Orthogonal-preserving and surjective cubic stochastic operators

Mukhamedov, Farrukh; Embong, Ahmad Fadillah; Rosli, Azizi

doi:10.1215/20088752-2017-0013

Cited by 17 publications

(4 citation statements)

References 12 publications

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“…Historically, a quadratic stochastic operator was originally introduced by Bernstein [1] back in 1942. A quadratic stochastic process associated with a cubic stochastic matrix is considered to be the simplest nonlinear Markov chain [7,12,13,14]. Indeed, it is noteworthy to mention that the rigorous establishment of the connection between Krause mean processes and nonlinear Markov chains was accomplished in a series of papers [2,15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Krause mean processes generated by off-diagonally positive doubly stochastic hyper-matrices

Saburov,

Saburov

2024

GJOM

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Back in 1974, DeGroot first introduced the concept of achieving consensus through iterative averaging processes using square stochastic matrices. A pivotal question arises regarding the feasibility of extending the classical DeGroot model from square stochastic matrices to higher-order stochastic hyper-matrices. In the paper, we introduce a novel mathematical model for opinion-sharing dynamics employing the Krause mean process that is generated by off-diagonally positive doubly stochastic hyper-matrices. The principal objective is to achieve consensus within the multi-agent system.

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Section: Introductionmentioning

confidence: 99%

Krause mean processes generated by off-diagonally positive doubly stochastic hyper-matrices

Saburov,

Saburov

2024

GJOM

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“…While most of the studied LV systems were quadratic, non‐quadratic LV systems such as cubic LV systems have been also studied as they appeared explicitly in modeling a phenomenon arising in an oscillating chemical reaction, called Lotka–Volterra–Brusselator model , 13 and in predator–prey models, which gave rise to periodic variations in the populations 14 . Some classes of discrete‐time cubic stochastic LV operators were also investigated in previous works 15,16 . Naturally, a more general LV systems have also been proposed to model the interaction between biochemical populations 11,17 …”

Section: Introductionmentioning

confidence: 99%

A class of bijective Lotka–Volterra operators and its application

Mukhamedov

Hee

Rosli

2023

Math Methods in App Sciences

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It is well known that any classical Lotka-Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

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“…We point out that the results on surjectivity of nonlinear Markov operators that we considered here open new insight to the theory of nonlinear operators 8 . In infinite-dimensional setting, it turns out that the surjectivity and orthogonal preserveness are not necessarily the same, while they coincide in the finite-dimensional case (see Mukhamedov and Embong [9][10][11][12][13]21,28,29,31 ).…”

Section: Introductionmentioning

confidence: 99%

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

Mukhamedov

Khakimov

Embong

2020

Math Methods in App Sciences

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In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.

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Orthogonal-preserving and surjective cubic stochastic operators

Cited by 17 publications

References 12 publications

Krause mean processes generated by off-diagonally positive doubly stochastic hyper-matrices

Krause mean processes generated by off-diagonally positive doubly stochastic hyper-matrices

A class of bijective Lotka–Volterra operators and its application

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

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