Related papers: Canonical forms, higher rank numerical range, conv…

A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations…

Versal deformation of a matrix A is a normal form to which all matrices A + E, close to A, can be reduced by similarity transformation smoothly depending on the entries of A + E. In this paper we discuss versal deformations and their use in…

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However…

This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…

We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller…

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…

In quasi-exactly solvable problems partial analytic solution (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are…

Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we…

We study convexity of image of a general multidimensional quadratic map. We split the full image into two parts by an appropriate hyperplane such that one part is compact, and formulate a sufficient condition for convexity of the compact…

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity…

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…

In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…

This paper is concerned with a covering problem of Euclidean space by a particular arrangement of cones that are not necessarily full and are allowed to overlap. The problem provides an equivalent geometric reformulation of the solvability…

The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…

An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the…

Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix…

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph…

Several new verifiable conditions are established for block matrices with scalar diagonal blocks to have the numerical range equal the convex hull of at most k ellipses where k by k is the size of the smaller diagonal block. For k = 2,…