一类矩阵多项式秩的恒等式

得到了一类矩阵多项式秩的恒等式,改进和推广了近期的一些结果.     

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence whose moment matrix is positive semidefinite and has finite rank is the sequence of moments of an -atomic nonnegative measure on . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure having degree at most if the principal submatrix of (indexed by all monomials of degree ) has full rank . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set , where and . If is positive semidefinite and has a flat extension such that all localizing matrices are positive semidefinite, then has an atomic representing measure supported by . We also review an application of this result to the problem of minimizing a polynomial over the set .

     

一类矩阵多项式的秩特征

给出了一类矩阵多项式的秩特征定理及它的多种证法.     

Morgan-Voyce多项式的一些新性质

本文证明了Morgan-Voyce多项式的零点在闭区间$[-4,0]$上是稠密的.本文也证明了Morgan-Voyce多项式系数的分布是渐近正态的,以及它的系数矩阵是全正的.     

两类递归图的Tutte多项式

本文用转移矩阵给出了C2×Pn和N2∨Pn的Tutte多项式的公式,根据公式设计了M ap le程序,该程序对每个固定的整数n 2都能计算出C2×Pn和N2∨Pn的Tutte多项式.     

实二次多项式正定性的判定

讨论n元实二次多项式f(x1,x2,…,xn)=(1 xT)A1x(x=(x1,…,xn)T)正定性的判定方法.     

车多项式对Ferrers矩阵的刻画

In this paper,we show that there exist precisely W(A) Ferrers matrices F(C1,C2,…,cm)such that the rook polynomials is equal to the rook polynomial of Ferrers matrix F(b1,b2,…,bm), where A={b1,b2-1,…,bm-m 1} is a repeated set,W(A) is weight of A.     

Invariant polynomials of matrices with prescibed entries

We study the invariant polynomials of an n by n matrix when n-1 arbitrary entries are prescribed.     

On tridiagonal conjugate-normal matrices

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.     

Bernoulli多项式的积分多项式

首次研究了 Bernoulli多项式的积分多项式 .首先 ,给出这类多项式的定义和基本性质 ;其次 ,建立两类幂和多项式的相互关系 ;最后 ,介绍上述结果在求解自然数幂和公式方面的应用 .     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

Almost Summing Polynomials

This paper introduces the class of almost summing homogeneous polynomials between Banach spaces. A characterization by means of vector‐valued sequences is proved, connections with the theory of absolutely summing polynomials are established and several examples are given. Some composition and coincidence theorems are obtained and the case of spaces with type or cotype is also investigated.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

纽结与多项式

In this paper, we deal with some corresponding relations between knots and polynomials by using the basic properties of knot polynomials （such as, some special values of knot polynomials, the Arf invariant and derivative of knot polynomials）. We give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than five, is the Jones polynomial of a certain knot.     

On the similarity of matrices of even order

We study invariants of simultaneous similarity of a pair of matrices of even order 2k over the field of complex numbers for the case in which all elementary divisors of the corresponding characteristic polynomial matrix are identical and their number is k.     

Positive Values of Polynomials

Mathematical Notes -     

Critical pairs of matrices for the degree of the minimal polynomial of the Kronecker sum

In [4] Dias da Silva and Hamidoune obtained a lower bound for the degree of the minimal polynomial of the Kronecker sum of two matrices in terms of the degrees of the minimal polynomials of the matrices. We determine the pairs of matrices for which this bound is reached.     

矩阵可对角化的零化多项式的判定

给定数域F上的方阵A,借助等价标准形和数学归纳法证明了如果存在数域F上互素的一次因式乘积的多项式是A的零化多项式,则A可对角化.