We consider a class of reflected backward doubly stochastic differential equations with time delayed generator (in short RBDSDE with time delayed generator), in this case generator at time t can depend on the values of a solution in the past. Under a Lipschitz condition, we ensure the existence and uniqueness of the solution.

He also defines, following Hardy, Littlewood and Polya [3] a partial ordering of w-dimensional real vectors: a<b if, and only if, there exists a d.s. matrix P such that a = Pb. In the above named article, the author investigates a conjecture of S. Kakutani to the effect that, if two d.s. matrices are such that Pi& <P3a for every real vector a, then Pi<P3. For this purpose he constructs a linear...

We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B = f (U) defined by Bi,j = |Ui,j| 2. For any U we define its defect, determined by the dimension of the image Df (TU U) of the space TU U tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, give...