Related papers: On the maximal spread of symmetric Bohemian matric…

The spread of a matrix is defined as the maximum of distances between any two eigenvalues of that matrix. In this paper we investigate spread maximization as a function on compact convex subset of the set of real symmetric matrices. We…

Let $A = [a_{i j}]_{i,j=1}^n$ be a nonnegative matrix with $a_{1 1} = 0$. We prove some lower bounds for the spread $s(A)$ of $A$ that is defined as the maximum distance between any two eigenvalues of $A$. If $A$ has only two distinct…

The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalue. The study of graph-related matrices has attracted considerable attention, leading to a substantial body of findings. In this…

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum…

We discuss some open problems concerning the maximal spread of coherent distributions. We prove a sharp bound on $\mathbb{E}|X-Y|^{\alpha}$ for $(X,Y)$ coherent and $\alpha \le 2$, and establish a novel connection between coherent…

Given a directed graph $G$, the spread of $G$ is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what $n$-vertex directed graph maximizes spread, and whether…

A matrix is called Bohemian if its entries are sampled from a finite set of integers. We determine the maximum absolute determinant of upper Hessenberg Bohemian Matrices for which the subdiagonal entries are fixed to be $1$ and upper…

Let $A$ be a $n\times n$ complex Hermitian matrix and let $\lambda(A)=(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising…

We calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank, generalizing a well-known result of Adams {\em et al.}

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduced this concept and study when…

The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread…

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers -- hence the name, from the acronym…

In matrix analysis, the \textit{Wielandt-Mirsky conjecture} states that $$ dist(\sigma(A), \sigma(B)) \leq \|A-B\|, $$ for any normal matrices $ A, B \in \mathbb C^{n\times n}$ and any operator norm $\|\cdot \|$ on $C^{n\times n}$. Here…

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a…

A set $X$ in the Euclidean space $\mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $\mathbb{R}^d$ is said to be maximal if there does…

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the 2- dimensional probability distribution P{Sn = x,max1?j?n Sn = a} of being at position x after n…

We construct an $S_3$-symmetric probability distribution on $\{(a,b,c) \in \mathbb{Z}_{\geq 0}^3 \: : \: a+b+c =n \}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\ldots,n\}$ with mean…

This article tackles the problem of existence and classification of maximal growth distributions on smooth manifolds. We show that maximal growth distributions of rank$>2$ abide by a full $h$-principle in all dimensions. We make use of M.…

Suppose $\left \{ X_{i,k}; 1\le i \le p, 1\le k \le n \right \} $ is an array of i.i.d.~real random variables. Let $\left \{ p=p_{n}; n \ge1 \right \} $ be positive integers. Consider the maximum interpoint distance $M_{n}=\max_{1\le i<…