Skip to main content
Log in

On One Regularity Condition for Quantum Quadratic Stochastic Processes

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We present necessary and sufficient conditions for the validity of a regularity condition for homogeneous quantum quadratic stochastic processes defined on von Neumann algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. S. N. Bernshtein, “Solution of a mathematical problem connected with the theory of heredity,” Uch. Zap. N.-I. Kaf. Ukr. Otd. Mat., No. 1, 83–115 (1924).

  2. S. M. Ulam, A Collection of Mathematical Problems [Russian translation], Nauka, Moscow (1964).

    Google Scholar 

  3. H. Kesten, “Quadratic transformations: a model for population growth. I, II,” Adv. Appl. Probab., No. 2, 1–82, 179–228 (1970).

  4. Yu. I. Lyubich, “Basic notions and theorems of evolutionary genetics of free populations,” Usp. Mat. Nauk, 26, No. 5, 51–116 (1971).

    Google Scholar 

  5. S. S. Vallander, “On the limit behavior of a sequence of iterations of certain quadratic transformations,” Dokl. Akad. Nauk SSSR, 202, No. 3, 515–517 (1972).

    Google Scholar 

  6. T. A. Sarymsakov and R. N. Ganikhodzhaev, “Ergodic principle for quadratic stochastic operators,” Izv. Akad. Nauk Uzb. SSR. Ser. Fiz.-Mat. Nauk, No. 6, 34–39 (1979).

  7. V. M. Maksimov, “Cubic stochastic matrices and their probability interpretations,” Teor. Ver. Primen., 41, Issue 1, 89–106 (1996).

    Google Scholar 

  8. T. A. Sarymsakov and N. N. Ganikhodzhaev, “Analytic methods in the theory of quadratic stochastic operators,” Dokl. Akad. Nauk SSSR, 305, No. 5, 1052–1056 (1989).

    Google Scholar 

  9. T. A. Sarymsakov and N. N. Ganikhodzhaev, “On the ergodic principle for quadratic processes,” Dokl. Akad. Nauk SSSR, 316, No. 6, 1315–1319 (1991).

    Google Scholar 

  10. T. A. Sarymsakov and N. N. Ganikhodzhaev, “Analytic methods in the theory of quadratic stochastic processes,” J. Theor. Probab., 3, No. 1, 51–70 (1990).

    Google Scholar 

  11. R. D. Janks, “Quadratic differential systems for interactive population models,” J. Different. Equat., 5, No. 3, 497–514 (1969).

    Google Scholar 

  12. Yu. I. Lyubich, Mathematical Structures in Population Genetics [in Russian], Naukova Dumka, Kiev (1983).

    Google Scholar 

  13. N. N. Ganikhodzhaev and F. M. Mukhamedov, “On quantum quadratic stochastic processes,” Dokl. Akad. Nauk Resp. Uzbekistan, No. 3, 13–16 (1997).

    Google Scholar 

  14. N. N. Ganikhodzhaev and F. M. Mukhamedov, “On quantum quadratic stochastic processes and ergodic theorems for such processes,” Uzb. Mat. Zh., No. 3, 8–20 (1997).

    Google Scholar 

  15. N. N. Ganikhodzhaev and F. M. Mukhamedov, “Ergodic properties of quantum quadratic stochastic processes,” Usp. Mat. Nauk, 53, No. 6, 243–244 (1998).

    Google Scholar 

  16. N. N. Ganikhodzhaev and F. M. Mukhamedov, “ Regularity conditions for quantum quadratic stochastic processes,” Dokl. Ros. Akad. Nauk, 365, No. 3, 301–303 (1999).

    Google Scholar 

  17. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, New York (1979).

    Google Scholar 

  18. A. N. Kolmogorov, “On analytic methods in probability theory,” Usp. Mat. Nauk, No. 5, 5–51 (1938).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mukhamedov, F.M. On One Regularity Condition for Quantum Quadratic Stochastic Processes. Ukrainian Mathematical Journal 53, 1657–1672 (2001). https://doi.org/10.1023/A:1015243910602

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015243910602

Keywords

Navigation