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Last Updated :2024/07/25

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Master

Affiliation (Master)

  • Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

  • Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Degree

  • Doctor of Science（University of Tokyo）

Profile and Settings

  • Name (Japanese)

    Honda

  • Name (Kana)

    Naofumi

  • Name

    200901078712049593

Achievement

Research Interests

  • algebraic analysis   代数解析学   

Research Areas

  • Natural sciences / Basic analysis

Research Experience

  • 2016/01 - Today Faculty of Science, Hokkaido University Department of Mathematics Professor

Education

  • 1988/04 - 1991/03  東京大学大学院
  • 1986/04 - 1988/03  東京大学大学院

Published Papers

  • Naofumi HONDA, Takeshi IZAWA, Tatsuo SUWA
    Journal of the Mathematical Society of Japan 75 (1) 229 - 290 0025-5645 2023/01/01 [Refereed]
  • Matthias Eller, Naofumi Honda, Ching-Lung Lin, Gen Nakamura
    Inverse Problems and Imaging 16 (6) 1529 - 1542 1930-8337 2022/12 [Refereed]
     

    <p style='text-indent:20px;'>A global unique continuation property (UCP) from the boundary for solutions to a viscoelastic system with a memory term is presented. The density and elasticity tensors are assumed to be real analytic. The tensors can be anisotropic and satisfy physically natural conditions such as full symmetry and strong convexity. The global UCP is given in terms of the travel time of the slowest wave of the viscoelastic system, which is the optimal description for the global UCP in our setup.</p>

  • Naofumi Honda, Ching-Lung Lin, Gen Nakamura, Satoshi Sasayama
    Journal of Inverse and Ill-posed Problems 30 (1) 5 - 21 0928-0219 2021/12/08 [Refereed]
     
    Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.
  • Hyperfunctions and Cech-Dolbeault cohomology in microlocal point of view
    N. Honda
    RIMS Koukyuroku 2101 (1) 7 - 12 2019/03 [Not refereed][Not invited]
  • On the Algebraic Study of Asymptotics
    N. Honda, L, Prelli
    Springer Proceedings in Mathematics & Statistics 256 (1) 227 - 238 2018/10 [Refereed][Not invited]
  • Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity
    CĂTĂLIN I. CÂRSTEA, N. HONDA, Gen NAKAMURA
    SIAM J. Math. Anal. 50 (3) 3291 - 3302 2018/07 [Refereed][Not invited]
  • Naofumi Honda, Kohei Umeta
    Journal of the Mathematical Society of Japan 70 (1) 111 - 139 1881-1167 2018 [Refereed][Not invited]
     
    We establish an edge of the wedge theorem for the sheaf of holomorphic functions with exponential growth at infinity and construct the sheaf of Laplace hyperfunctions in several variables. We also study the fundamental properties of the sheaf of Laplace hyperfunctions.
  • HONDA Naofumi, Kohei Umeta
    RIMS Koukyuroku 京都大学数理解析研究所 2020 (2020) 29 - 34 1880-2818 2017/04 [Not refereed][Not invited]
  • HONDA Naofumi, Luca Prelli
    RIMS Koukyuroku 京都大学数理解析研究所 2020 (2020) 18 - 28 1880-2818 2017/04 [Not refereed][Not invited]
  • HONDA Naofumi
    RIMS Koukyuroku 京都大学数理解析研究所 2020 (2020) 10 - 17 1880-2818 2017/04 [Not refereed][Not invited]
  • Takashi Aoki, Naofumi Honda, Susumu Yamazaki
    Journal of the Mathematical Society of Japan 69 (4) 1715 - 1801 1881-1167 2017 [Refereed][Not invited]
     
    A new symbol theory for pseudodifferential operators in the complex analytic category is given. Here the pseudodifferential operators mean integral operators with real holomorphic microfunction kernels. The notion of real holomorphic microfunctions had been introduced by Sato, Kawai and Kashiwara by using sheaf cohomology theory. Symbol theory for those operators was partly developed by Kataoka and by the first author and it has been effectively used in the analysis of operators of infinite order. However, there was a missing part that links the symbol theory and the cohomological definition of operators, that is, the consistency of the Leibniz-Hörmander rule and the cohomological definition of composition for operators. This link has not been established completely in the existing symbol theory. This paper supplies the link and provides a cohomological foundation of the symbolic calculus of pseudodifferential operators.
  • An invitation to Sato's postulates in micro-analytic S-matrix theory
    HONDA Naofumi, KAWAI Takahiro
    RIMS Koukyuroku Bessatsu B61 23 - 56 2017/01 [Refereed][Not invited]
  • A study of pinch points and cusps in the Landau-Nakanishi geometry
    Naofumi Honda, Takahiro Kawai
    Kokyuroku Bessatsu B57 195 - 234 2016/09 [Refereed][Not invited]
  • Multi-microlocalization
    Naofumi Honda, Luca Prelli, Susumu Yamazaki
    Kokyuroku Bessatsu B57 93 - 116 2016/09 [Refereed][Not invited]
  • Naofumi Honda, Luca Prelli, Susumu Yamazaki
    BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 144 (3) 569 - 611 0037-9484 2016 [Refereed][Not invited]
     
    The purpose of this paper is to establish the foundations of multi-microlocalization, in particular, to give the fiber formula for the multi-microlocalization functor and estimate of microsupport of a multi-microlocalized object. We also give some applications of these results.
  • Kernel functions and symbols of pseudodifferential operators of infinite order with an apparent parameter
    T. Aoki, N.Honda, S.Yamazaki
    RIMS Koukyuroku Bessatsu B52 175 - 192 2015/06 [Refereed][Not invited]
  • On the sheaf of Laplace hyperfunctions in several variables
    N. Honda, K. Umeta
    RIMS Koukyuroku Bessatsu B52 213 - 218 2015/06 [Refereed][Not invited]
  • On the geometric aspect of Sato's postulates on the S-matrix
    N. Honda, T. Kawai, H. P. Stapp
    RIMS Koukyuroku bessatsu B52 11 - 53 2015/06 [Refereed][Not invited]
  • Honda Naofumi, Kawai Takahiro, Stapp Henry P.
    RIMS Kokyuroku Bessatsu 京都大学 52 11 - 53 1881-6193 2014/11 [Refereed]
  • Aoki Takashi, Honda Naofumi, Yamazaki Susumu
    RIMS Kokyuroku Bessatsu 京都大学 52 193 - 211 1881-6193 2014/11 [Refereed]
  • Honda Naofumi, Umeta Kohei
    RIMS Kokyuroku Bessatsu 京都大学 52 213 - 217 1881-6193 2014/11 [Refereed]
  • Naofumi Honda, Joyce McLaughlin, Gen Nakamura
    INVERSE PROBLEMS 30 (5) 1 - 19 0266-5611 2014/05 [Refereed][Not invited]
     
    An inverse problem to identify unknown coefficients of a partial differential equation by a single interior measurement is considered. The equation considered in this paper is a strongly elliptic second order scalar equation which can have complex coefficients in a bounded domain with C-2 boundary. We are given a single interior measurement. This means that we know a given solution of the forward equation in this domain. The equation includes some model equations arising from acoustics, viscoelasticity and hydrology. We assume that the coefficients are piecewise analytic. Our major result is the local Holder stability estimate for identifying the unknown coefficients. If the unknown coefficient is a complex coefficient in the principal part of the equation, we assumed a condition which we name admissibility assumption for the real part and imaginary part of the difference of two complex coefficients. This admissibility assumption is automatically satisfied if the complex coefficients are real valued. For identifying either the real coefficient in the principal part or the coefficient of the 0th order of the equation, the major result implies global uniqueness for the identification.
  • A computer-assisted study of the Landau-Nakanishi Geometry
    Naofumi Honda, Takahiro Kawai
    RIMS Koukyuroku 1861 100 - 110 2014 [Not refereed][Not invited]
  • AOKI Takashi, HONDA Naofumi, YAMAZAKI Susumu
    RIMS Kokyuroku 京都大学 1861 156 - 170 1880-2818 2013/11
  • On kernel functions and symbols of analytic pseudo-differential operators
    T.Aoki, N.Honda, S.Yamazaki
    RIMS Koukyuroku 1835 21 - 37 2013/08 [Not refereed][Not invited]
  • On the number of the turning points of the second kind of the Noumi-Yamada systems with a large parameter
    T. Aoki, N. Honda, Y. Umeta
    RIMS Koukyuroku Bessatsu B37 1 - 30 2013/08 [Refereed][Invited]
  • Takashi Aoki, Naofumi Honda, Yoko Umeta
    Advances in Mathematics 235 496 - 524 0001-8708 2013/03/01 [Refereed][Not invited]
     
    We construct general formal solutions containing sufficiently many free parameters for the first Painlevé hierarchy with a large parameter from a viewpoint of the multiple-scale analysis. © 2012 Elsevier Ltd.
  • N. Honda, G. Nakamura, M. Sini
    Mathematische Annalen 355 (2) 401 - 427 0025-5831 2013 [Refereed][Not invited]
     
    We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle. We assume that the boundary of the obstacle is a non-analytic hypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only for n = 2, 3 in general. More precisely, we show that one measurement is enough for n = 2 and we need 3 linearly independent inputs for n = 3. However, in the case for the Helmholtz equation, we only need n - 1 linearly independent inputs, for any n ≥ 2. Here n is the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems. © 2012 Springer-Verlag.
  • Naofumi Honda, Luca Prelli
    ADVANCES IN MATHEMATICS 232 (1) 432 - 498 0001-8708 2013/01 [Refereed][Not invited]
     
    In this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions we construct different kinds of sheaves of multi-asymptotically developable functions, whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima. (C) 2012 Elsevier Inc. All rights reserved.
  • Naofumi Honda, Kohei Umeta
    JOURNAL OF MATHEMATICAL SCIENCES-THE UNIVERSITY OF TOKYO 19 (4) 559 - 586 1340-5705 2012 [Refereed][Not invited]
     
    We give a vanishing theorem of cohomology groups on a pseudoconvex open subset for holomorphic functions with exponential growth at infinity. As an application, we construct the sheaf of Laplace hyperfunctions and that with holomorphic parameters, and we also study several properties of these sheaves.
  • On the form of instanton-type solutions for equations of the first Painleve' hierarchy by multiple-scale analysis
    T. Aoki, N. Honda, Y. Umeta
    Rend. Sem. Mat. Univ. Politec. Torino 69 (4) 331 - 338 2012 [Refereed][Not invited]
  • Cohomology vanishing theorem and Laplace hyperfunctions with holomoprhic parameters
    H. Naofumi, K. Umeta
    Rend. Sem. Mat. Univ. Politec. Torino 69 (4) 347 - 353 2012 [Refereed][Not invited]
  • Naofumi Honda, Luca Prelli
    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 87 (5) 69 - 72 0386-2194 2011/05 [Refereed][Not invited]
     
    We extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions we construct different kinds of sheaves of multi-asymptotically developable functions, whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima.
  • Takashi Aoki, Naofumi Honda
    Journal of the Mathematical Society of Japan 63 (4) 1085 - 1119 0025-5645 2011 [Refereed][Not invited]
     
    The system of algebraic equations for the leading terms of formal solutions to the Noumi-Yamada systems with a large parameter is studied. A formula which gives the number of solutions outside of turning points is established. The number of turning points of the first kind is also given. © 2011 The Mathematical Society of Japan.
  • N. Honda, G. Morando
    BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 139 (3) 389 - 435 0037-9484 2011 [Refereed][Not invited]
     
    In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold X. Then, we define stratified ultradistributions of Beurling and Roumieu type on X. In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if X is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to X. Second, the tempered-stratified ultradistributions on the complementary of a 1-regular closed subset of X coincide with the sections of the presheaf of tempered ultradistributions.
  • The geometric structure of a virtual turing point and the motel of the Stokes geometry
    Naofumi Honda
    RIMS Kokyuroku Bessatsu B10 63 - 117 2009/12 [Refereed][Not invited]
  • Takashi Aoki, Naofumi Honda
    Algebraic Analysis of Differential Equations: From Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai 45 - 53 2008 [Refereed][Not invited]
     
    We consider the system of algebraic equations that defines the leading terms of formal solutions to the Noumi-Yamada equations of even order and prove that the polynomial sequence associated with the system is a regular sequence. © 2008 Springer Japan.
  • Naofumi Honda, Gen Nakamura, Roland Potthast, Mourad Sini
    ANNALI DI MATEMATICA PURA ED APPLICATA 187 (1) 7 - 37 0373-3114 2008/01 [Refereed][Not invited]
     
    This paper addresses the inverse obstacle scattering problem. In the recent years several non-iterative methods have been proposed to reconstruct obstacles (penetrable or impenetrable) from near or far field measurements. In the chronological order, we cite among others the linear sampling method, the factorization method, the probe method and the singular sources method. These methods use differently the measurements to detect the unknown obstacle and they require the use of many incident fields (i.e. the full or a part of the far field map). More recently, two other approaches have been added. They are the no-response test and the range test. Both of them use few incident fields to detect some informations about the scatterer. All the mentioned methods are based on building functions depending on some parameter. These functions share the property that their behaviors with respect to the parameter change drastically. The surface of the obstacle is located at most in the interface where these functions become large. The goal of this work is to investigate the relation between some of the non-iterative reconstruction schemes regarding the convergence issue. A given method is said to be convergent if it reconstructs a part or the entire obstacle by using few or many incident fields respectively. For simplicity we consider the obstacle reconstruction problem from far field data for the Helmholtz equation.
  • Takashi Aoki, Naofumi Honda
    Proceedings of the Japan Academy Series A: Mathematical Sciences 84 (3) 42 - 47 0386-2194 2008 [Refereed][Not invited]
     
    The notion of principally tame regular sequences is introduced for systems of polynomials with a weight vector. As an application, construction of formal solutions is discussed for the systems of nonlinear differential equations which belong to the fourth Painlevé hierarchy with a large parameter. © 2008 The Japan Academy.
  • Takashi Aoki, Naofumi Honda, Takahiro Kawai, Tatsuya Koike, Yukihiro Nishikawa, Shunsuke Sasaki, Akira Shudo, Yoshitsugu Takei
    Algebraic Analysis of Differential Equations: From Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai 29 - 43 2008 [Refereed][Not invited]
     
    Several aspects of the notion of virtual turning points are discussed its background, its relevance to the bifurcation phenomena of a Stokes curve, its importance in the analysis of the Noumi-Yamada system (a particular higher order Painlevé equation) and a concrete recipe for locating them. Examples given here make it manifest that virtual turning points are indispensable in WKB analysis of higher order linear ordinary differential equations with a large parameter. © 2008 Springer Japan.
  • Virtual turning points
    T.Aoki, N. Honda, T. Kawai, T. Koike, N. Nishikawa, S. Sasaki, A. Shudo, Y. Takei
    Algebraic Analysis of Differential Equations, Springer 29 - 44 2007 [Refereed][Not invited]
  • On the Stokes geometry of the Noumi-Yamada system
    Naofumi Honda
    RIMS Kokyuroku Bessatsu B2 45 - 72 2007 [Refereed][Not invited]
  • Degenerate Stokes geometry and some geometric structure underlying a virtual turning point
    Naofumi Honda
    RIMS Kokyuroku Bessatsu B5 15 - 49 2007 [Refereed][Not invited]
  • On the algebraic equations associated with Noumi-Yamada Systems
    T. Aoki, N.Honda
    RIMS Koukyuroku 1516 1 - 11 2006 [Not refereed][Not invited]
  • On the examples of Stokes geometry for Noumi-Yamada systems
    Naofumi Honda
    数理解析研究所 講究録 1516 24 - 167 2006 [Not refereed][Not invited]
  • Microlocal Stokes phenomena for holonomic modules
    Naofumi Honda
    Toward the Exact WKB Analysis of Differential Equations 33 - 39 2000 [Refereed][Not invited]
  • Regularity theorems for holonomic modules
    Naofumi Honda
    Proc.Banach Center publ. 33 85 - 91 1996 [Refereed][Not invited]
  • Microfunction Solutions of Holonomic Systems and Stokes Lines
    Naofumi Honda
    Structure of Solutions of Differential Equations, World Scientific 169 - 182 1996 [Refereed][Not invited]
  • N HONDA
    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 69 (5) 111 - 114 0386-2194 1993/05 [Refereed][Not invited]
  • HONDA Naofumi
    J. Fac. Sci. Univ. Tokyo Faculty of Science, The University of Tokyo 39 (2) 207 - 232 1992 [Refereed][Not invited]
  • N HONDA
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 27 (6) 923 - 943 0034-5318 1991/12 [Refereed][Not invited]
  • Naofumi Honda
    J. Fac. Sci. Univ. Tokyo Faculty of Science, The University of Tokyo 38 (2) 351 - 358 0040-8980 1991 [Refereed][Not invited]
  • N HONDA, P SCHAPIRA
    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 26 (3) 529 - 534 0034-5318 1990/10 [Refereed][Not invited]

MISC

Books etc

  • Virtual turning points, Springer briefs in Mathematical Physics, 4
    N. Honda, T.Kawai, Y.Takei (Joint work)
    Springer 2015/07
  • General topology
    Shuichi Jimbo, Naofumi Honda (Joint work)
    Suugaku Shyobou 2010/04

Presentations

  • Čech-Dolbeault cohomology and hyperfunctions  [Invited]
    HONDA Naofumi
    微分方程式の総合的研究 全体講演  2018/12
  • 仮想変わり点の幾何とストークス係数,  [Invited]
    本多 尚文
    2010年度日本数学会秋季大会函数方程式分科会特別講演  2010
  • 不確定特異点型極大過剰決定系の解の構造について  [Invited]
    本多 尚文
    1996年度日本数学会春季大会函数解析分科会特別講演  1996

Association Memberships

  • 日本数学会   

Research Projects

  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2021/04 -2024/03 
    Author : 本多 尚文
     
    本研究課題では、偏微分方程式系の多重強漸近展開可能解のより詳細な漸近挙動を解析する為にGevrey級の多重強漸近展開可能層を導入し、その性質を研究する。また、偏微分方程式系のGevrey級多重強漸近展開可能な解のなす層に対し順像定理や逆像定理を示すことで、異なる多重強漸近展開可能解の相互の関係を明らかにする。これらの結果を用いることで極大過剰決定系の多重強漸近展開可能解に関する存在定理等の基本的な性質を明らかにすることが目的である。 この目的を達成するには、基本的な道具である多重特殊化関手の理論の拡張と整備がまず必要である。実際、今までの理論では複数の部分多様体の配置にかなり強い条件を必要としていたが、本研究課題を実行するには、それを弱める必要があった。本年度は、Padova大学のLuca Prelliと伴にこの点に関して理論の整備と拡張をおこなった。特に、複数の部分多様体の配置が縮退している場合についての多重錘の幾何の特徴付けに成功した。この場合は、今までの幾つかの錘の交差によって幾何を記述する方法は用いることが出来ない。そこで、多様体に付随する或る種の単項式の生成する半群を準備し、この半群によって幾何的対象を記述した。この場合、記述された幾何が良い性質を持つかは自明のことではなくなり、研究が必要であった。最終的には、cohomology的に自明となる非常に良い性質を持った幾何が現れることを示す事が出来た。 漸近展開理論を展開するには、更に、このような集合上でのWhitney正則関数のコホモロジー消滅定理が必要であるが、その問題についても満足できる結果が得られた。 以上の結果については、現在Luca Prelliと論文を作成中である。
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2019/04 -2023/03 
    Author : 神保 秀一, 本多 尚文
     
    領域変形と楕円型方程式系におけるスペクトルの摂動問題を継続して研究している. 以下ケースごとの研究内容を記述する. (I) ストークス方程式の固有値問題についてアダマールの変分公式を研究協力者の牛越氏と計算した. 摩擦項付きスリップ条件下での公式を得たがその表現が非常に複雑過ぎて自然な最終形の公式となっているかどうかを不明で未だ検討中である. 論文を完成するところまで到達していない. (II) 弾性体の新道に関する作用素の固有値問題を解析している (i)細い弾性体の低周波モードの解析を行っている. 以前断面が極端なアスペクト比をもつケースを牛越氏と調べ漸近公式を得たが, 複数のそれらを組み立てて出来る立体の場合を協力者である牛越氏本多氏とともに計算している. (ii) バルクな弾性体に小さな穴をあけたときの固有値の摂動問題を伊東氏と研究している. ２次元の場合の結果を見通すことが出来て漸近公式を計算したが, 当初の目標である３次元弾性体の穴や亀裂がある場合の解析には全然届いていない. (III) ダブルY型グラフ上のアレン・カーン方程式のダイナミクスの研究を森田氏岩崎氏と協力して行った. 定常解の構造およびヘテロクリニック軌道に対応する時間全域解を調べた. その際定常解の安定性を解析する新しい手法を考案した. さらに一般のグラフについてダイナミクスの研究を行う. また, グラフ上のHeat Kernel の具体的表現を得る手法を俣野氏と協力して考案した.それによって一般星型グラフなどの単純ながらループを含む場合にも可能な計算法を得た.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2018/04 -2021/03 
    Author : HONDA NAOFUMI
     
    The kernel functions of analytic microdifferential operators are introduced by using local cohomology groups and their symbols theory are also developed. Several class of microdifferential operators can be introduced by considering the growth order of symbols such as Gevrey or Whitney classes. Our purpose is, by the theory of sheaves on subanalytic sites and Cech-Dolbeault cohomology theory, to formulate these kernel function from the viewpoint of algebraic analysis. We have succeeeded in constructing multi-microlocalization functor of morphisms and, as a result, formulating framework of these kernel functions with required growth order from the viewpoint of microlocal analysis.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2016/04 -2019/03 
    Author : JIMBO Shuichi, Ito Hiroya, Ushikoshi Erika
     
    Eigenfrequencies of an elastic body of uniform and isotropic material but with an extremely thin shape with non-uniform cross-section are studied. The distribution of eigenvalues and their structure was analyzed. The eigenfrequencies of the bending mode were proved to be very small for thinner limit and elaborate behavior were described by the aid of a certain 4-th order ODE operator with variable coefficients. The eigenfrequencies corresponding to the Stretching mode and the Torsion mode are also analyzed and the limiting behavors were described by a certain 2nd order ODE operator, respectively in the case that the thin domain is axissymmetric. The spectum of the elliptic opeator which arises as a vibration model in the geophysics was studied and it is proved that the essential spectrum is bounded in the comples plain while discrete spectrum is unbounded.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2015/04 -2018/03 
    Author : Honda Naofumi
     
    Multi-microlocalization is a method which enables us to microlocalize an object along several manifolds simultaneously. The purpose of this research is to extend this mothod to more general cases, and then, to establish a theory of multi-microlocal operators. In the first two years of our research, we studied generalization of multi-specialization and succeeded in constructing multi-specialization for a general family of submanifolds located suitably. During the rest of the period, we studied applications of Cech-Dolbeault cohomology theory to the theory of hyperfunctions. We had several interesting results, which suggested us that the theory is very effective for a construction of multi-microlocal operators too.
  • 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.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2016 -2018 
    Author : 中村 玄, 本多 尚文, 笹山 智司
     
    能動的サーモグラフィー, 光及び蛍光光トモグラフィー, バイブロサイス地盤解析法, MREやPVSのデータ解析法など幾つかの非破壊検査法に対する数学的にロジカルなインバージョン法の確立とその周辺研究を行い, 次の成果をあげた. 1)拡散方程式に対するinterior transmission problemのGreen関数の構成とその逆問題への応用 2)小介在物同定光トモグラフィー法に対するMUSIC法の確立 3)蛍光光トモグラフィーの数値的に有効なインバージョン法(有効なinitial guessの探索法)の研究 4)MREデータ解析のモデル方程式であるスカラーモデル方程式に対するLM法の収束性証明 5)定常均質等方弾性方程式に対する3つのスカラー関数だけで表現される特殊なヘルムホルツ分解の完全性の証明とそのPVS逆問題への応用 6)区分的に解析的な静・動非等方弾性方程式の境界値問題に対する一意性(バイブロサイス地盤解析法の数学的正当化)の証明 7)非整数階時間微分を持つ拡散方程式に対する一意接続定理の証明
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2013/04 -2017/03 
    Author : Uchida Motoo, Schapira Pierre
     
    The boundary value problems for sheaves and D-modules is formulated and the fundamental theorem is proved. We also considered the initial value problem for systems of micro differential equations and proved the fundamental Cauchy-Kowalevskaja-Kashiwara theorem (an extended version of Cauchy-Kowalevskaja theorem taking cohomology of any degree into consideration) in terms of microlocalization of sheaves.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2013/04 -2016/03 
    Author : Jimbo Shuichi, HONDA NAOFUMI, TONEGAWA YOSHIHIRO
     
    I studied spectra of elliptic operators for regularly or singularly deformed domain (Lame operator, Stokes operator, Maxwell operator). (i) I studied polynomial solutions, rational type solutions with their structures of homogeneous Stokes and Elastic equations (with H.Ito, N.Honda), (ii) I obtained spectral Hadamard variational formula of Stokes operator, Maxwell operator for regularly perturbed domain for Dirichlet and Slip type boundary condition (with E. Ushikoshi). I obtained an elaborate behaviors of eigenvalues for Maxwell operator, (iii) I studied elaborate behaviors of eigenvalues of Lame or Maxwell operators in a domain with small hole, (iv) I obtained elaborate behaviors of eigenfrequencies of elastic body composed of several thin rod.
  • 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) : 2011 -2013 
    Author : HONDA Naofumi, UCHIDA Motoo
     
    We study Stokes pheonmenon for a higher order linear differential equation with a large parameter, and we also study the same problems for a non-linear differential equation such as a Painlev'e hierarchy. A Stokes geometry for a higher order linear differential equation is quite different from one for a 2nd order linear differential equation becuase of existence of virtual turning points and new Sotkes curves. It is very complicated, and thus, possibility to succesively obtain a Stokes coefficient on each Stokes curve is quite uncertain. By using so called the depth function, in this study, we have shown that it is always possible to have all the Stokes coefficients succesively. We also have succeeded in constructing an instanton-type solution for the first Painlev'e hierarchy (PI)_m. This result is quite important becuase it contains sufficiently many free parametners, and hence, we can take a family of these solutions as a basis of solutions for a connection problem.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2010 -2012 
    Author : JIMBO Shuichi, HONDA Naofumi, NAKAMURA Gen
     
    I studied the eigenvalu problem of the Lame operator, which is obtained from the oscillation property of elastic body. I dealt with the compex domain which is a union several thin regions. The limit system when the thinnes goes to zero, is a 4th order ODE system with a complicated compatibility condtions on the verticies. I also dealt with the eigenvalue problem of a certain Lame operator with the low stiffness coefficient. I obtained the limit system, which is related with the eigenvalue problem of the Stokes operator in a fluid dynamical problem with the Dirichlet condition or the slip boundary condition.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2010 -2012 
    Author : NAKAMURA Gen, JIMBO Shuichi, HONDA Naofumi, KAWSHITA Misio, TANUMA Kazumi, WATANABE Michiyuki, SHIROTA Kenji
     
    Some studies were done and obtained sufficient results on the data analysis of elastography which measures the visco-elasticity of tissues in a living body non-invasively, and as their related studies, the LSM (linear sampling method) for parabolic equations and Carleman estimate for anomalous diffusion equations. We completed our recent year study on the dispersion formula of the speed of Rayleigh wave for half space depth dependent anisotrpic elastic media with residual stress. We gave an inversion scheme for detecting damage of connectors of steel-concrete composite beam.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2008 -2012 
    Author : YAMAZAKI Susumu, AOKI Takashi, HONDA Naofumi
     
    (1) If we impose an irregularity condition due to N. Honda for a system of analytic linear differential equations (D-Module), we can define non-characteristic initial and boundary values for the corresponding Gevrey function or ultradistribution solutions. Moreover, under a (weak) hyperbolicity condition, we can prove unique solvability theorems for Cauchy and boundary value problems.(2) For any regular-specializable system, we can define general boundary values for extensible distribution or ultradistribution solutions under an irregularity condition due to H. Tahara.(3) By a joint work with T. Aoki and N. Honda, we can establish new cohomological representation and symbol theory for pseudodifferetial operators of infinite order in analytic category.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2008 -2010 
    Author : HONDA Naofumi
     
    We study the geometry underlying a virtual turning point, which appears in exact WKB analysis. For this purpose, we have constructed a Riemann surface associated with a linear differential equation with a large parameter and a depth function which indicates dependency between new Stokes curves. As an application, we show that we can really obtain connection coefficients on all the new Stokes curves.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2007 -2009 
    Author : NAKAMURA Gen, HONDA Naofumi, TONEGAWA Yoshihiro, TAIRA Kazuaki, ISOZAKI Hiroshi, YAMAMOTO Masahiro, SHIROTA Kenji, WATANABE Michiyuki, OHE Takashi, TAKUWA Hideki
     
    For 1) inverse scattering problems, 2) thermography, 3) inverse problems for equations in fluids, some new reconstruction schemes and an framework which integrates several known reconstruction schemes are given.
  • 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 -2008 
    Author : JIMBO Shuichi, NAKAMURA Gen, TACHIZAWA Kazuya, HONDA Naofumi
     
    1. 典型的な2階楕円型作用素において, 特異的な領域変形の過程あるいは変数係数が特異摂動を受ける過程において, 固有値の漸近挙動を解析した. 扱った作用素はラプラス作用素, ラメ作用素, マクスウェルの作用素シュレディンガー作用素などである. 2. ジャンクションをもつ集合上のギンツブルク-ランダウ方程式の解構造を解析した. 分岐や安定性を調べた.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 2001 -2002 
    Author : 本多 尚文
     
    本研究はSchwartz超函数に対する緩増大関手を拡張しGevrey超函数に対しても同様のGevrey緩増大関手を構成する事を目的としている。Schwartz超函数に対する緩増大関手はM.Kashiwaraによって構成され、Riemann-Hilbert対応を具体的に与える関手となっている。このように、緩増大関手は確定特異点型の極大過剰決定系等の研究に於いて重要な道具となっている。一方、不確定特異点型の極大過剰決定系の研究に於いては、その解が指数的な発散を一般に伴うため、指数的な増大度を持つような緩増大関手の構成が望まれる。その構成は、Gevrey超函数の台の分解がSchwartz超函数の場合のように上手くいかず、本質的に困難な問題が存在する。本研究者はこの問題をGevrey超函数の概念を拡張する事によって解決する事を試みた。この拡張は対象となる超函数の台が特異点を持たない場合は既存のGevrey超函数に一致するようなものである。他方、超函数の台が特異点を持つ場合には、もはや既存の超函数には一致せず、一般にはより大きい空間となる。このような拡張されたGevrey超函数を用いることで、実2次元以下の場合はGevrey増大関手の構成に成功した。しかし、実3次元以上でこの拡張は、Gevrey超函数の台の分解に対して不十分である事を示す特異な例も見つかった。従って、より大きな拡張が必要になる。特異な例は、実3次元以上では拡張されたGevrey超函数の層のみならず、それをある意味含むような複体を直接考察する必要性を示唆していると考えられる。そこで、Gevrey超函数層を係数とするSubanalytic setsの複体を考察したが、まだ、最終的な構成までは至っていない。この問題を今後も考察し、最終的な構成に至る予定である。なお、実2次元以下の構成方法は論文を投稿中である。
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1998 -1999 
    Author : 山口 佳三, 中路 貴彦, 井上 純治, 上見 練太郎, 本多 尚文, 久保田 幸次
     
    本研究は、単純Lie環をその接触変換全体として持つ二階の系の内、特に例外型単純Lie環を無限小接触自己同型として持つ二階の系として、二階の過剰決定系のG_2-幾何学の研究を行うことである。 歴史的には、E.Cartanが、次の過剰決定系Rを不変にする無限小接触変換の成すリー環が例外型単純リー環G_2であることを見いだした。 R={(∂^2z)/(∂x^2)=1/2((∂^2z)/(∂y^2))^2,(∂^2z)/(∂x∂y)=1/3((∂^2z)/(∂y^2))^3} 本研究の目的は、E.Cartanが発見した上記の例外型単純リー環G_2に随伴する特別な過剰決定系をより深く理解するために、他の単純リー環に対しても随伴する過剰決定系を構成しexplicitに書き下ろそうとするものであった。 昨年度は、すべての例外型単純Lie群Gに対して、Boothy typeの接触多様体J=G/Pをもとに、E.CartanによるG_2モデル(例外単純リー環G_2を接触自己同型として持つ二階のsystem)の構成を他の例外型単純リー環の場合にも拡張し、古典型の場合への拡張も行った。 今年度は、昨年度の一般論の定式化を踏まえ、具体的な計算を行い古典型の場合に一部結果を得たが、例外型の場合を網羅的に実行するには至らなかった。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1997 -1999 
    Author : JEMBO Shuichi, HAYASHI Mikihiro, NAKAZI Takahiko, GIGA Yoshikazu, MORITA Yoshihisa, MIKAMI Tosio
     
    (I) Stable solutions and their domain dependency of the Ginzburg-Landau equation are studied. Solutions with vortices and topologically various kinds of solutions are (ii) Nonstationary complex Ginzburg-Landau equation and its time periodic solutions are studied. The stability of the solutions and dependence on the domains are investigated. Constructed pattern formation under the non-uniform environment is studied. (iii) Nonstationary Ginzburg-Landau equation and its dynamical system of singular perturbation problem is studied. Particularly, it is represented as a finite dimensional ODE and its formula is explicitly obtained. (iv) Homoclinic orbits arising in reaction-diffusion equations are studied. The bifurcation of the dynamical structure is studied. (v) Dynamical system arising in surface evolution equations such as mean curvature flow, surface diffusion equations are studied. Asymptotic behavior and geometrical property are analyzed. (vi) Global existence of solutions in wave equation system with component different propagation speeds is studied. (vii) Fast diffusion equations and their extinction phenomena are studied. (viii) Random crystalline evolution equation is studied (ix) The eigenvalue problem of the Laplacian and the semilinear eiliptic equations on a domain with partial degeneration are studied. More general cases of the singular perturbation of domains are dealt with and the results known before are extended.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1997 -1998 
    Author : 本多 尚文
     
    不確定特異点型の極大過剰決定系のstokes現象を研究した。超局所解析の立場から、マイクロ正則解に対するstokes現象を定式化した。定式化にあたり、平坦な解層を新たに導入し、ここへの解層からの写像をstokes写像として、定義した。 特に、stokes現象がε加群としての構造に、どの様に関わるかを考察し、一変数の場合に、形式的に同型なε加群をstokes写像の核から得られる情報によってε加群として分類出来る事を示した。 また、グレブナーベースの理論を用いる事で、極大過剰決定系の不確定特異点度を計算するアルゴリズムを考察し、具体的なシステムに対して、計算機を用いて計算を行った。ただし、余次元1の超平面に沿った場合のみであるので、高余次元への拡張が望まれる。
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1996 -1996 
    Author : 神保 秀一, 森田 善久, 本多 尚文, 泉屋 周一, 久保田 幸次, 儀我 美一
     
    1.ギンツブルグ-ランダウ方程式の安定解の構造: 領域を著しく変形したときに特徴的に現れる安定定常解を構成した.またさらに変形極限の方程式との関係を線型化固有値問題まで込めて導いた.次に,不均質媒質のモデルとされる変数係数のギンツブルグランダウ方程式の非一様安定解を構成した.また,非一様性によるゼロ点のピン止め効果により生じる安定定常解を構成し,さらに,与えられた点配置にたいし安定解のゼロ点の近似配置が可能であることを示した.また,近似配置を完全精密配置にできることを示す方法をなかば確立した. 2.複素ギンツブルグ-ランダウ方程式の作る力学系の構造: 流体現象(ベナ-ル対流)から導かれる非定常複素係数ギンツブルグランダウ方程式の解の挙動と周期解,不変集合の構成や分岐などを研究した,とくにS^1上においてはホップ分岐とともに安定な準周期解が生じることを示した.一般の有界領域の場合においては周期解を構成した.さらに,線型化安定性解析を行い,限られたパラメータ範囲での安定性を示した.さらにそれ以外の範囲におけるホップ分岐の可能性を探った.いまのところ,ゼロ点のない解について精密な挙動がわかっている.ゼロ点のある解については分岐によってゼロ点が運動する状況を解析した.また領域変形による力学系の不変集合の極限問題等の問題を考えた. 3.領域の特異変形と固有値問題: 有界領域から余次元2以上の部分多様体の管状近傍を取り除いてできる領域上のラプラシアンの固有値の摂動問題を研究した.すでにある小沢真,Courtois らの結果を一般化した.また,電磁場の固有振動の問題についても同様の研究を行った.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1996 -1996 
    Author : 山下 博, 平井 武, 本多 尚文, 山田 裕史, 齊藤 睦
     
    1.実半単純リー群Gの表現,より正確には、表現を微分して得られる展開環U(g)上のHarish-Chandra加群Hの随伴多様体ν(H)は、Riemann対称対(G,K)を複素化して得られる対(G_C, K_C)の接空間pにおけるべき零K_軌道からなる。研究代表者は、「各K_軌道O⊂ν(H)からケーリ-型変換と偏極化をとおしてH上に局所自由に作用するべき零部分環(群)n_oの存在」を示した昨年度からの研究を押しすすめ、Hが規約最高ウェイト表現の場合に、対応するべき零部分環n_oの具体的記述を与えた。この一連の研究結果をとりまとめた論文を日本数学会および数理解析研究所共同研究集会で口頭発表し、学会雑誌へ投稿した(京大行者明彦氏との共著)。 2.半単純リー群Gの極小べき零共役類に付随した極小ユニタリ表現H_mは、既約ユニタリ表現の分類問題とも深く関わる重要な表現である。(1)の成果をふまえて、G=SU(n,n)の極小表現Hmの一般化されたホイッタッカー模型を、HmをG/K上で実現するG_-不変な2階偏微分方程式系を用いて決定した(論文準備中)。さらに、極小表現のフォック模型を使って、U(n_o)-加群としてのHmの構造を明らかにした。この結果を任意の最高ウェイト加群に拡張することを目標とした研究を現在実施中である。 3.各研究分担者は、ホロノミックな不確定特異点型微分方程式系(本多)、多変数超幾何方程式(齊藤)、あるいは各種の群の表現に対するシューア・ワイルの相互律の研究(平井・山田)を各自押しすすめると同時に、これらののテーマが深く関わる上記2の研究実施の過程で、個人的な討論やセミナーをとおして本研究に常時参加した。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1995 -1996 
    Author : SUWA Tatsuo, OKA Mutsuo, HONDA Naofumi, KAWAZUMI Nariya, NAKAI Isao, ISHIKAWA Goo
     
    The research was done mainly on the indices and residues of vector fields and holomorphic singular foliations, the charactreistic classes of singular varieties, the Cech-de Rham cohomology theory and integration theory on stratified spaces. Let us be more specific. (1) Collaboration with J.Seade on the residue theorem for the Baum-Bott residues of foliations on open manifolds and its applications. The joint paper on this has been published in Mathematische Annalen. (2) In another collaboration with J.Seade, we investigated various indices of vector fields on varieties with isolated singularities and we obtained an "adjunction formula" for such varieties. The results are written in a joint paper. (3) As an application of the formula in (2), a formula for the Chem-Schwartz-MacPherson class of a local complete intersection variety with isolated singularities is obtained. The result has been published in C.R.Acad.Sci., Paris. (4) As a generalization of the formula in (2), in a collaboration with D.Lehmann and J.Seade, we introduced a generalized Milnor number and obtained a similar formula for varieties with possibly non-isolated singularities. The results are written in a joint paper. (5) In a joint work with B.Khanedani, we studied the invariants of singular holomorphic foliations on complex surfaces and obtained various formulas. The joint paper on these will appear in Hokkaido Math.J. (6) In a joint work with T.Honda, we proved a residue formula for meromorphic functions on complex surfaces and gave some applications. The results are written in a joint paper. (7) In a collaboration with J.-P.Brasselet, we studied the Nash modification associated with a sinular holomorphic foliation and, as an application, we proved a conjecture of Baum-Bott in some cases. The results are written in a joint paper.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1993 -1993 
    Author : 辻下 徹, 本多 尚文, 井上 昭彦, 三波 篤郎, 岡部 靖憲, 津田 一郎
     
    当研究課題に基づく本年度の研究において、情報の流れ・相関・同期化・コーヒーレンス等、神経系のいわゆる創発的挙動にかかわる概念を数学的に明確にすることに取り組んだ。これらは豊かで多様な意味を持っており、われわれの分析はその一面を捉え得たに過ぎないが、この分析を通して、コーヒーレンスの諸相を詳細に表現出来るようになる一方、創発的概念の多くが神経系を孤立系とみては意味を失うことが明瞭になった。 分析に用いた数学的概念は、加算無限個の有限値確率変数の組の持つ種々の情報理論的指標(エントロピーとそれから派生する種々の相互情報量)である。連続な時間や観測量を考慮することは、技術的煩雑さを別にすれば本質的な数学的困難があるとは思われないが、概念の分析という我々の目的には不要である。 われわれの枠組は、神経系の挙動全体の空間上の確率分布を土台とする。しかし、実験から得られる時系列は一般に複数の確率分布を決め、しかもその中に標準的と呼べるものはなく、各分布は各々神経系の挙動の一側面を表現している。このことが意味するところは、神経系の統合性・コーヒーレンスなどの概念が神経系の構造的のみに関するものではなく神経系内部に見られる現象のどの様相に注目するかという主観的因子にも強く依存した概念だ、ということである。 われわれの数学的枠組は簡単なものではあるが、それに基づく神経系の時空的挙動に関連する概念の分析は、神経系のみならず、いわゆる複雑な力学系を記述する際に用いられる諸概念が詳細な吟味を必要としていることを強く示唆している。
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