On doubly stochastic quadratic operators and Birkhoff's problem
  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. 
Rasul Ganikhodzhaev and Farruh Shahidi