On doubly stochastic quadratic operators and Birkhoff's problem

TL;DR

This paper introduces doubly stochastic quadratic operators, characterizes their properties, and explores an analogue of Birkhoff's theorem for these operators, expanding the understanding of their structure and convexity.

Contribution

It defines doubly stochastic quadratic operators, establishes their convex polytope structure, and investigates a Birkhoff-type theorem for this class, which is a novel extension.

Findings

Doubly stochastic quadratic operators are characterized by necessary and sufficient conditions.

The set of all such operators forms a convex polytope.

An analogue of Birkhoff's theorem is studied for these operators.

Abstract

In the present paper we introduce a concept of doubly stochastic quadratic operator. We prove necessary and sufficient conditions for doubly stochasticity of operator. Besides, we prove that the set of all doubly stochastic operators forms convex polytope. Finally, we study analogue of Birkhoff's theorem for the class of doubly stochastic quadratic operators.