Related papers: On sufficient conditions for the total positivity …

Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i…

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$.…

We show that for fixed $A,B$, hermitian nonnegative definite matrices, and fixed $k$ the coefficients of the $t^k$ in the polynomial $\tr (A+tB)^m$ is positive if $\tr AB >0$ and $m>N(A,B,k)$.

We consider the maximal p-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when p = 2, and we describe some maps which…

For an $n\times n$ balanced symmetric matrix $T=(t_{i,j})$ with positive elements satisfying $t_{i,i}= \sum_{j\neq i} t_{i,j}$ and certain bounding conditions, we propose to use the matrix $S=(s_{i,j})$ to approximate its inverse, where…

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show…

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Minc's conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some…

For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional…

Let $\Phi$ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex $2 \times 2$ matrices, and let $\Omega$ be any finite-dimensional completely positive map. For $p=2$ and…

Let $(M,I,J,K)$ be a hyperkaehler manifold, $\dim_\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping…

A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if…

Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The…

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\…

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set $S_n$ is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix $(S_n)_f$. From…

We prove that if $\mathbb{F}$ is a field of positive odd characteristic $p,$ and $m,$ and $n$ are positive integers such that $m\geq2,$ and $n\leq p,$ every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F})$ which is sum of…

For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ with positive elements satisfying certain bounding conditions, we propose to use a diagonal matrix $S=(s_{i,j})_{n\times n}$ to approximate the inverse of $T$, where…

Let $Q(z,w)=-\prod_{k=1}^n [(z-a_k)(\bar{w}-\bar{a}_k)-R_k^2]$. M. Putinar and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\leq i,j\leq n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We extend…

The long run behaviour of linear dynamical systems is often studied by looking at eventual properties of matrices and recurrences that underlie the system. A basic problem that lies at the core of many questions in this setting is the…

Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and…

We consider certain subsets of the space of $n\times n$ matrices of the form $K = \cup_{i=1}^m SO(n)A_i$, and we prove that for $p>1, q \geq 1$ and for connected $\Omega'\subset\subset\Omega\subset \R^n$, there exists positive constant…