発表論文

(i) 反応拡散方程式とそのシステム

  1. Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada,
    A free boundary oroblem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic properties of solutions and radial terrace solution,
    Preprint.
  2. Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada,
    A free boundary oroblem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I: Clasificaton of asymptotic behabior,
    Published in Online First of Discrete and Continuous Dynamical Systems, A, Jan. 2022.
  3. Mitsuki Kobayashi and Yoshio Yamada,
    Global existence of weak solutions to forest kinematic model with nonlinear degenerate diffusion,
    Advances in Mathematical Siences and Applicatins, Vol. 29 (2020), 187-209.
  4. Yoshio Yamada,
    Asymptotic properties of a free bounary problem for a reaction-diffusion equation with multi-stable nonlinearity,
    Rendiconti dell'Istituto di Matematica dell'Universita di Trieste, Vo.52 (2020), 65-89.
  5. Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada,
    Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity,
    SIAM Journal on Mathematical Analysis, Vol. 52 (2020), 65-103.
  6. Maho Endo, Yuki Kaneko and Yoshio Yamada,
    Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity,
    Disrete Continuous Dynamical Systems, A, Vol. 40 (2020), 3375-3394.
  7. M. Kobayashi, T. Suzuki and Y. Yamada,
    Lotka-Voltera systems with periodic orbits,
    Funkcialaj Ekvacioj, Vol. 62 (2019), 129-155.
  8. 遠藤真帆,兼子裕大,山田義雄,
    A free boundary problem for reaction-diffusion equation with positive bistable nonlinearity
    (Theoretical Developments to Phenomenon Analysis based on Nonlinear Evolution Equations),
    数理解析研究所講究録 2121巻, pp. 29-40, 2019年.
  9. M. Pierre, T. Suzuki and Y. Yamada,
    Dissipative reaction diffusion systems with quadratic growth,
    Indiana University Mathematics Journal, Vol. 68 (2019), 291-322.
  10. Yuki Kaneko and Yoshio Yamada,
    Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions,
    Journal of Mathematical Analysis and Applications, Vol.465 (2018), 1159-1175.
  11. Shanbing Li and Yoshio Yamada,
    Effects of cross-diffusion in the diffusion prey-predator model with a protection zone II,
    Journal of Mathematical Analysis and Applications, Vol. 461 (2018), 971-992.
  12. R. Yamamoto and Y. Yamada,
    Analysis of forest kinematic model with nonlinear degenerate diffusion,
    Advances in Mathematical Sciences and Applicaions, Vol. 25, No.1 (2016), 307-320. PDF file
  13. 兼子裕大,山田義雄,
    Spreading, vanishing and singularity for radially symmentric solutions of a Stefan-type free boundary problem,
    (Developments of the Theory of Evolution Equations as the Applications to the Analysis of Nonlinear Phenomena),
    数理解析研究所講究録 1997巻,pp. 86-95, 2016年.
  14. Yusuke Kawai and Yoshio Yamada,
    Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity,
    Journal of Differential Equations, Vol. 261 (2016), 538-572.
  15. Yusuke Yoshida and Yoshio Yamada,
    Asymptotic behavior of solutions for semilinear Volterra diffusion equations with spatial inhomogeneity and advection,
    Tokyo Journal of Mathematics Vol.39, No.1 (2016), 271-292.
  16. Yoshio Yamada
    Nonlinear diffusion equations with cross-diffusion: reaction-diffusion equations appearing in mathematical ecology, Sugaku Expositions Vol. 29, No. 2 (2016), 121-144. PDF file
  17. Y. Yamada,
    On logisitic diffusion equations with nonlocal interaction terms,
    Nonlinear Analysis, Theory, Methods & Applications, Vol. 118 (2015), pp.51-62.
  18. T. Suzuki and Y. Yamada,
    Global-in-time behavior of a Lotka-Volterra system with diffusion,
    Indiana University Mathematics Journal, Vol. 64 (2015), pp. 181-216.
  19. Y. Kaneko, K. Oeda and Y. Yamada,
    Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations,
    Funkcialaj Ekvacioj, Vol. 57 (2014), pp. 449-465.
  20. 山田義雄,
    On logistic equations with diffusion and nonlocal terms,
    (Progress in Qualitative Theory of Ordinary Differential Equations),
    数理解析研究所講究録 1901巻, pp. 121-134, 2014年.
  21. 兼子裕大,山田義雄,
    On a population model witha free boundary and related elliptic problems,
    (Progress in Qualitative Theory of Ordinary Differential Equations),
    数理解析研究所講究録 1901巻, pp. 69-78, 2014年.
  22. G. Karali, T. Suzuki and Y. Yamada,
    Global-in-time behavior of the solution to a Gierer-Meinhardt system,
    Discrete and Continuous Dynamical Systems, Vol. 33 (2013), pp. 2885-2900.
  23. 山田 義雄
    交差拡散を伴う非線形拡散方程式系--数理生態学に現れる反応拡散方程式系--,
    数学,64巻 (2012),pp. 384−406.
  24. K. Kuto and Y. Yamada,
    On limit systems for some population models with cross-diffusion,
    Discrete and Continuous Dynamical Systems B, Vol.17 (2012), pp. 2745-2769.
  25. 山田義雄,
    交差拡散を伴う数理生態学モデルについて,(非線形拡散の数理),
    数理解析研究所講究録 1810巻,pp. 39-58, 2012年.
  26. L. Latos, T. Suzuki and Y. Yamada,
    Transient and asymptotic dynamics for a prey-predator system with diffusion,
    Mathematical Methods in the Applied Sciences, Vol. 35 (2012), pp. 1101-1109.
  27. 兼子裕大,山田義雄,
    A free boundary problem modelling the invasion of soecies,
    (Analysis on Non-Equilibria and Nonlinear Phenomena: from the Evolution Equation Point of View),
    数理解析研究所講究録 1792巻, pp.108-117, 2012年.
  28. Y. Kaneko and Y. Yamada,
    A free boundary problem for a reaction-diffusion equation appearing in ecology, herh
    Advances in Mathematical Sciences and Applications, Vol. 21 (2011), pp. 467-492. PDF file
  29. K. Kuto and Y. Yamada,
    Positive solutions for Lotka-Volterra competition systems with large cross-diffusion,
    Applicable Analysis, Special Issue, Vol. 89 (2010), pp. 1037-1066.
  30. K. Kuto and Y. Yamada,
    Coexsitence problem for a prey-predator model with density-dpendent diffusion,
    Nonlinear Analysis; Theory, Methods & Applications Vol. 71 (2009), pp. e2223-e2232.
  31. K. Kuto and Y. Yamada,
    Limiting characterization of sataionary solutions for a prey-predator model with nonlinear diffusion of fractional type,
    Differential and Integral Equations Vol.22 (2009), pp.725-752.
  32. Y. Yamada,
    Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion,
    ``Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," Edited by Yihong Du, Hitoshi Ishii and Wei-Yuen Lin,
    pp. 282-299, World Scientific, 2009. PDF file
  33. Y. Yamada,
    Positive Solutions for Lotka-Volterra Systems with Cross-Diffusion,
    ``Handbook of Differential Equations: Stationary Partial Differential Equations," Vol. 6, Edited by M. Chipot, pp. 411-501, 2008.
  34. Y. Du and Y. Yamada,
    On the long-time limit of positive solutions to the degenerate logistic equation,
    Discrete and Contiunous Dynamical Systems, Special Issue, Ser. A, Vol. 25 (2009), pp. 123-132.
  35. M. Urano, K. Nakashima and Y. Yamada,
    Steady-states with transition layers and spikes for a bistable reaction-diffusion equation,
    Mathematical Approach to Nonlienar Phenomena, Analysis and Simulations, pp. 267-279, Gakuto Internat. Ser. Math. Sci. Appl. Vol. 23, 2005.
  36. M. Urano, K. Nakashima and Y. Yamada,
    Transition layers and spikes for a bistable reaction-diffusion equation,
    Adv. Math. Sci. Appl. Vol. 15(2005), pp. 683-707. PDF file
  37. M. Urano, K. Nakashima and Y. Yamada,
    Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity,
    Dynamical Systems and Differential Equations, Supplement Volume (2005), pp. 868-877.
  38. K. Kuto and Y. Yamada,
    Coexistence states for a prey-predator model with cross-diffusion,
    Dynamical Systems and Differential Equations, Supplement Volume (2005), pp. 536-545.
  39. K. Kuto and Y. Yamada,
    Multiple coexistence states for a prey-predator system with cross-diffusion,
    J. Differential Equations Vol. 197 (2004), pp. 315-348.
  40. Y. Yamada,
    Existence of global solutins for the Shigesada-Kawasaki-Teramoto model with cross- diffusion (Evolutin Equations and Asymptotic Analysis of Solutions), 数理解析研究所講究録 1358巻, pp. 24-33, 2004年.
  41. T. S. Choi, R. Lui and Y. Yamada,
    Existence of global solutions for the Shigesada-Kawasaki-teramoto model with strongly coupled cross-diffusion,
    Discrete Continuous Dynamical Systems Vol. 10 (2004), pp. 719-730.
  42. T. S. Choi, R. Lui and Y. Yamada,
    Existence of global solutions for the Shigesada-Kawasaki-teramoto model with weak cross-diffusion,
    Disscrete Continuous Dynamical Systems Vol. 9 (2003), pp. 1193-1200.
  43. T. Hirose and Y. Yamada,
    Multiple existence of positive solutions of competing species equations with diffusion and large interactions,
    Adv. Math. Aci. Appl. Vol. 12 (2002), pp. 435-453. PDF file
  44. Y. Yamada,
    Positive solutions for Lotka-Volterra competition system with diffusion,
    Nonlinear Anal. Vol. 47 (2001), Proceedings of the World Congress of Nonlienar Analysts, Catania, Sicily, Italy, 19-26 July 2000, pp. 6085-6096.
  45. 山田義雄,
    Multiple coexistence states for Lotka-Volterra competition model with diffusion,
    (Nonlinear Diffusive Systems: Dynamics and Asymptotics),
    数理解析研究所講究録 1178巻, pp. 167-180, 2000年.
  46. Y. Yamada,
    Coexistence states for Lotka-Volterra systems with cross-diffusion,
    Operator Theory and Its Applications,
    Fields Institute Communications, Vol. 25 (2000), pp. 551-564.
  47. T. Ichikawa and Y. Yamada,
    Some remarks on global solutions to quasilinear parabolic system with cross-diffusion,
    Funkcial. Ekvac. Vol. 43 (2000), pp. 285-301.
  48. 山田義雄,
    Sublinear term をもつ放物型方程式の解の一意性,(非線形発展方程式とその 応用),
    数理解析研究所講究録 1135巻, pp. 129-139, 2000年.
  49. A. Yoshida and Y. Yamada,
    Global attracitivity of a coexistence states for a certain class of
    reaction-difusion systems with 3 x 3 cooperative matrices,
    Adv. Math. Sci. Appl. Vol. 9 (1999), pp. 695-706.
  50. S. Takeuchi and Y. Yamada,
    Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian,
    Nonlinear Anal. Vol. 42 (2000), pp. 41-61.
  51. Y. Yamada,
    Coexistence states for some population models with nonlinearcross-diffusion, Biological Pattern Formation,
    Forma Vol. 12 (1997), pp. 153-166.
  52. K. Nakashima and Y. Yamada,
    Positive steady states for prey-predator models withcross-diffusion,
    Adv. Differential Equations Vol. 1 (1996), pp.1099-1122.
  53. K. Nakashima and Y. Yamada,
    On positive steady-states for some reaction-diffusion system,
    Adv. Math. Aci. Appl. Voo. 6 (1996), pp. 279-289. PDF file
  54. Y. Yamada,
    A certain class of reaction-diffusion systems with feedback effects,
    Adv. Math. Aci. Appl. Vol. 5 (1995), pp. 477-485. PDF file
  55. Y. Yamada,
    Global solutions for quasilinear parabolic systems with cross-diffusion effects,
    Nonlinear Anal. Vol. 24 (1995), pp. 1395-1412.
  56. H. Hoshino and Y. Yamada,
    Asymptotic behavior of global solutions for some reaction-diffusion systems,
    Nonlinear Anal. Vol. 23 (1994), pp. 639-650.
  57. H. Hoshino and Y. Yamada,
    Solvability and smoothing effect for semilinear parabolic equations,
    Funkcial. Ekvac. Vol. 34 (1991), pp. 475-494.
  58. Y. Yamada,
    Stability of steady-states for prey-predator diffusion equations with homogeneous Dirichlet conditions,
    SIAM J. Math. Anal. Vol.21 (1990), pp. 327-345.
  59. 星野弘喜,山田義雄,
    半線型高階放物型方程式の大域解について,(発展方程式とその応用),
    数理解析研究所講究録 698巻,pp. 14-35, 1989年.
・iA)応用数理に現れる放物型方程式に対する非線形問題

  1. M. Iida, Y. Yamada and S. Yotsutani,
    On the convergence rates for solutions of some chemical interfacial reaction problems,
    Osaka J. Math. Vol. 32 (1995), 373-396.
  2. M. Iida, Y. Yamada, E. Yanagida and S. Yotsutani,
    Exponential convergence of solutions for a mathematical model on chemical interfacial reactions,
    Adv. Math. Sci. Appl. Vol.3 (1993/94), 335-352.
  3. M. Iida, Y. Yamada and S. Yotsutani,
    Convergence of solutions of a chemical interacial reaction model,
    Funkcial. Ekvac. Vol. 36 (1993), 311-328.
  4. M. Iida, Y. Yamada and S. Yotsutani,
    Asymptotic behavior of solutions for a mathematical model on chemical interfacial reactions,
    Osaka J. Math. Vol. 29 (1992),483-495.
  5. T. Hishida and Y. Yamada,
    Global solutions for the heat convection equation in an exterior domain,
    Tokyo J. Math. Vol. 15 (1992), 135-151.
  6. 飯田雅人,山田義雄,四ツ谷晶二,
    界面における化学反応のモデル:[email protected]解の漸近挙動について,
    (発展方程式と非線型問題への応用),
    数理解析研究所講究録 755巻,pp. 171-184, 1991年.
  7. Y. Yamada and S. Yotsutani,
    A mathematical model on chemical interfacial reactions,
    Japan J. Appl. Math. Vol. 7 (1990), 369-398.
  8. Y. Yamada and S. Yotsutani,
    Note on chemical interfacial reaction models,
    Proc. Japan Aca. Ser. A Math. Sci. Vol. 62 (1987), 379-381.
  9. Y. Hashimoto, Y. Niikura and Y. Yamada,
    Stability and instability for semilinearparabolic equations with free boundary conditions,
    Nonlinear Anal. Vol. 8 (1984), 683-694.
  10. Y. Giga, Y. Hashimoto and Y. Yamada,
    Parabolic equations with free boundary conditions,
    Funkcial. Ekvac. Vol. 26 (1983), 263-279.
(B)ボルテラ型積分項をもつ非線形拡散方程式

  1. Y.Yoshida and Y. Yamada,
    Asymptotic behavior of solutions for semilinear Volterra diffusion equations with spatial inhomogeneity and advection,
    Tokyo Journal of Mathematics, Vol.39 (2016), No.1, 271-292.
  2. Y. Yamada,
    On logisitic diffusion equations with nonlocal interaction terms,
    Nonlinear Analysis, Theory, Methods & Applications, Vol. 118 (2015), pp.51-62.
  3. Y. Yamada,
    Asymptotic behavior of solutions for semilinear Volterra diffusion equations,
    Nonlinear Anal. Vol. 21 (1993), 227-239.
  4. Y. Yamada and Y. Niikura,
    Bifurcation of periodic solutions for nonlinear parabolic equations with infinite delays,
    Funkcial. Ekvac. Vol. 29 (1986), 309-333.
  5. Y. Yamada,
    Asymptotic stability for some systems of semilinear Volterra diffusion equations,
    J. Differential Equations Vol. 52 (1984), 295-326.
  6. Y. Yamada,
    On some semilinear Volterra diffusion equations arising in ecology,
    Equadiff 5 (Blatislava, 1981), 370-373,
    Teubner-Texte Math. Vol.47, Teubner, Leipzig, 1982.
  7. Y. Yamada,
    On a certain class of semi-linear Volterra diffusion equations,
    J. Math. Anal. Appl. Vol. 88 (1982), 433-451.
  8. 山田義雄,
    Volterra 型半線型拡散方程式について,(Mathematical Topics in Biology: '80 December),
    数理解析研究所講究録 420巻, pp. 77-92, 1981年.
(C)非線形波動方程式

  1. M. Hosoya and Y. Yamada,
    On some nonlinear wave equations. II. Global existence and energy decay of solutions,
    J. Fac. Sci. Univ. Tokyo Sect IA Math. Vol.38 (1991),239-250.
  2. M. Hosoya and Y. Yamada,
    On some nonlinear wave equations. I. Local existence and regularity of solutions,
    J. Fac. Sci. Univ. Tokyo Sect IA Math. Vol. 38 (1991),225-238.
  3. K. Nishihara and Y. Yamada,
    On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms,
    Funkcial. Ekvac. Vool. 33 (1990), 151-159.
  4. Y. Yamada,
    Some nonlinear degenerate wave equations,
    Nonlinear Anal. Vol. 11 (1987), 1155-1168.
  5. Y. Yamada,
    On some quasilinear wave equations with dissipative terms,
    Nagoya Math. J. Vol. 87 (1982), 17-39.
  6. Y. Yamada,
    Quasilinear wave equations and related nonlinear evolution equations,
    Nagoya Math. J. Vol. 84 (1981), 31-83.
  7. 山田義雄,
    準線型波動方程式について,・i応用科学に・ィける偏微分方程式の応用解析),
    数理解析研究所講究録 386巻, pp. 53-69,1980年.
  8. Y. Yamada,
    Some remarks on the equation $y_{tt} - \sigma(y_x)y_{xx} - y_{xtx} = f,
    Osaka J. Math. Vol. 17 (1980), 303-323.
(D)自由境界問題(放物型方程式)

  1. Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada,
    A free boundary oroblem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I: Clasificaton of asymptotic behabior,
    to appear in Discrete and Continuous Dynamical Systems, A.
  2. Yoshio Yamada,
    Asymptotic properties of a free bounary problem for a reaction-diffusion equation with multi-stable nonlinearity,
    Rendiconti dell'Instituto di Matematica dell'Universita di Trieste, Vo.52 (2020), 65-89.
  3. Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada,
    Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity,
    SIAM Journal on Mathematica Analysis, Vol. 52 (2020), 65-103.
  4. Maho Endo, Yuki Kaneko and Yoshio Yamada,
    Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity,
    Disrete Continuous Dynamical Systems A, Vol. 40 (2020), 3375-3394.
  5. 遠藤真帆,兼子裕大,山田義雄,
    A free boundary problem for reaction-diffusion equation with positive bistable nonlinearity
    (Theoretical Developments to Phenomenon Analysis based on Nonlinear Evolution Equations),
    数理解析研究所講究録 2121巻, pp. 29-40, 2019年.
  6. Y. Kaneko and Y. Yamada,
    Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions,
    Journal of Mathematical Analysis and Applications, Vol.465 (2018), 1159-1175
  7. 兼子裕大,山田義雄,
    Spreading, vanishing and singularity for radially symmentric solutions of a Stefan-type free boundary problem,
    (Developments of the Theory of Evolution Equations as the Applications to the Analysis of Nonlinear Phenomena),
    数理解析研究所講究録 1997巻,pp. 86-95, 2016年.
  8. Y.Kawai and Y.Yamada,
    Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity,
    Journal of Differential Equations, Vol. 261 (2016), 538-572.
  9. 兼子裕大,山田義雄,
    On a population model witha free boundary and related elliptic problems,
    (Progress in Qualitative Theory of Ordinary Differential Equations),
    数理解析研究所講究録 1901巻, pp. 69-78, 2014年.
  10. Y. Kaneko, K. Oeda and Y. Ymada,
    Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations,
    Funkcialaj Ekvacioj, Vol. 57 (2014), 449-465.
  11. Y. Kaneko and Y. Yamada,
  12. 兼子裕大,山田義雄,
    A free boundary problem modelling the invasion of soecies,
    (Analysis on Non-Equilibria and Nonlinear Phenomena: from the Evolution Equation Point of View),
    数理解析研究所講究録 1792巻, pp.108-117, 2012年.
    A free boundary problem for a reaction-diffusion equation appearing in ecology,
    Advances in Mathematical Sciences and Applications, Vol. 21 (2011), pp. 467-492.
  13. T. Aiki, H. Imai, N. Ishimura and Y. Yamada,
    Well-posedness of one-phase Stefan problems for sublinear heat equations.
    Nonlinear Anal. Vol. 51 (2002), 587-606.
  14. T. Aiki, H. Imai, N. Ishimura and Y. Yamada,
    One-phase Stefan problems for sublinear heat equations: Asymptotic behavior of solutions.
    Communications in Applied Analysis, Vol. 8 (2004), 1-15.
  15. T. Aiki, H. Imai, N. Ishimura and Y. Yamada,
    On one-phase Stefan problems for sublinear heat equations, Proceedings of the 3rd Asian Mathematical Conference 2000, 6-11,
    World Scientific, 2002.
  16. M. Mimura, Y. Yamada and S. Yotsutani,
    Free boundary problems for some reaction-diffusion equations,
    Hiroshima Math. J. Vol. 17 (1987), 241-280.
  17. M. Mimura, Y. Yamada and S. Yotsutani,
    Stability analysis for free boundary problems in ecology,
    Hiroshima Math. J. Vol. 16 (1986), 477-498.
  18. M. Mimura, Y. Yamada and S. Yotsutani,
    On a free boundary problem in ecology, Recent Topics in Nonlinear PDE,
    II (Sendai,1984), pp. 107-125,
    North-Holland Math. Stud.Vol. 128,
    North-Holland, Amsterdam-New York, 1985.
  19. M. Mimura, Y. Yamada and S. Yotsutani,
    A free boundary problem in ecology,
    Japan J. Appl. Math. Vol. 2 (1985), 151-186.
(E)非線形発展方程式

  1. Y. Yamada,
    Note on certain nonlinear evolution equations of second order,
    Proc. Japan Acad. Ser. A Math. Sci. Vol.55 (1979), 167-171.
  2. Y. Yamada,
    On the decay of solutions for some nonlinear evolution equations of second order,
    Nagoya Math. J. Vol. 73 (1979), 69-98.
  3. M. Otani and Y. Yamada,
    On the Navier-Stokes equations in noncylindrical domains:
    an approach by the subdifferential operator theory,
    J. Fac. Aci. Univ. Tokyo Sect.IA Math. Vol. 25 (1978), 185-204.
  4. Y. Yamada,
    Periodic solutions of certain nonlinear parabolic differential
    equations in domains with periodically moving boundaries.
    Nagoya Math. J. Vol.70 (1978), 111-123.
  5. 山田義雄,
    Subdifferential の発展方程式について,(発展系と自由境界問題),
    数理解析研究所講究録 264巻,pp. 176-192, 1976年.
  6. Y. Yamada,
    On evolution equations generated by subdifferential operators.
    J. Fac. Sci. Univ. Tokyo Sect.IA Math. Vol.23 (1976), 491-515.