Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是0,而链环的Jones多项式仅有可能的有理根是0和-1.给出厂作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有41,89,942,K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是O,而链环的Jones多项式仅有可能的有理根是0和-1.给出了作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有4_1,8_9,9_(42),K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

多项式零点保持线性映射

设H是维数大于2的复Hilbert空间,β(H)代表H上所有有界线性算子全体．假定Φ是从β(H)到其自身的弱连续线性双射．我们证明了映射Φ满足对所有的A,B∈β(H),AB=BA~*蕴涵Φ(A)Φ(B)=Φ(B)Φ(A)~*当且仅当存在非零实数c和酉算子U∈(?)(H),使得Φ(A)=cUAU~*对所有的A∈β(H)成立．     

李三系与 Laurent多项式代数F[t,t-1]

本文通过讨论Laurent多项式代数及其导子代数的对合自同构确定了一类具体的无限维单李三系,并且提供了一种利用Novikov代数上自然的李代数结构来构造李三系的方法.     

多项式函数零点重数的一个性质

研究探讨了多项式函数零点和零点的重数与函数D_k(f(x))=f~((k))(x)/f~((k+1))(x)之间的关系,得出了相应的结论.     

一种求多项式重零点的并行圆盘迭代法

本文给出一种三阶收敛的同时求多项式重零点的圆盘迭代法 ,并分析该法收敛的初始值条件 ,它改进了文献 [2 ]的结果 .     

亚纯函数多项式结合其导数的零点

研究p~t[f] aP~2[f](a≠0为常数）的零点问题．     

单零点多项式KNA算法的单调性

KNA 算法是计算多项式全部零点的单纯同伦算法。当多项式只有单零点时,本文证明。当计算达到某一深度后,KNA 算法是单调的,并且用多项式的系数给出开始出现单词性的深度。     

李三系与Laurent多项式代数F[t,t~(-1)]

本文通过讨论Laurent多项式代数及其导子代数的对合自同构确定了一类具体的无限维单李三系, 并且提供了一种利用Novikov代数上自然的李代数结构来构造李三系的方法.     

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481).     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair. This research is supported by grants from CNPq and FAPESP.     

On the Zeros of Orthogonal Polynomials: The Elliptic Case

Constructive Approximation - Let E = [–1, α] \cup [β, 1], –1 &;lt; α &;lt; β &;lt; 1, and let (pn) be orthogonal on E with respect to the weight function...     

Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces. J.M. Rodriguez supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 and MTM 2007-30904-E), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain. J.M. Sigarreta supported in part by a grant from M.E.C. (MTM 2006-13000-C03-02), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain.     

Ratio Asymptotics for Orthogonal Matrix Polynomials

Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind.     

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

     

Asymptotic Properties of Zeros of Hypergeometric Polynomials

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |z^k(1−z)|=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.     

Self-Inversive Polynomials with All Zeros on the Unit Circle

Conditions are given in the coefficients of a self-inversive polynomial under which all its zeros are on the unit circle.To my friend, Jean-Louis Nicolas at the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—30C15