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.     

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.     

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.     

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.     

EXISTENCE AND ITERATION OF POSITIVE SYMMETRIC SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM

In this paper,we consider the existence of symmetric solutions to a nonlinear second order multi-point boundary value problem,and establish corresponding iterative schemes based on the monotone iterative method.     

Superattracting cycles for some Newton type iterative methods

The paper is devoted to the analysis of certain dynamical properties of a family of iterative Newton type methods used to find roots of non-linear equations. We present a procedure for constructing polynomials in such a way that superattracting cycles of any prescribed length occur when these iterative methods are applied. This paper completes the study begun in Amat, Bermúclez, Busquier, et al., (2009).     

ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATIONS

The existence, uniqueness and non-symmetric iterative approximation of solutions for a class of systems of mixed monotone operator equations are discussed. As an appplication, we utilize the results presented in this paper to study the existence and uniqueness problems of common solutions for a class of systems of functional equations arising in dynamic programming of multistage decision processes and a class of systems of nonlinear integral equations. The results obtained in this paper not only answer an open question suggested in [3] but also generalize the corresponding results of.     

SAGBI BASIS UNDER COMPOSITION Ⅱ

In this paper we mainly discuss that SAGBI basis under composition of polynomials.Poly-nomial composition is the operation of replacing the variables of a polynomial with otherpolynomials.The main question of this paper is:Does there exists a decision procedurethat will determine whetber a given composition commutes with SAGBI basis computationunder a given term ordering?We will give a better answer(using elementary rational rowtransformations of matrice).     

CLOSED SMOOTH SURFACE DEFINED FROM CUBIC TRIANGULAR SPLINES

In order to construct closed surfaces with continuous unit normal, we introduce a news pline space on an arbitrary closed mesh of three-sided faces. Our approach generalizes an idea of Goodman and is based on the concept of ‘Geometric continuity‘ for piecewise polynomial parametrizations. The functions in the spline space restricted to the faces are cubic triangular polynomials. A basis of the spline space is constructed of positive functions which sum to 1. It is also shown that the space is suitable for interpolating data at the midpoints of the faces。     

On Monotone Method for Second Order Periodic Boundary Value Problems and Periodic Solutions of Functional Difference Equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions to second order periodic boundary value problems and periodic solutions of functional difference equations. We obtain some new results under the lower solution a and upper solution βwith α≤β.     

INTEGRAL BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

By developing a comparison result and using the monotone iterative technique, we obtain the existence of the minimal and the maximal solutions to an integral boundary value problem for first order impulsive integro-differential equations.     

The growth of H-harmonic functions on the Heisenberg group

We discuss the relationship between the frequency and the growth of H-harmonic functions on the Heisenberg group.Precisely,we prove that an H-harmonic function must be a polynomial if its frequency is globally bounded.Moreover,we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.     

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of P~1 in Genus 0

In this paper, we study genus 0 equivariant relative Gromov–Witten invariants of P1 whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.     

SOME LEAST SQUARES ESTIMATES OF THE AUTOREGRESSIVE MODELS

In this paper, we present some iterative methods for solving lth order autoregressive models, prove global convergence for l=1 case, and the numerical results of new algorithms seem to be more efficient than the ones of Cochrane-Orcutt iterative method.