Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions

  • Yoshimasa Nakamura


Gradient systems on the manifolds of Gaussian and multinomial distributions are shown to be completely integrable Hamiltonian systems. The corresponding flows converge exponentially to equilibrium. A Lax representation of the gradient systems is found.

Key words

integrable gradient systems integrable Hamiltonian systems Lax representation Gaussian distributions multinomial distributions information geometry 


  1. [1]
    R. Abraham and J.E. Marsden, Foundations of Mechanics. Benjamin/Cummings, Reading, 1978.MATHGoogle Scholar
  2. [2]
    S. Amari, Differential-Geometrical Methods in Statistics. Lecture Notes in Statist. Vol. 28, Springer-Verlag, Berlin, 1985.MATHGoogle Scholar
  3. [3]
    S. Amari, O.E., Barndorff-Nielsen, R.E. Kass, S.L. Lauritzen and C.R. Rao, Differential Geometry in Statistical Inferences. IMS Lecture Notes-Monograph Ser. Vol. 10, Inst. Math. Statist., Hayward, 1987.Google Scholar
  4. [4]
    D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories. Trans. Amer. Math. Soc.,314 (1989), 527–581.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A.M. Bloch, A completely integrable Hamiltonian system associated with line fitting in complex vector spaces. Bull. Amer. Math. Soc. (New Ser.),12 (1985), 250–254.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry. Phys. Rep.,66 (1980), 213–393.CrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Fujiwara, Dynamical systems on statistical models. State of Art and Perspectives of Studies on Nonlinear Integrable Systems (eds. Y. Nakamura, K. Takasaki and K. Nagatomo), RIMS Kokyuroku Vol. 822, Kyoto Univ., Kyoto, 1993, 32–42.Google Scholar
  8. [8]
    M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Pure and Appl. Math. Vol. 5, Academic, New York, 1974.MATHGoogle Scholar
  9. [9]
    J. Moser, Integrable Hamiltonian Systems and Spectral Theory. Lezioni Fermiane, Pisa, 1981.MATHGoogle Scholar
  10. [10]
    M. Toda, Studies of a non-linear lattice. Phys. Rep.,8 (1975), 1–125.CrossRefMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 1993

Authors and Affiliations

  • Yoshimasa Nakamura
    • 1
  1. 1.Department of MathematicsGifu UniversityGifuJapan

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