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Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

Research Project

Project/Area Number 16340043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

OTANI Mitsuharu  Waseda University, Faculty of Science and Engineering, Professor (30119656)

Co-Investigator(Kenkyū-buntansha) YAMADA Yoshio  Waseda University, Faculty of Science and Engineering, Professor (20111825)
TANAKA Kazunaga  Waseda University, Faculty of Science and Engineering, Professor (20188288)
ISHII Hitoshi  Waseda University, Faculty of Education and Integrated Arts and Sciences, Professor (70102887)
KENMOCHI Nobuyuki  Chiba University, Faculty of Education, Professor (00033887)
OZAWA Tohru  Hokkaido University, Graduate School of Science, Professor (70204196)
西原 健二  早稲田大学, 政治経済学術院, 教授 (60141876)
林 仲夫  大阪大学, 大学院理学研究科, 教授 (30173016)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥17,490,000 (Direct Cost: ¥16,200,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2007: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2006: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2005: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2004: ¥4,300,000 (Direct Cost: ¥4,300,000)
KeywordsNonlinear Evolution Equation / Nonlinear Elliptic Equation / Nonlinear PDE / Method of Variation / Subdifferential Operator / 関数方程式 / 関数解析 / 非線形現象
Research Abstract

(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever
(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very … More important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long. Less

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (66 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (52 results) (of which Peer Reviewed: 13 results) Presentation (8 results) Book (6 results)

  • [Journal Article] Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space2008

    • Author(s)
      H.Ishii
    • Journal Title

      Ann.Inst.H.Poincare Anal.Non Lineaire 25

      Pages: 231-266

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space2008

    • Author(s)
      H. Ishii
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Linearite Vol.25

      Pages: 231-266

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Positive Solutions for Lotka-Volterra Systems with Cross-Diffusion2008

    • Author(s)
      Y.Yamada
    • Journal Title

      Handbook of Differential Equations:Stationary Partial Differential Equations 6

      Pages: 411-501

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space2008

    • Author(s)
      H.Ishii
    • Journal Title

      Ann.Inst.H.Poincare Anal.Non Lineaire 25

      Pages: 231-266

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence of nontrivial solutions for elliptic equations with supercritical nonlinearity in exterior domains2007

    • Author(s)
      S.Hashimoto and M.Otani
    • Journal Title

      Discrete and Continuous Dynamical Systems 19

      Pages: 323-333

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] L∞-Energy Method, Basic tools and usage2007

    • Author(s)
      M.Otani
    • Journal Title

      Differential Equatiorns, Chaos and Variational Problems, Ed.by Vasile Staicu, Birkhauser 75

      Pages: 357-376

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Infinite dimensional attractors for evolution equations with p-Laplacian and their Kolmogorov entropy2007

    • Author(s)
      M.Efendiev and M.Otani
    • Journal Title

      Differential and Integral Equations 25

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exponential attractors for a quasilinear parabolic equation2007

    • Author(s)
      K.Matsuura and M.Otani
    • Journal Title

      Dynamical Systems, Differential Equations and Applications Supplement2007

      Pages: 713-720

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians2007

    • Author(s)
      H.Ishii, and H.Mitake
    • Journal Title

      Indiana Univ.Math.J. 56

      Pages: 2159-2183

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Monotonicity and compactness methods for nonlinear variational inequalities,2007

    • Author(s)
      N.Kenmochi
    • Journal Title

      HANDBOOK OF DIFFERENTIAL EQUATIONS edited by M.Chipot 4

      Pages: 205-298

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Phase separation model of Penrose-Fife type with Signorini boundary condition2007

    • Author(s)
      A.Ito, N.Kenmochi and M.Niezg\'odka
    • Journal Title

      Adv.Math.Sci.Appl. 17

      Pages: 337-356

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A generalization of the weighted Strichartz estimates for the wave equations and an application to self-similar solutions2007

    • Author(s)
      J.Kato, M.Nakamura and T.Ozawa
    • Journal Title

      Commun.Pure Appl.Math. 60

      Pages: 164-186

    • NAID

      120000970574

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On small amplitude solutions to the generalized Boussinesq equations2007

    • Author(s)
      Y.Cho, T.Ozawa
    • Journal Title

      Discrete and Continuous Dynamical Systems 17

      Pages: 691-711

    • NAID

      120006459470

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity2007

    • Author(s)
      Y.Naito, T.Suzuki
    • Journal Title

      J.Differential Equations 232

      Pages: 176-211

    • Related Report
      2006 Annual Research Report
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation2006

    • Author(s)
      P.Felmer, S.Martinez and K.Tanaka
    • Journal Title

      Arch.Rat.Mech.Anal. 182

      Pages: 333-366

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation2006

    • Author(s)
      P. Felmer, S. Martinez, K. Tanaka
    • Journal Title

      Arch. Rat. Mech. Anal. Vol.182

      Pages: 333-366

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] equation, Mathematical Approach to Nonlinear Phenomena : Modelling, Analysis and Simulations,2006

    • Author(s)
      M.Urano, K.Nakashima, Y.Yamada
    • Journal Title

      Internat. Ser. Math. Sci. Appl. 23

      Pages: 267-279

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      J. Math. Anal. Appl. 313

      Pages: 598-610

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global asymptotics for the damped wave equation with absorption in higer dimensional space2006

    • Author(s)
      K.Nishihara
    • Journal Title

      J.Math.Soc.Japan 58

      Pages: 805-836

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Domain and range of the modified wave operator for Schrodinger equations with a critical nonlinearity2006

    • Author(s)
      N.Hayashi, P.I.Naumkin
    • Journal Title

      Comm.Math.Phys. 267

      Pages: 477-492

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Damped wave equation with a critical nonlinearity2006

    • Author(s)
      N.Hayashi, E.I.Kaikina, P.I.Naumkin
    • Journal Title

      Trans.Amer.Math.Soc. 358

      Pages: 1165-1185

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global attractors for nonlinear wave equations with some nonlinera dissipations2006

    • Author(s)
      Mitsuhiro Nakao
    • Journal Title

      J.Differential Equations 227

      Pages: 204-229

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Existence of positive solutions of higher-order quasilinear ordinary differential equations2006

    • Author(s)
      Manabu Naito
    • Journal Title

      Annali di Matematica Pura ed Applicata 186

      Pages: 59-84

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      Kenji NISHIHARA, H.Zhao
    • Journal Title

      J.Math.Anal.Appl. 313

      Pages: 598-610

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Existence and asymptotic behavior of positive solutions of higher-order quasilinear ordinary differential equations2006

    • Author(s)
      Manabu NAITO
    • Journal Title

      Mathematische Nachrichten 279

      Pages: 198-216

    • Related Report
      2005 Annual Research Report
  • [Journal Article] L∞-energy method and its applications to some nonlinear parabolic systems2005

    • Author(s)
      Mitsuharu OTANI
    • Journal Title

      Gakuto International Series, Mathematical Sciences and Applications 22

      Pages: 233-244

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces2005

    • Author(s)
      Goro AKAGI, Mitsuharu OTANI
    • Journal Title

      Journal of Differential Equations 209

      Pages: 392-415

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Transition layers and spikes for a bistable reaction-diffusion equation2005

    • Author(s)
      Michio URANO, Kimie NAKASHIMA, Yoshio YAMADA
    • Journal Title

      Advances in Mathematical Sciences and Applications 15

      Pages: 683-707

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Coexistence states for a prey-predator model with cross-diffusion2005

    • Author(s)
      Kousuke KUTO, Yoshio YAMADA
    • Journal Title

      Dynamical Systems and Differential Equations (Edited by S.Hu, X.Lu and W.Xie)

      Pages: 536-545

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Equations with initial data decaying slowly near infinity2005

    • Author(s)
      Ryo IKEHATA, Kenji NISHIHARA
    • Journal Title

      Math.Sci.Appl. 22

      Pages: 265-275

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Asymptotics for sub critical non convective type equations2005

    • Author(s)
      Nakao HAYASHI, E.I.KAIKINA, P.I.NAUMKIN
    • Journal Title

      Mathematical Methods Applied Science 28

      Pages: 275-308

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Optimal consumption and portfolio choice with stopping2005

    • Author(s)
      Shigeaki KOIKE, H.MORIMOTO
    • Journal Title

      Funkcialaj Ekvacioj 48

      Pages: 183-202

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Long time behavior of solutions for a nonlinear evolution equation2005

    • Author(s)
      Kei MATSUURA
    • Journal Title

      Nonlinear Analysis 63

      Pages: 1199-1206

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The principle of symmetric criticality for non-differentiable mappings2004

    • Author(s)
      J.Kobayashi and M.Otani
    • Journal Title

      Journal of Functional Analysis 124

      Pages: 428-449

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Evolution inclusions governed by subdifferentials in reflexive Banach spaces2004

    • Author(s)
      Goro AKAGI, Mitsuharu OTANI
    • Journal Title

      Journal of Evolution Equation 4

      Pages: 519-541

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Time-dependent constraint problems arising from macroscopic critical-state models for type-II superconductivity and their approximations2004

    • Author(s)
      Goro AKAGI, Mitsuharu OTANI
    • Journal Title

      Advances in Mathematical Sciences and Applications 14

      Pages: 683-712

    • Related Report
      2005 Annual Research Report
  • [Journal Article] L∞-energy method and its applications2004

    • Author(s)
      Mitsuharu OTANI
    • Journal Title

      Nonlinear partial differential equations and their applications, GAKUTO Internat.Ser.Math.Sci.Appl. 20

      Pages: 505-516

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The principle of symmetric criticality for non-differentiable mappings2004

    • Author(s)
      Jun KOBAYASHI, Mitsuharu OTANI
    • Journal Title

      Journal of Functional Analysis 214

      Pages: 428-449

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Multiple existence and stability of steady states for a prey-predator system with cross-diffusion2004

    • Author(s)
      Kousuke KUTO, Yoshio YAMADA
    • Journal Title

      Nonlocal Elliptic and Parabolic Problems, Banach Center Publications 66

      Pages: 199-210

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain2004

    • Author(s)
      Hitoshi ISHII, H.SATO
    • Journal Title

      Nonlinear Anal. 57

      Pages: 1077-1098

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Motion of a graph by R-curvature2004

    • Author(s)
      Hitoshi ISHII, T.MIKAMI
    • Journal Title

      Arch.Ration.Mech.Anal. 171

      Pages: 1-23

    • NAID

      120000955918

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations2004

    • Author(s)
      Kenji NISHIHARA, T.YANG, H.ZHAO
    • Journal Title

      SIAM J.Math.Anal. 35

      Pages: 1561-1597

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Global stability of strong rarefaction waves of the Jin-Xin relaxation model for the p-system2004

    • Author(s)
      Kenji NISHIHARA, H.ZHAO, Y.ZHAO
    • Journal Title

      Comm.Partial Differential Equations 29

      Pages: 1607-1634

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Damped wave equation in the subcritical case2004

    • Author(s)
      NakaoHAYASHI, E.I.KAIKINA, P.I.NAUMKIN
    • Journal Title

      J.Differential Equations 207

      Pages: 161-194

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Damped wave equation with super critical nonlinearities2004

    • Author(s)
      Nakao HAYASHI, E.I.KAIKINA, P.I.NAUMKIN
    • Journal Title

      Differential Integral Equations 17

      Pages: 637-652

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Asymptotic behaviorof solutions for parabolic equations associated with p-Laplacian as p tends to infinity2004

    • Author(s)
      Goro AKAGI
    • Journal Title

      Nonlinear Partial Differential Equations and Their Applications, GAKUTO International Series 20

      Pages: 418-429

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The principle of symmetric criticality for non-differentiable mappings

    • Author(s)
      J. Kobayashi, M. Otani
    • Journal Title

      Journal of Functional Analysis vol.214

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Existence of nontrivial solutions or some elliptic equations with supercritical nonlinearity in exterior domains

    • Author(s)
      Satoshi Hashimoto, Mitsuharu Otani
    • Journal Title

      Discrete and Continuous Dynamical System (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation

    • Author(s)
      P.Felmer, S.Martinez, K.Tanaka
    • Journal Title

      Arch. Rat. Mech. Anal (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians

    • Author(s)
      Hitoshi Ishii
    • Journal Title

      Comm. PDE (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean nspace August

    • Author(s)
      Hitoshi Ishii
    • Journal Title

      Annales de 1'IHP-Analyse (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Stationary isothermic surfaces for unbounded domains

    • Author(s)
      R.Magnanini, S.Sakaguchi
    • Journal Title

      Indiana University Mathematics Journal (to appear)

    • Related Report
      2006 Annual Research Report
  • [Presentation] L∞-energy method-its basic tools and applications-2008

    • Author(s)
      Mitsuharu Otani
    • Organizer
      Mathematical Society of Japan Invited Special Lecture
    • Place of Presentation
      Tohoku University
    • Year and Date
      2008-09-23
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 境界条件に非線形性を持つ楕円型の解の存在について2008

    • Author(s)
      原田 潤一、大谷 光春
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-03-21
    • Related Report
      2007 Annual Research Report
  • [Presentation] A Positive solution of a scalar field equation:-Δu=g(x,u) in R-N2007

    • Author(s)
      K.Tanaka
    • Organizer
      First Chile-Japan Workshop on Nonlinear Elliptic and Parabolic PDE
    • Place of Presentation
      Univ.de Chile
    • Year and Date
      2007-10-26
    • Related Report
      2007 Annual Research Report
  • [Presentation] L∞-エネルギー法について-その基礎と応用-2007

    • Author(s)
      大谷 光春
    • Organizer
      日本数学会 企画特別講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] L∞-エネルギー法について-その基礎と応用-2007

    • Author(s)
      大谷 光春
    • Organizer
      日本数学会企画特別講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] 擬共形不変な非線型シュレディンガー力程式の波動作用素2007

    • Author(s)
      小澤 徹
    • Organizer
      応用数理学会年会総合講演
    • Place of Presentation
      北海道大学
    • Year and Date
      2007-09-17
    • Related Report
      2007 Annual Research Report
  • [Presentation] Asymptotic solutions of Hamilton-Jacobi equations for large time and relatedtopics2007

    • Author(s)
      H.Ishii
    • Organizer
      6th InternationalCongress on Industrial and Applied Mathematics
    • Place of Presentation
      チューリッヒ,スイス
    • Year and Date
      2007-07-19
    • Related Report
      2007 Annual Research Report
  • [Presentation] Population models of interacting species with cross and self diffusion2007

    • Author(s)
      Y.Yamada
    • Organizer
      Ecosummit 2007
    • Place of Presentation
      Beijing,China
    • Year and Date
      2007-05-23
    • Related Report
      2007 Annual Research Report
  • [Book] Variational Problems and Applications2007

    • Author(s)
      Chjan C.Lim, M.Otani and J.Shi
    • Total Pages
      467
    • Publisher
      Discrete and Continuous Dynamical Systems-Series A
    • Related Report
      2007 Annual Research Report
  • [Book] 理工系のための「微分積分I」2007

    • Author(s)
      鈴木 武, 山田 義雄, 柴田 良弘, 田中 和永
    • Total Pages
      245
    • Publisher
      内田老鶴圃
    • Related Report
      2007 Annual Research Report
  • [Book] 理工系のための「微分積分II」2007

    • Author(s)
      鈴木 武, 山田 義雄, 柴田 良弘, 田中 和永
    • Total Pages
      504
    • Publisher
      1
    • Related Report
      2007 Annual Research Report
  • [Book] Mathematical Approach to Nonlinear Phenomena:Modelling, Analysis and Simulations2005

    • Author(s)
      T.Aiki, N.kenmochi, M.Niezgodka and M.Otani
    • Total Pages
      327
    • Publisher
      Gakkotosho Co.,Ltd.
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Mathematical Approach to Nonlinear Phenomena : Modelling, Analysis and Simulations2005

    • Author(s)
      T. Aiki, N. kenmochi, M. Niezgodka, M. Otani
    • Publisher
      Gakkotosho Co., Ltd.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Nonlinear Partial Differential Equations and Their Applications, International Series vol.202004

    • Author(s)
      N.KENMOCHI, Mitsuharu OTANI, S.Zheng
    • Total Pages
      534
    • Publisher
      GAKKOTOSHO
    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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