Let T be an arbitrary n × n matrix with real entries. We explicitly find the closest (in Frobenius norm) matrix A to T , where A is n × n with real entries, subject to the condition that A is “generalized doubly stochastic” (i.e. Ae = e and eA = e , where e = (1, 1, ..., 1) , although A is not necessarily nonnegative) and A has the same first moment as T (i.e. eT1 Ae1 = e T 1 Te1). We also expl...

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.

Let m,n ≥ 1 be integers. Define Tm,n to be the transportation polytope consisting of the m × n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case Bn = Tn,n is the much-studied Birkhoffvon Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determ...

Motivated by the applications in routing in data centers, we study the problem of expressing an n × n doubly stochastic matrix as a linear combination using the smallest number of (sub)permutation matrices. The Birkhoff-von Neumann decomposition theorem proves that there exists such a decomposition, but does not give a representation with the smallest number of permutation matrices. In particul...

The set of bistochastic or doubly stochastic N × N matrices form a convex set called Birkhoff’s polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff’s polytope. For N = 3 we present fairly complete results. For N = 4 partial results are obtained. An interesting difference between the two cases is that there is a ball...

Finite simple, unipotent Bol loops have recently been identified and constructed using group theory. However, the purely group-theoretical constructions of the actual loops are indirect, somewhat arbitrary in places, and rely on computer calculations to a certain extent. In the spirit of revisionism, this paper is intended to give a more explicit combinatorial specification of the smallest simp...

The characteristic functional (c.fl.) of a doubly stochastic Poisson process (DSPP) is studied and it provides us the finite dimensional distributions of the process and so its moments. It is also studied the case of a DSPP which intensity is a narrow-band process. The Karhunen–Loève expansion of its intensity is used to obtain the probability distribution function and a decomposition of this P...

To address different uncertainties associated with patient recruitment in multicentre clinical trials a mixed Poisson model is elaborated (patients arrive at different clinical centres according to delayed doubly stochastic Poisson processes with gamma distributed rates). Analytic approach to modeling and predicting the recruitment at the initial and ongoing stages is suggested. A Bayesian appr...