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Yokes and symplectic structures. (English) Zbl 0943.37027

Summary: A relationship between yokes and symplectic forms is established and explored. It is shown that normalized yokes correspond to certain symplectic forms. A method of obtaining new yokes from old is given, motivated partly by the duality between the Hamiltonian and Lagrangian formulations of conservative mechanics. Some variants of this construction are suggested.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
62E20 Asymptotic distribution theory in statistics
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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[1] Abraham, R.; Marsden, J. E., Foundations of Mechanics (1978), Addison-Wesley: Addison-Wesley Redwood City
[2] Amari, S.-I., Differential-Geometrical Methods in Statistics, (Lecture Notes in Statistics, vol. 28 (1985), Springer: Springer Heidelberg) · Zbl 0701.62008
[3] Arnol’d, V. I.; Givental’, A. B., Symplectic geometry, (Arnol’d, V. I.; Novikov, S. P., Dynamical Systems IV: Symplectic Geometry and its Applications. Dynamical Systems IV: Symplectic Geometry and its Applications, Encyclopaedia of Mathematical Sciences, vol. 4 (1990), Springer: Springer Berlin), 1-136 · Zbl 0780.58016
[4] Barndorff-Nielsen, O. E., Likelihood and observed geometries, Ann. Statist., 14, 856-873 (1986) · Zbl 0632.62028
[5] Barndorff-Nielsen, O. E., Differential geometry and statistics: some mathematical aspects, Indian J. Math., 29, 335-350 (1987) · Zbl 0663.62013
[6] Barndorff-Nielsen, O. E., Parametric Statistical Models and Likelihood, (Lecture Notes in Statistics, vol. 50 (1988), Springer: Springer Heidelberg) · Zbl 0830.62002
[7] Barndorff-Nielsen, O. E.; Cox, D. R., Inference and Asymptotics (1994), Chapman & Hall: Chapman & Hall London · Zbl 0826.62004
[8] Blæsild, P., Yokes and tensors derived from yokes, Ann. Inst. Statist. Math., 43, 95-113 (1991) · Zbl 0782.62007
[9] Eguchi, S., Second order efficiency of minimum contrast estimation in a curved exponential family, Ann. Statist., 11, 793-803 (1983) · Zbl 0519.62027
[10] Friedrich, T., Die Fisher-Information und symplectische Strukturen, Math. Nachr., 153, 273-296 (1991) · Zbl 0792.62003
[11] Marsden, J. E., Lectures on Mechanics, (London Mathematical Society Lecture Note Series, vol. 174 (1992), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0197.20901
[12] Murray, M. K.; Rice, J. W., Differential Geometry and Statistics (1993), Chapman & Hall: Chapman & Hall London · Zbl 0628.22003
[13] Nakamura, Y., Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions, Japan. J. Ind. Appl. Math., 10, 179-189 (1993) · Zbl 0814.58021
[14] Nakamura, Y., Gradient systems associated with probability distributions, Japan. J. Ind. Appl. Math., 11, 21-30 (1994) · Zbl 0811.58036
[15] Skovgaard, I. M., On the density of minimum contrast estimators, Ann. Statist., 18, 779-789 (1990) · Zbl 0709.62029
[16] Weinstein, A., Lectures on Symplectic Manifolds, (AMS Regional Conference Series in Mathematics, vol. 29 (1977), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0406.53031
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