Updated on 2024/06/20

写真a

 
ISHIMURA Naoyuki
 
Organization
Faculty of Commerce Professor
Other responsible organization
Commerce Course of Graduate School of Commerce, Master's Program
Commerce Course of Graduate School of Commerce, Doctoral Program
Contact information
The inquiry by e-mail is 《here
External link

Degree

  • 博士(数理科学) ( 東京大学 )

  • 修士(理学) ( 東京大学 )

Education

  • 1989.3
     

    The University of Tokyo   Graduate School, Division of Science   master course   completed

  • 1987.3
     

    The University of Tokyo   Graduate School, Division of Science   master course   withdrawn before completion

  • 1986.3
     

    The University of Tokyo   Faculty of Science   graduated

  • 1982.3
     

    徳島市立高等学校   理数科   graduated

Research History

  • 2015.4 - Now

    中央大学商学部教授

  • 2005.4 - 2015.3

    一橋大学大学院経済学研究科教授

  • 1998.4 - 2005.3

    一橋大学大学院経済学研究科助教授

  • 1996.4 - 1998.3

    一橋大学経済学部助教授

  • 1992.4 - 1996.3

    東京大学大学院数理科学研究科助手

  • 1989.4 - 1992.3

    東京大学理学部数学教室助手

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Professional Memberships

  • 日本ファイナンス学会

  • Australian Mathematical Society

Research Interests

  • "Mathematical finance, Nonlinear Analysis"

Research Areas

  • Natural Science / Mathematical analysis  / 数学解析

Papers

  • Insurance design for the loss of epidemic outbreaks involving Cramer-Lundberg model Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura, Koichiro Takaoka, Chenwei Sun

    International Journal of Mathematics for Industry   16   2024

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

    DOI: 10.1142/S2661335224500151

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  • Efficient numerical computation of the integral equation for the ruin probability Reviewed

    Hiroko Soutome, Naoyuki Ishimura, Hitoshi Imai

    IEEE XPlore: 2023 27th International Computer Science and Engineering Conference (ICSEC)   37 - 41   2023.12

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

    DOI: 10.1109/icsec59635.2023.10329787

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  • Remarks on a copula‐based conditional value at risk for the portfolio problem Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura

    Intelligent Systems in Accounting, Finance and Management   30 ( 3 )   150 - 170   2023.8

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

    Summary

    We deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula‐based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula‐based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results.

    DOI: 10.1002/isaf.1540

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  • Efficient numerical computation of the ruin probability Reviewed

    Hiroko Soutome, Naoyuki Ishimura, Hitoshi Imai

    Proceedings of the 54th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Nara, Oct. 14-15, 2022   35 - 40   2023.3

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    File: SSS22_06.pdf

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  • Global in space numerical computation of the ruin probability Reviewed

    Hiroko Soutome, Naoyuki Ishimura, Hitoshi Imai

    Advances in Mathematical Sciences and Applications   31 ( 2 )   397 - 406   2022.12

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    File: no0165.pdf

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  • Long memory estimation in a non-Gaussian bivariate process Reviewed

    Ledys Llasmin Salazar Gomez, Soledad Torres, Jozef Kiseľák, Felix Fuders, Naoyuki Ishimura, Yasukazu Yoshizawa, Milan Stehlík

    Applied Mathematics and Computation   420   126871 - 126871   2022.5

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

    DOI: 10.1016/j.amc.2021.126871

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  • On a determination formula for the estimation of copula-based Value at Risk Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura, Koichiro Takaoka

    Proceedings of the 53rd International Symposium on Stochastic Systems Theory and its Applications, Kusatsu, Oct. 30-31   86 - 92   2022.3

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:The Institute of Systems, Control and Information Engineers  

    File: SSS21Proc_14.pdf

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  • Copula-based estimation of Value at Risk for the portfolio problem Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura

    Proceedings of the Forum "Math-for-Industry" 2018 (Cheng J., Dinghua X., Saeki O., Shirai T. (eds))   35   1 - 13   2021

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

    DOI: 10.14458/RSU.res.2020.4

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  • A determination formula on the copula-based estimation of Value at Risk for the portfolio problem Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura

    Proceedings of RSU International Research Conference 2020 (https://rsucon.rsu.ac.th/proceeding/article/2527)   1236 - 1246   2020.5

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  • Value at Risk for the portfolio problem with copulas Reviewed

    Andres Mauricio Molina Barreto, Naoyuki Ishimura, Yasukazu Yoshizawa

    in "Empowering Science and Mathematics for Global Competitiveness," Edited by Y. Rahmawanti and P.C. Taylor   371 - 376   2019.6

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:CRC Press (Taylor and Francis Group), London  

    DOI: 10.1201/9780429461903

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  • Time evolution of copulas and foreign exchange markets Reviewed

    Ivan Kupka, Jozef Kisel'ak, Naoyuki Ishimura, Yasukazu Yoshizawa, Ledys Salazar, Milan Stehlik

    Information Sciences   467   163 - 178   2018.7

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  • Evolution of multivariate copulas in continuous and discrete processes Reviewed

    Yasukazu Yoshizawa, Naoyuki Ishimura

    Intelligent Systems in Accounting, Finance and Management   25   44 - 59   2018.4

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  • Numerical limit and its application to a blow-up problem related to default risk Reviewed

    Yuya Sasaki, Hiroko Soutome, Hitoshi Imai, Naoyuki Ishimura

    Advances in Mathematical Sciences and Applications   26 ( 1 )   29 - -38   2017.4

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

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  • On the convergence of discrete processes with multiple independent variables Reviewed

    Naoyuki Ishimura, Naohiro Yoshida

    ANZIAM JOURNAL   58 ( 3-4 )   379 - 385   2017.4

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

    We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito's formula converges to the corresponding continuous formula.

    DOI: 10.1017/S1446181116000389

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  • Evolution of copulas in discrete processes with application to a numerical modeling of dependence relation between exchange rates Reviewed

    Naoyuki Ishimura, Yasukazu Yoshizawa

    Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)   10187   359 - 366   2017

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

    Copulas are known to provide a flexible tool for analyzing the dependence structure between random events. Here we apply the newly introduced notion of evolution of copulas to real data of exchange rates so that we ensure the quality of practically employing our theory. Results show that our algorithm provides a prospective handy method in computational finance.

    DOI: 10.1007/978-3-319-57099-0_39

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  • Remarks on the optimal portfolio problem in discrete variables with multiple stochastic processes Reviewed

    Naohiro Yoshida, Naoyuki Ishimura

    International Journal of Modeling and Optimization   6 ( 2 )   96 - 99   2016.4

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  • Risk estimation model on epidemic outbreaks for an insurer Reviewed

    Naoyuki Ishimura, Daniel Komadel, Yasukazu Yoshizawa

    ASTIN Colloquium 2016 (http://actuaries.org/lisbon2016/papers/19.NaoyukiIshimuraPaper.pdf)   2016

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  • Estimate on the risk of epidemic outbreaks for an insurer: a simple stochastic model Reviewed

    Chiaki Ishii, Naoyuki Ishimura, Takahiko Fujita

    International Journal of Modeling and Optimization   5   161 - -165   2015

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  • Evolution of copulas and its applications Reviewed

    Naoyuki Ishimura

    Academie van Belgique,|rn|http://www.afmathconf.ugent.be/FormerEditions/Proceedings2014.pdf   85 - 89   2014

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  • Note on the measures of dependence in terms of copulas Reviewed

    Ayumi Ida, Naoyuki Ishimura, MasaAki Nakamura

    INTERNATIONAL CONFERENCE ON APPLIED ECONOMICS (ICOAE 2014)   14   273 - 279   2014

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:ELSEVIER SCIENCE BV  

    The dependence structure among each risk factors has been an important topic for researches both from theoretical and applied standpoints. To measure such dependence, several characteristic quantities have been already introduced and widely employed, which include, for instance, the population version of Kendall's tau (tau) and/or Spearman's rho (rho). Copulas, on the other hand, are well known tools for understanding the dependence relation among random variables, and the above t and. are expressed in terms of copulas. In this note, we generalize these expressions. We also compute the extended formula for the Archimedean copulas as well as its generalized copulas, and pursue the possibility of its applications. (C) 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

    DOI: 10.1016/S2212-5671(14)00712-6

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  • On traveling wave solutions to a Hamilton-Jacobi-Bellman equation with inequality constraints Reviewed

    Naoyuki Ishimura, Daniel Sevcovic

    JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   30 ( 1 )   51 - 67   2013

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KINOKUNIYA CO LTD  

    The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct monotone traveling wave solutions and identify parametric regions for which the traveling wave solution has a positive or negative wave speed.

    DOI: 10.1007/s13160-012-0087-8

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  • A model of the instantaneous interest rate in discrete processes Reviewed

    Naoyuki Ishimura, Takahiko Fujita, MasaAki Nakamura

    INTERNATIONAL CONFERENCE ON APPLIED ECONOMICS (ICOAE) 2013   5   355 - 360   2013

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    The modeling of the term structure of interest rates is one of important topics for researches in financial economics. Here we consider a model of the instantaneous interest rate in discrete processes, which may be regarded as a discrete version of the usual continuous process models. We compute the price of the zero-coupon bond. Our methodology of analysis is based on the framework of discrete stochastic calculus. (C) 2013 The Authors. Published by Elsevier B.V.

    DOI: 10.1016/S2212-5671(13)00042-7

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  • Discrete stochastic calculus and its applications: an expository note Reviewed

    Takahiko Fujita, Naoyuki Ishimura, Norihisa Kawai

    Advances in Mathematical Economics   16   119 - 131   2012

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  • Evolution of multivariate copulas in discrete processes Reviewed

    Naoyuki Ishimura, Yasukazu Yoshizawa

    INTERNATIONAL CONFERENCE ON APPLIED ECONOMICS (ICOAE)   1   186 - 192   2012

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:ELSEVIER SCIENCE BV  

    Copulas provide a key ingredient in the field of quantitative risk management because of their flexibility upon investigating dependence relations among random variables. Except for several examples, however, copulas are mainly concerned with the static problems, not with the time-dependent situations. In this paper, on the other hand, we deal with the evolution of copulas in discrete processes with multivariate cases, which extend our previous results on the time evolution of copulas and are expected to enhance the applicability of one method to financial mathematics. (C) Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Organising Committee of ICOAE 2012

    DOI: 10.1016/S2212-5671(12)00022-6

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  • Evolution of bivariate copulas in discrete processes Reviewed

    Yasukazu Yoshizawa, Naoyuki Ishimura

    JSIAM Letters   3   77 - 80   2011

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:日本応用数理学会  

    A copula function makes a bridge between multivariate joint distributions and univariate marginal distributions, and provides a flexible way of describing nonlinear dependence among random circumstances. We introduce a new family of bivariate copulas which evolves according to the discrete process of heat equation. We prove the convergence of solutions as well as the measure of dependence. Numerical experiments are also performed, which shows that our procedure works substantially well.

    DOI: 10.14495/jsiaml.3.77

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  • Risk preference under stochastic environment Reviewed

    Naoyuki Ishimura, Masaaki Nakamura

    BMEI 2011 - Proceedings 2011 International Conference on Business Management and Electronic Information   1   668 - 670   2011

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    We introduce a quantity describing the risk preference of the optimal decision problem under random environment, which extends the Arrow-Pratt coefficient of relative risk aversion for the utility function. We show the existence of well-behaved solutions to the evolution equation of the quantity. © 2011 IEEE.

    DOI: 10.1109/ICBMEI.2011.5917024

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  • K. Ecker: Regularity Theory for Mean Curvature Flow, Progr. Nonlinear Differential Equation Appl., 57, Birkhäuser, 2004年,xiv + 165ページ.

    石村 直之

    数学   63 ( 1 )   129 - 132   2011

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    DOI: 10.11429/sugaku.0631129

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  • On time-dependent bivariate copulas Reviewed

    Naoyuki Ishimura, Yasukazu Yoshizawa

    Theoretical and Applied Mechanics Japan   59   303 - 307   2011

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  • A note on the time evolution of bivariate copulas Reviewed

    Naoyuki Ishimura, Yasukazu Yoshizawa

    FOURTH INTERNATIONAL CONFERENCE FINANCIAL AND ACTUARIAL MATHEMATICS - FAM-2011   3 - 7   2011

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    Dependence properties between random events are important issues not only for researches but also from the viewpoint of risk management. For example, the dependence relation between exchange rates is worth further investigation both for academics and for insurance industries. To discuss such association problems, Kendall's tau tau and Spearman's rho rho are known measures of concordance between random variables. On the other hand, a copula is well employed tool for analyzing dependence relations, and the formula of tau and rho in terms of copula is already known. In our previous study, we introduce the time evolution of bivariate copulas, which is provided as a solution of diffusion equation. This paper is concerned with the convergence in time of tau and rho under the evolution of copulas. It is shown that any dependence structure converges to the one of independence as the time tends to infinity.

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  • Traveling wave solutions to the nonlinear evolution equation for the risk preference Reviewed

    Naoyuki Ishimura, Sakkakom Maneenop

    JSIAM Letters   3   25 - 28   2011

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:日本応用数理学会  

    A singular nonlinear partial differential equation (PDE) is introduced, which can be interpreted as the evolution of the risk preference in the optimal investment problem under the random risk process. The unknown quantity is related to the Arrow-Pratt coefficient of relative risk aversion with respect to the optimal value function. We show the existence of monotone traveling wave solutions and the nonexistence of non-monotone such solutions, which are suitable from the standpoint of financial economics.

    DOI: 10.14495/jsiaml.3.25

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  • Numerical Solution of a Nonlinear Evolution Equation for the Risk Preference Reviewed

    Naoyuki Ishimura, Miglena N. Koleva, Lubin G. Vulkov

    NUMERICAL METHODS AND APPLICATIONS   6046   445 - +   2011

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

    A singular nonlinear partial differential equation (PDE) for the risk preference was derived by the first, author in previous publications. The PDE is related to the Arrow-Pratt coefficient of relative risk aversion. In the present paper, we develop a Rothe-Bellman & Kalaba quasilinearization method on quasi-uniform space mesh to numerically investigate such PDE. Numerical experiments are discussed.

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  • Remarks on the nonlinear Black-Scholes equations with the effect of transaction costs Reviewed

    Naoyuki Ishimura

    Asia-Pacific Financial Markets   17 ( 3 )   241 - 259   2010

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    We deal with the solvability and a weak formulation of nonlinear partial differential equations of Black-Scholes type for the pricing of options in the presence of transaction costs. Examples include the Hoggard-Whalley-Wilmott equation, which is introduced to model portfolios of European type options with transaction costs based on the idea of Leland. The cost structure gives rise to nonlinear terms. We show the existence and the uniqueness of solutions both in the classical and the weak sense, where the notion of weak solutions is introduced. © 2010 Springer Science+Business Media, LLC.

    DOI: 10.1007/s10690-010-9115-3

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  • Global in space numerical computation for the nonlinear Black-Scholes equation Reviewed

    Naoyuki Ishimura, Hitoshi Imai

    Nonlinear Models in Mathematical Finance   219 - 242   2009

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  • Numerical treatment of nonlinear partial differential equations for the risk preference Reviewed

    Kushida Masahiro, Naoyuki Ishimura, Hitoshi Imai

    Theoretical and Applied Mechanics Japan   57   487 - 492   2009

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  • Space-Precise Computation of a Singular Nonlinear Evolution Equation for the Risk Preference Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Masahiro Kushida

    IMECS 2009 Proceedings   2135 - 2139   2009

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  • A NOTE ON THE OPTIMAL PORTFOLIO PROBLEM IN DISCRETE PROCESSES Reviewed

    Naoyuki Ishimura, Yuji Mita

    KYBERNETIKA   45 ( 4 )   681 - 688   2009

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    We deal with the optimal portfolio problem in discrete-time setting. Employing the discrete Ito formula, which is developed by Fujita, we establish the discrete Hamilton-Jacobi-Bellman (d-HJB) equation for the value function. Simple examples of the d-HJB equation are also discussed.

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  • AN ARBITRAGE APPROACH TO THE PRICING OF CATASTROPHE OPTIONS INVOLVING THE COX PROCESS Reviewed

    Takahiko Fujita, Naoyuki Ishimura, Daichi Tanaka

    HITOTSUBASHI JOURNAL OF ECONOMICS   49 ( 2 )   67 - 74   2008.12

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    We investigate file valuation and hedging of catastrophe options, whose claim arrival process is modeled by the Cox process or a doubly stochastic Poisson process. Employing the non-arbitrage principle we obtain closed form formula for the pricing of the option. Various hedging parameters are also computed.

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  • Existence of periodic traveling wave solutions for the Ostrovsky equation Reviewed

    Naoyuki Ishimura, Tetsu Mizumachi

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES   31 ( 14 )   1646 - 1652   2008.9

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:JOHN WILEY & SONS LTD  

    We are concerned with the Ostrovsky equation, which is dervied from the theory of weakly nonlinear long surface and internal waves in shallow water under the presence of rotation. On the basis of the variational method, we show the existence of periodic traveling wave solutions. Copyright (C) 2008 John Wiley Sons, Ltd.

    DOI: 10.1002/mma.990

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  • Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem Reviewed

    Ryo Abe, Naoyuki Ishimura

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   84 ( 1 )   11 - 14   2008.1

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

    We are concerned with the solvablity of certain nonlinear partial differential equation (PDE), which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions.

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  • Global in space simulation for the black-scholes equation incorporating transaction costs Reviewed

    Zhenyu Jin, Hideo Sakaguchi, Naoyuki Ishimura, Hitoshi Imai

    Theoretical and Applied Mechanics Japan   56   445 - 450   2008

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Science Council of Japan  

    We propose a sophisticated method to numerically compute the partial differential equations (PDEs) of Black-Scholes type as they are. The method overcomes the difficulty of infinite domains and unbounded values of the solution. As an application we deal with the PDE with the effect of transaction costs.

    DOI: 10.11345/nctam.56.445

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  • Numerical Treatment of a Singular Nonlinear Partial Differential Equation Arising in the Optimal Investment Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Masahiro Kushida

    Thai Journal of Mathematics   5 ( 2 )   321 - 326   2007

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  • On the Hoggard-Whalley-Wilmott equation for the pricing of options with transaction costs Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Ikumi Mottate, Masaaki Nakamura

    Asia-Pacific Financial markets   13 ( 4 )   315 - 326   2007

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  • Computational technique for treating the nonlinear Black-Scholes equation with the effect of transaction costs Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Hideo Sakaguchi

    KYBERNETIKA   43 ( 6 )   807 - 815   2007

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

    We deal with numerical computation of the nonlinear partial differential equations (PDEs) of Black-Scholes type which incorporate the effect of transaction costs. Our proposed technique surmounts the difficulty of infinite domains and unbounded values of the solutions. Numerical implementation shows the validity of our scheme.

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  • Exact solutions of a model for asset prices by K. Takaoka Reviewed

    Naoyuki Ishimura, Toshihiko Sakaguchi

    Asia-Pacific Financial markets   11 ( 4 )   445 - 451   2006.12

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

    DOI: 10.1007/s10690-006-9022-9

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  • Bifurcations of steady states for the Eguchi-Oki-Matsumura model of phase separation Reviewed

    Naoyuki Ishimura, Kenichi Nakamura, Masaaki Nakamura

    Applicable Analysis   85 ( 6-7 )   831 - 843   2006.6

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Taylor & Francis  

    DOI: 10.1080/00036810600787972

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  • Singular perturbation problem for steady state solutions to a model equation of phase separation Reviewed

    T Hanada, N Ishimura, M Nakamura

    ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK   85 ( 12 )   896 - 903   2005.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:WILEY-V C H VERLAG GMBH  

    A model equation for phase separation introduced by Eguchi-Oki-Matsumura (EOM) is investigated. The behavior of steady state solutions is analyzed when the parameter, which depends on the ratio of the surface energy, tends to zero. The problem is a kind of singular perturbations in the sense that the coefficient of the highest order derivative is very small. Analytical convergence results are established. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

    DOI: 10.1002/zamm.200410216

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  • Self-similar solutions for the kinematic model equation of spiral waves Reviewed

    JS Guo, N Ishimura, CC Wu

    PHYSICA D-NONLINEAR PHENOMENA   198 ( 3-4 )   197 - 211   2004.11

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

    We present a class of self-similar solutions of the kinematic model equation, introduced by V.A. Davydov, A.S. Mikhailov, and V.S. Zykov. This equation is designed to describe the dynamics of spiral waves in excitable media. In this model, the sharply located spiral fronts are regarded as planar curves. If the tip neither grows nor retracts in the tangential direction and if their normal velocity (with the eikonal approximation) is assumed to possess no driving force, then the kinematic equation admits self-similar solutions with nonzero curvature. We show the global structure of both forward and backward self-similar solutions, which implies mathematically the existence of various types of spiral waves. (C) 2004 Elsevier B.V. All rights reserved.

    DOI: 10.1016/j.physd.2004.08.028

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  • An elementary approach to the analysis of exact solutions for the Navier-Stokes stagnation flows with slips Reviewed

    N Ishimura, TK Ushijima

    ARCHIV DER MATHEMATIK   82 ( 5 )   432 - 441   2004.5

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

    We deal with the exact solutions of the Navier-Stokes equations for stagnation flows with slips. The problem becomes the solvability of certain third-order ordinary differential equations (ODEs). Reducing the order of ODEs, we exhibit another elementary proof of the existence and asymptotic behavior of solutions. Numerical investigations are also provided.

    DOI: 10.1007/s00013-003-0111-y

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  • Stable finite difference scheme for a model equation of phase separation Reviewed

    T Hanada, N Ishimura, M Nakamura

    APPLIED MATHEMATICS AND COMPUTATION   151 ( 1 )   95 - 104   2004.3

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    A stable, conservative, and practical finite difference scheme is presented in order to solve numerically a model equation of phase separation in some binary alloys. The model system is introduced by Eguchi, Oki, and Matsumura in the attempt of theoretically describing such phenomena, and extends the well-known Cahn-Hilliard equation, even numerical solutions are hard to obtain for this dynamics. Our scheme utilizes a Lyapunov functional, that is, the free energy of the system, whose effectiveness is demonstrated by numerical implementations. (C) 2003 Elsevier Inc. All rights reserved.

    DOI: 10.1016/S0096-3003(03)00325-4

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  • On the Eguchi-Oki-Matsumura equation for phase separation in one space dimension Reviewed

    T Hanada, N Ishimura, M Nakamura

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   36 ( 2 )   463 - 478   2004

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    Eguchi-Oki-Matsumura equations are introduced to describe the dynamics of pattern formation that arises from phase separation in some binary alloys. The model extends the well-known Cahn-Hilliard equation and consists of coupled two functions; one is the local concentration and the other is the local degree of order. We show the existence of a solution, its asymptotic pro. le, and in part the structure of steady state solutions. Computational studies are also given.

    DOI: 10.1137/S0036141003400124

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  • One-phase Stefan problems for sublinear equations: Asymptotic behavior of solutions Reviewed

    Toyohiko Aiki, Hitoshi Imai, Naoyuki Ishimura, Yoshio Yamada

    Communications in Applied Analysis   8   1 - 15   2004

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  • On blowing-up solutions of the Blasius equation Reviewed

    N Ishimura, S Matsui

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   9 ( 4 )   985 - 992   2003.7

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

    The Blasius equation is a well-known third-order nonlinear ordinary equation, which arises in certain boundary layer problems in the fluid mechanics. In this note, we investigate the behavior of blowing-up solutions for related initial value problems.

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  • Well-posedness of one-phase Stefan problems for sublinear heat equations Reviewed

    T Aiki, H Imai, N Ishimura, Y Yamada

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   51 ( 4 )   587 - 606   2002.11

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  • Remarks on third-order ODEs relevant to the Kuramoto-Sivashinsky equation Reviewed

    N Ishimura

    JOURNAL OF DIFFERENTIAL EQUATIONS   178 ( 2 )   466 - 477   2002.1

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    We are interested in the third-order ordinary differential equations (ODES) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited. (C) 2002 Elsevier Science.

    DOI: 10.1006/jdeq.2001.4018

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  • Spiral solutions for a weakly anisotropic curvature flow equation Reviewed

    Yoshikazu Giga, Naoyuki Ishimura, Yoshihito Kohsaka

    Advances in Mathematical Sciences and Applications   12   393 - 408   2002

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  • On steady solutions of the Kuramoto-Sivashinsky equation Reviewed

    N Ishimura

    NAVIER-STOKES EQUATIONS: THEORY AND NUMERICAL METHODS   223   45 - 51   2002

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    This article surveys our recent studies on the Kuramoto-Sivashinsky equation. Main topics include: (i) a simple proof of the nonexistence of monotonic global solutions; (ii) the existence of solutions which blow up on bounded intervals.

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  • On one-phase Stefan problems for subkinear heat equations Reviewed

    Toyohiko Aiki, Hitoshi Imai, Naoyuki Ishimura, Yoshio Yamada

    Third Asian Mathematical Conference 2000   6 - 11   2002

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  • Numerical computation of Lyapunov exponents related to attractors in a free boundary problem Reviewed

    H Imai, T Takeuchi, SS Shanta, N Ishimura, T Aiki

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   47 ( 6 )   3823 - 3833   2001.8

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  • 自由境界問題の数値解法 Reviewed

    Naoyuki Ishimura, Hitoshi Imai, Toshiki Takeuchi

    一橋論叢   Vol.126 ( No.4 )   419 - 428   2001.1

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    本論文は,無限精度数値計算手法を実際問題である数理ファイナンスに現れるアメリカン・プット・オプションの価格が従う偏微分方程式に適用したものである.なお,この偏微分方程式は自由境界問となる.本論文では,Black-Scholes偏微分方程式の自由境界問題の枠組みに,無限精度数値計算手法を適用した結果,定式化において現れる満期状態での適合条件の不連続生を連続になるように補正すれば,うまく数値計算を実行することができることがわかった.

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  • Nonexistence of monotonic solutions of some third-order ode relevant to the Kuramoto-Sivashinsky equation Reviewed

    N Ishimura, M Nakamura

    TAIWANESE JOURNAL OF MATHEMATICS   4 ( 4 )   621 - 625   2000.12

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    We study the third-order ordinary differential equations (ODEs) which are relevant to the steady state of the Kuramoto-Sivashinsky equation, and/or to a model of dendritic growth of needle crystals. We show that there is no monotonic solution for certain range of parameter.

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  • Note on steady solutions of the Eguchi-Oki-Matsumura equation Reviewed

    T Hanada, N Ishimura, M Nakamura

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   76 ( 9 )   146 - 148   2000.11

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    Eguchi-Oki-Matsumura (EOM) equations are introduced to model theoretically the kinetics of ordering which accompanies phase separation in some binary alloys. Numerical analysis shows that EOM equations admit several steady states. Employing the variational structure associated with the? free energy, we prove that there really exist non-trivial steady solutions for EOM equations.

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  • Analysis on One-Phase Stefan Problems for Semilinear Parabolic Equations with the Dirichlet Boundary Condition Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Toyohiko Aiki

    Gakuto International Series, Mathematical Sciences and Applications   14   176 - 183   2000

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  • A note on the nonexistence of monotonic steady solutions of the Kuramoto-Sivashinsky equation Reviewed

    Naoyuki Ishimura, Masaaki Nakamura

    Annali dell'Universita di Ferrara   46 ( 1 )   175 - 180   2000

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    We present a simple proof of the nonexistence of monotonic solutions derived from the Kuramoto-Sivashinsky equation. Although our method restricts the range of parameters, it has the advantage of applicability to other equations, which includes a model of dendritic growth of needle crystals. © 2000 Università degli Studi di Ferrara.

    DOI: 10.1007/BF02837296

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  • A crystalline motion of spiral-shaped curves with symmetry Reviewed

    H Imai, N Ishimura, T Ushijima

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   240 ( 1 )   115 - 127   1999.12

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    We study the evolution of spiral-shaped polygonal curves by crystalline curvature. Crystalline curvature is known to extend the notion of ordinary curvature to the special class of nonsmooth curves. With the assumption of symmetry, we show that the motion of our spirals can be analyzed up to the rest state, possibly beyond singularities, (C) 1999 Academic Press.

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  • Remarks on the blow-up criterion for the 3-D Bousslnesq equations Reviewed

    N Ishimura, H Morimoto

    MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES   9 ( 9 )   1323 - 1332   1999.12

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    We consider the problem of blow-up of smooth, solutions for the 3-D Boussinesq equations. Owing to the viscosity, we prove that the maximum norm of the gradient of vorticity controls the breakdown of the solutions; the scalar temperature function is shown to be irrelevant to the breakdown.

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  • Motion of spirals by crystalline curvature Reviewed

    H Imai, N Ishimura, T Ushijima

    RAIRO-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE   33 ( 4 )   797 - 806   1999.7

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    Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

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  • A Direct Approach to an Inverse Problem Reviewed

    Hitoshi Imai, Toshiki Takeuchi, Masaaki Nakamura, Naoyuki Ishimura

    Gakuto International Series, Mathematical Sciences and Applications   12   223 - 232   1999

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  • Mathematics Education for Undergraduates who are Studying Social Sciences

    Ishimura Naoyuki

    The Hitotsubashi review   120 ( 3 )   396 - 402   1998.9

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  • Self-similar solutions for the Gauss curvature evolution of rotationally symmetric surfaces Reviewed

    N Ishimura

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   33 ( 1 )   97 - 104   1998.7

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  • Shape of spirals Reviewed

    N Ishimura

    TOHOKU MATHEMATICAL JOURNAL   50 ( 2 )   197 - 202   1998.6

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    The structure of spiral-like self-similar solutions for the plane curve shortening flow is investigated.

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  • On the interior derivative blow-up for the curvature evolution of capillary surfaces Reviewed

    K Asai, N Ishimura

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   126 ( 3 )   835 - 840   1998.3

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    We give examples of the interior derivative blow-up solutions for the curvature evolution of capillary surfaces over a bounded domain in R(N).

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  • On the structure of steady solutions for the kinematic model of spiral waves in excitable media Reviewed

    Ryo Ikota, Naoyuki Ishimura, Tomohiko Yamaguchi

    Japan Journal of Industrial and Applied Mathematics   15 ( 2 )   317 - 330   1998

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    Spiral waves are commonly observed in biological and chemical systems. Representing each wave front by a single curve, Brazhnik, Davydov, and Mikhailov introduce a kinematic model equation. The aim of this paper is to provide a detailed analysis for the steady state solutions of these equations. The existence of asymptotically Archimedean solutions is analytically shown.

    DOI: 10.1007/BF03167407

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  • Characterization on the long time behavior of the 2D Navier-Stokes equations Reviewed

    Naoyuki Ishimura, Masaaki Nakamura

    Pitman Research Notes in Mathematics   388   38 - 44   1998

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  • On the Finite Determination of the Solutions for the MHD Equations Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura

    Gakuto International Series, Mathematical Sciences and Applications   11   126 - 135   1998

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  • Convergence of Attractors for Simplified Magnetic Benard System Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura

    Gakuto International Series, Mathematical Sciences and Applications   11   136 - 144   1998

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  • Uniqueness for unbounded classical solutions of the MHD equations Reviewed

    N Ishimura, M Nakamura

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES   20 ( 7 )   617 - 623   1997.5

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:JOHN WILEY & SONS LTD  

    The uniqueness for unbounded classical solutions of the magnetohydrodynamic (MHD) equations in the whole space is investigated. Under suitable growth condition, it is shown that the solution to the initial value problem is unique. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

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  • Happy under Graduates

    Ishimura Naoyuki

    The Hitotsubashi review   117 ( 4 )   640 - 643   1997.4

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  • International Conference on Navier-Stokes Equations, Theory and Numerical Methods(Conference Reports)

    Ishimura Naoyuki

    Bulletin of the Japan Society for Industrial and Applied Mathematics   7 ( 4 )   322 - 322   1997

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    Language:Japanese   Publisher:The Japan Society for Industrial and Applied Mathematics  

    DOI: 10.11540/bjsiam.7.4_322

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  • Analytical approach to estimating the dimension of attractors Reviewed

    T Hakamada, Imai, I, N Ishimura

    APPLIED MATHEMATICS AND OPTIMIZATION   34 ( 1 )   29 - 36   1996.7

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    A mathematically rigorous procedure to estimate the Hausdorff dimension of the attractor is given. The method is based on invoking the Kaplan-Yorke-type estimate, through extending the argument of Constantin, Foias, and Temam.

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  • Convergence of attractors for the simplified magnetic Benard equation Reviewed

    Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura

    European Journal of Applied Mathematics   7 ( 1 )   53 - 62   1996.2

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  • CURVATURE EVOLUTION OF PLANE-CURVES WITH PRESCRIBED OPENING ANGLE Reviewed

    N ISHIMURA

    BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY   52 ( 2 )   287 - 296   1995.10

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    We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same asymptotics.

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  • 磁気ベナール問題のカオス Reviewed

    石村直之, 今井仁司, 中村正彰

    日本物理学会誌   50 ( 9 )   697 - 703   1995.9

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    DOI: 10.11316/butsuri1946.50.697

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  • INERTIAL MANIFOLDS FOR BURGERS ORIGINAL MODEL SYSTEM OF TURBULENCE Reviewed

    N ISHIMURA, OHNISHI, I

    APPLIED MATHEMATICS LETTERS   7 ( 3 )   33 - 37   1994.5

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    The existence of inertial manifolds for Burgers' original mathematical model system of turbulence is investigated. The system consists of two equations and enjoys the characteristic quantity: the Reynolds number. Our object in this article is to express the existence in terms of this Reynolds number. The difficulty of first order derivatives is circumvented by the method originally due to M. Kwak.

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  • On the simplified magnetic Benard problem -dimension estimate of the attractor Reviewed

    Naoyuki Ishimura, Masaaki Nakamura

    Advances in Mathematical Sciences and Applications   4   241 - 247   1994

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  • Dimension estimate of the global attractor for forced oscillation systems Reviewed

    Yuji Hattori, Naoyuki Ishimura, Isamu Ohnishi, Makoto Umeki

    Japan Journal of Industrial and Applied Mathematics   10 ( 3 )   351 - 366   1993.10

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    We deal with the largest compact invariant set X for forced oscillation systems. As in the Lorenz system X may contain a strange attractor. Invoking the Kaplan-Yorke formula we give an upper bound for the dimension of X analytically and compare it with numerical results. It is, observed that both agree rather well when the damping coefficient is small. © 1993 JJIAM Publishing Committee.

    DOI: 10.1007/BF03167279

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  • LIMIT SHAPE OF THE CROSS-SECTION OF SHRINKING DOUGHNUTS Reviewed

    N ISHIMURA

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   45 ( 3 )   569 - 582   1993.7

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  • Existence of symmetric capillary surfaces via curvature evolution Reviewed

    Naoyuki Ishimura

    Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics   40 ( 2 )   419 - 427   1993

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:東京大学理学部  

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  • On the mean curvature flow of thin doughnuts Reviewed

    Kazushi Ahara, Naoyuki Ishimura

    Lecture Notes in Numerical and Applied Analysis   12   1 - 33   1993

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  • Remarks on the asymptotic behavior for elliptic equations with critical growth Reviewed

    Naoyuki Ishimura

    Differential and Integral Equations   6 ( 6 )   1253 - 1264   1993

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  • Linear discrete model for shortening polygons Reviewed

    Kazushi Ahara, Naoyuki Ishimura, Kazumasa Ikeda

    Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics   39 ( 2 )   365 - 377   1992

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  • Generalized ground states for quasilinear elliptic equations Reviewed

    Naoyuki Ishimura

    Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics   38 ( 1 )   137 - 147   1991

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  • Nonlinear eigenvalue problem associated with the generalized capillarity equation Reviewed

    Naoyuki Ishimura

    Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics   37 ( 2 )   457 - 466   1990

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Books

  • 数理物理学の方法(下)

    R.クーラント, D.ヒルベルト(著, 藤田宏, 石村直之( Role: Joint translator共訳のため担当部分抽出不可能)

    丸善出版  2019.9  ( ISBN:9784621304020

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    Total pages:385   Language:Japanese   Book type:Scholarly book

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  • 東京大学工学教程 基礎系数学 非線形数学

    吉田善章, 永長直人, 石村直之, 西成活裕( Role: Joint author第4章「分岐・アトラクタ―・カオス」担当)

    丸善出版  2016.1 

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    Total pages:212   Responsible for pages:113-158   Language:Japanese   Book type:Scholarly book

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  • 確率微分方程式入門―数理ファイナンスへの応用

    ( Role: Sole author)

    共立出版  2014.6 

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  • 数理物理学の方法(上)

    R.クーラント, D.ヒルベルト, 藤田宏, 高見穎郎, 石村直之( Role: Joint translator共訳のため担当部分抽出不可能)

    丸善出版  2013.1  ( ISBN:9784621065259

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  • Primary大学ノート よくわかる基礎数学

    藤田岳彦( Role: Joint author担当:5講,6講,コラム 積分法,数列,どちらが先?ニュートンとライプニッツ)

    実教出版  2012.12 

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  • 統合リスクマネジメント

    ニール.A.ドハーティ, 森平爽一郎, 米山高生( Role: Joint translator担当:16章 事例研究:大災害リスクの証券化)

    中央経済社  2012.1 

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  • Primary大学ノート よくわかる線形代数

    藤田岳彦( Role: Joint author担当:第1章 ベクトル)

    実教出版  2011.12 

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  • Primary大学ノート よくわかる微分積分

    藤田岳彦( Role: Joint author担当:全体の調整)

    実教出版  2011.12 

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  • Primary大学ノート 基礎数学

    藤田岳彦, 石村直之, 藤岡敦( Role: Joint author担当:2,4章 関数,確率統計)

    実教出版  2007.9 

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  • Primary大学ノート 微分積分

    藤田岳彦, 石村直之, 藤岡敦( Role: Joint author担当:1,2,3章 数列と極限,関数,微分法)

    実教出版  2007.1 

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  • Primary大学ノート 線形代数

    藤田岳彦, 石村直之, 藤岡敦( Role: Joint author担当:1,2章 行列,連立一次方程式)

    実教出版  2007.1 

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  • 基礎コース 経済数学

    武隈愼一, 石村直之( Role: Joint author担当:1,2,4,6,10章 数学の基礎概念,線形代数,微分と積分,最適化問題,動学最適化問題)

    新世社  2003.12 

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  • ワークブック 微分積分

    藤田岳彦, 石村直之( Role: Joint author担当:1~11章 関数と極限値~積分とその応用その3)

    講談社  2003.9 

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  • 偏微分方程式入門―数理ファイナンスとともに

    ( Role: Sole author)

    神戸大学数学教室  2003.3 

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  • パワーアップ 微分方程式

    ( Role: Sole author)

    共立出版  2001.11 

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MISC

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Presentations

  • Insurance design for the loss of epidemic outbreaks involving Cramer -Lundberg model

    Naoyuki Ishimura

    International Conference on Computational Finance (ICCF24, CWI Amsterdam)  2024.4 

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    Event date: 2024.4    

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  • Efficient numerical computation of the ruin probability

    Naoyuki Ishimura

    5th European Actuarial Journal Conference (Tartu, Estonia)  2022.8 

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    Event date: 2022.8    

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  • Mathematical models for estimating the risk of epidemic outbreaks

    2018 8th International Conference on Applied Physics and Mathematics (ICAPM 2018) (Invited talk)  2018.1 

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  • 社会現象を記述する微分方程式

    2017年度東京大学大学院数理科学研究科公開講座「現象を記述する微分方程式」(招待講演)  2017.11 

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    Language:Japanese   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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  • Discrete Stochastic Calculus and its Applications

    2017 7th International Conference on Applied Physics and Mathematics (ICAPM 2016)  2017.1 

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    Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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  • Recent Development of the Theory of Copulas and its Applications

    2016 6th International Conference on Applied Physics and Mathematics (ICAPM 2016) (Invited talk)  2016.1 

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Works

Research Projects

  • Study on nonlinear problems arising in mathematical finance

    Grant number:20K03737  2020.4 - 2024.3

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

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    Grant amount: \4030000 ( Direct Cost: \3100000 、 Indirect Cost: \930000 )

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  • Study on nonlinear partial differential equations arising in the mathematical finance

    Grant number:15K04992  2015.4 - 2019.3

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

    ISHIMURA NAOYUKI

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    Grant amount: \4550000 ( Direct Cost: \3500000 、 Indirect Cost: \1050000 )

    Through the study of nonlinear partial differential equations arising in the mathematical finance, we establish several results on nonlinear analysis. To be precise, we propose a risk estimation model for an insurer due to the epidemic outbreaks and estimate its risk. Next, we analyze the optimal portfolio problem in discrete processes with two independent random variables; we derive the discrete analogue of the famous Ito formula and prove the convergence from discrete to continuous setting. Finally, we apply the evolution of copulas to the analysis of foreign exchange rate and show the validity of the concept of evolution of copulas.

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  • Financial Engineering to ERM: Theoretical and Empirical Investigation

    Grant number:24243031  2012.10 - 2016.3

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

    SHIBA Tsunemasa, TANAKA Katsuto, ISHIMURA Naoyuki, HONDA Toshio, KUROZUMI Eiji, YAMAMOTO Yohei, ISHIHARA Tsunehiro, YAMADA Masahiro, YAMADA Toshiro, SHIMOTSU Katsumi, TAKAHASHI Hajime, WATANABE Toshiaki, KARIYA Takeaki, FUJITA Takahiko

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    Grant amount: \43810000 ( Direct Cost: \33700000 、 Indirect Cost: \10110000 )

    Our research in theoretical and empirical aspects of enterprise risk management (ERM), was conducted in several different groups. The groups kept in close contact with each other, however. We formed the following groups: macroeconomics and financial time series group, financial engineering and copula group, financial high frequency data group, and nonparametric/semiparametric statistics group. Among the above listed areas of research, copula seems to stand out as the most promising important tool for ERM.
    In each of the four years, we invited top-rated researchers from around the world to hold an international conference, open to the public. These conferences contributed to our understanding of the above areas, and served as great opportunities to present our ongoing research.

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  • Study on the nonlinear partial differential equations arising in applied field

    Grant number:21540117  2009 - 2012

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

    ISHIMURA Naoyuki

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

    The study is performed on nonlinear partial differential equations (PDEs) arising in applied field with the purpose of deeply understanding the phenomena represented by the PDEs. We consider the Hamilton-Jacobi-Bellman equation in the optimal investment problem. We derive the slightly general nonlinear PDEs for the risk preference and analyze the equation. Moreover, we study the copula function, which describes the nonlinear relation among random variables. We introduce the time evolution of copulas.Both results are worth advancing the understandings of economics phenomena.

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  • Mathematical analysis of the nonlinear systems arising in the industry.

    Grant number:21540147  2009 - 2011

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

    NAKAMURA Masaaki, ISHIMURA Naoyuki, IMAI Hitoshi, MIZUTANI Akira, CHEN Yungang

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    Grant amount: \4290000 ( Direct Cost: \3300000 、 Indirect Cost: \990000 )

    We have obtained the following results.1. We showed the effectiveness of the multiple precision arithmetic to approximate the solutions of PDE‘s which have transition phase and the blow up in finite time.2. We derived the default risk model equation and analyze theoretically and numerically.3. We showed the existence and the finite dimensional property of the attractors in magnetic Benard problem4. We applied the multi-precision computation to the equation with delay term,which is hard to compute, and showed that in the case the solution is analytic, it is effective.

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  • Study on Navier-Stokes equations and the related nonlinear differential equations

    Grant number:18540222  2006 - 2007

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

    MASUDA Kyuya, ISHIMURA Naoyuki

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    Grant amount: \1150000 ( Direct Cost: \1000000 、 Indirect Cost: \150000 )

    In the period 2006-2007, Masuda studied the dynamical behavior in time of solution of a model describing the phase transition in binary in alloys. This model is described by a system of partial differential equations of the forth order, proposed by Eguchi-Oki-Matsumura(1984).
    Masuda succeeded in showing the existence of maximal attractor and inertial set for the so-called Eguch-Oki-Matsumra Equation and reported the results in the International Congress held at Athens.
    The Hoggard-Whalley-Willmott equation is introduced to mode portforios of European type options incorporating transaction costs. The model gives rise to a nonlinear parabolic partial differential equation, whose nonlinearity reflects the presence of transaction costs. Ishimura showed analytically the existence of solutions which are deviced to effectively handle an infinite domain and unbounded solution.
    Also Ishimura deal with numerical computation of solution. Numerical computation shows the validity of the scheme proposed by Ishimura.
    Ishimura is concerned with the solvability of certain partial differential equations, which is derived from the optimal Investment problem under the random risk process. The equations describe the evolution of the Arrow-Pratt coefficient of absolute risk aversion woth respect to the optimal value function.
    Employing the fixed point approach combined with the convergence argument Ishimura shows the existence of solution.

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  • Study on the nonlinear phenomena with moving boundaries

    Grant number:16540184  2004 - 2006

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

    ISHIMURA Naoyuki

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    Grant amount: \2700000 ( Direct Cost: \2700000 )

    This research project is mainly concerned with the next two subjects, among the popular research topics of nonlinear phenomena involving the moving boundaries. 1 : Study on the phase separation. 2 : Study on the exercise boundary in mathematical finances.
    1.The author has been tackling the analysis of a model equation proposed by Professor T.Eguchi, which describes the phase separation in terms of two unknown variables. One is the local concentration and the other is the local degree of order. The model consists of coupled two evolution equations ; one is 4^<th> order and hence the analysis is tough. We concentrate on the one-dimensional case and we obtain fairy a lot of results. The first one is one the existence of solutions and the structure of steady state solutions. Analysis of a singular perturbation problem and a bifurcation problem then follows. We publish these papers in various journal, which are rather well estimated by anonymous referees.
    2.The author gives a set of exact solutions for the model equation proposed by K.Takaoka, which extends the famous Black-Scholes analysis. Then the author tries to solve the partial differential equations with the effect of transaction costs. The paper is submitted now.
    The research of the phase separation still continues. The direction with respect to the dynamical system seems promising and the work is now in progress.

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  • Research on the Navier-Stokes equation and the related topics on the nonlinear differential equations

    Grant number:15540215  2003 - 2005

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

    MASUDA Kyuya, MORIMOTO Hiroko, HIROSE Munemitsu, ISHIMURA Naoyuki

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    Grant amount: \1900000 ( Direct Cost: \1900000 )

    Masuda considered the discrete Lax pair for discrete Toda equation. Flashka developed the inverse scattering method for the solution of the Toda equation. On the other hand, Hirota proposed time-discrete Toda equation. Hirota-Tsujimoto-Imai constructed the disrete Lax pair. But this Lax pair does not have symmery property. Masuda constructed symmetric discrete Lax pair in the frame work of funcitonal analysis.
    Masuda considered the degenerate nonlinear parabolic equation of the porous medium type and showed the comparison porperty for solutions of the degenerate nonlinear parabolic equations.
    Morimoto considered a steady Navier-Stokes equations on a 2-D bounded domain, symmetric with respect to the y-axis. The boundary has several connected components, intersecting the y-axis. The boundary value is symmetric with respect to the y-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier-Stokes equations was established by Amick and Fujita. Fujita proved a key lemma concerning the solenoidal extension of the boundary value by virtual method. Morimoto gave a proof by a method different from Fujita.
    Ishimura considered Eguchi-Oki-Matsumura equations which describes the dynamics of pattern formation that arises from phase separation in some binary alloys. The model extends the well-known Cahn-Hilliard equation and consists of coupled two functions ; one is the local concentration and the other is the local degree of order. Ishimura showed the existence of a solutions, its symptotic profile, and in part the structure of steady state solutions.
    Ishimura deals with the exact solutions for stagnation flows with slip/ The problem becomes the solvability of certain third-order differential equations(ODEs). Reducing the order of ODEs, Ishimura exhibit another elementary proof of the existence and asymptotic behavior of solutions.

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  • Research of the Navier-Stokes exterior problem by using dual semigmups and the Lorentz spaces

    Grant number:13640157  2001 - 2004

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

    YAMAZAKI Masao, SHIBATA Yoshihiro, TANAKA Kazunaga, FUJITA Takahiko, ISHIMURA Naoyuki

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    Grant amount: \3800000 ( Direct Cost: \3800000 )

    In a joint work with Yoshihiro Shibata, we obtained a sufficient condition on time-independent external forces for the unique existence of a stationary solution in a certain class of the Navier-Stokes equation in exterior domains of dimension n greater than or equal to 3 by using the duality between the Lorentz spaces and real interpolation. Our class is a natural generalization of the so-called physically reasonable solutions, and our suffirient condition gives a unified view for the case with zero velocity at infinity and the case with non-zero velocity at infinity.
    Next, in a joint work with Yuko Enomoto and Yoshihiro Shibata, we verified the stability in the weak-Ln space of the stationary solution above for time-evolution under small initial perturbation in the weak-Ln space, and showed that the smallness above can be taken uniformly in the velocity at infinity of the stationary solution.
    Furthermore, by using real interpolation for sublinear operators, we generalized these results for time-dependent external forces, and obtained a sufficient condition for the unique existence of the corresponding time-periodic or almost periodic solutions. We also showed the stability of these solutions in weak-Ln spaces under perturbations on the external forces and initial data uniform in the velocity at infinity of the solutions.
    On the other hand, as a preparation for generalized the results above for general unbounded domains, we generalized the Lp-theory on the boundary value problem for the Stokes equation in a layer domain, in a joint work with Takayuki Abe for higher-order Sobolev spaces and Besov spaces, and obtained a sufficient condition on the external forces for the unique existence of the solution of the boundary value problem. In particular, we showed that the uniqueness of the solution fails in the case p=infinity, and that the Poiseuille flow can be characterized as the solution with zero as the external forces and boundary values.

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  • 応用数学の日本・イタリア研究集会のための企画調査研究

    Grant number:15634006  2003    

    日本学術振興会  科学研究費助成事業  基盤研究(C)  日本大学

    中村 正彰, 大森 克史, 今井 仁司, 石村 直之, 岡田 正巳

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    Grant amount: \2500000 ( Direct Cost: \2500000 )

    本年度の研究計画に基づき研究を実施し,次のような成果を得た。
    ・イタリア応用数理学会会長のフィレンツェ大学M.Primicerio教授と研究連絡を実施し、次のような方向で開催をする事に合意した。
    1.2004年5月の3日間、日本で開催する。候補地として葉山をあげた。
    2.参加者は1講演50分、講演者は両国15-20人とする。
    3.テーマは金融工学、ナノテク・ナノサイ、非線形現象、PDEとその近似を基本項目とする。
    ・上のテーマのための下記の分野の調査と研究を実施し成果を挙げた。
    Navier-Stokes方程式を基礎とする摩擦境界条件などの非線形問題の数学的および数値的解析に成功した。
    二種の金属合金の相分離を記述する江口・沖松村方程式の1次元定常解の構造を数値的かつ理諭的に解析した。
    多倍長計算手法と選点法を用いた離散化手法を組み合わせた高精度計算手法の開発と応用に成功した。
    適切でない可能性のある逆問題へのスペクトル選点法を用いた多倍長高精度計算法の応用。
    数理金融工学における種々のオプションに現れる自由境界問題の数値的理論的な解析。
    有限要素法,差分法、モルタル法、領域分割法を用いた手法による各種流体問題の解析に成果をあげた。
    燃焼合成反応におけるヘリカル波の出現は平面進行波からの安定分岐であることの解析。
    2流体問題に対する質量保存的有限要素数値シミュレーションスキーム、と界面の収束性の誤差評価。
    二重指数関数型変換とsinc近似を用いた不定積分の数値積分法の効率の高さの解析。
    競合拡散系、Stefan様問題、非線形拡散系など数理生物学の諸問題の応用解析。
    仮想領域法と差分法を用いた、オイル汚染など環境Eco systemに現れる流体問題の数値解析。
    仮想領域法と混合有限要素法を用いたradiation problemの数値解析

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  • Study on the global behavior of solutions for the fluid equation

    Grant number:13640206  2001 - 2002

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

    ISHIMURA Naoyuki, TAKAOKA Koichiro, YAMAZAKI Masao, MORIMOTO Hiroko

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    Grant amount: \1000000 ( Direct Cost: \1000000 )

    We have undertaken our research projects mainly on the following two subjects.
    (1) Results are obtained for the analysis on the structure of solutions to the steady state of the Kuramoto-Sivashinsky (KS) equation and/or to the Blasius equation. Both equations are related to the fluid dynamics and have the similar third-order differential operator. By use of the monotonicity, the reduction of the third-order equation into the second-order one is performed. In view of this reduction, the existence of blowing-up solutions for the steady state of the KS equation is proved. These kind of solutions have not been mentioned in the literature so far. Moreover, an elementary existence proof of blowing-up solutions for the Blasius equation is also given, which may shed light on the validity of the Blasius equation itself with regard to the Prandtl boundary layer theory.
    (2) Free boundary problems arise in a wide variety of nonlinear sciences, including one-phase fluid flow problem. Here we are concerned with the pricing of American put option. We present an exact integral formula for the solution.

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  • Study on Navier-Stokes equations and related nonlinear partial differential equations

    Grant number:12640223  2000 - 2002

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

    MASUDA Kyuya, KATSURADA Masashi, KONNO Reiji, MORIMOTO Hiroko, ISHIMURA Naoyuki, TANI Atsushi

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    Grant amount: \3300000 ( Direct Cost: \3300000 )

    Masuda succeeded in classifying completely the minimal surfaces of constant Gaussian curvature in two-dimensional complex form
    Morimoto showed the existence of solutions of two dimensional stationary Navier Stokes equations with general outlet boundary condition for some symmetric domain of channel type. Konno showed the non-existence of positive eigen-value for Schroedinger operator in the square integrable function space on unbounded domain. Ishimura and Nakamura found some important properties for the Kuramoto-Shivashinsky equations. Tani showed the existence of solutions of Navier-Stokes equations under the general slip boundary conditions.

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  • Research of the Navier-Stokes equations by using the theory of Fourier analysis and semigroup theory

    Grant number:11640156  1999 - 2000

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

    YAMAZAKI Masao, YAMADA Hiromichi, MACHIDA Hajime, IWASAKI Shiro, ISHIMURA Naoyuki, FUJITA Takahiko

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    Grant amount: \2300000 ( Direct Cost: \2300000 )

    We studied the Navier-Stokes equation on the whole space or on exterior domains. We have already studied the unique existence and the stability under initieal perturbations of stationary solutions with external forces independent of time. In this research we considered the case where external force depends on the time-variable, and studied the unique existence and the stability of solutions. This research generalizes similar researches on time-periodic solutions and solutions almost periodic in time.
    On the whole space we employ the Morrey spaces as the space of solutions, and succeeded in generalizing the results for usual L^p-spaces obtained by Professors Hideo Kozono, Mitsuhiro Nakao and Yasushi Taniuchi. On exterior domains we employ the weak-L^p spaces as the space of solutions, and we succeeded in relaxing the assumptions on the external forces very much, and firstly obtained conditions sufficient for the unique existence of solutions in 3-dimensional exterior domains.
    The results for the Morrey spaces can be obtained in a manner similar to that employed in our previous study. On the other hand, in the proof of the results for weak-L^p spaces, it is essential to show that the integral of functions with values in a Banach space converges, where the integral is considered to diverge in general. In order to show this fact, we first consider the family of the Lorentz spaces which generalizes the weak-L^pA spaces, and we improved estimates of L^p-L^q type, which is often employed in previous studies, by using real interpolation, and the we used the duality property between the Lorentz spaces.

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  • 相転移と自由境界問題

    Grant number:11894003  1999    

    日本学術振興会  科学研究費助成事業  基盤研究(C)  北海道大学

    儀我 美一, 石井 仁司, 西田 孝明, 西浦 康政, 石村 直之, 友枝 謙二

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    Grant amount: \2600000 ( Direct Cost: \2600000 )

    物質は温度、圧力などの環境が異なると液体や固体などの異なる相を呈する。ひとつの相から他の相へかわる過程は、相転移とよばれ、これを解き明かすことは、現代科学の主テーマのひとつである。物質の2相が共存し、混じり合っていないとき、その相境界の動きを調べることは、相転移のしくみを知る上で重要である。現象を微分方程式を用いてマクロに定式化できても、相境界の形状は未知で方程式を解くことによって決定しなければならない。境界の形状があらかじめ決まっていないという意味では自由境界問題と呼ばれている。
    本研究では、この分野の今後の重点的研究目標を設定するために、この分野で既に大きな寄与のある数学的理論「粘性解」や、また重要な課題である「界面ダイナミクスと特異性形成」とその関連の会議を4回ほど開催した。これらの会議には、数学分野研究者以外の研究者や、外国人研究者にも参加してもらい、各テーマについて深く討論した。1999年11月に千葉で開催された国際研究集会「自由境界問題(FBP'99)」では「界面ダイナミクス」についての分科会を分担者の人が企画し、さまざまな問題を提起した。外国人研究者より代表者の著書「非線形偏微分方程式」の内容についての質問が多かった。そのため英訳の作成を開始した。また解説を中心とした論文を数編とりまとめ学術交換の便をはかった。
    このような試みを通じて、重要なテーマがうかびあがってきた。代表的なものをふたつあげる。偏微分方程式の初期値境界値問題は数多く研究されているが、その境界条件に未知関数の時間微分がふくまれている場合は、重要であるにもかかわらず研究がすすんでいない。特に方程式の拡散効果が退化している場合の研究はない。また研究代表者の研究分野のひとつである特異性の強い拡散方程式は画像処理の問題との関連もあり、さらに深く調べる必要性を認識した。これらの活動をもとに来年度の特定領域研究(代表者 三村 昌春(広島大))の分担者として自由境界問題の会議を開催することとなった。

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  • Studies on the structure of the lattice of clones consisting of multiple-valued logical functions

    Grant number:10640109  1998 - 1999

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

    MACHIDA Hajime, FUJITA Takahiko, YAMADA Hiromochi, IWASAKI Shiro, ISHIMURA Naoyuki, YAMAZAKI Masao

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    Grant amount: \2200000 ( Direct Cost: \2200000 )

    A clone is a set of k-valued logical functions which is closed under composition and contains all the projections. The set of all clones consisting of k-valued logical functions is denoted by LィイD2kィエD2. Whereas the structure of LィイD22ィエD2 is completely determined, our knowledge about the structure of LィイD2kィエD2 for k > 2 at present is very little. The main objective of this research is to clarify the structure of LィイD2kィエD2 and we have obtained the following results.
    1. The structure of LィイD23ィエD2 as a metric space
    We have introduced a metric into the lattice LィイD2kィエD2 of clones and showed that LィイD2kィエD2 is a compact metric space. Moreover, we constructed continuous mappings, based on the meet operation, from LィイD23ィエD2 onto LィイD22ィエD2 and studied the images of maximal clones in LィイD23ィエD2 and those of some clones being accumulation points under such mappings.
    2. Minimal clones in LィイD2kィエD2 and related topics
    Since the classification of minimal clones is far from complete, the study of various properties of minimal clones are very important. We have studied a particular problem concerning minimal clones: Given a pair (CィイD21ィエD2, CィイD22ィエD2) of minimal clones, we call it gigantic pair if the union CィイD21ィエD2∪CィイD22ィエD2 generates the whole set of functions. We proved a characterization theorem of gigantic pairs and showed that gigantic pairs exist for most k's.
    3. Study of hyperclones
    Recently, I. G. Rosenberg initiated the study of hyperclones. We continued his work and proved the following: The cardinality of the lattice of all hyperclones on the set {0,1} is of continuum. This is interesting as the cardinality of the lattice of all (ordinary) clones on {0, 1} is countable.
    4. Study of partial clones consisting of partial functions
    We investigated the following problems on partial clones : (1) The minimal number of maximal partial clones whose meet is the trivial partial clone. (2) The minimal number of minimal partial clones whose join is the clone of all partial operations. This is a joint work with Professors L. Haddad and I.G. Rosenberg.

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  • Study on Navier-Stokes equations and related nonlinear partial differential equations

    Grant number:10440055  1998 - 1999

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

    MASUDA Kyuya, HASEGAWA Fumio, KONNO Reiji, MORIMOTO Hiroko, KATSURADA Masashi, KANEKO Satiomi

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    Grant amount: \8800000 ( Direct Cost: \8800000 )

    (1) Masuda's Results : He could show the existence of solutions of Navier-Stokes equations in continuous function space.
    (2) The result of Masuda and Kenmotsu : They could classify completely the minimal surfaces of constant Gaussian curvature in two-dimensional complex form.
    (3) Morimoto's results : she could show the existence of solutions of two dimensional stationary Navier-Stokes equations with general outlet boundary with general outlet boundary condition forsome symmetric domain. Also, she could show the existence of solutions for stationary Boussinesq equations with general outlet boundary condition.
    (4) Tani's results : he could show the classical solvability of the Stefan problem in a viscous incompressible fluid flow. Also, he could solve a free boundary problem for an incompressible ideal fluid in two space dimensions.
    (5) The result of Ishimura and Nakamura : They could show the convergence of attracters for simplified magenetic Benard system.

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  • Mathematical Studies of melting, solidification and growth phenomena in material science

    Grant number:09354001  1997 - 1999

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

    MIURA Masayasu, SAKAMOTO Kunimochi, FUNAKI Naohisa, OHTA Takao, KIMURA Masato, OKUZONO Tohru

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    Grant amount: \15900000 ( Direct Cost: \15900000 )

    Towards mathematical understanding of melting, solidification and growth processes in material sciences, we have carried theoretical research from experiments, modeling and complementarily computer analysis. Since the head investigator had moved from the university of Tokyo to Hiroshima University at the second year of the research term, some investigators had to he altered from the original members but there was no trouble to carry out the research plan. Mimura has mainly studied pattern formation arising in nonlinear non equilibrium systems. In particular, has investigated dendritic patterns in biological and chemical systems, in order to reveal the universality in mechanism of such patterns in natural sciences. Ohta has studied dynamics of micro phase separation, as the basic theory in polymer science. Ishikawa has experimentally studied dendritic growth in material process and in particular, and has studied the effect of micro gravity on growth of colloid crystal. Oharu has developed nonlinear semi group theory to extend basic theory of free boundary problems. Funaki has studied interfacial dynamics from probabilistic approach. Matano, Yanagida and Kimura have investigated analytically and numerically mean curvature equations which describe interfaces. Yamada and Kusano have developed numerical methods to solve reaction-diffusion systems on a sphere. Tsujikawa and Mimura has studied growth process in biological systems by using singular limit methods. Sakamoto has developed singular perturbation methods and established the internal layer theory in higher dimensions. Onda has made fractalization of the surface of shapes and has obtained super water -repellent surfaces from theoretic and experimenrtal standing points. Ishimura has analyzed spiral patterns which arises in growth process in materials by using curvature flow theory. Okuzono has analyzed dynamics of two phase flow in droplet dissipative systems. Ueyama has given theoretical understanding of self-replication process in reaction-diffusion systems by using computer aided analysis. The above results have been reported in several conferences inside and outside of Japan. Most of them were talked at the Applied Mathematics Meeting in Japan which is held every year and were published in their proceedings.

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  • 応用領域にあらわれる非線型偏微分方程式の研究

    Grant number:09740139  1997 - 1998

    日本学術振興会  科学研究費助成事業  奨励研究(A)  一橋大学

    石村 直之

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    Grant amount: \500000 ( Direct Cost: \500000 )

    1. 界面方程式の研究
    曲率流、あるいは一般に、曲率に依存して発展する界面の運動は、物質科学を含む応用領域科学において近年その重要性がますます認識されてきている。今年度はこれら界面の曲率に依存する発展が、特異なpatternに結び付く場合を、具体的にはらせん模様を現出する場合を考察した。自然界に見られるらせん模様は、その形成の仕方から多くはArchimedesらせんでよく近似される。単純な曲率流の自己相似解としてはArchimedesらせんは得られないことをまず示した。Russiaの物理学者達が導出した運動学的界面方程式の場合には、漸近的にArchimedesらせんを導く解が存在することを示した。
    2. 流体方程式の研究
    多くの流体方程式は、現在でも応用科学における基礎方程式である。しかし数学としてそれらは難しい対象である。3次元Navier-Stokes方程式の時間大域解の存在は今でも未解決で、そのため大域解が存在するための十分条件が広く議論されている。温度が有界ならば問題は肯定的であるというのはよく知られている。今年度は3次元Boussinesg方程式に関して大域解が存在するための十分条件を考察した。温度と流体との相互作用を記述するのがBoussinesg方程式であるが、そこでも温度が問題の支配的な量であることが示された。

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  • Research on the Navier-Stokes equations by interpolation spaces and perturbation theory.

    Grant number:09640164  1997 - 1998

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

    YAMAZAKI Masao, ISHIMURA Naoyuki, FUJITA Takehiko, YAMADA Hiromichi, MACHIDA Hajime, IWASAKI Shiro

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    Grant amount: \1900000 ( Direct Cost: \1900000 )

    We studied the existence, the uniqueness and the stability of stationary solutions of the Navier-Stokes equations in exterior domains, which is regarded as a model describing the motion of fluids, by functional analytic methods.
    For n-dimensional spaces, the function space L^n is mainly employed in previous works on this direction. We showed during the period from April, 1997 to March, 1998 that, when the space dimension is 3, solutions in the function space L^3 exist only in very limited cases, and hence we must consider a some-what larger space L^<3, *> as the function space in which solutions exist. Moreover, for the Navier-Stokes equation in the whole space, we gave a condition on the external forces sufficient for the unique existence of small stationary solutions belonging to the Morrey spaces, which is strictly larger than the space L^<n, *>. We furthermore showed that the stationay solutions above are stable under small initial perturbation in function spaces which contain distributions other than Radon measures.
    We studied the exterior problem of the Navier-Stokes equations in the space for n<greater than or equal>3 during the period from April, 1998 to March, 1999. We then showed the unique existence of small stationary solutions in the function space under the condition that the external forces are given as the first order derivatives of potentials small in the function space L^<n/2, *>. We also showed that the stationary solutions above are stable under small initial perturbations in the function space L^<n, *>. This result is applicable to external forces more general than those which can be treated by previous results obtained by potential theoretical methods, and is applicable to the 3-dimensional case which could not be treated by functional analytic methods before.

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  • Vertex operator algebras with group action

    Grant number:09640019  1997 - 1998

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

    YAMADA Hiromichi, ISHIMURA Naoyuki, YAMAZAKI Masao, FUJITA Takahiko, MACHIDA Hajime, IWASAKI Shiro

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

    In 1997 a subalgebra having order three symmetry of a vertex operator algebra associated with a rank two root lattice of type A was discovered and its properties were studied. In particular its automorphism group was determined and its simple modules were classified. The results were written in a joint paper Ternary codes and vertex operator algebras. Relations between this vertex operator algebra and the Monster module still remain to be studied. Moreover, Borwein type identity was studied from a point of view of vertex operator algebra. The results were written in a joint paper Borwein identity and vertex opertor algebras.
    In 1998 all highest weight vectors with weight at most two of a vertex operator algebra associated with a root lattice of type A were classified. Highest weight vectors are important, because once they are known then a vertex operator algebra can be decomposed into a direct sum of simple modules for a tensor product of Virasoro vertex operator algebras. Although not all highest weight vectors are known, much information about the structure of the vertex operator algebra can be obtained from highest weight vectors with weight at most two. For example, a vertex operator algebra having a symmetry of order 4 was constructed by using these highest weight vectors. Recently all highest weight vectors were classified in the case of rank three root lattice of type A.In order to generalize this result to a root lattice of arbitrary rank, a new idea would be required.

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  • 物理学および工学と関連する非線型偏微分方程式の研究

    Grant number:08740133  1996    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  一橋大学

    石村 直之

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    Grant amount: \1000000 ( Direct Cost: \1000000 )

    1.曲率流方程式の研究
    曲率流方程式を引き続き論じた.パターン形成の観点から,スパイラル形の解が存在するのかどうかを調べて,ほぼ決定的な結果を得た.拡張型の自己相似解のクラスにおいてのみ,これらスパイラル形の解が可能である.しかも,それら解の回転数は有限であり,したがってアルキメデスら線ではあり得ない.このことは,BZ反応などの実際現象にあらわれるスパイラル形パターンの解析に極めて有効な知見を与える.それら現象の理解のため,曲率を含んだ複雑な非線型方程式が提出されている.現在,これら方程式の解析を強力に進めている.
    2.時間無限大での解の挙動の研究
    この課題においても,今年度は重要な進歩があった.MHD方程式は,ナヴィエ・ストークス方程式を含んだ複雑な方程式であるが,荷電粒子の運動をよく記述し,応用性の極めて高いものである.このMHD方程式の解は,2次元の場合,その時間無限大での挙動が,有限モードの有限個の点における情報のみで完全に決定される,というのが主結果である.アトラクタの次元有限性は古くから知られていたが,その特徴付けを、ほぼ完全に与えたことになる.現在,他の方程式への適用を考察中である.

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  • Hardy空間と補間理論を用いる偏微分方程式の研究

    Grant number:08640181  1996    

    日本学術振興会  科学研究費助成事業  基盤研究(C)  一橋大学

    山崎 昌男, 石村 直之, 藤田 岳彦, 山田 裕理, 山崎 秀記, 岩崎 史郎

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

    1.空間1次元における伝播速度無限大の2×2双曲型保存系の連続な大域解の一意性
    空間1次元における2×2双曲型保存系は,一般には連続な解を持たず,いわゆる弱解の範囲にしか解を持たないことが知られている.この保存系が連続な大域解をただ一つ持つためには,伝播速度が有限の場合には,初期値がある種の単調性をみたすことが必要十分であることが知られているが,伝播速度が無限大の場合の条件は知られていなかった.この論文では伝播速度が無限大の場合には,上の単調性のみでは一般には解は一意的には定まらないことを示し,さらに解が一意的に定まるための,初期速度の増大度に関する必要十分条件を与えた.
    2.全空間におけるNavier-Stokes方程式及び半線型熱方程式の初期値問題
    全空間におけるNavier-Stokes方程式及び半線型熱方程式の初期値問題を,Radon速度にならないdistributionを含むような広い函数空間で考え,時間局所的な一意解,および時間大域的な一意解を持つための十分条件を考え,結果としてRadon速度にならないdistributionを初期値とする上の方程式の解が存在することを示した.また,適当なMorrey空間に属するNavier-Stokes方程式の定常解が十分に小さいとき,その定常解はもとのMorrey空間で安定であることを示した.応用として,Navier-Stokes方程式及び半線型熱方程式の,新たな自己相似解のクラスを構成した.

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  • 物理学工学と関連する非線型偏微分方程式の研究

    Grant number:07740140  1995    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  東京大学

    石村 直之

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    Grant amount: \1100000 ( Direct Cost: \1100000 )

    今年度は主に,磁気流体力学の解析において結果を得た。
    まず,磁気ベナ-ル問題と呼ばれる磁気熱対流現象について,その簡略化された方程式を,この数年研究してきた。長時間後の振舞いを特徴付ける次元の評価,および,磁場を変化させたときの収束の問題,等のまとめを解説論文として日本物理学会誌に発表した。
    次に、磁気流体力学方程式の全空間における古典解の一意性を考察した。ある解の集合を設定し,その集合においては解は一意であることを示した。そこで圧力が重要な役割を持つことに注意を向けた。実際の数値計算等において,圧力項は見過せないものであるが,数学理論においては,長らく陰に隠れていたといってよい。我々の研究は,圧力項が数学としても実は重要であることを示唆している。
    曲率流の研究においては,一次元拡大自己相似解の構造を完全決定した。

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  • 幾何学と関連する非線型偏微分方程式の研究

    Grant number:06740098  1994    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  東京大学

    石村 直之

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    Grant amount: \900000 ( Direct Cost: \900000 )

    1.平均曲率流方程式の研究
    1次元全領域において、開き角を指定したときの曲率流方程式を論じた。既にK.EckerおよびG.Huiskenによる一般論において、解の存在また適当な条件のもとでの自己相似解への収束が調べられている。我々の問題に対しても、解の存在、自己相似解への収束に関しては、この一般論の適用が改善をもって可能である。我々の研究の新しさは、自己相似解の構造を決定したことにある。自己相似解とは、簡単にいえば、時間によって形を変えない解のことで、特異点発生の機構に関して重要な役割をはたす。近年の曲率流方程式の研究においても、しばしば課題になってきた。我々の問題においては、自己相似解は一意に決まり、豊かな構造を持っていることが示された。
    2.消散型方程式の長時間後のふるまいの研究
    引き続き、簡略化された磁気ベナ-ル問題を論じた。磁場を0に近付けたとき、そのアトラクタがローレンツアトラクタにハウスドルフ上半収束することを示した。数値計算の比較と鋭い対照をみせた。

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  • 非線形現象に関連した非線形偏微分方程式における解の特異性の生成とその性質

    Grant number:06640205  1994    

    日本学術振興会  科学研究費助成事業  一般研究(C)  東京大学

    堤 誉志雄, 小松 彦三郎, 石村 直之, 山田 道夫, 片岡 清臣, 俣野 博

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    非線形偏微分方程式において、解の特異性が非線形性によってどのような相互作用をするのかということを調べるのは、極めて重要な問題の一つである。それは、解の時間大域的存在や有限時刻での爆発の問題と密接に関連している。空間1次元の特殊な3次の非線形性を持つ非線形シュレデインガー方程式は完全可積分系となり、その解は時刻無限大でも非線形効果が消えず、解は摂動を受けていない自由解には近付かないことが知られている。空間1次元の場合、3次の非線形性は線形散乱理論で云うところの長距離ポテンシャルに相当しており、この事実自体は自然なことである。しかし最近、非線形波動方程式について、従来長距離ポテンシャルに相当すると考えられていた場合でも、ある特別な非線形項に対しては解の特異性が相殺し、時刻無限大で解は自由解に近付くことが分かってきた。非線形シュレデインガー方程式に対しても、特別な非線形項の場合は波動方程式の時と同様、解の特異性が相殺し時刻無限大で非線形効果が消えることが予想される。そこで、今年度は、どのような3次の非線形項に対して、解の特異性が相殺し時刻無限大で解が自由解に近付くかを調べた。
    また、最近は、単に理論的に解析するだけではなく、数値計算によって偏微分方程式を調べるということも、応用上重要な問題となっている。今回非圧縮性ナヴィエ・ストークス方程式の分岐問題について装置実験を行い、2次分岐の発生やカタストロフ理論におけるカスプ点に相当するものが存在することを捕らえた。このような数値解析が、ナヴィエ・ストークス方程式の解の正則性や特異性の解析に役立つことが期待される。

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  • 幾何学と関連した非線型偏微分方程式の研究

    Grant number:05740082  1993 - 1994

    日本学術振興会  科学研究費助成事業  奨励研究(A)  東京大学

    石村 直之

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    Grant amount: \900000 ( Direct Cost: \900000 )

    1.平均曲率流方程式の研究。
    平均曲率流の方法を毛細管方程式に適用して、解の存在定理の簡単な別証明を得た。平均曲率流方程式は、幾何学の動機づけにより研究されることが多かったが、それが、毛細管方程式に対する汎関数の勾配流として自然に拡張できることより今回の研究となった。今のところ軸対称な領域の場合のみに結果を得ているが、一般の領域に対しても拡張できるかどうかは、さらに考察中である。方法はa priori estimateを上手く行うことにあり、特に計算機の助けは必要としなかった。
    2.慣性多様体の存在証明
    Burgersのもとの乱流の簡略化方程式に対する慣性多様体の存在証明を得た。慣性多様体とは、散逸糸方程式系の長時間後の解のふるまいを統一的にとらえようとする努力の中から生まれた概念である。多様体の存在が証明されれば、解の長時間後のふるまいは、多様体上の常微分方程式系により完全に記述されることになる。今回の方程式における困難は、それが低階微分項を含むということにあった。存在証明は、固有値の間隔の広さを必要とするため、低階微分項は悪影響を与えるのである。新たに項を導入するという技巧を用いてこの困難を回避した。糸はReynolds numberの概念がある。多様体の次元と、このReynolds numberとの関係を調べたが、まだ改善すべき点は多い。

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  • GEOMETRIC STUDY OF VARIATIONS ON INFINITE DIMENSIONAL MANIFOLDS

    Grant number:05452006  1993 - 1994

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for General Scientific Research (B)  THE UNIVERSITY OF TOKYO

    OCHIAI Takushiro, ISHIMURA Naoyuki, OTSU Yukio, TSUBOI Takashi, MATANO Hiroshi, MATSUMOTO Yukio

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    Grant amount: \6400000 ( Direct Cost: \6400000 )

    We have studied some geometry on infinite dimensional manifolds. Especially we have studied manifolds given by some functional spaces on finite dimensional manifolds, and as results we have obtained important contribution to study topological properties of the original manifolds.
    More precisely, we have studied various functionals derived from finite dimensional manifolds and their variations. The main idea behind this setting is to generalize the Morse theory well studied so far. We have reduced the study of Morse theory to that of dynamical systems through the gradient vector fields. Thorough this view point, we have been able to obtain some contribution to the study of infinite dimensional dynamical systems. As results of our methods, we have been able to clarify some of topological and geometrical properties of infinite and finite dimensional manifolds desclibed as follows.
    At first we studied some topology of the space of Riemannian metrics on finite dimensional manifolds. We have done this study from three view points explained as follows. The first view point is to study the variations of the several (integral geometric) invariants derived from metrics. The critical values of these invariants are given as solutions of various partial differetial equations. So we have been able to adopt the method of various problems and partial differetial equations. The second view point is to consider the limits of various geometric inequaliteis derived from the curvatures of Riemannian metrics. Although many of the limits of these inequalities cannot be characteraized from partial differential equations, we have found that instead usual Riemannian geometric methods are usufull. The third view point is to observe that the spaces concered are the classifying spaces of (infinite dimentional) Lie groups of the diffeomorphisms of manifolds. Therefore we have found that the usual methods in the study of the classifying spaces of foliated product are very useful.
    Moreover we have studied the topology of the space of the moduli of the given geometric structures on finite dimensional manifolds, and that of the family of smooth mappings between two finite dimentional manifolds. This problem contains the study of submanifolds. For these problems, we have taken the method of variations of various functionals.

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  • 変分問題に関連した非線型偏微分方程式の研究

    Grant number:03740075  1991    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  東京大学

    石村 直之

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    Grant amount: \900000 ( Direct Cost: \900000 )

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  • Study of Nonlinear Partial Differential Equations from the View Point of Functional Analysis

    Grant number:01460004  1989 - 1990

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for General Scientific Research (B)  University of Tokyo

    MASUDA Kyuya, SHOJI Mayumi, ISHIMURA Naoyuki, FUKAYA Kenji, KATAOKA Kiyomi, MATANO Hiroshi

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    Grant amount: \6200000 ( Direct Cost: \6200000 )

    We have studied nonlinear partial differential equations from many aspects in the period 1989 May-1991 March. Masuda succeeded first in showing the existence of stable periodic solutions of reaction-diffusion systems of some type, the most important class of nonlinear parabolic equations. He also considered the heat equation in the form u=*u+a (x) u^p and showed any non-negative solution blows up in a finite time if a (x) is posive at some point in the domain considered. Matano considered the nonlinear heat equation in some one-dimensional space interval with Dirichlet boundary condition.
    He could show the remarkable result that any solutions with a continuous function as the initial data blows up at a (at most) finite number of points.Ishimura showed the existence and nonexistence of multi-valued solutions of nonlinear elliptic equations of the form *u=u^p. shouji made research into the permanent progressive waves and analyzed the bifurcation of progressive waves. Fukaya made research into nonlinear differential equations in differential geometry. He clarified the structure of small ball in the Riemannian manifold with sectional curvature less than one in absolute value.
    Kataoka succeeded in showing the hypoellipticity of degenerate elliptic equations.

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