Nonsingular sign patterns and the orthogonal group

Nonsingular Group Actions

This paper deals with measurable stationary symmetric stable random fields indexed by R and their relationship with the ergodic theory of nonsingular R-actions. Based on the phenomenal work of Rosiński (2000), we establish extensions of some structure results of stationary SαS processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different beh...

Maharam Extension for Nonsingular Group Actions

Nonsingular Group Actions and Stationary Sαs Random Fields

This paper deals with measurable stationary symmetric stable random fields indexed by Rd and their relationship with the ergodic theory of nonsingular Rd-actions. Based on the phenomenal work of Rosiński (2000), we establish extensions of some structure results of stationary SαS processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different b...

Minimum ranks of sign patterns via sign vectors and duality

A sign pattern matrix is a matrix whose entries are from the set {+,−, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any m×n sign pattern A with minimum rank n− 2, rational realization of the minimum rank is possible. This is done using a new ap...

Square ice, alternating sign matrices and classical orthogonal polynomials

The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be der...