Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional. Theorem 2, the main result announced in this note, gives an ex...

We initiate a study of determinantal representations with symmetry. We show that Grenet’s determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomia...

In the paper, using two inversion theorems for the Stirling numbers and binomial coefficients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two differentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive ...

This paper presents some results that complement (2). We believe our results are of new pattern concerning determinantal inequalities. Let us fix some notation. The matrices considered here have entries from the field of complex numbers. X ′,X ,X∗ stand for transpose, (entrywise)conjugate, conjugate transpose of X , respectively. For two n -square Hermitian matrices X ,Y , we write X > Y to mea...

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

In this survey we review two topics concerning determinantal (or fermion) point processes. First, we provide the construction of diffusion processes on the space of configurations whose invariant measure is the law of a determinantal point process. Second, we present some algorithms to sample from the law of a determinantal point process on a finite window. Related open problems are listed.

The paper contains an exposition of recent as well as sufficiently old results on determinantal random point fields . We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of determinantal random point field and a criterion for the weak convergence of its the distribution. In the second section we proceed with the examples of the de...

Given a graph whose edges are labeled by ideals of commutative ring R with identity, generalized spline is vertex labeling the elements such that difference labels on adjacent vertices lies in ideal associated to edge. The set splines has and an R-module structure. We study module structure where base greatest common divisor domain. give basis criteria for cycles, diamond graphs trees using det...