多项式族的根分布不变性

本文研究多项式族的根分布不变性问题.我们首先提出了多项式族根分布的广义剔零原理,给出了参数空间中鲁棒稳定性的复边界定理和复棱边定理,并基于广义剔零原理得到了参数空间和系数空间中关于根分布的相应结论.另外,对于系数空间中鲁棒稳定性中实棱边定理,我们证明了它对稳定区域的要求还可放宽.对于一些更具几何特征的凸多面体和特定的稳定区域,棱边定理还可进一步改进,使所需检验的棱边数目与凸多面体的棱边数目无关.最后,我们给出了检验棱边根分布的Nyquist型图示方法.     

多项式凸多面体的稳定性及严格正实性

对于由区间多项式的凸组合描述的不确定系统,它的Hurwitz稳定性可由某个仅由顶点和棱边构成的子集来保证,且此集合的大小与系统的维数无关。     

关于一类多项式族的鲁棒稳定性研究

本文对包括区间多项式族和菱形多项式族的一类多项式族的鲁棒稳定性进行了研究.我们给出并证明了其中几个可用有限检验来判断的多项式族的Hurwitz稳定的实例;同时举例说明了有限检验对所有这一类多项式族并不总是可行的.     

结构摄动下多项式的鲁棒稳定性

本文对与Ｋｈａｒｉｔｏｎｏｖ多项多族相对偶的菱形族式项式的一类结构摄动下的鲁棒稳定性进行了详细的研究，提出了在此结构摄动下，菱形族形族多项式稳定当且仅当检验有限个顶点多项式或有限个边多项式的稳定性定理，而菱形族定量只是本定理的推论，另外，对低阶多项式族进行了讨论。     

复多项式族鲁棒稳定性的边界定理

依据文献[1]的思想和理论,我们曾得出了复系数多项式值映射,D等价等概念和边界定理,棱边定理等结论,这些结论是可以改进的.本文利用文献[1]的思想和方法给出了多项式边界定理的一般形式,改进了文献[2]中的某些结果.利用这一边界定理可以给出已有的关于多项式族鲁棒稳定性定理简洁的证明,同时为有关问题的进一步研究提供了有效的工具.     

二元多项式矩阵Smith型的进一步结果（英文）

系统等价在二维系统研究中发挥重要作用,它与二元多项式矩阵等价问题密切相关.文章主要研究几类二元多项式矩阵与其Smith型等价问题,得到一些新的结果及这几类矩阵分别与其Smith型等价的判别准则.这些准则可以通过计算给定多项式矩阵的低一阶子式生成理想的Gr(o|")bner基进行检验.     

Dynamic Responses Analysis of Bridges With Uncertain Parameters Under Moving Loads北大核心CSCD

针对具有不确定参数桥梁在移动荷载作用下的动力响应分析,首次建立了移动荷载作用下桥梁响应分析的多项式维数分解法.将结构的不确定参数视为独立的随机变量,构造了结构动力响应关于不确定参数的随机函数;进而采用一组变量数目逐次增加的成员函数实现结构动力响应的维数分解,并利用Fourier多项式展开推导成员函数的近似显式表达.通过降维积分方法降低概率空间内的积分维度,高效地实现了展开系数的计算.在数值算例中,进行了具有不确定参数桥梁在移动荷载作用下的响应估计,并与Monte-Carlo模拟进行对比,验证了该文方法的精确性和效率.     

一族具有大稳定域的显式方法的讨论

本文讨论由生成多项式确定的二步一阶含有二个参数的一族显式格式. 令     

二维格的覆盖半径

求格的覆盖半径是一个经典的困难问题,当格的维数不固定时,这个问题还没有非确定性的多项式时间的算法.已知的算法都是通过求Voronoi cell来计算覆盖半径,对于二维格,文章利用高斯算法给出了一个确定性的多项式时间的算法来求覆盖半径以及deep holes.     

多项式簇的稳定性分析

本文利用值映射的概念在系数空间里讨论了多项式簇的Hurwitz稳定性的鲁棒性,给出了判别多项式簇是Hurwitz稳定的原象定理,同时应用这个定理证明了边界定理,棱边定理,Харитонов定理和菱形簇定理.     

NP-complete problems for systems of Linear polynomial’s values divisibilities

The paper studies the algorithmic complexity of subproblems for satisfiability in positive integers of simultaneous divisibility of linear polynomials with nonnegative coefficients. In the general case, it is not known whether this problem is in the class NP, but that it is in NEXPTIME is known. The NP-completeness of two series of restricted versions of this problem such that a divisor of a linear polynomial is a number in the first case, and a linear polynomial is a divisor of a number in the second case is proved in the paper. The parameters providing the NP-completeness of these problems have been established. Their membership in the class P has been proven for smaller values of these parameters. For the general problem SIMULTANEOUS DIVISIBILITY OF LINEAR POLYNOMIALS, NP-hardness has been proven for its particular case, when the coefficients of the polynomials are only from the set {1, 2} and constant terms are only from the set {1, 5}.     

The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space

We consider an extremal problem for continuous functions that are nonpositive on a closed interval and can be represented by series in Legendre polynomials with nonnegative coefficients. This problem arises from the Delsarte method of finding an upper bound for the kissing number in the three-dimensional Euclidean space. We prove that the problem has a unique solution, which is a polynomial of degree 27. This polynomial is a linear combination of Legendre polynomials of degrees 0, 1, 2, 3, 4, 5, 8, 9, 10, 20, and 27 with positive coefficients; it has simple root 1/2 and five double roots in (?1, 1/2). We also consider the dual extremal problem for nonnegative measures on [?1, 1/2] and, in particular, prove that an extremal measure is unique.     

Polynomial zerofinders based on Szegő polynomials

The computation of zeros of polynomials is a classical computational problem. This paper presents two new zerofinders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegő polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial.     

On Reduction of Quaternionic Polynomials and Their Identical Equality

We show that any quaternionic polynomial with one variable can be represented in such a way that the number of its terms will be not larger than a certain number depending on the degree of the polynomial. We study also some particular cases where this number can be made even smaller. Then we use the above-mentioned representation to study how to check whether two given quaternionic polynomials with one variable are identically equal. We solve this problem for all linear polynomials and for some types of nonlinear polynomials.     

Quasi-permutation polynomials

A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite’s criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established. Different types of quasi-permutation polynomials and the problem of counting quasi-permutation polynomials of fixed degree are investigated.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993     

Solving polynomial least squares problems via semidefinite programming relaxations

A polynomial optimization problem whose objective function is represented as a sum of positive and even powers of polynomials, called a polynomial least squares problem, is considered. Methods to transform a polynomial least square problem to polynomial semidefinite programs to reduce degrees of the polynomials are discussed. Computational efficiency of solving the original polynomial least squares problem and the transformed polynomial semidefinite programs is compared. Numerical results on selected polynomial least square problems show better computational performance of a transformed polynomial semidefinite program, especially when degrees of the polynomials are larger.     

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.     

Computing the Invariant Polynomials of a Polynomial Matrix. II

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the W and V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles.