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

In this paper we propose a new scaling method to study the Schur complements of $SDD_{1}$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the…

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In…

A matrix $T \in \M_n(\C)$ is \emph{UECSM} if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we…

The Birkhoff polytope $\mathcal{B}_d$ consisting of all bistochastic matrices of order $d$ assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset $\mathcal{U}_d$ of…

We introduce the notions of $d$-orthostochastic, $d$-unistochastic, and $d$-qustochastic matrices. These are the particular cases of $F^d$-bistochastic matrices where $F$ is real or complex numbers or quaternions. The concept is motivated…

In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $\text{div}[\mathbf{A}(x) \nabla u] = \text{div}{\mathbf{F}(x)}$, where the matrix $\mathbf{A}$ is just measurable and its skew-symmetric part…

Applying the $A_4$ symmetry in the scenario of unimodular second scheme of trimaximal $TM_2$ mixing matrix, where the charged lepton mass matrix is diagonal and the nature of neutrinos are Majorana, we investigate and analyze feasible two…

We show that the parameters of the quark mass matrices M_u and M_d of a 4 texture zero model, exhibit an interesting hierarchical regularity we have called equal spacing rule, and that this regularity leads us to a further observation that…

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{\em Adv.\ Appl.\ Math.} {\bf…

We are interested in entire solutions for the semilinear biharmonic equation $\Delta^{2}u=f(u)$ in $\R^N$, where $f(u)=e^{u}$ or $-u^{-p}\ (p>0)$. For the exponential case, we prove that any classical entire solution verifies $-\Delta u>0$…

A nonnegative multidimensional matrix is called polystochastic if the sum of its entries over each line is equal to $1$. In this paper we overview known results on positiveness of the permanent of polystochastic matrices and prove that the…

The spectral study of nonnegative and more specifically stochastic matrices is an important topic in matrix theory. In this paper, we prove a conjecture, formulated by Ran and Teng, which states that the second largest eigenvalue of an…

We consider finite simple graphs. Given a graph $H$ and a positive integer $n,$ the Tur\'{a}n number of $H$ for the order $n,$ denoted ${\rm ex}(n,H),$ is the maximum size of a graph of order $n$ not containing $H$ as a subgraph. Erd\H{o}s…

Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$. The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated…

In this paper, we explore quadratures for the evaluation of $B^T \phi(A) B$ where $A$ is a symmetric positive-definite (s.p.d.) matrix in $\mathbb{R}^{n \times n}$, $B$ is a tall matrix in $\mathbb{R}^{n \times p}$, and $\phi(\cdot)$…

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…

This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0…

An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines,…