Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


In this paper we construct some families of three-dimensional evolution algebras which satisfies Chapman-Kolmogorov equation. For all of these chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

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1. Baur E. Das Wesen und die Erblichkeitsverhaltnisse der Varietates albomarginatae hort. von Pelargonium zonale. Zeitschr f ind Abst u Vererbungs, Vol. 1, 330–351 (1909).

2. Casas J.M., Ladra M., Rozikov U.A. A chain of evolution algebras. Linear algebra Appl., Vol. 435, Issue 4, 852–870 (2011).

3. Casas J.M., Ladra M., Rozikov U.A. Markov processes of cubic stochastic matrices: Quadratic stochastic processes. Linear algebra Appl., Vol. 575, 273–298 (2019).

4. Dmitriev V.P. Quantum mechanics as a theory of a non-Markov random process. Dokl. Akad. Nauk SSSR, Vol. 292, Issue 5, 1101–1105 (1987) (in Russian).

5. Hänggi P., Thomas H., Time evolution, correlations, and linear response of non-Markov processes. Z. Phys. B., Vol. 26, 85–92 (1977).

6. Imomkulov A.N. Isomorphicity of two dimensional evolution algebras and evolution algebras corresponding to their idempotents. Uzb. Math. Jour., Issue 4, 63–73 (2018).

7. Imomkulov A.N. Classification of a family of three dimensional real evolution algebras. TWMS J. Pure Appl. Math., Vol. 10, Issue 2, 225–238 (2019).

8. Imomkulov A.N.,Rozikov U.A. Approximation of an algebra by evolution algebras. arXiv preprint, arXiv: 1910.03708.

9. Van Kampen N.G. Remarks on non-Markov processes. Braz. J. Phys., Vol. 28, Issue 2, 90–96 (1998).

10. Ladra M., Rozikov U.A. Flow of finite-dimensional algebras. J. Algebra, Vol. 470, 263–288 (2017).

11. Lyubich Y.I. Mathematical structures in population genetics. Springer-Verlag, Berlin, (1992).

12. Mendel G., Experiments in plant-hybridization. The Electronic Scholarly Publishing Project, http://www.esp.org/foundations/ genetics/classical/gm-65.pdf (1865).

13. Maksimov V.M. Cubic stochastic matrices and their probability interpretations. Theory Probab. Appl., Vol. 41, Issue 1, 55–69 (1996).

14. Murodov Sh.N. Classification dynamics of two-dimensional chains of evolution algebras. International Journal of Mathematics, Vol. 25, Issue 02, 1450012-1-23 (2014).

15. Omirov B.A.,Rozikov U.A.,Tulenbayev K.M. On real chains of evolution algebras. Linear and Multilinear Algebra, Vol. 63, Issue 1, 586–600 (2015).

16. Rozikov U.A., Murodov Sh.N. Chain of evolution algebras of chicken population. Linear Algebra Appl., Vol. 450, 186–201 (2014).

17. Rozikov U.A.,Murodov Sh.N. Dynamics of two-dimensional evolution algebras. Lobachevskii Jour. Math., Vol. 34, Issue 4, 344–358 (2013).

18. Rozikov U.A., Tian J.P. Evolution algebras generated by Gibbs measures. Lobachevskii Jour. Math., Vol. 32, Issue 4, 270–277 (2011).

19. Suhov Y., Kelbert M. Probability and Statistics by Example, vol. II, Markov Chains: a Primer in Random Processes and Their Applications. Cambridge Univ. Press, Cambridge, (2008).

20. Tian J.P. Evolution algebras and their applications, Lecture Notes in Mathematics, 1921. Springer-Verlag, Berlin, (2008).

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