Related papers: Higher-rank Numerical Ranges and Kippenhahn Polyno…

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a…

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=\lambda$ categorical in regular $\kappa >\lambda^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>\lambda}$…

We generalise the Elliptical Range Theorem to characterise the numerical range of matrices belonging to a subspace of the space of \(3 \times 3\) matrices. Using Specht's Theorem, which characterizes when two matrices are unitarily…

In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of free resolution algorithms have been given in both cases. In this present work,…

We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of "low description complexity". Specifically, we consider $m\times n$ random matrices…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the…

In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two…

It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an…

We discuss parallel (additive) and sequential (multiplicative) variants of overlapping Schwarz methods for the Helmholtz equation in $\mathbb{R}^d$, with large real wavenumber and smooth variable wave speed. The radiation condition is…

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}\lambda_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-\sigma|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm}…

Let $\Gamma$ denote a finite (strongly) connected regular (di)graph with adjacency matrix $A$. The {\em Hoffman polynomial} $h(t)$ of $\Gamma=\Gamma(A)$ is the unique polynomial of smallest degree satisfying $h(A)=J$, where $J$ denotes the…

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We…

The Lie-algebraic method approximates differential operators that are formal polynomials of {1,x,d/dx} by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the n-dimensional…

There are two fundamental problems motivated by Silverman's conversations over the years concerning the nature of the exact values of canonical heights of $f(z)\in\bar{\mathbb{Q}}(z)$ where $f$ has degree $d\geq 2$. The first problem is the…

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy…

We study the rank of the adjacency matrix $A$ of a random Erdos Renyi graph $G\sim \mathbb{G}(n,p)$. It is well known that when $p = (\log(n) - \omega(1))/n$, with high probability, $A$ is singular. We prove that when $p = \omega(1/n)$,…

Let $n$ and $k$ be two positive integers with $k\leq n$ and $C$ an $n \times n$ matrix with nonnegative entries. In this paper, the rank-$k$ numerical range in the max algebra setting is introduced and studied. The related notions of the…