄭実成他: "量子コンピューティング最小置換回数問題への応用"電子情報通信学会技術研究報告,信学技報. Vol00.No.586. 49-54 (2001)

[Publications] 澤江隆一, 坂田年男: "整数行列のエシェロン形式と3元表上の酔歩"日本計算機統計学会第14回大会論文集. 106-109 (2000)

[Publications] 坂田年男, 澤江隆一: "Hypergeometric Systems, Weyl Algebra, Creation Operator and Sequential Conditional Inference of Contingency Tables"シンポジウム・統計的実験における理論的基礎とその応用報告集. 41-51 (2000)