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Ergodic Theory and Arithmetical Simulation of Random Processes

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Cybernetics and Systems Analysis Aims and scope

Abstract

The relationships between arithmetical simulation of random processes, ergodic theory, and optimization are analyzed. Some new results are considered and their possible applications to optimization problems are described.

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REFERENCES

  1. A. G. Postnikov, Arithmetical Simulation of Random Processes [in Russian], Nauka, Moscow (1960).

    Google Scholar 

  2. I. V. Sergienko and N. Z. Shor, “Academician V. S. Mikhalevich as a scientist and science organizer,” Kibern. Sist. Analiz, No. 1, 77–100 (2000).

    Google Scholar 

  3. A. G. Postnikov and I. I. Pyatetskii–Shapio, “Bernoulli-normal sequences of signs,” Izv. AN SSSR, Ser. Mat., 21, 501–514 (1957).

    Google Scholar 

  4. L. P. Postnikova, Geometry of Numbers and Diophantine Approximations [in Russian], Inst. Fiz. Mat. AN LSSR, Vilnius (1974).

    Google Scholar 

  5. Ð. Weyl, “On a uniform distribution of numbers modulo one,” in: Selected papers [Russian translation], Nauka, Moscow (1984), pp. 58–93.

    Google Scholar 

  6. E. Borel, Lecon sur la theorie des functions, IHES, Paris (1914).

    Google Scholar 

  7. H. Lebesgue, “Sur certaines demonstrations d'existence,” Bull. Soc. Math. France, 45, 132–144 (1917).

    Google Scholar 

  8. H. Steinhaus, “Uber einige prinzipiielle Fragen der mathematischen Statistik,” in: Ber. Tagung Wahr. und Math. Statistik, Springer, Berlin (1956), pp. 211–134.

    Google Scholar 

  9. W. Sierpinski, “Demonstration elementaire du theoreme de E. Borel,” Bull. Soc. Math. France, 45, 127–132 (1917).

    Google Scholar 

  10. G. Polya and G. Sege, Problems and Theorems from Analysis. I [Russian translation], Nauka, Moscow (1978).

    Google Scholar 

  11. I. Venn, The Logic of Chance, King College, London (1888).

    Google Scholar 

  12. R. Mises, Probability and Statistics [in Russian], GIZ, Moscow (1930).

    Google Scholar 

  13. A. Church, “On the concept of random sequence,” Bull. Amer. Math. Soc., 46, 130–135 (1940).

    Google Scholar 

  14. N. M. Korobov, “On some problems of uniform distribution,” Izv. AN SSSR, Ser. Mat., 14, 215–231 (1950).

    Google Scholar 

  15. A. N. Kolmogorov, “Logical fundamentals of information and probability theories,” Probl. Peredachi Inform., 5, Issue 3, 3–11 (1969).

    Google Scholar 

  16. D. G. Champernowne, “The construction of the decimals normal in the scale of ten,” J. London Math. Soc., 8, 254–260 (1933).

    Google Scholar 

  17. O. A. Gelfond and Yu. V. Linnik, Elementary Methods in the Analytical Theory of Numbers [in Russian], Fiz. Mat. GIZ, Moscow (1962).

    Google Scholar 

  18. D. L. Doob, Probability Processes [Russian translation], Mir, Moscow (1956).

    Google Scholar 

  19. A. G. Postnikov, “Solving a system of finite-difference equations for the Dirichlet problem using a normal sequence of signs,” Dokl. AN SSSR, 123, No. 3, 407–409 (1958).

    Google Scholar 

  20. A. H. Copeland and P. Erdos, “Note on normal numbers,” Bull. Amer. Math. Soc., 45, 857-860 (1946).

    Google Scholar 

  21. H. Davenport and P. Erdos, “Note on normal decimals,” Can. J. Math., 4, 58–63 (1952).

    Google Scholar 

  22. I. I. Pyatetskii-Shapiro, “Laws of distribution of fractional parts of exponential functions,” Izv. AN SSSR, Ser. Mat., 15, 47–52 (1951).

    Google Scholar 

  23. Yu. M. Ermolyev and N. Z. Shor, “The method of random search for two-stage stochastic problems and its generalizations,” Kibernetika, No. 1, 90–92 (1968).

    Google Scholar 

  24. Yu. M. Ermolyev, Methods of Stochastic Programming [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  25. L. P. Postnikova, “The quantitative form of the Borel problem,” Acta Arithmetica, 21, 235–250 (1972).

    Google Scholar 

  26. A. N. Kolmogorov, Main Concepts of Probability Theory [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  27. A. G. Postnikov, An Introduction to the Analytical Theory of Number [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  28. G. Cramer and M. Litbetter, Stationary Random Processes [Russian translation], Mir, Moscow (1967).

    Google Scholar 

  29. P. R. Halmos, Measure Theory, Van Nostrand, Princeton, New Jersey (1950).

  30. I. P. Kornfeld, Ya. G. Sinai, and S. V. Fomin, The Ergodic Theory [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  31. I. I. Gikhman, “To the problem on the number of intersections of the boundary of a given domain by a random function,” in: Trans. Conf. on the Probability Theory and Math. Statistics [in Russian], (1958), pp. 247–262.

  32. I. N. Kovalenko, “On a limiting theorem for determinants in the class of Boolean functions,” DAN SSSR, 161, No. 3, 517–519 (1965).

    Google Scholar 

  33. I. N. Kovalenko and A. A. Levitskaya, “The asymptotic behavior of the number of solutions of the system of random linear equations over a finite field and a finite ring,” DAN SSSR, 221, No. 4, 778–781 (1975).

    Google Scholar 

  34. I. N. Kovalenko, A. A. Levitskaya, and M. N. Savchuk, Selected Problems of the Probability Theory of Combinations [in Russian], Naukova Dumka, Kiev (1986).

    Google Scholar 

  35. K. de Leeuw, E. F. Moore, C. E. Shannon, and H. Shapiro, “Computability by probabilistic machines,” in: C. E. Shannon and J. McCarthy (eds.), Automata Studies, Princeton Univ. Press, Princeton, NJ (1955), pp. 183–212.

    Google Scholar 

  36. H. S. Shapiro, “Extremal problems for polynomials and power series,” Sc. M. Theses, Massachusetts Inst. of Technology (1951).

  37. J. H. Loxton and A. J. van der Poorten, “Arithmetic properties of automata: regular sequences,” J. Reine Angew. Math., 392, 57–69 (1988).

    Google Scholar 

  38. J. S. Byrnes, “On polynomials with coefficients of modulus one,” Bull. London Math. Soc., 9, 171–176 (1977).

    Google Scholar 

  39. T. W. Korner, “On a polynomial of J. S. Byrnes,” Bull. London Math. Soc., 2, 219–224 (1980).

    Google Scholar 

  40. J. S. Byrnes and H. S. Shapiro, “Shapiro sequences, Reed-Muller codes, and functional equations,” http://www.prometheus-inc.com/public/pons/RM.pdf.

  41. N. M. Glazunov, “On validated numerics, category theory and computer algebra framework for simulation and computation in theoretical physics,” Nuclear Instruments and Methods in Physics Research, Section A, 502, No. 2–3, 654–656 (2003).

    Google Scholar 

  42. L. Bartholdi, R. Grigorchuk, and V. Nekrashevych, “From fractal groups to fractal sets,” in: P. Grabner and W. Woess (eds.), Fractals in Graz 2001, Birkhauser, Basel (2003), pp. 25–118.

    Google Scholar 

  43. N. M. Glazunov, “About moduli spaces, equidistribution, estimates, and rational points of algebraic curves,” Ukr. Mat. Zh., 53, No. 9, 1174–1183 (2001).

    Google Scholar 

  44. N. Z. Shor, Nondifferentiable Optimization and Polynomial Problems, Kluwer Academ. Publ., Boston (1998).

    Google Scholar 

  45. R. Kannan, “Rapid mixing in Markov chains,” ICM2002, 3, 674–683 (2003).

    Google Scholar 

  46. P. S. Knopov, Optimal Estimates of Stochastic Systems Parameters [in Russian], Naukova Dumka, Kiev (1981).

    Google Scholar 

  47. C. Deninger, “Some analogies between number theory and dynamical systems on foliated spaces,” Doc. Math. J. DMV Extra, 163–186 (1999).

  48. P. D'Ambros, G. Everest, R. Miles, and T. Ward, “Dynamical systems arising from elliptic curves,” Colloq. Math., 84 85, Pt. 1, 95–107 (2000).

    Google Scholar 

  49. N. M. Glazunov, “Critical lattices, elliptic curves and their possible dynamics,” in: Proc. Voronoi Conf. on Analytic Number Theory and Spatial Tessellation, Kyiv, Sept. 22–28, 2003, Inst. of Math. (2003), p. 29.

  50. H. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, Berlin (1980).

    Google Scholar 

  51. H. Zieschang and E. Fogt, Surfaces and Discontinuous Groups [Russian translation], Nauka, Moscow (1987).

    Google Scholar 

  52. L. Bers, “Finite dimensional Teichmuller spaces and generalizations,” Bulletin AMS, 5, No. 2, 131–172 (1981).

    Google Scholar 

  53. N. M. Glazunov, “Category aspects of complex manifolds and problems of mirror symmetry,” Probl. Program., No. 3–4, 104–110 (2002).

    Google Scholar 

  54. W. M. Goldman, “Ergodic theory on moduli space,” Annals of Math., 146, 475–507 (1997).

    Google Scholar 

  55. D. Pickrell and E. Xia, “Mapping class group actions on representation varieties, I. Ergodicity,” Prepr. University of Arizona (1998).

  56. M. Shiffer, “Extreme problems and variational methods in a conformal mapping,” in: International Mathematical Congress in Edinburgh [in Russian], Fiz.-Mat. GIZ, Moscow (1962), pp. 192–218.

    Google Scholar 

  57. F. Gering, “Some problems of the theory of quasiconformal mappings,” in: International Congress of Mathematicians in Berkeley [in Russian], Mir, Moscow (1991), pp. 114–138.

    Google Scholar 

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Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    N. M. Glazunov & N. Z. Shor

  2. Moscow Pedagogical University, Moscow, Russia

    L. P. Postnikova

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  1. N. M. Glazunov
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  2. L. P. Postnikova
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  3. N. Z. Shor
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Glazunov, N.M., Postnikova, L.P. & Shor, N.Z. Ergodic Theory and Arithmetical Simulation of Random Processes. Cybernetics and Systems Analysis 40, 527–536 (2004). https://doi.org/10.1023/B:CASA.0000047874.19986.96

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  • DOI: https://doi.org/10.1023/B:CASA.0000047874.19986.96

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