Related papers: A family of determinants associated with a square …
We prove several evaluations of determinants of matrices, the entries of which are given by the recurrence $a_{i,j}=a_{i-1,j}+a_{i,j-1}$, or variations thereof. These evaluations were either conjectured or extend conjectures by Roland…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
We give one more proof of the fact that symplectic matrices over real and complex fields have determinant one. While this has already been proved many times, there has been lasting interest in finding an elementary proof. Our result is…
In these notes we focus a bit on the complex case for some families of matrices and equivalences between them.
We obtain sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum of banded, in general non-normal, matrices of finite or infinite size. Each inclusion set is the union of the pseudospectra of certain…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…
The paper presents the classification of matrix valued superpotentials corresponding to shape invariant systems of Schr\"odinger equations. All inequivalent irreducible matrix superpotentials realized by matrices of arbitrary dimension with…
We establish a relation between the known parametrization of a family of irreducible representations of a Weyl group and Springer's correspondence. We outline a parametrization of unipotent character sheaves on a connected reductive group…
Given a real, symmetric matrix S, we define the slice through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some…
We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas…
The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a…
We generalize the concept of Pascal matrices to matrices associated with sets of points by considering multidimensional binomial coefficients as entries. We study their properties and prove that the infinite matrix associated with the set…
In this note we study a family of algebras with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between…
We consider sentence-definable and diagram-definable subfamilies of given families of theories, calculi for these subfamilies, as well dynamics and characteristics of these subfamilies with respect to rank and degree.
In the one-parameter family of power-law maps of the form $f_a(x)=-|x|^{\alpha}+a,$ $\alpha >1,$ we give examples of mutually related dynamically determined quantities, depending on the parameter $a$, such that one is a Pick function of the…
We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and…
In this paper we explore a family of congruences over $\N^\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of…
We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.