On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

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, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

Existence and Construction of H 1-Splines of Class C k on a Three Directional Mesh

Let be the uniform triangulation generated by the usual three directional mesh of the plane and let H ₁ be the regular hexagon formed by the six triangles of surrounding the origin. We study the space of piecewise polynomial functions in C ^k (R ²) with support H ₁ having a sufficiently high degree n, which are invariant with respect to the group of symmetries of H ₁ and whose sum of integer translates is constant. Such splines are called H ₁-splines. We first compute the dimension of this space in function of n and k. Then we prove the existence of a unique H ₁-spline of minimal degree for any fixed k0. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of this spline.     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

水解聚丙烯酰胺分子量的表征

在 2 5℃、0 .5mol/LNaCl水溶液中 ,测定了 40种水解聚丙烯酰胺样品的特性粘数 ,求出了Mark -Houwink方程常数K、α值。在水解度 2～ 40 %之间 ,发现K、α对水解度具有强烈的依赖性 ,并且特征粘数与水解度的平方根成直线关系。利用本文所求的 K、α值预测本研究的水解聚丙烯酰胺样品分子量 ,其误差在± 8%之间 ;而对文献数据进行处理 ,其平均误差为 11.12 %。     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

Periodic solutions of damped differential systems with repulsive singular forces

We consider the periodic boundary value problem for the singular differential system: where , , and . The singular potential is of repulsive type in the sense that as . Under Habets-Sanchez's strong force condition on at the origin, the existence results, obtained by coincidence degree in this paper, have no restriction on the damping forces . Meanwhile, some quadratic growth of the restoring potentials at infinity is allowed.

     

Lusternik-Schnirelmann 定理的推广

本文根据Brouwer映射度的理论和微分拓扑的基本方法推广了Lusternik_Schnirelmann定理     

在Lp（s）空间中多元的Bernstein－Kantorovich算子的逼近性质

本文对定义在单形上的Ｂｅｒｎｓｔｅｉｎ－Ｋａｎｔｏｒｏｖｉｃｈ（以下简记为Ｂ－Ｋ）算子，给出了几个弱型不等式，得到了     

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.