Hermite插值多项式在不同基下的显式表示

本文利用对偶基的概念,导出了 Ｈｅｒｍ ｉｔｅ 插值多项式在不同基下的显式表示,这给人们对 Ｈｅｒｍ ｉｔ插值多项式在不同基下从一种表示转换到另一种表示带来极大的方便     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

Orthogonal Algebraic Polynomial Schauder Bases of Optimal Degree

For any fixed we construct an orthonormal Schauder basis for C[-1,1] consisting of algebraic polynomials with The orthogonality is with respect to the Chebyshev weight.     

Structure Analysis of Polynomial Modules with Pommaret Bases

We show how Pommaret bases of submodules of free polynomial modules can be used to extract structural information like the dimension, the depth or the Castelnuovo‐Mumford regularity. In particular, we discuss the syzygy theory of Pommaret bases and its use for the construction of free resolutions.     

Rearrangements of the Haar System that Preserve BMO

Let D denote the collection of dyadic intervals in the unitinterval. Let be a rearrangement of the dyadic intervals. Westudy the induced operator where h_I is the L normalized Haar function. We find geometricconditions on which are necessary and sufficient for T to bebounded on BMO. We also characterize the rearrangements of theL^p normalized Haar system in L^p. 1991 Mathematics Subject Classification:42C20, 43A17, 47B38.     

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

We describe a novel approach to the computation of free resolutions and of Betti numbers of polynomial modules based on a combination of the theory of involutive bases with algebraic discrete Morse theory. This approach allows for the first time to compute Betti numbers (even single ones) without determining a whole resolution which in many cases drastically reduces the computation time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

量子代数${\cal U}_q(f(K,H))$的量子伴随作用

Let H = uq（sl（2）） or u（sl（2））. By means of the standard basis of polynomial algebras, the Glebsch-Gordan formula and quantum Clebsch-Gordan formula are proved by a unified method, and the explicit formula of the decomposition of V（1）^n into the direct sum of simple modules is given in this paper.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

二维多项式本征应变边界积分方程及其数值验证

针对弹性介质中的椭圆形异质体,给出了低阶多项式分布的二维本征应变边界积分方程和相应的Eshelby张量的定义．以边界元分域法为参照,利用含有单个异质体的弹性介质对提出的计算模型和算法进行了数值验证．结果表明该算法取得较大的改进,其计算效率高于传统的边界元法,计算精度则高于采用常数本征应变的计算模型．     

On a Polynomial Fractional Formulation for Independence Number of a Graph

In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independent set problem.     

From Bounded Families of Localized Cosines to Bi-Orthogonal Riesz Bases via Shift-Invariance

The notion of bi-inner product functionals generated by two Bessel sequences {f _n } and {g _n } of functions from L ² was introduced in our earlier work as a vehicle to identify dual frames and bi-orthogonal Riesz bases of L ². The objective was to find conditions under which P is a constant mu<iple of the inner product <f,g> of L ². A necessary and suffici condition derived in is that P is both spatial shift-invariant and phase shift-invariant. A<hough these two shift-invariance properties are, in general, unrelated, it could happen that one is a consequence of the other for certain clases of Bessel sequences {f _n } and {g _n }. In this paper, we show that, indeed, for localized cosines with two-overlapping windoes (i.e., only adjacent window functions are allowed to overlap), spatial shift-invariance of P is already sufficient to guarantee that P is a constant mu<iple of the inner product, while phase shift-invariance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is a consequence of spatial shift-invariance, but the converse is not valid. As an application, we also show that two families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonal Riesz bases of L ², if and only if P is spatial shift-invariant. In addition, we apply this resu< to generalize a resu< on characterization of dual localized cosine bases in our earlier work in to the mu<ivariate setting. A method for computing the dual windows is also given in this paper.     

分担多项式的亚纯函数的进一步结果(英文)

In this paper,we use the theory of value distribution and study the uniqueness of meromorphic functions.We will prove the following result：Let f（z）and g（z）be two transcendental meromorphic functions,p（z）a polynomial of degree k,n≥max{11,k＋1}a positive integer.If fn（z）f（z）and gn（z）g（z）share p（z）CM,then either f（z）=c1ec p（z）dz, g（z）=c2e ？c p（z）dz ,where c1,c2 and c are three constants satisfying（c1c2） n＋1 c2=-1 or f（z）≡tg（z）for a constant t such that tn＋1=1.     

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada, and by the MITACS Network of Centres of Excellence.     

On Some Properties of Polynomial Bases of Subspaces over the Field of Rational Functions in Several Variables

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a free polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.     

亚纯函数及其导数多项式的值分布

本文主要得到亚纯函数及其导数的多项式的零点的定量估计，推广并改进了Ｗ．Ｋ．Ｈａｙｍａｎ及敖海龙等人的有关结果．