Updated on 2024/10/07

写真b

 
KAKEI Saburou
 
*Items subject to periodic update by Rikkyo University (The rest are reprinted from information registered on researchmap.)
Affiliation*
College of Science Department of Mathematics
Graduate School of Science Doctoral Program in Mathematics
Graduate School of Science Master's Program in Mathematics
Title*
Professor
Degree
博士（工学） （ 東京大学 ）
Research Theme*

  • 主要研究テーマである「可積分系」とは、物理をはじめとする諸分野に現れる「良い」モデルの背後にある数学的構造を研究するものである。また、非線型現象全般にも興味をもっていて、2つのテーマの接点であるソリトン方程式を中心に研究を進めている。

    • Research Interests

  • Integrable systems

  • Solitons

  • Special functions

  • Nonlinear phenomena

    • Campus Career*
    • 4 2010 - Present 
      College of Science   Department of Mathematics   Professor
    • 4 2010 - Present 
      Graduate School of Science   Master's Program in Mathematics   Professor
    • 4 2010 - Present 
      Graduate School of Science   Doctoral Program in Mathematics   Professor
    • 4 2007 - 3 2010 
      College of Science   Department of Mathematics   Associate Professor
    • 4 2007 - 3 2010 
      Graduate School of Science   Master's Program in Mathematics   Associate Professor
    • 4 2007 - 3 2010 
      Graduate School of Science   Doctoral Program in Mathematics   Associate Professor
    • 4 2003 - 3 2007 
      College of Science   Department of Mathematics   Associate Professor (as old post name)
    • 4 2005 - 3 2007 
      Graduate School of Science   Master's Program in Mathematics   Associate Professor (as old post name)
    • 4 2005 - 3 2007 
      Graduate School of Science   Doctoral Program in Mathematics   Associate Professor (as old post name)
    • 4 2001 - 3 2003 
      College of Science   Department of Mathematics   Lecturer

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    Research Areas

    • Natural Science / Mathematical analysis  / Integrable Systems

    • Natural Science / Applied mathematics and statistics

    • Natural Science / Basic mathematics

    Research History

    • 4 2010 - Present 
      RIKKYO UNIVERSITY   Graduate School of Science Field of Study: Mathematics   Professor

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    • 4 2010 - Present 
      RIKKYO UNIVERSITY   Graduate School of Science Field of Study: Mathematics   Professor

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    • 4 2010 - Present 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Professor

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    • 4 2007 - 3 2010 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Associate Professor

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    • 4 2003 - 3 2007 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Associate Professor (as old post name)

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    • 4 2001 - 3 2003 
      RIKKYO UNIVERSITY   College of Science Department of Mathematics   Lecturer

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    • 4 1999 - 3 2001 
      Waseda University   School of Science and Engineering

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    • 4 1998 - 3 1999 
      日本学術振興会   日本学術振興会特別研究員

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    • 4 1995 - 3 1998 
      The University of Tokyo   Graduate School of Mathematical Sciences

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    Education

    • 4 1992 - 3 1995 
      The University of Tokyo   The Graduate School of Engineering   Department of Applied Physics

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      Country： Japan

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    • 4 1990 - 3 1992 
      The University of Tokyo   The Graduate School of Engineering   Department of Applied Physics

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      Country： Japan

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    • 4 1988 - 3 1990 
      The University of Tokyo   Faculty of Engineering   Department of Applied Physics

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      Country： Japan

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    Committee Memberships

    • 4 2006 - 3 2007 
      日本応用数理学会   評議員

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      Committee type：Academic society

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    • 9 2006 
      日本物理学会   JPSJ編集委員

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      Committee type：Academic society

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    • 4 2004 - 3 2005 
      日本数学会   代議員

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      Committee type：Academic society

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    • 11 2002 - 10 2003 
      日本物理学会   領域11世話人

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      Committee type：Academic society

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    Papers

    • Toda lattice hierarchy and soliton equations on square lattice Peer-reviewed International journal

      Saburo Kakei

      Journal of Physics A: Mathematical and Theoretical54 ( 7 ) 074001 - 074001   19 2 2021

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：IOP Publishing  

      DOI： 10.1088/1751-8121/ab88e2

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      Other Link： https://iopscience.iop.org/article/10.1088/1751-8121/ab88e2/pdf

    • Wronski-Pfaffian expression of soliton solutions for DKP equation and network

      Shinya Kido, Yasuyuki Watanabe, Yuta Tanaka, Saburo Kakei, Kenichi Maruno

        29AO-S7 ( 1 ) 42 - 48   3 2018

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1957506

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    • Solitons and networks

      Yuta Tanaka, Shinya Kido, Yasuyuki Watanabe, Saburo Kakei, Kenichi Maruno

        29AO-S7 ( 1 ) 131 - 137   3 2018

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1957523

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    • Derivation of fundamental relation for misanthrope model

      Mariko Kanamaru, Saburo Kakei

        29AO-S7 ( 1 ) 151 - 156   3 2018

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)  

      DOI： 10.15017/1957526

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    • Linearization of the box-ball system: an elementary approach Peer-reviewed

      S. Kakei, J.J.C. Nimmo, S. Tsujimoto, R. Willox

      Journal of Integrable Systems3   1 - 32   2018

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    • Elliptic function solutions for difference equations that have conserved quantity expressed in terms of hypergeometric functions

      Ryohei Kubo, Saburo Kakei

        28AO-S6 ( 1 ) 133 - 138   3 2017

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1832822

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    • GL(3)-type Atiyah-Ward ansatz and monodromy preserving deformation

      Yoshiteru Adachi, Saburo Kakei

        28AO-S6 ( 1 ) 125 - 132   3 2017

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)  

      DOI： 10.15017/1832821

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    • Attempt to discretize term structure models

      Arisa Sato, Saburo Kakei

        28AO-S6 ( 1 ) 119 - 124   3 2017

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1832820

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    • Toda lattice hierarchy and Goldstein-Petrich flows for plane curves Peer-reviewed

      Kenji Kajiwara, Saburo Kakei

      Comment. Math. Univ. St. Pauli64 ( 1 ) 29 - 45   5 2015

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      Language：English   Publishing type：Research paper (scientific journal)  

      A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of τ-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

      DOI： 10.14992/00011497

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    • Hele-Shaw problem with surface tension

      Masayuki Nomiyama, Saburo Kakei

        26AO-S2 ( 32 ) 176 - 181   3 2015

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1807783

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    • Structure of soliton solutions for coupled modified KdV equation

      Takeshi Nakagawa, Saburo Kakei

        26AO-S2 ( 29 ) 157 - 162   3 2015

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1807779

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    • Stability of planar vortex equilibria

      Taku Ueno, Saburo Kakei

        26AO-S2 ( 27 ) 151 - 156   3 2015

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1807778

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    • Jeu de taquin slide and ultra-discrete KP equation

      Yosuke Katayama, Saburo Kakei

        26AO-S2 ( 24 ) 133 - 138   3 2015

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1807774

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    • Tau functions and partition functions of rhombus tilings of a hexagon

      Takashi Tamura, Saburo Kakei

        25AO-S2 ( 26 ) 163 - 168   3 2014

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1448884

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    • Tau functions and partition functions of symmetric 6-vertex model under domain-wall boundary condition

      Takuya Watanabe, Saburo Kakei

        25AO-S2 ( 25 ) 157 - 162   3 2014

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1448883

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    • Toda hierarchy and motion of plane curves governed by modified KdV hierarchy

      Saburo Kakei

        25AO-S2 ( 2 ) 8 - 13   3 2014

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/1448842

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    • Generalized Tzitzeica equation and equi-centro-affine surfaces

      Hiromasa Mitani, Saburo Kakei, Ralph Willox

        24AO-S3 ( 20 ) 128 - 133   3 2013

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/27180

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    • Symmetry of coupled Euler-Poisson-Darboux equations

      Akihiro Misawa, Saburo Kakei

        23AO-S7 ( 31 ) 202 - 207   3 2012

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/23477

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    • Inverse ultra-discretization of piecewise linear FitzHugh-Nagumo equation

      Takeshi Okawa, Saburo Kakei

        23AO-S7 ( 30 ) 196 - 201   3 2012

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/23476

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    • Max-plus recurrence equations and inverse ultra-discretization

      Hiroshi Nanba, Saburo Kakei

        22AO-S8 ( 39 ) 248 - 253   3 2011

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/23423

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    • Discrete BKP equation and Yang-Baxter maps International coauthorship

      Saburo Kakei, Jonathan, J.C. Nimmo, Ralph Willox

        21ME-S7 ( 21 ) 142 - 148   3 2010

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)   Publisher：Research Institute for Applied Mechanics, Kyushu University  

      DOI： 10.15017/18711

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    • Yang-Baxter Maps from the Discrete BKP Equation Peer-reviewed

      Saburo Kakei, Jonathan J. C. Nimmo, Ralph Willox

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS6   028   2010

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：NATL ACAD SCI UKRAINE, INST MATH  

      We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

      DOI： 10.3842/SIGMA.2010.028

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    • YANG-BAXTER MAPS AND THE DISCRETE KP HIERARCHY Peer-reviewed

      S. Kakei, J. J. C. Nimmo, R. Willox

      GLASGOW MATHEMATICAL JOURNAL51A ( A ) 107 - 119   2 2009

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

      We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the Ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.

      DOI： 10.1017/S0017089508004825

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    • The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials Peer-reviewed

      Saburo Kakei, Michitomo Nishizawa, Yoshihisa Saito, Yoshihiro Takeyama

      SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS5   010   2009

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：NATL ACAD SCI UKRAINE, INST MATH  

      We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra gl(N). The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.

      DOI： 10.3842/SIGMA.2009.010

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    • Difference equations with an invariant expressed in terms of the hypergeometric function Peer-reviewed

      Takeshi Sakaki, Saburo Kakei

      Transactions of the Japan Society for Industrial and Applied Mathematics17 ( 4 ) 455 - 462   12 2007

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      Language：Japanese   Publishing type：Research paper (scientific journal)   Publisher：The Japan Society for Industrial and Applied Mathematics  

      We introduce a method for constructing difference equations with an invariant expressed as hypergeometric functions. Examples including the equations associated with the arithmetic-geometric mean and the arithmetic-harmonic mean are given.

      DOI： 10.11540/jsiamt.17.4_455

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    • Discrete stochastic processes and q-hypergeometric functions (jointly worked) Peer-reviewed

      Takeshi Ga, Saburo Kakei, Yasushi Kitane

        17 ( 4 ) 463 - 468   12 2007

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      Language：Japanese   Publishing type：Research paper (scientific journal)  

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    • The sixth Painleve equation as similarity reduction of (gl)over-cap(3) generalized Drinfel'd-Sokolov hierarchy Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      LETTERS IN MATHEMATICAL PHYSICS79 ( 3 ) 221 - 234   3 2007

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

      Scaling symmetry of (gl) over cap (n)-type Drinfel'd-Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painleve VI equation. We also discuss Weyl group symmetry of the hierarchy.

      DOI： 10.1007/s11005-007-0144-4

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    • A q-analogue of gl<SUB>3</SUB> hierarchy and q-Painlevé VI (共著) Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      Journal of Physics A: Mathematical and General39 ( 39 ) 12179 - 12190   29 9 2006

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      DOI： 10.1088/0305-4470/39/39/S11

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    • Toroidal Lie algebra and bilinear identity of the self-dual Yang-Mills hierarchy Peer-reviewed

      Saburo Kakei

      Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete201   137 - 146   2006

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      Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：SPRINGER  

      Bilinear identity associated with the self-dual Yang-Mills hierarchy is discussed by using a fermionic representation of the toroidal Lie algebra. sl(2)(tor)

      DOI： 10.1007/978-1-4020-3503-6_13

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    • SOLUTIONS OF A DERIVATIVE NONLINEAR SCHRODINGER HIERARCHY AND ITS SIMILARITY REDUCTION Peer-reviewed

      Saburo Kakei, Tetsuya Kikuchi

      GLASGOW MATHEMATICAL JOURNAL47A   99 - 107   6 2005

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

      The hierarchy structure of a derivative nonlinear Schrodinger equation is investigated in terms of the Sato-Segal-Wilson formulation. Special solutions are constructed as ratios of Wronski determinants. Relations to the Painleve IV and the discrete Painleve I are discussed by applying a similarity reduction.

      DOI： 10.1017/S0017089505002326

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    • Affine lie group approach to a derivative nonlinear Schrodinger equation and its similarity reduction Peer-reviewed

      S Kakei, T Kikuchi

      INTERNATIONAL MATHEMATICS RESEARCH NOTICES2004 ( 78 ) 4181 - 4209   2004

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：OXFORD UNIV PRESS  

      DOI： 10.1155/S1073792804141573

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    • Similarity reduction of the modified Yajima-Oikawa equation Peer-reviewed

      Tetsuya Kikuchi, Takeshi Ikeda, Saburo Kakei

      Journal of Physics A: Mathematical and General36 ( 45 ) 11465 - 11480   14 11 2003

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      We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A2(1)The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom 2, and admits a group of Bäcklund transformations isomorphic to the affine Weyl group of type A2(1). We show that the system is equivalent to a two-parameter family of the fifth Painlevé equation.

      DOI： 10.1088/0305-4470/36/45/008

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    • Hierarchy of (2+1)-dimensional nonlinear Schrodinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras Peer-reviewed

      S Kakei, T Ikeda, K Takasaki

      ANNALES HENRI POINCARE3 ( 5 ) 817 - 845   2002

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

      The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schrodinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

      DOI： 10.1007/s00023-002-8638-1

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    • A differential-difference system related to toroidal Lie algebra Peer-reviewed

      Saburo Kakei, Yasuhiro Ohta

      Journal of Physics A: Mathematical and General34 ( 48 ) 10585 - 10592   7 12 2001

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      Language：English   Publishing type：Research paper (scientific journal)  

      We present a novel differential-difference system in (2 + 1)-dimensional space-time (one discrete, two continuum), arising from Bogoyavlensky's (2 + 1)-dimensional Korteweg-de Vries hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex-operator representation of the toroidal Lie algebra, s-fraktur sign and l-fractur signtor2.

      DOI： 10.1088/0305-4470/34/48/322

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    • Integrable systems and random permutations

      Takahiro Shiota, Saburo Kakei

        1221   195 - 198   7 2001

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      Language：Japanese  

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    • Symmetry, discretization, and ultra-discretization of coupled KP hierarchy

      Saburo Kakei

        1221   199 - 208   7 2001

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      Language：Japanese   Publishing type：Research paper (conference, symposium, etc.)  

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    • Algegraic structure of coupled KP hierarchy

      Saburo Kakei

        ( 11ME-S4 ) 178 - 182   3 2000

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

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    • Algebraic aspects of quantum Calogero models (共著)

      Saburo Kakei, Yusuke Kato

      Special Functions: Proceeding of the International Workshop   125 - 139   15 1 2000

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    • Dressing method and the coupled KP hierarchy Peer-reviewed

      S Kakei

      PHYSICS LETTERS A264 ( 6 ) 449 - 458   1 2000

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

      The coupled KP hierarchy, introduced by Hirota and Ohta, is investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy. (C) 2000 Elsevier Science B.V. All rights reserved.

      DOI： 10.1016/S0375-9601(99)00848-8

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    • Experiment on shock wave in a random LC circuit Peer-reviewed

      H Asano, S Kakei, S Ishiwata, S Watanabe

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN68 ( 10 ) 3208 - 3213   10 1999

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOCIETY JAPAN  

      Shock wave in a random system has been investigated theoretically and experimentally. A nonlinear LC ladder circuit with random inhomogeneity of linear inductor in series branch has been studied. The circuit is equivalent to a random Toda lattice. It is shown theoretically that wave propagation in the system is described by the Burgers equation in the case of strong randomness with weak nonlinearity. The transition from an oscillatory shock wave to a shock wave has been observed on a random LC ladder circuit experimentally. The velocity and width of shock wave agree with the theoretical prediction from the Burgers equation in a region of strong randomness.

      DOI： 10.1143/JPSJ.68.3208

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    • Orthogonal and symplectic matrix integrals and coupled KP hierarchy Peer-reviewed

      S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN68 ( 9 ) 2875 - 2877   9 1999

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOCIETY JAPAN  

      Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a tau-function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.

      DOI： 10.1143/JPSJ.68.2875

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    • Coupled KP hierarchy and matrix integrals

      Saburo Kakei

        ( 10ME-S1 ) 69 - 74   5 1999

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

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    • Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials Peer-reviewed

      S Kakei

      JOURNAL OF MATHEMATICAL PHYSICS39 ( 9 ) 4993 - 5006   9 1998

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

      Several properties of the multivariable Hermite and Laguerre polynomials associated with the quantum Calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine Hecke algebra. As applications, raising operators and shift operators for the polynomials are constructed in a unified manner. (C) 1998 American Institute of Physics.

      DOI： 10.1063/1.532505

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    • Algegraic structure of quantum Calogero models

      Saburo Kakei

        ( 9ME-S2 ) 29 - 34   5 1998

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

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    • Quantum Calogero models and the degenerate double affine Hecke algebra

      筧 三郎

      京都大学 数理解析研究所 講究録1036   65 - 87   4 1998

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      Language：English   Publishing type：Research paper (bulletin of university, research institution)   Publisher：Kyoto University  

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    • Reply to the remark by W. Malfliet on "Shock wave in a nonlinear LC circuit" Peer-reviewed

      S Watanabe, S Ishiwata, J Ehara, S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN67 ( 2 ) 702 - 702   2 1998

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

      DOI： 10.1143/JPSJ.67.702

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    • An orthogonal basis for the B<SUB>N</SUB>-type Calogero model Peer-reviewed

      Saburo Kakei

      Journal of Physics A: Mathematical and General30 ( 15 ) L535 - L541   7 8 1997

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      Language：English   Publishing type：Research paper (scientific journal)  

      DOI： 10.1088/0305-4470/30/15/009

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    • On long-range interaction models of 1/r<SUP>2</SUP>-type

      Saburo Kakei

        1005   72 - 94   6 1997

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

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    • Common algebraic structure for the Calogero-Sutherland models Peer-reviewed

      Saburo Kakei

      Jornal of Physics A: Mathematical and General29 ( 24 ) L619 - L624   21 12 1996

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      Language：English   Publishing type：Research paper (scientific journal)  

      DOI： 10.1088/0305-4470/29/24/002

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    • Toda lattice hierarchy and Zamolodchikov's conjecture Peer-reviewed

      S Kakei

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN65 ( 2 ) 337 - 339   2 1996

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

      In this letter, we show that the Fredholm determinant D(lambda;t), introduced by Zamolodchikov in his study of 2D polymers, is the continuum limit of the soliton solution for the Toda lattice hierarchy with a 2-periodic reduction condition.

      DOI： 10.1143/JPSJ.65.337

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    • BILINEARIZATION OF A GENERALIZED DERIVATIVE NONLINEAR SCHRODINGER-EQUATION Peer-reviewed

      S KAKEI, N SASA, J SATSUMA

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN64 ( 5 ) 1519 - 1523   5 1995

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

      A generalized derivative nonlinear Schrodinger equation, iq(t)+q(xx)+2i gamma\q\(2)q(x)+2i(gamma-1)q(2)q(x)(*)+(gamma-1)(gamma-2)\q\(4)q=0, is studied by means of Hirota's bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed.

      DOI： 10.1143/JPSJ.64.1519

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    • Bilinear structure of nonlinear Schrodinger-type equations (jointly worked)

      Saburo Kakei, Narimasa Sasa, Junkichi Satsuma

        889   101 - 112   11 1994

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

      CiNii Article

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    • A generalization of Grassmann hierarchy (jointly worked)

      Saburo Kakei, Junkichi Satsuma

        868   85 - 97   4 1994

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      Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

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    • MULTISOLITON SOLUTIONS OF A COUPLED SYSTEM OF THE NONLINEAR SCHRODINGER-EQUATION AND THE MAXWELL-BLOCH EQUATIONS Peer-reviewed

      S KAKEI, J SATSUMA

      JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN63 ( 3 ) 885 - 894   3 1994

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      Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

      Multi-soliton solutions of a coupled system of the nonlinear Schrodinger equation and the Maxwell-Bloch equations are given. Solutions of the system are explicitly constructed as a quotient of Casorati-type determinants. By using explicit form of the soliton solutions, the influence of the MB-term on the speed of soliton is evaluated.

      DOI： 10.1143/JPSJ.63.885

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    Misc.

    • 高校の微積分から大学の微積分へ／“悪魔の階段” にまつわるあれこれ

      筧三郎

      数学セミナー62 ( 5 ) 6 - 11   4 2023

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

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    • 線形代数の持つ力 Invited

      筧三郎

      数学セミナー60 ( 4 ) 12 - 17   3 2021

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • “1/6公式” から超幾何関数へ

      筧三郎

      数学セミナー56 ( 4 ) 36   3 2017

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • Dynamics of discrete curves and discrete integrable systems

      Kenji Kajiwara, Hiromasa Mitani, Saburo Kakei

        1   2 2013

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      Language：Japanese   Publishing type：Rapid communication, short report, research note, etc. (bulletin of university, research institution)   Publisher：Research Center for Mathematical Physics, Rikkyo University  

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    • 線形性とは? (特集 微分積分・線形代数を深く学ぼう)

      筧 三郎

      数学セミナー48 ( 6 ) 8 - 11   6 2009

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      Language：Japanese   Publisher：日本評論社  

      CiNii Article

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    • Yang-Baxter maps from the discrete KP hierarchy (Recent Trends in Integrable Systems)

      Kakei Saburo, Willox Ralph, Nimmo Jonathan J.C.

      RIMS Kokyuroku1650   162 - 172   5 2009

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      Language：Japanese   Publisher：Kyoto University  

      CiNii Article

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      Other Link： http://hdl.handle.net/2433/140767

    • 座談会 戸田格子40年の軌跡 (特集 戸田格子40年)

      薩摩 順吉, 井ノ口 順一, 筧 三郎

      数学セミナー47 ( 3 ) 30 - 36   3 2008

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      Language：Japanese   Publisher：日本評論社  

      CiNii Article

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    • 連続性

      筧三郎

      数学セミナー46 ( 6 ) 26 - 30   1 6 2007

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • Random matrices and integrable systems

      Saburo Kakei

      Mathematical Sciences ( 524 ) 24 - 29   1 2 2007

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：Saiensu sha Co. Ltd. Publishers  

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    • ベクトルとベクトル空間

      筧三郎

      数学セミナー45 ( 4 ) 27 - 31   1 4 2006

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • 運動方程式からの微分積分

      筧三郎

      数学セミナー44 ( 4 ) 26 - 29   1 4 2005

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • 可積分系という「おもちゃ」

      筧三郎

      数学セミナー42 ( 4 ) 47 - 49   1 4 2003

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：日本評論社  

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    • Quantum Calogero models

      Saburo Kakei

      Mathematical Sciences ( 405 ) 54 - 59   1 3 1997

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      Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：Saiensu sha Co. Ltd. Publishers  

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    • 4p-A-6 Bilinear Structure of a Generalized Nonlinear Schrodinger Equation

      KAKEI Saburo, SASA Narimasa, SATSUMA Junkichi

      Abstracts of the meeting of the Physical Society of Japan. Sectional meeting1994 ( 4 ) 95 - 95   16 8 1994

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      Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

      CiNii Article

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    • 30a-Z-14 Coupled System of Nonlinear Schrodinger Equation and Self-Induced Transparency Equation

      Kakei Saburo, Satsuma Junkichi

      Abstracts of the meeting of the Physical Society of Japan. Annual meeting48 ( 4 ) 80 - 80   16 3 1993

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      Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

      CiNii Article

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    • 28a-G-7 Soliton equations related to 3-component KP hierarchy

      Kakei Saburo, Satsuma Junkichi

        1992 ( 4 ) 101 - 101   14 9 1992

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      Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

      CiNii Article

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    • 30a-ZD-8 超対称ソリトン方程式について

      筧 三郎, 薩摩 順吉

      年会講演予稿集47 ( 4 ) 111 - 111   12 3 1992

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      Language：Japanese   Publisher：一般社団法人日本物理学会  

      CiNii Article

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    Books and Other Publications

    • Vector Analysis

      Saburo Kakei, Gen Yoneda（ Role： Joint author）

      8 2018  （ ISBN:4781914268 ）

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      Language：Japanese Book type：Textbook, survey, introduction

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    • Various aspects of nonlinear mathematical models : continuous, discrete, ultra-discrete, and beyond

      （ Role： Compilator）

      3 2015 

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      Total pages：208  

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    • Applied Mathematics Handbook

      （ Role： Contributor ,  KP hierarchy）

      10 2013  （ ISBN:9784254111415 ）

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      Total pages：xiii, 685p   Language：Japanese

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    • Handbook of Self-Organization

      （ Role： Contributor）

      11 2009  （ ISBN:9784860432645 ）

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      Language：Japanese Book type：Dictionary, encyclopedia

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    • 理工系のための 解く! 線形代数

      筧三郎, 西成活裕（ Role： Joint author）

      講談社サイエンティフィク  16 11 2007  （ ISBN:406155767X ）

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      Language：Japanese Book type：Textbook, survey, introduction

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    • この数学書がおもしろい

      数学書房編集（ Role： Contributor ,  具体例へのこだわり）

      数学書房  3 2006 

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      Language：Japanese Book type：General book, introductory book for general audience

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    • 数学ガイダンス hyper

      数学セミナー編集部（ Role： Contributor ,  hyper鼎談；数学を学ぶ心構え、数学の勉強法）

      日本評論社  3 2005 

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      Language：Japanese Book type：General book, introductory book for general audience

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    • 応用数学ハンドブック

      藤原毅夫, 平尾公彦, 久田俊明, 広瀬啓吉（ Role： Contributor ,  線形代数）

      丸善  3 2005  （ ISBN:4621075292 ）

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      Language：Japanese Book type：Dictionary, encyclopedia

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    • 工科系 線形代数

      筧 三郎（ Role： Sole author）

      数理工学社  10 2002  （ ISBN:4901683020 ）

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      Language：Japanese Book type：Textbook, survey, introduction

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    Research Projects

    • KP階層のさまざまな拡張と諸分野への応用

      日本学術振興会  科学研究費助成事業 基盤研究(C) 

      筧 三郎

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      4 2023 - 3 2028

      Grant number：23K03137

      Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

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    • 可積分系の離散的方法を基盤とした非線形波動解析のための計算アルゴリズムの開発

      日本学術振興会  科学研究費助成事業 

      丸野 健一, 太田 泰広, 筧 三郎

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      4 2018 - 3 2023

      Grant number：18K03435

      Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

      今年度は主として以下の問題に取り組んだ．
      (1) 解構造を保存する離散化，自己適合移動格子スキームの構築と実装に関する研究．大変形問題に対する数値計算法として有効な自己適合移動格子スキームは，これまで非零境界条件には適用が困難であったが，前年度にその困難の克服ができる実用的手法を探索し，最終的にそれを見出すことに成功した．本年度はそれをさらに前進させ，より多様な方程式に適用できるように理論を整備した．また，これまで困難であった多成分短パルス方程式の解構造を保ち空間と時間を共に離散化する全離散化に成功した．また，前年度から行なっている感染症の数理モデルの解構造を保存する離散化にも引き続き取り組んだ．
      (2) Pfaffian解を持つHungry Lotka-Volterra型方程式およびBKP方程式についての研究．Pfaffian解を持つHungry Lotka-Volterra型方程式のソリトン解を構成し，ソリトン相互作用の詳細を調べた．また，前年度から引き続き２次元ソリトンパターンの分類問題，特にBKP方程式のソリトン解の分類に適した形の解の表示を用いBKP方程式のソリトン相互作用を調べる研究も行ったが，この成果をPfaffian解を持つHungry Lotka-Volterra型方程式のソリトン相互作用の研究に有用であることがわかった．
      (3) 遅延ソリトン方程式の研究 離散KP方程式から遅延ソリトン方程式およびソリトン解を構成する方法についての研究を行い，新たな可積分な遅延ソリトン方程式を見出した．
      新型コロナウイルスの感染拡大によって研究の進捗状況に遅れがでているが，次年度に上記の結果をまとめ，論文として出版する予定である．

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    • 諸分野に現れるタウ関数の構造

      日本学術振興会  科学研究費助成事業 基盤研究(C) 

      筧 三郎

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      4 2016 - 3 2023

      Grant number：16K05184

      Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）

      本研究では，可積分系の研究における「タウ関数」という視点を広げるとともに，その概念と諸分野との関連を考察することで，可積分系の世界をさらに広げていくことを目標としている。2019年度は，離散可積分系と関連するテーマを研究し，昨年度からの研究を発展する形での成果が得られた。特に，格子KdV方程式，格子ブシネスク方程式とKP階層のタウ関数との関係について，昨年度では証明が完成していなかった部分をまとめることができた。具体的には，オランダのグループによって研究されてきたタイプの格子KdV方程式，格子ブシネスク方程式が，以前の梶原(九大IMI)との共同研究で得られた離散戸田階層の定式化から得られることを示すことができ，それに伴い，戸田階層のタウ関数との関係も確立することができた。また，戸田階層の代数関数解について，高崎の先行結果を利用することで，格子KdV方程式，格子ブシネスク方程式の解を構成することができた。この場合，よく知られているように，タウ関数はリーマン・テータ関数を用いて表される。上記の結果を証明する際には膨大な代数的計算が必要となるが，その部分には，計算機代数システム Risa/Asir を利用した。上記の結果については論文を投稿中である。2019年度に参加した研究集会でも，これらの結果を紹介した。
      また，昨年度からの引き続きのテーマである「Jeu de taquin と離散戸田格子」，「交通流モデルとGKZ超幾何系」という話題については，進展はあったものの，最終的な解決までには至っていない。本年度も引き続き研究を続けていこうと考えている。

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    • Construction of time discrete geometric models based on discrete differential geometry

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      Kajiwara Kenji, MARUNO Ken-ichi, KAKEI Saburo, HIROSE Sampei, Broadbridge Philip

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      4 2016 - 3 2018

      Grant number：16K13763

      Grant amount：\2210000 （ Direct Cost: \1700000 、 Indirect Cost：\510000 ）

      In this project, the following three topics have been studied:(a)discrete models of curve deformation with extension and external force, (b)time discrete model of deformation of layer, (c) discrete model of one-dimensional elastic rod. As to (a) we succeeded in constructing a new discrete model of the curve shortening equation and discrete local induction equation. For (b) we obtained the bilinearization of the complex Dym equation describing the Hele-Shaw flow, and also succeeded in discretization of the Broadbridge-White model for the one-dimensional soil water infiltration and its numerical simulation. Regarding (c) we succeeded in formulating the integrable discrete model of the Euler's elastic curves by the discrete variational principle. Further we have shown that the elastic curves in the similarity geometry are nothing but the log-aesthetic curves used in the industrial design, and gave a sound mathematical foundation for their generalization.

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    • Extension of special functions from the viewpoint of integrable systems

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) 

      KAKEI Saburo, SAITO Yoshihisa, WILLOX Ralph

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      4 2011 - 3 2016

      Grant number：23540252

      Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）

      This research project aims to extend the theory of integrable systems and special functions. We have focused mainly on applications to differential geometry and combinatorics, and obtained the following results;
      (1) Goldstein and Petrich introduced a hierarchy of equations that governs isoperimetric deformation of plane curves. We have revealed that the Goldstein-Petrich hierarchy is indeed a reduced case of the Toda lattice hierarchy. (2) We have studied rigged configurations from the viewpoint of soliton cellular automaton. We have obtained alternative proof of the linearization property of the time-evolution, which is simpler than the known proofs.
      We have obtained still other preliminary results that are related to applications of integrable systems to differential geometry and combinatorics. We will investigate them further in near future.

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    • 古典可積分系と量子可積分系の接点の探求

      日本学術振興会  科学研究費助成事業 

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      4 2007 - 3 2010

      Grant type：Competitive

      可積分系の研究は，大別すれば「古典可積分系」，「量子可積分系」の２つに分類できる。両者の間には単に「正準量子化」でつながること以上の結びつきがあり，より深い意味での「量子・古典対応」が存在していると思われる。そこで本研究では，この「量子・古典対応」の背後にある数学的構造を解明することを目的とする。

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    • 可積分系における楕円ワイル群対称性

      日本学術振興会  科学研究費助成事業 

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      4 2004 - 3 2007

      Grant type：Competitive

      これまでの可積分系研究の中で見出されてきたワイル群の構造を，「楕円ワイル群」というキーワードの下に拡張し，新たな可能性を切り開くことを目的とする。

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    • A quasi-triangular structure in Kashiwara-Miwa model

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      YAMADA Yuji, SHIRAISHI Junichi, KAKEI Saboro

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      2005 - 2007

      Grant number：17540204

      Grant amount：\3730000 （ Direct Cost: \3400000 、 Indirect Cost：\330000 ）

      We study the classification of the solutions to the reflection equation for Cremmer-Gervais R-matrix in N=3 case. The R-matrix of Cremmer-Gervais type is obtained from Uq(sl_N) through the theory of quasi-triangular Hopf algebras of Drinfeld as solutions to the Yang-Baxter equation. There are only two types of the R-matrices which are obatained through the theory of quasi-triangular Hopf algebras. One is the series of Belavin' s R-matrices, and the other is the series of Cremmer-Gervais type. The algebraic structures of the solutions to the reflection equations are not well understood. We have only two cases in which all solutions to the reflection equation are known, (1) N=2 Belavin' s elliptic R-matrix (eight-vertex case), and (2) N=3 trigonomerically degenerated Belavin' s R-matix case. In order to understand the algebraic structure in solutions to the reflection equation, we study the case of N=3 Cremmer-Gervais R-matix case (with Kohei MOTEGI). The N=2 Cremmer-Gervais R-matrix is only the degenerated N=2 Belavin' s elliptic R-matrix.
      The solution space of the reflection equation to the N=3 Cremmer-Gervais R-matrix obtained through this study is described by the rational surface in projective spaces. In the case of the N=3 trigonometric R-matrix, the parameter space of the solution of the reflection equation is the Segre three-fols in $P^5(C)$. In our case there appeared two parameters spaces. One is the same Segre three-fold. But the other new parameter space is embedded in the projective space$P^10$.

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    • Mordell-Weil Lattices and Cycles on Algebraic Surfaces

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      SHIODA Tetsuji, AOKI Noboru, KAKEI Saburo

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      2005 - 2007

      Grant number：17540044

      Grant amount：\3670000 （ Direct Cost: \3400000 、 Indirect Cost：\270000 ）

      The following results have been accomplished on Algebraic cycles on K3 surfaces which forms the main theme of this research proposal :
      (i) We have determined, for an example of a K3 surface attaining the maximal Mordell-Weil rank 18, the structure of its Mordell-Weil lattice, explicit generators of rational points, together with the splitting field.[2]
      (ii) More generally, we have clarified the structure of the transcendental lattice, Neron-Severi lattice and the Mordell-Weil lattice, of the singular K3 surfaces which arise via the cyclic base change of degree up to six from Inose's elliptic K3 surface. Ref.[3] and in preparation, and (3).
      (iii) Classical Kummer surfaces attached to genus two Jacobian varieties are studied from the viewpoint of Mordell-Weil lattices, which establishes the close relationship of automorphisms of a genus two curve with the singular fibres and generators of the Mordell-Weil lattice in question. Ref.[4].
      (iv) In a joint work with M.Kuwata, we have determined the elliptic parameters and the defining equations of elliptic fibrations on the Kummer surface of the product of two non-isogenous elliptic crves., which has been classified by Oguiso by geometric method. Ref.[6].
      [The rest is omitted in this English summary because of the space limitation.]

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    • Relation between automorphic forms and zeta functions associated with prehomogeneous vector spaces

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      SATO Fumihiro, HIGA Tatsuo, KAKEI Saburo, IBUKIYAMA Tomoyoshi, HIRONAKA Yumiko, KIMURA Tatsuo

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      2004 - 2007

      Grant number：16340012

      Grant amount：\16380000 （ Direct Cost: \15300000 、 Indirect Cost：\1080000 ）

      The main problems we investigated in this research project are
      (1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms
      (2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :
      (1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.
      (2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras.

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    • 高次元における非線形可積分系の構造

      日本学術振興会  科学研究費助成事業 

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      4 2002 - 3 2004

      Grant type：Competitive

      従来のソリトン理論では，KdV方程式，非線形シュレディンガー方程式のような(1+1)次元の時空における偏微分方程式，KP方程式，Davey-Stewartson方程式のような(2+1)次元の時空における偏微分方程式が研究されてきた。本研究では，特に「トロイダル・リー代数」の対称性に注目し，高次元における非線形可積分系の構造を解明することを目的とする。

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    • パフィアン型ソリトン方程式の構造と諸分野への応用

      日本学術振興会  科学研究費助成事業 

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      4 2000 - 3 2002

      Grant type：Competitive

      KdV方程式，KP方程式に代表されるソリトン方程式の理論においては，特殊解が行列式で表わされることが本質的であった。そこで本研究では，パフィアンで表わされる特殊解をもつソリトン方程式に対して，その構造と諸分野への応用を探ることを目的とする。

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    • 擬リーマン対称空間上の数論的解析学

      日本学術振興会  科学研究費助成事業 

      佐藤 文廣, 宇澤 達, 筧 三郎, 荒川 恒男, 落合 啓之

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      2001 - 2002

      Grant number：13874007

      Grant amount：\1800000 （ Direct Cost: \1800000 ）

      1.この研究でまず問題としたのは,(A)擬リーマン対称空間上の数論的アイゼンシュタイン級数とポアッソン変換から得られる佐藤超関数としてのアイゼンシュタイン級数の関係,(B)概均質ベクトル空間に付随する(ディリクレ級数としての)ゼータ関数と超関数値ゼータ関数の関係,という2点を解明することであった.これについて,まだ十分な理解に到達したとはいえないが,概均質ベクトル空間のゼータ関数を弱球等質空間のアイゼンシュタイン級数と読み替えることにより,(B)の問題を(A)の問題に関連付ける研究が進行中である.
      2.擬リーマン対称空間上の具体的な数論的関数としてArakawa-SuzukiによるSiegel Eisenstein級数のKoecher-Maass級数の不定値類似を昨年に引き続き研究した.これについては,概均質ベクトル空間のゼータ関数の立場から理論を再構成するとともに,パラメータに関する解析接続についてより精密な結果を得ることができた.また,量指標つきのKoecher-Maass級数への拡張の研究が現在進行中である.(上野隆彦(立教大)との共同研究)
      3.関連研究として,分担者荒川は,不定値整数係数原始的2元2次形式の類数の和の評価を,基本単数のノルムが1と-1の場合に分離して,評価を精密化することに成功した.そのために,even Maass波動形式の空間のセルバーグ跡公式の応用として,群GL(2,Z)に関する素測地線定理を証明した.

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    • Applicaticns of Ultradiscretization Method

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      TAKAHASHI Daisuke, SHINZAWA Nobuhiko, NISHINARI Katsuhiro, MATSUKIDAIRA Junta

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      1999 - 2002

      Grant number：11555023

      Grant amount：\6900000 （ Direct Cost: \6900000 ）

      Ultradiscretization is a method to discretize a dependent variable of difference equation by using an non-analytic limit of exp type. Combination of this method and differentiation enables a series of discretizing process from differential to ultradiscrete via difference. We apply these methods to various equations and try to find their application in this research.
      First, we succeeded to obtain a model which show a traffic jamming process by ultradiscretizing the well-known Burgers equation. The ultradiscrete analogue to Burgers equation is a rule-184 cellular automaton.
      Next, we proposed a new digital equation which can show a pattern formation process. This digital equation is given by the max-plus algebra which is a base of the ultradiscretization method.
      Finally, we ultradiscretized some nonlinear integrable differenece equations of second order and analyzed their solution structure. Ultradiscrete solutions are similar to the mod function. We clarified the behavior of the solutions from a view point of the dynamical system theory and obtained explicit solutions.

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    • Faddeev-Hopf結び目の研究

      日本学術振興会  科学研究費助成事業 

      村上 斉, 本間 泰史, 落合 啓之, 牛島 顕, 鈴木 達夫, 筧 三郎

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      2000 - 2001

      Grant number：12874012

      Grant amount：\1700000 （ Direct Cost: \1700000 ）

      今年度は,結び目が(電磁)流体力学などにどのように現れるかについて研究を行った.どのような結び目も,ひもの交差を許せばほどけることが容易にわかる.この事実を使うことによって任意の結び目が渦流として表されることが知られている.渦流を決定する方程式を考察することによって,結び目の(原理的には全ての)位相的な性質が得られるはずであるが,今のところ知られているのは「ねじれ」を示す量くらいである.
      一方,結び目理論の有限型不変量(Vassiliev不変量)の言葉で言うと,「ねじれ」は最も簡単な1次の不変量とみなすことができる.現在2次以上の不変量も様々な視点から考察が続けられており,研究代表者および研究分担者もこの方面での貢献が多い.また,有限型不変量は,Kontsevich積分を通して量子不変量とも関係が深く多くの研究者の注目を集めている.
      そこで今後の研究課題として,次のようなことが考えられる.
      1.渦流の立場から2次以上のVassiliev不変量を定義できないか.また,それらに対する流体力学的観点からの意味付けができないか.他の量子不変量ではどうか.
      2.結び目解消数(上述のような,結び目をほどくためのひもの交差の必要最小数)を渦流の言葉で説明できないか.(高分子化学や生物学(DNA)においても結び目解消数が興味をもたれていることに注意.)
      3.3次元多様体の研究,に応用できないか.(任意の3次元多様体は結び目・絡み目を使って記述できることに注意.)
      残念ながら新しい結果を得るまでには到らなかったが,今後の研究課題が明確になったという意味で実りある研究であった.

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    • 量子可積分系と多変数直交多項式

      早稲田大学  受託研究（一般受託研究） 

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      4 1999 - 3 2000

      Grant type：Competitive

      特定課題研究

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    • 量子可積分系の代数的構造の究明と物理的情報の抽出

      日本学術振興会  科学研究費助成事業 特別研究員奨励費 

      筧 三郎

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      1998 - 1998

      Grant number：98J03759

      Grant amount：\1200000 （ Direct Cost: \1200000 ）

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    • Theory of Integrable Systems and Its Application

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      Grant type：Competitive

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    • 可積分系の理論とその応用

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      Grant type：Competitive

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