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NAKAMURA Yayoi

    Department of Science　Associate Professor
Last Updated :2024/04/23

Researcher Information

URL

J-Global ID

Research Interests

  • D-加群   孤立特異点   代数的局所コホモロジー   計算代数解析   ホロノミック系   b-関数   代数的局所コホモロジー類   グロタンディック双対性   Milnor algebra   スタンダード基底   超幾何微分方程式   グレブナー基底   多重ゼータ値   特異摂動   変わり点   スペクトル分解   複素解析幾何   

Research Areas

  • Natural sciences / Basic analysis
  • Natural sciences / Geometry

Academic & Professional Experience

  • 2009/04 - Today  Kindai UniversityFaculty of Science and Engineering准教授
  • 2004/04 - 2009/03  Kindai University理工学部理学科講師

Published Papers

  • Kentaro Ihara; Yusuke Kusunoki; Yayoi Nakamura; Hitomi Saeki
    Canadian J. Math. 76 (1) 1 - 17 2024/02 [Refereed]
  • Yayoi NAKAMURA; Yoshitaka SASAKI
    Lithuanian Mathematical Journal Springer Science and Business Media LLC 63 (3) 382 - 395 0363-1672 2023/09 [Refereed]
  • Yusuke Kusunoki; Yayoi Nakamura
    Research in Number Theory 8 (4) 2022/11 [Refereed]
  • Y. Kusunoki; Y. Nakamura; Y. Sasaki
    Acta Arithmetica Institute of Mathematics, Polish Academy of Sciences 195 (2) 131 - 148 0065-1036 2020/06 [Refereed]
  • Y.Kusunoki; Y. Nakamura; Y. Sasaki
    Commentarii Mathematici Universitatis Sancti Pauli 67 (1) 49 - 64 2019/07 [Refereed]
  • Yayoi Nakamura
    JOURNAL OF ALGEBRA AND ITS APPLICATIONS WORLD SCIENTIFIC PUBL CO PTE LTD 15 (6) 1650115  0219-4988 2016/08 [Refereed]
     
    The b-function and related invariants of the singularity z(1)(4) + z(2)(p) + z(1)z(2)(p-1) = 0 are determined. We exploit the theory of computer algebra and Weyl algebra for the study of local b-function by T. Yano in 1970s.
  • Radius problems for certain strongly close-to-convex functions
    Shimoda, Yutaka; Nakamura, Yayoi; Owa, Shigeyoshi
    PanAmer. Math. J. 23 (2) 53 - 58 2013/01 [Refereed]
  • Shinichi, Tajima; Yayoi, Nakamura
    Publ. Res. Inst. Math. Sci. Kyoto University 48 (1) 21--43 - 43 0034-5318 2012/03 [Refereed]
  • Yayoi Nakamura; Shinichi Tajima
    SINGULARITIES IN GEOMETRY AND TOPOLOGY: STRASBOURG 2009 EUROPEAN MATHEMATICAL SOC 20 103 - + 2012 [Refereed]
     
    The role of the weighted degree of algebraic local cohomology classes in the computation of b-function is discussed. The result which is similar to quasihomogeneous cases is observed for semiquasihomogeneous isolated singularities with L(f) = 2.
  • Shinichi Tajima; Yayoi Nakamura
    JOURNAL OF SYMBOLIC COMPUTATION ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD 44 (5) 435 - 448 0747-7171 2009/05 
    Hypersurface isolated singularities are considered in the context of algebraic analysis. A method for Computing relative Cech cohomology representations of algebraic local cohomology classes supported at the isolated singular point is described. An effective method to test membership and a method to compute the normal form for the Jacobi ideal are presented. The main purpose of this paper is to provide ail effective algorithm for computing annihilating ideals, in the ring of partial differential operators, of the algebraic local cohomology class that generates the dual vector space to the local Milnor algebra. (C) 2008 Elsevier Ltd. All rights reserved.
  • Shinichi Tajima; Yayoi Nakamura; Katsusuke Nabeshima
    Advanced Studies in Pure Mathematics Mathematical Society of Japan 56 341  0920-1971 2009/03 [Refereed]
  • Shanmugam, T. N; Owa, Shigeyoshi; Ramachandran, C; Sivasubramanian, S; Nakamura, Yayoi
    J. Math. Inequal. ELEMENT 3 (1) 31 - 41 1846-579X 2009/01 [Refereed]
     
    In the present investigation, the authors obtain sharp upper bounds for certain coefficient inequalities for linear combination of Mocanu alpha-convex p-valent functions. The results are extended to functions defined by convolution.
  • An algorithm to compute parametric standard bases using algebraic local cohomology for zero-dimensional ideals
    K., Nabeshima; Y., Nakamura; S., Tajima
    Proceedings of the Joint Conference of ASCM 2009 and MACIS 2009, COE Lecture Notes Kyushu Univ. Press 22 123-126  2009/01 [Refereed]
  • Yayoi Nakamura
    RIMS Kokyuroku Bessatsu Kyoto Univ. B5 7 - 14 2008
  • Sufficient conditions for starlikeness and convexity in |z|<1/2
    Nunokawa, Mamoru; Owa, Shigeyoshi; Nakamura, Yayoi; Hayami, Toshio
    JIPAM. J. Inequal. Pure Appl. Math. 9 (2) 2008 [Refereed]
  • Two points-distortion theorems for multivalued starlike functions
    Polatog̃lu, Yaşar; Yavuz, Emel; Owa, Shigeyoshi; Nakamura, Yayoi
    Int. J. Math. Anal. 2 (17-20) 799 - 806 2008 [Refereed]
  • The univalence conditions for a general integral operator
    Breaz, Daniel; Nakamura, Yayoi; Owa, Shigeyoshi
    Int. J. Open Probl. Comput. Sci. Math. 1 (1) 43 - 51 2008 [Refereed]
  • On holonomic D-modules associated with isolated singularities of a Reiffen surface
    Yayoi NAKAMURA; Shinichi Tajima
    Sovrem. Mat. Prilozh. 54 124 - 132 2008 [Refereed]
  • Alawiah Ibrahim; Shigeyoshi Owa; Maslina Darus; Yayoi Nakamura
    BANACH JOURNAL OF MATHEMATICAL ANALYSIS BANACH MATHEMATICAL RESEARCH GROUP 2 (2) 16 - 22 1735-8787 2008 [Refereed]
     
    For analytic functions f in the open unit disc U, a generalization operator D(lambda) f(z) of Salagean operator is introduced. Some properties for D(lambda) f(z) are discussed in the present paper.
  • Shinichi Tajima; Yayoi Nakamura
    Adv. Stud. Pure Math. Mathematical Society of Japan 46 105 - 117 0920-1971 2006
  • S Tajima; Y Nakamura
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIV 41 (1) 1 - 10 0034-5318 2005/03 
    The purpose of this paper is to study hypersurface isolated singularities by using partial differential operators based on D-modules theory. Algebraic local cohomology classes supported at a singular point that constitute the dual space of the Milnor algebra are considered. It is shown that an isolated singularity is quasi-homogeneous if and only if an algebraic local cohomology class generating the dual space can be characterized as a solution of a holonomic system of first order partial differential equations.
  • Computational aspects of Grothendieck local residues
    Sshinichi Tajima; Yayoi Nakamura
    Seminaires et Congres SMF 10 287 - 305 2005
  • Unimodal singularities and differential operators
    Yayoi Nakamura; Shinichi Tajima
    Seminaires et Congres SMF 10 191 - 208 2005
  • On the dual space of the Tjurina algebra attached to a semiquasihomogeneous isolated singularity
    Yayoi Nakamura; Shinichi Tajima
    Banach Center Publications 65 261 - 272 2004
  • Y Nakamura; S Tajima
    MATHEMATICAL SOFTWARE, WORLD SCIENTIFIC PUBL CO PTE LTD 158 - 168 2002
  • Local cohomology classes and dual bases for quasihomogeneous isolated singularities
    Yayoi Nakamura
    Proceedings of the eighth international colloquium on complex analysis Shandong Science and Technology Pres 213 - 218 2001
  • Y Nakamura; S Tajima
    PROCEEDINGS OF THE SECOND ISAAC CONGRESS, VOLS 1 AND 2 SPRINGER 7 809 - 817 2000 
    The local residue has been presented since the early days of several complex variables and plays a fundamental role in complex analysis and geometry. Nevertheless, in many cases, there are very few practical tools in the computational point of view. The purpose of this note is to give a new algorithm for computing the local residue with the viewpoint of the theory of D-modules. For a given regular sequence of holomorphic functions, consider the algebraic local cohomology group with support at the set of common zeros of these functions. If the set of common zeros consists of finitely many points, the cohomology classes with support at each zero can be characterized as a solution of a system of certain linear partial differential equations. In this note, a system of such linear partial differential equations is constructed by using the transformation formula. And, on the basis of the properties of the image of adjoint operators of the system, an algorithm for computing local residues is provided.
  • Y Nakamura
    Lecture Notes in Pure and Appl. Math. MARCEL DEKKER 214 345 - 351 0075-8469 2000
  • Yayoi Nakamura; Shinichi Tajima
    Kyushu J. of Mathematics Faculty of Mathematics, Kyushu University 54 (1) 127 - 136 1340-6116 2000
  • Nakamura Yayoi; Tajima Shinichi
    Josai Mathematical Monographes Josai University 2 149 - 158 1344-7777 2000 
    In a previous paper [4] , we studied residues of a rational function frorn a view pointof D-modules. We considered the system of linear differential equations for the algebraic10cal cohomology class associated to a given rational function. In particular, we gave therea description of the kernel space of the residue map induced by the cohomology class interms of adjoint differential operators.In this paper, we present algorithms for computing residues of a rational functionaccording to the results obtained in [4] . By exploiting properties of adjoint differentialoperators, we reduce the computation of residues of a rational function to that of a rationalfunction which has simple poles.In the first section, we recall some facts about D-modules and the algebraic localcohomology groups for the one dimensional case. In the second section, we briefly recall themain results obtained in [4] . In the third section, we describe two algorithms for computingresidues of a rational function. We utilize a formula for a squarefree denominator andproperties of a logarithmic derivative in each algorithms.NLA99: Computer Algebra : The Fourth Symposium on Nonlinear Analysis, September 16-18, 1999 Josai University, edited by Kiyoko Nishizawa (Josai University), Tomokatsu Saito (Sophia University), Teluhiko Hilano (Kanagawa Inst. Tec.)
  • Nakamura Yayoi
    Josai Mathematical Monographes Josai University 2 139 - 148 1344-7777 2000 
    We studied the residue pairing induced by an algebraic local cohomology class froma view point of the theory of D-modules in [2] and [3] . For a cohomology class of onedimensional case, we constructed a linear differential operator of order one which was thetheoretical foundation of an algorithm for computing residues (cf. [3]). On the other hand,in the theory of quasihomogeneous singularities, it is known that linear partial differentialoperators of order one determined by weights play an important role.In this paper, we look at a differential operaots of order I associated to an algebraic localcohomology class. First, we consider the normal forms of quasihomogeneous polynomials.Then we provide a method for computing a presentation of a cohomology class for aquasihomogeneous polynomial. Next, we provide a method for computing linear partialdifferential operators of order one associated to a cohomology class of general n dimensionalcase.NLA99: Computer Algebra : The Fourth Symposium on Nonlinear Analysis, September 16-18, 1999 Josai University, edited by Kiyoko Nishizawa (Josai University), Tomokatsu Saito (Sophia University), Teluhiko Hilano (Kanagawa Inst. Tec.)

Books etc

  • 井原, 健太郎; 鄭, 仁大; 中村, 弥生 (数学); 松井, 優 培風館 2023/03 9784563012441 iv, 169p
  • 井原, 健太郎; 鄭, 仁大; 中村, 弥生 (数学); 松井, 優 培風館 2023/03 9784563012458 iv, 176p
  • 青木, 貴史; 大野, 泰生; 佐久間, 一浩; 中村, 弥生 (数学) 培風館 2013/02 9784563003982 iv, 148p
  • 青木, 貴史; 大野, 泰生; 佐久間, 一浩; 中村, 弥生 (数学) 培風館 2012/12 9784563003975 iv, 135p
  • 青木, 貴史; 大野, 泰生; 尾崎, 学; 佐久間, 一浩; 中村, 弥生 (数学) 培風館 2009/04 9784563003869 iv, 133p
  • 青木, 貴史; 大野, 泰生; 尾崎, 学; 佐久間, 一浩; 中村, 弥生 (数学) 培風館 2009/04 9784563003852 iv, 144p
  • 微分積分学
    青木貴史; 大野泰生; 尾﨑学; 佐久間一浩; 中村弥生 J.B.企画 2006/04
  • 線形代数学
    青木貴史; 大野泰生; 尾﨑学; 佐久間一浩; 中村弥生 J.B.企画 2004/04

Conference Activities & Talks

  • A residue calculus for few variants of MZV  [Invited]
    中村弥生
    RIMS研究集会 Zeta functions and their representations  2023/03
  • An interpolation function of truncated multiple zeta values  [Invited]
    中村弥生
    RIMS研究集会 多重ゼータ値の諸相  2019/11
  • 多重ポリログ関数の解析接続の明示式について  [Not invited]
    Yayoi NAKAMURA
    名古屋大学大学院多元数理研究科解析数論セミナー  2018/04
  • Analytic continuation of multiple polylogarithm via residues  [Not invited]
    Yayoi NAKAMURA
    第11回ゼータ若手研究集会  2018/02
  • On an invariant of b-operator for Reiffen’s (p,4) isolated singularity  [Not invited]
    Yayoi NAKAMURA
    a workshop “Algebra, Algorithms and Algebraic Analysis”  2013
  • Algebraic local cohomology and b-function for hypersurface isolated singularities  [Not invited]
    Yayoi NAKAMURA
    ボルドー大学数学教室セミナー  2007/12
  • Annihilating ideals for an algebraic local cohomology class  [Not invited]
    中村弥生; 田島慎一
    Computer Algebra and Applications  2006/09
  • An annihilating ideal for algebraic local cohomology classes  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 Efficient computation of Groebner bases and mathematical algorithms based on it  2006/08
  • Invariants of isolated singularities via holonomic systems  [Not invited]
    中村弥生; 田島慎一
    International Conference on Differential Equations and Dynamical Systems  2006/07
  • 零次元代数多様体に台を持つ代数的局所コホモロジー類のannihilatorの計算法について  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 グレブナー基底の理論的有用性と実践的有効性  2006/06
  • invariant of hypersurface isolated singularities  [Not invited]
    中村弥生; 田島慎一
    セビリア大学数学教室セミナー  2006/03
  • Invariants of semiquasihomogeneous isolated singularities  [Not invited]
    中村弥生; 田島慎一
    Theoretical Effectivity and Practical Effectivity of Groebner bases  2005/08
  • On weighted-degrees for algebraic local cohomologies associated with semiquasihomogeneous singularities  [Not invited]
    中村弥生; 田島慎一
    Singularities Franco-Japonaises  2004/08
  • Hermite-Jacobi 多変数再生核の計算代数解析
    中村弥生; 田島慎一
    再生核の理論の応用  2003/10
  • Grothendieck 留数の代数解析と計算アルゴリズム  [Invited]
    中村弥生; 田島慎一
    日本数学会秋季総合分科会函数論分科会  2003/09
  • A method for constructing holonomic systems attached to a zero-dimensional algebraic local cohomology class  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 超局所解析の展望  2003/08
  • Unimodal singularities に付随したホロノミック系  [Not invited]
    Yayoi NAKAMURA
    複合領域科学の新展開に関する第１回国際シンポジウム  2003/03
  • 代数的局所コホモロジー類の annihilator の構成法について  [Not invited]
    Yayoi NAKAMURA
    OpenXM committer's meeting  2003/02
  • Differential operators and algebraic local cohomologies attached to isolated singularities  [Not invited]
    中村弥生; 田島慎一
    実・複素特異点のトポロジーII  2002/12
  • A method to compute Grothendieck residue  [Not invited]
    中村弥生; 田島慎一
    Seminor on Grothendieck residue  2002/11
  • 代数的局所コホモロジー類のannihilator とsemiquasihomogeneous singularities  [Not invited]
    Yayoi NAKAMURA
    RIMS研究集会 多変数函数論の萌芽的研究  2002/11
  • Unimodal singularities, algebraic local cohomologies and differential operators  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 多変数函数論の萌芽的研究  2002/11
  • 代数的局所コホモロジー類のannihilator とsemiquasihomogeneous singularities  [Not invited]
    Yayoi NAKAMURA
    計算代数セミナー  2002/10
  • D-加群における孤立特異点と代数的局所コホモロジー  [Not invited]
    Yayoi NAKAMURA
    第45回函数論シンポジウム  2002/08
  • A method for constructing holonomic systems for algebraic local cohomology classes with support on a zero dimensional variety  [Not invited]
    中村弥生; 田島慎一
    International Cong- ress of Mathematic- al Software  2002/08
  • Unimodal singularities and differential operators  [Not invited]
    中村弥生; 田島慎一
    Singularites Franco-Japonaises  2002/08
  • D-加群における孤立特異点と代数的局所コホモロジー  [Not invited]
    Yayoi NAKAMURA
    代数幾何セミナー  2002/07
  • A study of semiquasihomo- geneous singularities by using holonomic system  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 特異点とNewton図形  2001/11
  • 代数的局所コホモロジー類の満たすホロノミック系の構成法について(2)  [Not invited]
    Yayoi NAKAMURA
    RIMS研究集会 Computer Algebra - Algorithms, Implementations and Applications -  2001/11
  • On the dual space of the Milnor algebra attached to a non quasi homogeneous isolated singularity  [Not invited]
    田島慎一; 中村弥生
    Polish-Japanese Singularity Theory Work Days  2001/09
  • ホロノミック系を用いた多変数留数計算アルゴリズム(2)  [Not invited]
    Yayoi NAKAMURA
    第 40 回多変数函数論サマーセミナー  2001/08
  • Semiquasihomogeneous singularity に付随した代数的局所コホモロジー類とholonomic 系  [Not invited]
    中村弥生; 田島慎一
    第44回函数論シンポジウム  2001/07
  • Milnor algebra に付随したholonomic 系について  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 微分方程式論における積分公式とTwisted cohomology  2001/05
  • Unimodal 例外型特異点における代数的局所コホモロジー類  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 微分方程式の漸近解析と超局所解析  2001/04
  • 2変数 unimodal 例外型 singularity に付随する局所コホモロジーの計算  [Not invited]
    中村弥生; 田島慎一
    日本数学会函数論分科会  2001/03
  • Milnor algebraの双対基底と微分作用素  [Not invited]
    Yayoi NAKAMURA
    特異点論ワークショップ  2000/11
  • 代数的局所コホモロジー類の満たすホロノミック系の構成法について  [Not invited]
    中村弥生; 田島慎一
    数式処理における理論と応用の研究  2000/11
  • 擬斉次孤立特異点に関する双対基底の計算  [Not invited]
    中村弥生; 田島慎一
    日本数学会秋季総合分科会函数論分科会  2000/09
  • Local cohomology classess and dual bases for quasihomogeneous isolated singularities  [Not invited]
    中村弥生; 田島慎一
    The eighth interna- tional conference on finite or infi- nite dimensional complex analysis  2000/08
  • 擬斉次孤立特異点に対する双対基底の計算  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 D-加群のアルゴリズム  2000/06
  • 有理関数の極におけるローラン展開の計算法 II--アルゴリズムの実装--  [Not invited]
    中村弥生; 田島慎一
    Risa Consortium  2000/03
  • 最小多項式を用いた有理型関数の留数計算に関するアルゴリズム  [Not invited]
    中村弥生; 田島慎一
    Risa Consortium  1999/11
  • Construction of the system of differential operators in an algorithm for computing the residues  [Not invited]
    Yayoi NAKAMURA
    The fourth symposium on Nonlinear Analysis and Applications  1999/09
  • D-加群を用いた多変数留数値計算 (shape lemma を満たす場合)  [Not invited]
    中村弥生; 田島慎一
    数式処理学会  1999/05
  • How to compute Grothendieck residues  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 超局所解析と複素領域の偏微分方程式系  1998/11
  • 微分作用素を用いた留数計算 VI, VII  [Not invited]
    中村弥生; 田島慎一
    日本数学会秋季総合分科会函数論分科会  1998/10
  • グレブナ基底による多変数留数の計算アルゴリズム  [Not invited]
    中村弥生; 田島慎一
    日本応用数理学会年会  1998/09
  • 多変数留数計算について  [Not invited]
    中村弥生; 田島慎一
    日本数式処理学会大会  1998/06
  • 微分作用素を用いた留数計算 IV  [Not invited]
    中村弥生; 田島慎一
    日本数学会函数論分科会  1998/03
  • On asymptotic estimates for coefficients of divergent solutions to second order non-homogeneous linear ordinary differential equations  [Not invited]
    Yayoi NAKAMURA
    RIMS研究集会 複素解析と超局所解析  1997/12
  • Grothendieck residue calculus and holonomic D-modules  [Not invited]
    中村弥生; 田島慎一
    解析的偏微分方程式の研究  1997/11
  • 微分作用素を用いた留数計算  [Not invited]
    中村弥生; 田島慎一
    RIMS研究集会 数式処理における理論と応用の研究  1997/11
  • 多変数留数計算とホロノミックな D-加群  [Not invited]
    中村弥生; 大阿久敏則; 田島慎一
    日本数学会函数論分科会  1997/10
  • 多項式係数を持つ非斉次線形常微分方程式の形式解の係数に関する評価  [Not invited]
    Yayoi NAKAMURA
    RIMS研究集会 複素領域における偏微分方程式  1997/09
  • 微分作用素を用いた留数計算 III  [Not invited]
    中村弥生; 田島慎一
    日本数学会函数論分科会  1997/09
  • 微分作用素を使って留数計算を楽しむ法  [Not invited]
    田島慎一; 中村弥生
    函数論サマーセミナー  1997/08
  • Multidimensional residue calculus and holonomic D-modules  [Not invited]
    田島慎一; 大阿久俊則; 中村弥生
    特異点と複素解析幾何  1997/07

MISC

Research Grants & Projects

  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2018/04 -2023/03 
    Author : 青木 貴史; 中村 弥生; 鈴木 貴雄
     
    まず、本年度出版された論文に関して概要を述べる。「研究発表」項目第1番目の論文では、大きなパラメータを持つ超幾何微分方程式に対してヴォロス係数を定義し、一般的な形で明示式を与えた。2番目の論文では、大きなパラメータを持つ超幾何関数・合流型超幾何関数と、これらが満たす微分方程式のWKB解のボレル和との関係を一般的な場合に完全に記述した。応用として、これらの関数のパラメータに関する漸近展開公式を与えた。3番目の論文では、一般化されたq-ガルニエ系に対するLax形式を与えた。本年度実施した研究では、次のような成果が得られた。一般化超幾何関微分方程式に対する原点および無限遠点におけるヴォロス係数を定義し，その具体形を与えた。さらに、一般化超幾何微分方程式の間に成り立つ退化図式と、ヴォロス係数に対する極限操作が整合していることを見出した。多変数の場合への研究にも着手した。その原型として、大きなパラメータを持つエアリーの微分方程式のWKB解をホロノミック系の立場から見直し、接続公式の初等的な別証明を与えた。従来の標準的理論では、超幾何関数の接続公式を用いていた証明に対して、新たに得られた証明は、代数関数の接続のみを用いる。この立場から、多変数の場合の最も基本的なものとして大きなパラメータを持つパーシー積分が満たすホロノミック系（パーシー系と呼ぶ）について研究を行い、エアリーの微分方程式と同様の構造があることを見出した。これにより、パーシー系のWKB解のリサージェンスが証明できた。これらの成果を記載した論文は、現在執筆中である。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2014/04 -2018/03 
    Author : AOKI TAKASHI; HONDA Naofumi; KAWAI Takahiro; TAKEI Yoshitsugu; YAMAZAKI Susumu; KOIKE Tatsuya; UMETA Yoko
     
    Introducing a large parameter in the 3 parameters contained in the Gauss hypergeometric differential equation, we can construct the WKB solutions which are formal solutions to the equation. The construction is done algebraically and elementarily, however, these formal solutions are divergent in general and do not have analytic sense. We may apply the Borel resummation method to the formal solutions and can construct analytic solutions and bases of the solution space. On the other hand, the Gauss hypergeometric differential equation has standard bases of solutions expressed by the hypergeometric function. In this research, we have obtained linear relations between these two classes of bases. As an application, asymptotic expansion formulas with respect to the large parameter of the Gauss hypergeometric function have been obtained. At the same time, we have some formulas which describe the parametric Stokes phenomena of the WKB solutions.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2012/04 -2017/03 
    Author : SUWA Tatsuo; ASUKE Taro; OHMOTO Toru; OKA Mutsuo; TAKEUCHI Kiyoshi; TAJIMA Shinichi; NAKAMURA Yayoi; YOKURA Shoji
     
    The localization theory of characteristic classes developed by the principal investigator turned out to be very effective in a wide range of problems related to characteristic classes mainly in complex analytic geometry. During the period, we obtained the following results. (1) As to the degeneracy loci problem of vector bundle homomorphisms, we constructed a universal localization. (2) We generalized the Lefschetz coincidence point formula. For this, the local and global homology classes we introduced played key roles. (3) We developed the theory of relative Bott-Chen cohomology and give some applications. (4) We discovered a simple way of expressing Sato hyperfunctions. For this we strengthened the theory of relative Dolbeault cohomology. We also gave simple expressions of fundamental operations on the hyperfunctions.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2012/04 -2015/03 
    Author : TAJIMA Shinichi; OHARA Katsuyoshi; NABESHIMA Katsusuke; NAKAMURA Yayoi; OAKU Toshinori
     
    Complex analytic properties of hypersurface isolated singularities are considered in the context of algebraic analysis. An algorithm is constructed for computing Tjurina stratifications, the parameter dependency of Tjurina numbers, standard bases of relevant ideal quotients of semi quasihomogeneous hypersurface isolated singularities with deformation parameters. Polar varieties and logarithmic vector fields associated with hypersurface isolated singularities are studied. A new effective method is obtained for computing logarithmic vector fields associated with hypersurface isolated singularities. Exact methods for computing generalized eigenvectors of given matrices are studied. An efficient method is constructed for conmuting annihilating polynomials of unit vectors. Efficient algorithms are constructed and also implemented in a computer algebra system for computing eigenvectors and also generalized eigenvectors.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2010/04 -2014/03 
    Author : AOKI Takashi; SUZUKI Takao; IZUMI Shuzo; MATSUI Yutaka; NAKAMURA Yayoi; HONDA Naofumi; KAWAI Takahiro; TAKEI Yoshitsugu; KOIKE Tatsuya
     
    In this research, we have investigated the global properties of solutions to differential equations with a large parameter from the view point of the exact WKB analysis. There are three main results. Firstly, we have constructed the exponential-asymptotic (instanton-type) solutions, namely general formal solutions, to the equations which belong to the first Painleve hierarchies. Secondly, we have classified the topological types of the Stokes curves of the Gauss equation in terms of the parameters of the equation. Thirdly we have defined and computed explicit forms of the Voros coefficients of Gauss equation with a large parameter and obtained the Borel sums go them. We have obtained the formulas that describe parametric Stokes phenomena of WKB solutions.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2009 -2011 
    Author : NAKAMURA Yayoi
     
    Studying algebraic local cohomologies and holonomic systems attached to several types of isolated singularities from the view point of computational algebraic analysis, formulas concerningμ-constant deformation of singularities and jumps of roots of b-functions are given under some conditions for systems of differential operators. And a result on differential orders and a construction of holonomic systems which is useful for studies of Logarithmic Comparison Theorem is given.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2009 -2011 
    Author : TAJIMA Shinichi; NAKAMURA Yayoi; OHARA Katsuyoshi; MATSUI Yutaka
     
    Algebraic local cohomology classes attached to hypersurface isolated singularities are considered in the context of algebraic analysis. A new algorithm of computing parametric algebraic local cohomology classes is derived. A new framework to study logarithmic vector fields and associated holonomic D-modules is constructed. An efficient algorithm to compute spectral decomposition of square matrices is derived by analyzing resolvent.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2009 -2011 
    Author : SUWA Tatsuo; OHMOTO Toru; OKA Mutsuo; KAWAZUMI Nariya; TAKEUCHI Kiyoshi; TAJIMA Shinichi; NAKAMURA Yayoi; YOKURA Shoji; YOSHIKAWA Kenichi
     
    (1) Concerning the localization theory of Atiyah classes, with collaboration of M. Abate, F. Bracci and F. Tovena, we established the following fundamental theories : (1) a simple definition of Atiyah classes suitable for the localization theory,(2) Cech-Dolbeault cohomology theory,(3) introduction of the complex analytic Thom class,(4) proof of a Bott type vanishing theorem in terms of Atiyah forms. (2) Concerning the degeneracy loci problem of a homomorphism of vector bundles, with collaboration of T. Ohmoto, we started to try to prove the Thom-Porteous formula localized at the degeneracy loci. This is done by constructing a universal localization of a Schur polynomial of Chern. It is a vast generalization of the Thom class of a vector bundle.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2007 -2008 
    Author : NAKAMURA Yayoi
     
    本研究では、特異点にデフォーメーションを施した際の代数的局所コホモロジーの変化を解析することにより、特異点の性質を研究した.擬斉次孤立特異点に関して、ミルナー代数の単項基底に注目したデフォーメーションを行った場合の代数的局所コホモロジーの変化の解析を行った.また、デフォーメーションによるコホモロジーの重みのジャンプ幅と特異点のb-関数との対応を解析することにより、b-関数の変化とモダリティーのとの間に成り立つ関係を得た.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2006 -2008 
    Author : AOKI Takashi; HONDA Naofumi; OHNO Yasuo; NAKAMURA Yayoi; MATSUI Yutaka; HONDA Naofumi; NAKAMURA Yayoi
     
    大きなパラメータを自然な形で含む連立非線型微分方程式系の形式解を構成するためには,主要部を決定する代数方程式系を解く必要がある.方程式の階数や方程式の個数が大きい場合は代数方程式系が複雑なものとなり,一見したところでは主要部が決定可能かどうかの判定は困難である.本研究では,この間題に関して主要部が決定可能であることを保証する幾つかの条件を与えた.これらの条件を実際の例に適用して重要な方程式系に対する形式解の存在が証明された.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2005 -2007 
    Author : TAJIMA Shinichi; YOSHIHARA Hisao; KOJIMA Hideo; TAKEUCHI Kiyoshi; NAKAMURA Yayoi
     
    Hypersurface isolated singularities and associated residues currents are considered in the context of algebraic analysis. ・An efficient algorithm that computes bases of a dual vector space of a Milnor algebra associated to a singular point has been constructed. ・A new method for computing standard bases of a zero-dimensional ideal in a power series ring has been proposed. The key ingredient in this approach is the concept of algebraic local cohomology and the Grothendieck local duality. ・An algorithm for construction holonomic D-modules attached to hypersurface isolated singularities has been derived and the structure of these holonomic D-modules have been investigated. ・An algorithmic method for computing homological indices of holomorphic vector fields has been proposed.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2005 -2006 
    Author : 中村 弥生
     
    超平面孤立特異点に付随する代数的局所コホモロジーとそのannihilatorである偏微分作用素からなるホロノミック系を用いることにより、ホロノミック系の不変量と特異点の不変量との関係について調べた。1970年代における斉藤恭司氏による研究などにより、特異点と微分作用素系とが結べつけられることが知られているが、申請者はこれまでの研究において、偏微分方程式系をD-加群の範疇で捉えなおすことにより、ホロノミック系の不変量と特異点の古典的不変量との関係を示唆する結果を得ていた。本研究ではさらに、矢野環氏によるb-関数の研究の中で導入されている不変量と我々の導入した不変量との関係について調べた。 その結果、正則ドラム複体が原点においてexactとはならないことで知られているReiffenの特異点に関して、我々の導入したホロノミック系の不変量とintegral closureとの間に成り立つ関係式を得た。 またこれまで困難であった、特異点の定義方程式がパラメーターを含んでいる場合のホロノミック系の計算を可能とするアルゴリズムを導出し、プログラミングを行った。 これらの研究成果について、ホロノミック系の不変量に関して、7月に行われたロシアでの国際研究集会で発表し、ホロノミック系の構成に関して、9月にスペインで行われた国際研究集会で発表した。これらの研究成果に関して論文を執筆し、投稿した。b-関数とホロノミック系に関する考察についての研究過程を12月に京都大学数理解析研究所で行われた研究会で報告し、現在論文を執筆中である。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2003 -2005 
    Author : AOKI Takashi; IZUMI Shuzo; OHNO Yasuo; NAKAMURA Yayoi; YAMAZAKI Susumu
     
    The purpose of this project was to establish the fundamental theory of the exact WKB analysis for microdifferential equations of infinite order and develop its applications. We have obtained the following results : (1)We introduced the notion of microdifferential operators of WKB type and we showed that for operators of WKB type, we can construct exact WKB solutions. The notions of turning points and Stokes curves can be defined as well as the case of differential operators with a large parameter. We proved that in a neighborhood of a turning point, such an operator can be decomposed into the product of two operators and the equation corresponding to the operator is reduced to an equation of finite order. Thus local theory for WKB solutions is exactly the same as in the case of equations of finite order. Thus, if the turning point is simple, then the equation is reduced to the Airy equation. We have found that, at least locally, the order of the equation is irrelevant and that the degree of the turning point is essential for the connection problem of the WKB solutions. (2)As an application of connection problems of differential equations, we have obtained new families of relations that hold among multiple zeta values. There are two ways of defining multiple zeta values. Both are defined by the Euler sum of products of reciprocals of powers of positive integers : one is defined by sums over indices of powers with strict inequalities and another with non-strict inequalities. We constructed a generating function made of the latter and showed that the function is a unique solution of an inhomogeneous ordinary differential equation of Fuchsian type. Solving this equation directly by using power series or integration, we obtained some families of relations of multiple zeta values. That is, we showed that sums of multiple zeta values with non-strict inequalities, which we call multiple zeta-star values, with fixed weight and height can be expressed as a rational multiple of Riemann zeta values. (3)To have the complete description of the Stokes geometry of a given higher-order differential equation is very difficult problem in general. We found that the notion of virtual turning points is crucial to understand the Stokes geometry and the connection problem for the equation. For example, an equation of higher order with a deformation parameter, we have a family of Stokes curves. We know that not only the Stokes curves but also the so called new Stokes curves are indispensable to describe the Stokes geometry. If the deformation parameter changes, the Stokes geometry also changes and we observed that sometimes the role of ordinary Stokes curves and new Stokes curves interchange each other. This phenomenon can be well understood by using the notion of virtual turning points.

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