On a class of separable quadratic stochastic operators (original) (raw)

Quadratic Stochastic Operators and Processes: Results and Open Problems

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

On Volterra and orthoganality preserving quadratic stochastic operators

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Moreover, we provide its generalization in continuous setting. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.

Infinite Dimensional Orthogonal Preserving Quadratic Stochastic Operators

arXiv: Functional Analysis, 2017

In the present paper, we study infinite dimensional orthogonal preserving quadratic stochastic operators (OP QSO). A full description of OP QSOs in terms of their canonical form and heredity coefficient's values is provided. Furthermore, some properties of OP QSOs and their fixed points are studied.

On xi-(s) quadratic stochastic operators on two-dimensionalsimplex and their behavior

2013

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We study ( ) -QSO defined on 2D simplex. We first classify ( ) -QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.

ON DYNAMICS OF ℓ-VOLTERRA QUADRATIC STOCHASTIC OPERATORS

International Journal of Biomathematics, 2010

We introduce a notion of -Volterra quadratic stochastic operator defined on (m − 1)dimensional simplex, where ∈ {0, 1, . . . , m}. The -Volterra operator is a Volterra operator if and only if = m. We study structure of the set of all -Volterra operators and describe their several fixed and periodic points. For m = 2 and 3, we describe behavior of trajectories of (m − 1)-Volterra operators. The paper also contains many remarks with comparisons of -Volterra operators and Volterra ones.

Doubly stochasticity of quadratic operators and Extremal symmetric matrices

2010

In the present paper we study geometrical and algebraical structures of a certain type of convex sets which plays an important role in the theory of doubly stochastic quadratic operators. More precisely, we describe the set of extreme points of that convex set, and represent its every element AAA in the following form A=fracX+Xt2A=\frac{X+X^t}{2}A=fracX+Xt2, where XXX is a substochastic matrix, and XtX^tXt denotes its transpose.

ON DYNAMICS OF ξ S QUADRATIC STOCHASTIC OPERATORS

International Journal of Modern Physics: Conference Series, 2012

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 2Dsimplex 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 nonhomogeneous geometric quadratic stochastic operators

Turkish Journal of Mathematics

In this paper, we construct a nonhomogeneous geometric quadratic stochastic operator generated by 2partition ξ on countable state space X = Z *. The limiting behavior of such operator is studied. We have proved that such operator possesses the regular property.

On Quadratic Stochastic Operators Having Three Fixed Points

Journal of Physics: Conference Series, 2016

We knew that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix starting from any initial point from the simplex converges to a unique fixed point. However, in general, the similar result for a quadratic stochastic operator associated with a positive cubic stochastic matrix does not hold true. In this paper, we provide an example for the quadratic stochastic operator with positive coefficients in which its trajectory may converge to different fixed points depending on initial points.

On some properties of quadratic stochastic processes

1990

In this paper we prove that every measurable quadratic stochastic process X : R N x Q->■ R is continuous and has the form N x (x ,-) = E x ix j Yu (•) (a-e-), 'j= i where x = (x 1, ..., x N) e Rv and YitJ:Q-> R are random variables. Moreover, we give a proof of the stability of the quadratic stochastic processes.