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Full-Text Articles in Dynamic Systems
Ganikhodjaev's Conjecture On Mean Ergodicity Of Quadratic Stochastic Operators, Mansoor Saburov, Khikmat Saburov
Ganikhodjaev's Conjecture On Mean Ergodicity Of Quadratic Stochastic Operators, Mansoor Saburov, Khikmat Saburov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
A linear stochastic (Markov) operator is a positive linear contraction which preserves the simplex. A quadratic stochastic (nonlinear Markov) operator is a positive symmetric bilinear operator which preserves the simplex. The ergodic theory studies the long term average behavior of systems evolving in time. The classical mean ergodic theorem asserts that the arithmetic average of the linear stochastic operator always converges to some linear stochastic operator. While studying the evolution of population system, S.Ulam conjectured the mean ergodicity of quadratic stochastic operators. However, M.Zakharevich showed that Ulam's conjecture is false in general. Later, N.Ganikhodjaev and D.Zanin have generalized Zakharevich's example …
Sarymsakov Cubic Stochastic Matrices, Mansoor Saburov, Khikmat Saburov
Sarymsakov Cubic Stochastic Matrices, Mansoor Saburov, Khikmat Saburov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
The class of Sarymsakov square stochastic matrices is the largest subset of the set of stochastic, indecomposable, aperiodic (SIA) matrices that is closed under matrix multiplication and any infinitely long left-product of the elements from any of its compact subsets converges to a rank-one (stable) matrix. In this paper, we introduce a new class of the so-called Sarymsakov cubic stochastic matrices and study the consensus problem in the multi-agent system in which an opinion sharing dynamics is presented by quadratic stochastic operators associated with Sarymsakov cubic stochastic matrices.