Related papers: The higher rank numerical range of matrix polynomi…

The present paper deals with the problem of computing (or at least estimating) the LW-number $\lambda(n)$, i.e., the supremum of all $\gamma$ such that for each convex body $K$ in $\mathbb{R}^n$ there exists an orthonormal basis…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

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,…

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid…

For a polynomial $P$ of degree $n$ and an $m$-tuple $\Lambda=(\lambda_1,\dots,\lambda_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $\Lambda$ is $D_P(\Lambda)=(\delta_{ij})_{i\in [1,m],j\in[0,n]}$, where…

In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is the…

Consider an $n \times n$ matrix polynomial $P(\lambda)$. An upper bound for a spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was…

After defining in detail the Lambert $W$-function branches, we give a large number of exact identities involving (infinite) symmetric functions of these branches, as well as geometrically convergent series for all the branches. In doing so,…

Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{\la_1,...,\la_k\}$ such that the linear combination $A+\la_1B_1+\la_2B_2+...+\la_kB_k$ has rank smaller…

In this paper we introduce and study two generalizations of Lah numbers, analogous to the Stirling numbers with higher level - a combinatorial one (Lah numbers with higher level) and an algebraic one (Lah numbers of order $s$). We define…

In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…

We consider level crossing in a matrix family $H=H_0+\lambda V$ where $H_0$ is a fixed $N\times N$ matrix and $V$ belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing…

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the…

The work considers the set $\Lambda_n^k$ of all $n\times n$ binary matrices having the same number of $k$ units in each row and each column. The article specifically focuses on the matrices whose rows and columns are sorted…

Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew…

For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e…

Let $a(\lambda)$ and $b(\lambda)$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $\lambda$. We study whether there exist infinitely many…

We consider the set $\mathcal{M}_n(\mathbb{Z}; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening…