Related papers: A note on Erd\H{o}s matrices and Marcus\unicode{x2…

A complete description of 4-by-4 matrices $\begin{bmatrix}\alpha I & C \\D & \beta I\end{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is…

The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the…

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…

The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C\\0&B] and [A &0\\0 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996).…

In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials…

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we…

Given a symmetric matrix $M\in \{0,1,*\}^{D\times D}$, an $M$-partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no non-edge to a $1$. We give a computer-assisted proof that,…

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…

$ \newcommand{\schs}{\scriptstyle{\mathsf{S}}_1} $For all $n \ge 1$, we give an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices…

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by…

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that…

In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture,…

Let $n,\alpha\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$ nilpotent, that satisfy $AB-BA=A^{\alpha}$ or…

Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the…

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer…

We give an effective bound of the joint spectral radius $\rho(\Sigma)$ for a finite set $\Sigma$ of nonnegative matrices: For every $n$, \[ \sqrt[n]{\left(\frac{V}{UD}\right)^{D} \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\in\Sigma}(A_1\dots…

The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of…

Let $\| A \|_{\max} := \max_{i,j} |A_{i,j}|$ denote the maximum magnitude of entries of a given matrix $A$. In this paper we show that $$\max \left\{ \|U_r \|_{\max},\|V_r\|_{\max} \right\} \le \frac{(Cr)^{6r}}{\sqrt{N}},$$ where $U_r$ and…

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of this space where the maximal dimension is attained. Extensions of this result in the direction of linear…