DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respectto which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zerofree interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu-tion that a Markov system, under an additional assumption, is dense if and only if the maxi-mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.     

非扩张映射的不动点与变分包含问题的公共解

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inclusion for an inverse-strongly monotone mapping and a maximal monotone mapping in a real Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using the result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space.     

On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical？ We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.     

WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS

The purpose of this paper is to discuss representations of high order C⁰finite element spaces on simplicial meshes in any dimension.When computing with high order piecewise polynomials the conditioning of the basis is likely to be important.The main result of this paper is a construction of representations by frames such that the associated L²condition number is bounded independently of the polynomial degree.To our knowledge,such a representation has not been presented earlier.The main tools we will use for the construction is the bubble transform,introduced previously in[1],and properties of Jacobi polynomials on simplexes in higher dimensions.We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.     

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.     

REAL PIECEWISE ALGEBRAIC VARIETY

We give definitions of real piecewise algebraic variety and dimension.By using the techniques of real radical ideal,P-radical ideal,affine Hilbert polynomial,Bernstein-net from of polynomials on simplex,and decomposition of semi-algebraic set,etc.,we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in C^μ spline ring.     

STRONG UNIQUENESS OF GENERALIZED P0LYNOMIAL OF BEST APPROXIMATION HAVING BOUNDED COEFFICIENTS

Let the set of generalized polynomials haviug bounded coefficients bewhere g_1, g_2,…, g_n are linearly independent continuous functions defined on the interval [a, b], α_jβ_j, are extended real numbers satisfying α_j< ∞, β_j>-∞ and α_j≤β_j.Assume that f is a continuousfunction defined on a compact set X[a, b]. In the paper, we first give the sufficient conditions forthe polynomial of best uniform approxmation to f from K being unique and strongly unique.Furthermore, we give two forms of necessary and sufficient conditions for the best approximationto be strongly unique.     

OPTIMALLY ACCURATE PETROV-GALERKIN METHOD OF FINITE ELEMENTS

We perform analysis for a finite elements method applied to the singular self-adjoint problem.This method uses continuous piecewise polynomial spaces for the trial and the test spaces.We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjomt problem by approximating driving terms by Lagrange piecewise polynomials,linear,quadratic and cubic.Wt measure the erroe in max norm.We show that method is optimal of the first order in the error estimate,We also give numerical results for the Galerkin approximation.     

A GENERALIZED RESULT ON THE POLYNOMIAL-LIKE ITERATIVE EQUATION

Most results on the polynomial-like iterative equation are given under the condition that the given function is monotone,while a work by L.Liu and X.Gong gets nonmonotone PM solutions with height 1 when the given function is of the same case.Removing the condition on height for the given function,we first give a method to assert the nonexistence of C⁰solutions,then present equivalent conditions for the existence of PM solutions with finite height.Finally,as an application of the equivalent conditions,we construct the PM solutions in the case that the given function has one fort.     

Iterative Weighted Semiparametric Least Squares Estimation in Repeated Measurement Partially Linear Regression Models

Consider a repeated measurement partially linear regression model with an unknown vector parameter β, an unknown function g(.), and unknown heteroscedastic error variances. In order to improve the semiparametric generalized least squares estimator (SGLSE) of β, we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that it improves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given to determine the number of iterations. We also show that when the number of replicates is less than or equal to two, the IWSLSE can not improve upon the SGLSE. These results are generalizations of those in [2] to the case of semiparametric regressions.     

Iterative roots of piecewise monotonic functions with finite nonmonotonicity height

It is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function. In this paper, we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts. By investigating the trajectory of possible continuous roots, we give a general method to find all iterative roots of those functions with finite nonmonotonicity height.     

一维映射迭代根的非单调性及光滑性

迭代根问题是动力系统嵌入流问题的弱问题,是动态插值方法的基础.然而,即使是对一维映射,迭代根的非单调性和全局光滑性都是困难的问题.本文介绍这方面的若干新结果,尤其是关于严格逐段单调连续函数的连续迭代根的存在性和构造,以及迭代根局部光滑与全局光滑的新进展.最后给出多项式迭代根这类既严格逐段单调又具光滑性的迭代根的存在条件及计算方法.     

Non-monotonic iterative roots extended from characteristic intervals

It has been treated as a difficult problem to find iterative roots of non-monotonic functions. For some PM functions which do not increase the number of forts under iteration a method was given to obtain a non-monotonic iterative root by extending a monotone iterative root from the characteristic interval. In this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of PM functions.     

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order may be bounded from below by with , and that this bound may be improved at most by a factor . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations. Received December 1, 1997 / Revised version received February 25, 1999 / Published online 16 March 2000     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

C0 conjugacy between PM functions with nonmonotonicity height equal to 1

In this paper, a topological classification of piecewise monotone functions(abbreviated as PM functions) with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.     

Sturm's Method in Counting Roots of Random Polynomial Equations

The problem of finding the probability distribution of the number of zeros in some real interval of a random polynomial whose coefficients have a given continuous joint density function is considered. An algorithm which enables one to express this probability as a multiple integral is presented. Formulas for the number of zeros of random quadratic polynomials and random polynomials of higher order, some coefficients of which are non-random and equal to zero, are derived via use of the algorithm. Finally, the applicability of these formulas in numerical calculations is illustrated.     

Geometry of the locus of polynomials of degree 4 with iterative roots

We study polynomial iterative roots of polynomials and describe the locus of complex polynomials of degree 4 admitting a polynomial iterative square root.