Class Of Quadratic Stochastic Operators Research Articles

On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators

It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space <img src=image/13491747_01.gif>. We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space <img src=image/13491747_01.gif> and considered for a special case of the parameter Ɛ. We found that the mixing quadratic stochastic operator is a regular transformation for <img src=image/13491747_02.gif> and is a non-regular for <img src=image/13491747_03.gif>. Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter Ɛ by the parameter <img src=image/13491747_04.gif>, where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy <img src=image/13491747_04.gif>. We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.

On Quadratic Stochastic Operators Corresponding to Cyclic Groups

We introduce a new class of quadratic stochastic operators corresponding to cyclic groups. We study the set of fixed points and prove that almost all (w.r.t. Lebesgue measure) trajectories of such operators converge to the center of the simplex. For the cyclic groups of order 2n we show that for any subgroup corresponding quadratic stochastic operator is a regular operator.

On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called b-bistochastic q.s.o. defined on a finite dimensional simplex. We include several properties of b-bistochastic q.s.o. and their dynamical behaviour. One of the main findings in this paper is the description on the uniqueness of the fixed points. Besides, we list the conditions on strict contractive b-bistochastic q.s.o. on low-dimensional simplices, and it turns out that, the uniqueness of the fixed point does not imply its strict contractivity. Finally, we associate non-homogeneous Markov measures with b-bistochastic q.s.o. The defined measures were proven to satisfy the mixing property for regular b-bistochastic q.s.o. Moreover, we show that non-homogeneous Markov measures associated with a class of b-bistochastic q.s.o on one-dimensional simplex, meet the absolute continuity property.

B— Bistochastic Quadratic Stochastic Operators and Their Properties

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called 6-bistochastic q.s.o. We study several descriptive properties of b- bistochastic q.s.o. It turns out that, upper triangular stochastic matrix defines a linear b-stochastic operator. This allowed us to find some sufficient conditions on cubic stochastic matrix to be a b—bistochastic q.s.o.

Prevalence Problem in the Set of Quadratic Stochastic Operators Acting on $$L^{1}$$ L 1

This paper is devoted to the study of the problem of prevalence in the class of quadratic stochastic operators acting on the L1 space for the uniform topology. We obtain that the set of norm quasi-mixing quadratic stochastic operators is a dense and open set in the topology induced by a very natural metric. This shows the typical long-term behaviour of iterates of quadratic stochastic operators.

On mixing in the class of quadratic stochastic operators

We study different types of limit behavior of quadratic stochastic operators acting on ℓ1 (or ℓd1) spaces in both strong and uniform topologies. The main motif of the paper is to express the uniform and strong asymptotic stability of the quadratic stochastic operator in terms of convergence of the associated (linear) nonhomogeneous Markov chain. We also examine which type of uniform convergence of iterates of the quadratic stochastic operator is prevalent.

ON DYNAMICS OF ξS QUADRATIC STOCHASTIC OPERATORS

In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2D-simplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.

On the extreme points of the set of bistochastic operators

In [1], a class of quadratic stochastic operators mapping the finite-dimensional simplex into itself was singled out, and these operators were called bistochastic quadratic operators. They are closely related to the notion of majorization and are applied not only to problems in population genetics [1], [2], but also to the problems in economics [3]. In mathematical economics, the bistochastic quadratic operator is called thewelfare operator. Bistochastic quadratic operators were first studied in [1], where a necessary and sufficient condition for the bistochasticity of operators was obtained. This theorem will be presented below and used throughout the paper. In the present paper, it is shown that the set of bistochastic quadratic operators is a convex polyhedron. Therefore, an analog of Birkhoff’s theorem on the extreme points of the set of bistochastic matrices [4] is of interest. In this paper, the problem is partially solved and, more precisely, we obtain a sufficient condition for points to be extreme and, for the two-dimensional simplex, a necessary and sufficient condition as well. Moreover, we find the number of extreme points of the set of bistochastic quadratic operators in the two-dimensional simplex. Let us now pass to some necessary notions.

QUADRATIC STOCHASTIC OPERATORS, LYAPUNOV FUNCTIONS, AND TOURNAMENTS

A class of quadratic stochastic operators acting in a finite-dimensional simplex is distinguished that has trajectory at any point of the simplex behaving in a nonregular fashion as a rule. For the discrete dynamical systems generated by such operators the existence of a Lyapunov function of the form is proved, and an algorithm for finding the numbers is indicated. Upper estimates are obtained for the set of limit points of the trajectories. It is proved that the “negative” trajectories exist and converge. The question of the number of isolated fixed points of the operators in the distinguished class is considered. The connection between discrete dynamical systems and the theory of tournaments is also studied. Bibliography: 12 titles.