Related papers: Studying the divisibility of power LCM matrics by …

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring $M_n(\mathbb{Z})$ proposed by Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Zhao in 2014. We present…

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by…

Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and…

For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive…

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of…

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite set of distinct positive integers. Throughout this article we assume that the set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with…

Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n)_{n\geqslant1}$ be a sequence of nonzero elements in $R$. We prove that $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for all $n,m\geqslant1$ if and only if…

A matrix $M$ over the finite field $ \mathbb{F}_q $ is called \emph{maximum distance separable} (MDS) if all of its square submatrices are non-singular. These MDS matrices are very important in cryptography and coding theory because they…

We study the existence over small fields of Maximum Distance Separable (MDS) codes with generator matrices having specified supports (i.e. having specified locations of zero entries). This problem unifies and simplifies the problems posed…

The invertibility of LCM matrices and their Hadamard powers have been studied a lot over the years by many authors. Bourque and Ligh conjectured in 1992 that the LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots,…

This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given $m\times n$ binary matrix $M$, the GM-MDS conjecture, due to…

Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of discrete optimization problems, and subsequently, machine learning and control theory problems. On the one hand, SCCs are…

Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of…

Given $A\subseteq GL_2(\mathbb{F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying \[ \max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll…

We address two sets of long-standing open questions in probability theory, from a computational complexity perspective: divisibility of stochastic maps, and divisibility and decomposability of probability distributions. We prove that finite…

Let $\mathbb{F}_q$ be the finite field of $q$ elements, where $q=p^{m}$ with $p$ being a prime number and $m$ being a positive integer. Let $\mathcal{C}_{(q, n, \delta, h)}$ be a class of BCH codes of length $n$ and designed $\delta$. A…

In this paper, we study the eigenvalues of the GCD matrix $(S_n)$ and the LCM matrix $[S_n]$ defined on $S_n=\{1,2,\ldots,n\}$. We present upper and lower bounds for the smallest and the largest eigenvalues of $(S_n)$ and $[S_n]$ in terms…

A new necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L.A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The new…

Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in SICOMP 2008) proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We…

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not.…