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Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions. (English) Zbl 0814.58021

Summary: 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.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] R. Abraham and J.E. Marsden, Foundations of Mechanics. Benjamin/Cummings, Reading, 1978. · Zbl 0393.70001
[2] S. Amari, Differential-Geometrical Methods in Statistics. Lecture Notes in Statist. Vol. 28, Springer-Verlag, Berlin, 1985. · Zbl 0559.62001
[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. · Zbl 0694.62001
[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. · Zbl 0671.90046
[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. · Zbl 0604.58032 · doi:10.1090/S0273-0979-1985-15365-0
[6] T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry. Phys. Rep.,66 (1980), 213–393. · doi:10.1016/0370-1573(80)90130-1
[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.
[8] M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Pure and Appl. Math. Vol. 5, Academic, New York, 1974. · Zbl 0309.34001
[9] J. Moser, Integrable Hamiltonian Systems and Spectral Theory. Lezioni Fermiane, Pisa, 1981. · Zbl 0527.70022
[10] M. Toda, Studies of a non-linear lattice. Phys. Rep.,8 (1975), 1–125. · Zbl 0298.55009 · doi:10.1016/0370-1573(75)90018-6
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