显拟凹函数的若干新性质

本文提出了显拟凹函数的若干新性质．这些新性质是用水平集、上水平集及其相对内部、相对边界的性质与它们之间的关系来表述的．     

On Duality for a Class of Quasiconcave Multiplicative Programs

Multiplicative programs are a difficult class of nonconvex programs that have received increasing attention because of their many applications. However, given their nonconvex nature, few theoretical results are available. In this paper, we study a particular case of these programs which involves the maximization of a quasiconcave function over a linear constraint set. Using results from conjugate function theory and generalized geometric programming, we derive a complete duality theory. The results are further specialized to linear multiplicative programming.     

Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems

For three-objective maximization problems involving continuous, semistrictly quasiconcave functions over a compact convex set, it is shown that the set of efficient solutions is connected. With that, an open problem stated by Choo, Schaible, and Chew in 1985 is solved.     

Approximation of Smooth Functions on the Semiaxis by Entire Functions of Bounded Half-Degree

In this paper, we study the pointwise approximation of functions defined on the real semiaxis and having an rth derivative bounded almost everywhere. The approximation is performed by means of entire functions of bounded half-degree, which were introduced by S. N. Bernstein. An asymptotically sharp estimate for pointwise approximation of this class of functions is obtained.     

Contractibility of the Efficient Set in Strictly Quasiconcave Vector Maximization

In this paper, we investigate the contractibility of the efficient frontier in a vector maximization problem defined by a continuous vector-valued strictly quasiconcave function and a convex compact set D in ^p . It is shown that the efficient frontier is contractible if one of the components of g is strongly quasiconcave on X. This work extends a result by Sun (see Ref. 1), which confirms the connectedness of the efficient frontier.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omit a Function II

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ? ?, all of whose zeros have multiplicity at least k, where k ?? 2 is an integer. Let h(z) ? 0 and ?? be a meromorphic function on D. Assume that the following two conditions hold for every f ?? F: $$ \begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $$ Then F is normal on D.     

On the Orders of Nonlinear Approximations for Classes of Functions of Given Form

Suppose that Δ ₊ ^s is the set of functions x: I → ℝ on a finite interval I such that the divided differences [x; t ₀,..., t _s ] of order _s ∈ ℕ of these functions are nonnegative for all collections from (s +1) different points t ₀,..., t _s ∈ I. For all s ∈ ℕ and 1 ≤ p ≤ ∞, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of L _p for classes , where B _p is the unit ball in L _p . We also establish the asymptotics of pseudodimensional widths in L _p of these classes of functions.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 98–114.Original Russian Text Copyright © 2005 by V. N. Konovalov.     

几种辅助函数方法及应用

本通过举例说明的方法说明了寻求辅助函数的几种方法及其在解题中的作用。     

Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains

We study the contractibility of the efficient solution set of strictly quasiconcave vector maximization problems on (possibly) noncompact feasible domains. It is proved that the efficient solution set is contractible if at least one of the objective functions is strongly quasiconcave and any intersection of level sets of the objective functions is a compact (possibly empty) set. This theorem generalizes the main result of Benoist (Ref.1), which was established for problems on compact feasible domains.The authors thank Dr. T. D. Phuong, Dr. T. X. D. Ha, and the referees for helpful comments and suggestions.     

与几何凹函数相关的函数的准线性研究

先介绍四个与几何凹函数定积分有关的不等式,后相应定义了四个函数,证明了它们具有某种准线性.作为应用,由此可得到这些函数的单调性,使得这四个不等式在理论上得到推广和完善,并且部分解决了一个公开问题.     

A Property of Piecewise Smooth Functions

Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

New Representation Formula for Convex Functions on Separable Normed Spaces

We derive a new representation formula for lower-semicontinuous convex functions on separable normed spaces. As a consequence of this formula, we obtain a C -approximation method for convex functions which are not necessarily differentiable.     

Analytical representation of concave and quasiconcave homogeneous functions

The formula which implements bijection between the class of concave linearly homogeneous functions defined on the nonnegative orthant of an arithmetic space and the simpler class of concave functions defined on the standard (probabilistic) simplex is presented. Two generalizations of this formula for analytical representation of quasiconcave homogeneous function are also proposed. These formulas particularly extend opportunities of modelling production objects and consumption.     

分担函数集的正规定则

本文将亚纯函数正规性与分担函数集相结合,探讨了亚纯函数族F正规与F中任意两个函数f与g分担函数的个数,f与其导数f'分担函数的个数及函数f的重值的个数三者之间的关系,给出了一个综合判定亚纯函数族正规的充分条件.     

两个函数的拓扑极大极小定理

A topological minimax theorem involving two functions is derived. It generalizes Greco and Horvath＇minimax theorem given by them in 2002.     

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.     

锥有效拟凹集的有效点集的连通性

本文引进了锥有效拟凹集的概念，讨论了R ^m-严格拟凹与R ^m－有效拟凹的关系，证明了一个紧集为锥有效拟凹且其有效点集为闭集时，这个有效点集是连通的。