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Scientific Bulletin of Namangan State University

Abstract

In this paper described some quadratic operators which map the dimensional simplex of idempotent measures to itself. Such operators are divided to two classes: the first class contains all - cubic matrices with nonpositive entries which in each dimensional th matrix contains exactly one non-zero row and exactly one non-zero column; the second class contains all - cubic matrices with non-positive entries which has at least one quadratic zero-matrix. These matrices play a role of the stochastic matrices in the case of idempotent measures. For both classes of quadratic maps we find fixed points and their characters. And also, we find trajectories of quadratic maps which map to itself.

First Page

10

Last Page

18

References

[1] Shiryaev, A.N. (1996), Probability, 2 nd Ed. Springer. [2] Ganikhodzhayev, R.N., Mukhamedov, F.M., and Rozikov, U.A. (2011), Quadratic stochastic operators and processes: Results and Open Problems, Infin. Dim. Anal., Quantum Probab. Related Topics. 14(2), 279-285. [3] Akian,M. (1999), Densities of idempotent measures and large deviations, Trans. Amer. Math. Soc., 351(4), 4515-4543. [4] Casas, J.M., Ladra, M., and Rozikov, U.A. (2011), A chain of evolution algebras, Linear Algebra. Appl., 435(4), 852-870. [5] Del Moral, P. and Doisy, M. (1999), Maslov idempotent probability calculus, II. Theory Probab. Appl., 44, 319-332. [6] Litvinov, G.L. and Maslov, V.P. (2003), Idempotent Mathematics and Mathematical Physics, Vienna. [7] Zarichnyi, M.M. (2010), Spaces and maps of idempotent measures, Izvestiya: Mathematics. 74(3), 481-499. [8] Rozikov, U.A. and Karimov, M.M. (2013), Dinamics of linear maps of idempotent measures, Lobachevskii Journal of Mathematics, 34(1), 20-28. [9] Juraev, I.T. and Karimov, M.M. (2019), Quadratic Operators Defined on a Finite- dimensional Simplex of Idempotent Measures, Journal of Discontinuity, Nonlinearity and Complexity, 8(3), 279-286.

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