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We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…
By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$.
The long run behaviour of linear dynamical systems is often studied by looking at eventual properties of matrices and recurrences that underlie the system. A basic problem that lies at the core of many questions in this setting is the…
We prove two inequalities regarding the ratio $\det(A+D)/\det A$ of the determinant of a positive-definite matrix $A$ and the determinant of its perturbation $A+D$. In the first problem, we study the perturbations that happen when positive…
Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are…
This paper is devoted to a generalization of a Hadamard type inequality for the permanent of a complex square matrix. Our proof is based on a non-trivial extension of a technique used in Carlen, Lieb and Loss (Methods and Applications of…
We argue that there should exist a "noncommutative Fourier transform" which should identify functions of noncommutative variables (say, of matrices of indeterminate size) and ordinary functions or measures on the space of paths. Some…
Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of…
The purpose of this paper is to present a collection of interesting generating functions for partitions which have connections to positive definite binary quadratic forms. In establishing our results we obtain some new Bailey pairs.
This note introduces a sufficient Linear Matrix Inequality (LMI) condition for the ultimate boundedness of a class of continuous-time dynamical systems with conic uncertain/nonlinear terms.
About last 70s, Haynsworth [6] used a result of the Schur complement to refine a determinant inequality for positive definite matrices. Haynsworth's result was improved by Hartfiel [5]. We extend their result to a larger class of matrices,…
We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.
Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are…
The aim of this paper is to prove an improved version of the bounded differences inequality for matrix valued functions, by developing the methods of Mackey et al.: "Matrix Concentration Inequalities via the Method of Exchangeable Pairs".…
In this article we study the causality of non-homogeneous linear singular discrete time systems whose coefficients are square constant matrices. By assuming that the input vector changes only at equally space sampling instants we provide…
This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random…
Given two positive definite matrices $A$ and $B$, a well known result by Gelfand, Naimark and Lidskii establishes a relationship between the eigenvalues of $A$ and $B$ and those of $AB$ by means of majorization inequalities. In this work we…
We are interested in ordering the elements of a subset A of the non-zero integers modulo n in such a way that all the partial sums are distinct. We conjecture that this can always be done and we prove various partial results about this…
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of…