Related papers: New result and some open problems on the primitive…

In this paper, we present a necessary and sufficient condition for a nonnegative tensor to be a primitive one, show that the exponent set of nonnegative primitive tensors with order $m(\ge n)$ and dimension $n$ is $\{k| 1\le k\le…

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

In this paper, we show that an order $m$ dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order $m$ dimension 2 strongly primitive tensors and the bound of the…

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \ldots M_{i_m}$ is positive (i.e. has all its entries $>0$). The length of…

In this paper, we first introduce the invertibility of even-order tensors and the separable tensors, including separable symmetry tensors and separable anti-symmetry tensors, defined respectively as the sum and the algebraic sum of rank-1…

A nonnegative matrix $A$ is said to be {\it primitive} if for some positive integer $m$, entries in $A^m$ are positive, notationally represented as $A^m>0.$ The smallest such $m$ is called the {\it exponent} of $A$, denoted $exp(A).$ For…

An algorithm for finding the eigenvalue of a nonnegative irreducible tensor was recently proposed by Michael Ng, Liqun Qi, and Guanglu Zhou in {\it Finding the largest eigenvalue of a nonnegative tensor}. However, the authors did not prove…

We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^(k-1)). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^(r/2) - O(n^(r/2)-1) when r is odd. The tensors are constructed from a simple recursive…

We define three types of upper (and lower) triangular blocked tensors, which are all generalizations of the triangular blocked matrices. We study some basic properties and characterizations of these three types of triangular blocked…

Let $(M_n)_{n\geq0}$ be the Mersenne sequence defined by $M_n=2^n-1$. Let $\omega(n)$ be the number of distinct prime divisors of $n.$ In this short note, we present a description of the Mersenne numbers satisfying $\omega(M_n)\leq3$.…

Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension…

A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains…

It is well-known that a symmetric matrix with its entries $\pm1$ is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of…

Let m be a positive integer and A an elementary abelian group of order q^r with r greater than or equal to 2 acting on a finite q'-group G. We show that if for some integer d such that 2^{d} is less than or equal to (r-1) the dth derived…

Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise as constitutive tensors of proportionality between basic physics quantities. The…

Inspired by Dickson's classification of regular ternary quadratic forms, we prove that there are no primitive regular $m$-gonal forms when $m$ is sufficiently large. In order to do so, we construct sequences of primes that are inert in a…

We study the primitive divisors of the terms of $(\Delta_n)_{n \geq 1}$, where $\Delta_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms…

Tensor type data are used recently in various application fields, and then a typical rank is important. Let $3\leq m\leq n$. We study typical ranks of $m\times n\times (m-1)n$ tensors over the real number field. Let $\rho$ be the…

In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\…

Let $n$ be a primitive non-deficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1, p_2 \cdots p_k$ are distinct primes. We prove that there exists an $i$ such that $$p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$$ We conjecture…