Updated on 2024/09/20
博士(数理科学) ( 東京大学 )
修士(理学) ( 東京大学 )
The University of Tokyo Graduate School, Division of Science master course completed
The University of Tokyo Graduate School, Division of Science master course withdrawn before completion
The University of Tokyo Faculty of Science graduated
徳島市立高等学校 理数科 graduated
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
日本ファイナンス学会
Australian Mathematical Society
"Mathematical finance, Nonlinear Analysis"
Natural Science / Mathematical analysis / 数学解析
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 2450015(9 pages) 2024
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
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
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
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
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
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
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
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
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
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
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
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
On the convergence of discrete processes with multiple independent variables Reviewed
Naoyuki Ishimura, Naohiro Yoshida
ANZIAM JOURNAL 58 ( 3-4 ) 379 - 385 2017.4
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
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
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
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
Evolution of copulas and its applications Reviewed
Naoyuki Ishimura
Academie van Belgique,|rn|http://www.afmathconf.ugent.be/FormerEditions/Proceedings2014.pdf 85 - 89 2014
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
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
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
Discrete stochastic calculus and its applications: an expository note Reviewed
Takahiko Fujita, Naoyuki Ishimura, Norihisa Kawai
Advances in Mathematical Economics 16 119 - 131 2012
Evolution of multivariate copulas in discrete processes Reviewed
Naoyuki Ishimura, Yasukazu Yoshizawa
INTERNATIONAL CONFERENCE ON APPLIED ECONOMICS (ICOAE) 1 186 - 192 2012
Evolution of bivariate copulas in discrete processes Reviewed
Yasukazu Yoshizawa, Naoyuki Ishimura
JSIAM Letters 3 77 - 80 2011
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
石村 直之
数学 63 ( 1 ) 129 - 132 2011
On time-dependent bivariate copulas Reviewed
Naoyuki Ishimura, Yasukazu Yoshizawa
Theoretical and Applied Mechanics Japan 59 303 - 307 2011
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
Traveling wave solutions to the nonlinear evolution equation for the risk preference Reviewed
Naoyuki Ishimura, Sakkakom Maneenop
JSIAM Letters 3 25 - 28 2011
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
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
Global in space numerical computation for the nonlinear Black-Scholes equation Reviewed
Naoyuki Ishimura, Hitoshi Imai
Nonlinear Models in Mathematical Finance 219 - 242 2009
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
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
A NOTE ON THE OPTIMAL PORTFOLIO PROBLEM IN DISCRETE PROCESSES Reviewed
Naoyuki Ishimura, Yuji Mita
KYBERNETIKA 45 ( 4 ) 681 - 688 2009
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
On blowing-up solutions of the Blasius equation Reviewed
N Ishimura, S Matsui
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 9 ( 4 ) 985 - 992 2003.7
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
Remarks on third-order ODEs relevant to the Kuramoto-Sivashinsky equation Reviewed
N Ishimura
JOURNAL OF DIFFERENTIAL EQUATIONS 178 ( 2 ) 466 - 477 2002.1
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
On steady solutions of the Kuramoto-Sivashinsky equation Reviewed
N Ishimura
NAVIER-STOKES EQUATIONS: THEORY AND NUMERICAL METHODS 223 45 - 51 2002
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
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
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
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
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
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
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
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
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
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
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
Mathematics Education for Undergraduates who are Studying Social Sciences
Ishimura Naoyuki
The Hitotsubashi review 120 ( 3 ) 396 - 402 1998.9
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
Shape of spirals Reviewed
N Ishimura
TOHOKU MATHEMATICAL JOURNAL 50 ( 2 ) 197 - 202 1998.6
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
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
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
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
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
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
Ishimura Naoyuki
The Hitotsubashi review 117 ( 4 ) 640 - 643 1997.4
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
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
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
CURVATURE EVOLUTION OF PLANE-CURVES WITH PRESCRIBED OPENING ANGLE Reviewed
N ISHIMURA
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 52 ( 2 ) 287 - 296 1995.10
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
INERTIAL MANIFOLDS FOR BURGERS ORIGINAL MODEL SYSTEM OF TURBULENCE Reviewed
N ISHIMURA, OHNISHI, I
APPLIED MATHEMATICS LETTERS 7 ( 3 ) 33 - 37 1994.5
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
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
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
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
On the mean curvature flow of thin doughnuts Reviewed
Kazushi Ahara, Naoyuki Ishimura
Lecture Notes in Numerical and Applied Analysis 12 1 - 33 1993
Remarks on the asymptotic behavior for elliptic equations with critical growth Reviewed
Naoyuki Ishimura
Differential and Integral Equations 6 ( 6 ) 1253 - 1264 1993
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
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
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
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 ）
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
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
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 ）
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
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
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
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
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
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
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
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
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
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
パワーアップ 微分方程式
（ Role： Sole author）
共立出版 2001.11
On an extension of Value at Risk and its applications
Chuo Business Review ( 42 ) 3 - 17 2023.2
Blow-up Behavior of a System of Nonlinear Ordinary Differential Equations
( 130 ) 1 - 8 2013.2
現象から方程式を創り出す--立式で鍛える論理的思考法(16)ブラック・ショールズ方程式
石村 直之
数理科学 49 ( 5 ) 71 - 76 2011.5
Book Review: Regularity Theory for Mean Curvature Flow
K.Ecker
2011.1
デリバティブ再入門（5）
2009.10
デリバティブ再入門（4）
2009.9
デリバティブ再入門（3）
2009.8
デリバティブ再入門（2）
2009.7
デリバティブ再入門（1）
2009.6
Discrete Ito Formula and its Applications (Numerical Analysis : Theory, Methods and Applications)
Fujita Takehiko, Ishimura Naoyuki
RIMS Kokyuroku 1638 130 - 136 2009.4
Ishimura Naoyuki
Bulletin of the Japan Society for Industrial and Applied Mathematics 17 ( 4 ) 336 - 341 2007.12
Ishimura Naoyuki
Bulletin of the Japan Society for Industrial and Applied Mathematics 17 ( 3 ) 240 - 245 2007.9
Ishimura Naoyuki
Bulletin of the Japan Society for Industrial and Applied Mathematics 17 ( 2 ) 165 - 170 2007.6
Ishimura Naoyuki
Bulletin of the Japan Society for Industrial and Applied Mathematics 17 ( 1 ) 14 - 19 2007.3
ISHIMURA Naoyuki, MIZUMACHI Tetsu
RIMS Kokyuroku 1464 58 - 65 2006.1
経済学に現れる非線形問題 (フォーラム:現代数学のひろがり 非線形問題への招待)
石村 直之
数学のたのしみ : have fun with mathematics 2006 92 - 106 2006
Ishimura Naoyuki, Nakamura Masaaki, Hanada Takao
RIMS Kokyuroku 1416 85 - 93 2005.2
Exact American option pricing and the free boundary problem
Takeuchi Toshiki, Ishimura Naoyuki, Imai Hitoshi
NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan 52 ( 0 ) 529 - 529 2003
Eguchi-Oki-Matsumura Equation for Phase Separation: Numerically Guided Approach
Takao Hanada, Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura
Proceedings of International Conference on Inverse Problems -Recent Developments in Theories & Numerics 433 - 442 2003
On a method in analysis of the American put option
石村 直之, 竹内 敏己, 今井 仁司
Proceedings of the Annual Conference of the Japan Society for Industrial and Applied Mathematics 2002 ( 0 ) 31 - 31 2002
A New Numerical Technique for the Option Pricing Problem of American Type
Toshiki Takeuchi, Naoyuki Ishimura, Hitoshi Imai
World Multiconference on Systemics, Cybernetics and Informatics Proceedings 2 233 - 236 2001
Aiki Toyohiko, Imai Hitoshi, Ishimura Naoyuki, Yamada Yoshio
RIMS Kokyuroku 1178 36 - 47 2000.12
Numerical Computation of Lyapunov Exponents Related to Attractors in a Free Boundary Problem
Hitoshi Imai, Toshiki Takeuchi, Shewli Shamim Shanta, Naoyuki Ishimura
NIFS-PROC 46 21 - 29 2000
Numerical Computation of Attractors in Two-Phase Stefan Problems
Toshiki Takeuchi, Hitoshi Imai, Shewli Shamim Shanta, Naoyuki Ishimura
RIMS Kokyuroku 1145 220 - 228 2000
Convergence of Fractal Dimensions of Attractors for the Simplified Magnetic Benard System
Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura
Proceedings of Third China-Japan Seminar on Numerical Mathematics 73 - 81 1998
曲率流方程式 (《特集》非線形現象の解析--非線形偏微分方程式の追究)
石村 直之
数理科学 35 ( 11 ) 30 - 35 1997.11
Convergence of Attractors for Simplified Magnetic Benard System
Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura
RIMS Kokyuroku 989 56 - 64 1997
NAKAMURA Masaaki, IMAI Hitoshi, ISHIMURA Naoyuki
Proceedings of the conference on computational engineering and science 1 ( 1 ) 159 - 162 1996.5
Numerical computation of attractors for the simplified magnetic Benard system using spectral collocation methods
IMAI Hitoshi, NAKAMURA Masaaki, ISHIMURA Naoyuki, SHINOHARA Yoshitane
IEICE technical report. Nonlinear problems 96 ( 72 ) 5 - 12 1996.5
ISHIMURA NAOYUKI
RIMS Kokyuroku 951 117 - 119 1996.5
Minimal Surfaces II, Boundary Regularity
U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab
1995.1
Magnetic Benard System and its Simplified Model
Hitoshi Imai, Naoyuki Ishimura, Masaaki Nakamura
Lecture Notes in Numerical and Applied Analysis（Vol.14），Proceedings of the Second Japan-China Seminar on Numerical Mathematics 14 79 - 92 1995
Remarks on the Asymptotic Behaviour for Elliptic Equations with Critical Growth
ISHIMURA NAOYUKI
RIMS Kokyuroku 770 21 - 25 1991.11
Insurance design for the loss of epidemic outbreaks involving Cramer -Lundberg model
Naoyuki Ishimura
International Conference on Computational Finance (ICCF24, CWI Amsterdam) 2024.4
Efficient numerical computation of the ruin probability
Naoyuki Ishimura
5th European Actuarial Journal Conference (Tartu, Estonia) 2022.8
Mathematical models for estimating the risk of epidemic outbreaks
2018 8th International Conference on Applied Physics and Mathematics (ICAPM 2018) (Invited talk) 2018.1
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
Discrete Stochastic Calculus and its Applications
2017 7th International Conference on Applied Physics and Mathematics (ICAPM 2016) 2017.1
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
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
Grant amount： \4030000 （ Direct Cost: \3100000 、 Indirect Cost： \930000 ）
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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
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
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.
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
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.
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
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.
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
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。
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
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.
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
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.
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
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.
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
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。
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
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.
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
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.
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
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.
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
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.
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
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。
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
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。
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
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。
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
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。
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
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.
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
Grant amount： \900000 （ Direct Cost: \900000 ）
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
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.
2024 Freshman Seminar Ⅰ Department
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2024 Basic Seminar (Economics) Graduate school
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