On b-bistochastic quadratic stochastic operators

Farrukh Mukhamedov,Ahmad Fadillah Embong

doi:10.1186/s13660-015-0744-y

Abstract

In the present paper we introduce a new notion of order called b-order. Then we define a bistochasticity quadratic stochastic operator (q.s.o.) with respect to the b-order, and call it a b-bistochastic q.s.o. We include several properties of the b-bistochastic q.s.o. and descriptions of all b-bistochastic q.s.o. defined on a two dimensional simplex.

Highlights

The history of quadratic stochastic operators can be traced back to Bernstein’s work [ ].Nowadays, scientists are interested in these operators, since they have a lot of applications, especially in population genetics [, ]
Scientists are interested in these operators, since they have a lot of applications, especially in population genetics [, ]
The quadratic stochastic operators were used as a crucial source of analysis for the study of dynamical properties and modelings in many different fields such as biology [ – ], physics [, ], economics, and mathematics [, – ]

Summary

Introduction

The history of quadratic stochastic operators can be traced back to Bernstein’s work [ ]. The concept of majorization was established in [ ] even though the idea was introduced much earlier by Lorenz [ ], Dalton [ ], and Schur [ ] This kind of theory was very important from an economic point of view, which resulted from the gaps in the income or wealth distribution in society. In [ , ], the necessary and sufficient conditions were given for the bistochasticity of a q.s.o. In general, the descriptions of such a kind of operators are still an open problem. In [ ] a q.s.o. was introduced and studied with the property V (x) ≺ x for all x ∈ Sn– Let V be a b-bistochastic q.s.o. defined on Sn– , the following statements hold:.

This implies kn

Findings

Lemma one finds that