给定主曲率函数的一类特殊曲面的位置向量场

本文给出了 Ｒ３,Ｒ２．１内给定主曲率函数的一类特殊曲面在定义域范围内的位置向量场,从而完满地解决了［１］中所讨论的问题．     

Approximation of Vector Valued Functions

In this paper, we extend the well-known result of Lipschitz approximation of lower semi-continuous functions to a class of lattice valued vector functions. We then use this approximation to get convergence results and we give two applications to the law of large numbers and to an ergodic theorem.     

Smooth Convex Approximation to the Maximum Eigenvalue Function

In this paper, we consider smooth convex approximations to the maximum eigenvalue function. To make it applicable to a wide class of applications, the study is conducted on the composite function of the maximum eigenvalue function and a linear operator mapping ^m to , the space of n-by-n symmetric matrices. The composite function in turn is the natural objective function of minimizing the maximum eigenvalue function over an affine space in . This leads to a sequence of smooth convex minimization problems governed by a smoothing parameter. As the parameter goes to zero, the original problem is recovered. We then develop a computable Hessian formula of the smooth convex functions, matrix representation of the Hessian, and study the regularity conditions which guarantee the nonsingularity of the Hessian matrices. The study on the well-posedness of the smooth convex function leads to a regularization method which is globally convergent.     

Smooth Approximation of Orlicz-Sobolev Maps Between Manifolds

The problem of density of smooth functions in Orlicz-Sobolev spaces of maps between compact manifolds, without topological restrictions, is addressed. Our contribution refines earlier results in the literature via ad hoc Orlicz space techniques.     

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.     

Normal Multiresolution Approximation of Curves

A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.     

一种新的光滑支持向量机的构造及其应用

光滑支持向量机模型是一个无约束、可微的最优化模型,人们可应用快速的最优化方法求解,从而降低计算复杂性.在前人工作的基础上研究基于样条函数的光滑支持向量机,采用广义三弯矩方法构造出六次样条光滑函数,分析了其性能及与正号函数的逼近精度,实现了求解六次样条光滑支持向量机的算法,与其它光滑支持向量机进行了比较,取得了较好的结果.最后将其应用于心脏病模型诊断,实验结果显示具有较高的精确度.     

Convenient Vector Spaces of Smooth Functions

By a convenient vector space is meant a locally convex IR-vector space which is separated, bornological and Mackey-complete. The theory of such spaces, initiated in [Kr 82], [Fr 82], and [FGK 83], has evolved into a book [FK 88]. In the preliminaries below we outline the principal features of this theory relevant to this paper. We are concerned mainly with questions about the reflexiveness of spaces C^∞(X, ?) for various X and matters closely related to this.     

On the Energy of a Unit Vector Field

The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M T ₁ M, where the unit tangent bundle T ₁ M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S ⁵,S ⁷,..., and an estimate on the index is obtained.     

Uniform Approximation of Curvature of Smooth Planar Curves

We estimate the error of curvature approximation for graphs of functions of the class W ^r for r ≥ 3 in the uniform metric.     

The Simultaneous Approximation Order by Hermite Interpolation in a Smooth Domain

Let D be a smooth domain in the complex plane. In D consider the simultaneous approximation to a function and its ith (0 ≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper. Received May 25, 2000, Revised November 3, 2000, Accepted December 7, 2000     

Smooth Approximation of Sobolev Functions on Planar Domains

We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case.     

光滑Jordan.曲线上一类极值多项式的逼近性质

关于Bieberbach多项式的逼近性质已有许多精彩结果,然而Jordan曲线上极值多项式的逼近性质却很少被考察.本文得到了C^1+α光滑Jordan曲线上一类极值多项式的一些逼近结果.     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.     

Best Approximation by Normal and Conormal Sets

The aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements—conormal sets—in the non-negative orthant ^I₊ of a finite-dimensional coordinate space ^I endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities f_α(x)0 (αA), x ^I₊ (respectively, f_α(x)0 (αA), x ^I₊), where f_α is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets.     

Smooth and Semismooth Newton Methods for Constrained Approximation and Estimation

In the article, we show that the constrained L ₂ approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.