The sequence h0, h1, . . . , hs is called the h-vector of the ring. Orginally the question was to decide whether a given sequence can arise as the h-vector of some ring. In this sense the validity of the conjecture would imply that log-concavity was a necessary condition on the h-vector. It is now known however [12, 3] that Stanley’s conjecture is not true in general. Several natural weakenings...

Determinantal representation for the generating function of plane partitions with fixed volumes diagonal parts is investigated in limiting cases.

Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate are definite determinantals. Indeed, general, it is difficult problem to decide whether given polynomial determinantal, if is, of paramount interest determine determinantal representation polynomial. We provide necessary sufficient condition for t...

The goal of this paper is to quantitatively describe some statistical properties of higher-dimensional determinantal point processes with a primary focus on the nearest-neighbor distribution functions. Toward this end, we express these functions as determinants of NxN matrices and then extrapolate to N-->infinity . This formulation allows for a quick and accurate numerical evaluation of these q...

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vande...

The theory of determinantal point processes has its roots in work in mathematical physics in the 1960s, but it is only in recent years that it has been developed beyond several specific examples. While there is a rich probabilistic theory, there are still many open questions in this area, and its applications to statistics and machine learning are still largely unexplored. Our contributions are...

Tavenas has recently proved that any nO(1)-variate and degree n polynomial in VP can be computed by a depth-4 ΣΠ[O( p n)]ΣΠ[ p n] circuit of size 2O( p n log n) [Tav13]. So to prove VP 6= VNP, it is sufficient to show that an explicit polynomial ∈ VNP of degree n requires 2ω( p n log n) size depth-4 circuits. Soon after Tavenas’s result, for two different explicit polynomials, depth-4 circuit s...

We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute. This note is devoted to the functions of determinantal rank and fooling set number, which are receiving attention in different applications, see [1, 3] and references therein. The purpose of this note is to give an NP-completeness proof for those functions, th...