Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f. We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.     

Internal sets and internal functions in Colombeau theory

This paper is devoted to the study of generalized functions as pointwise functions (so-called internal functions) on certain sets of generalized points (so-called internal sets). We treat the case of the Colombeau algebras of generalized functions, for which these notions have turned out to constitute a fundamental technical tool. We provide general foundations for the notion of internal functions and internal sets and prove a saturation principle. Various applications to Colombeau algebras are given.     

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.     

Meromorphic Functions of Derivatives Covering Certain Finite Sets at the Same Points

ＭｅｒｏｍｏｒｐｈｉｃＦｕｎｃｔｉｏｎｓｏｆＤｅｒｉｖａｔｉｖｅｓＣｏｖｅｒｉｎｇＣｅｒｔａｉｎＦｉｎｉｔｅＳｅｔｓａｔｔｈｅＳａｍｅＰｏｉｎｔｓ￥ＷｅｎＺｈｏｎｇｌｉａｎｇ（ＷｅｎｚｈｏｕＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ，Ｚｈｅｊｉａｎｇ，３２５００...     

Large and Moderate Deviations of Random Upper Semicontinuous Functions

We firstly discuss the topological properties of the space of upper semicontinuous functions, and then we obtain large deviation principles for random upper semicontinuous functions under various topologies. Finally, we prove moderate deviation principles for random sets and random upper semicontinuous functions.     

超整函数族Julia集的Hausdorff维数

Stallard曾经用一族特殊的整函数说明了:超越整函数的Julia集的Hausdorff维数可以无限接近1.本文证明了该函数族的随机迭代的Julia集的Hausdorff维数也可无限接近于1.另一方面,对任意自然数M及任意实数d∈(1,2),本文给出了M个元素的整函数族其随机迭代的Julia集的Hausdorff维数等于d.     

Piecewise affine functions and polyhedral sets ∗

In this paper we present a number of characterizations of piecewise affine and piecewise linear functions defined on finite dimesional normed vector spaces. In particular we prove that a real-valued function is piecewise affine [resp. piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets[resp..Polyhedral cones]. Also,We show that the collection of all piecewise affine[resp.piecewise linear] functions. Furthermore, we prove that a function is piecewise affine[resp.piecewise linear] if it can be represented as a difference of two convex [resp.,sublinear] polyhedral fucntions.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

Concerning Nikodym-type sets in 3-dimensional curved spaces

We investigate maximal functions involving averages over geodesics in three-dimensional Riemannian manifolds. We first show that one can easily extend the Euclidean results of Bourgain and Wolff if one assumes constant curvature. These results need not hold if this assumption is dropped. Nonetheless, we formulate a generic geometric condition which allows favorable estimates. Curiously, this condition ensures that one is in some sense as far as possible from the constant curvature case. Assuming this, one can prove dimensional estimates for Nikodym-type sets which are essentially optimal. Optimal estimates for the related maximal functions are still open though.

     

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.     

集合上的递归函数

本文在Ｊｅｎｓｅｎ和Ｋａｒｐ工作的基础上引进了集合上的递归函数的概念．研究了递归集函数的初步性质，讨论了递归集函数与Ｊｅｎｓｅｎ和Ｋａｒｐ定义的原始递归集函数及递归数论函数之间的关系，并给出了ＺＦＣ的可定义集模型上递归集函数的范式定理．     

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.     

复解析动力系统发展概况

本文综述了由有理映照、整函数、亚纯函数、Ｃ＾＊上解析自映照及它们的子函数族分别所生成的复解析动力系统的一些结果。这些结果大多数是近十年来所取得的。这些成果均被分类归纳于如下４节：稳定集；Ｊｕｌｉａ集；参数空间；渐近值对复解析动力系统的影响。     

Properties of Restricted NCP Functions for Nonlinear Complementarity Problems

In this paper, we study restricted NCP functions which may be used to reformulate the nonlinear complementarity problem as a constrained minimization problem. In particular, we consider three classes of restricted NCP functions, two of them introduced by Solodov and the other proposed in this paper. We give conditions under which a minimization problem based on a restricted NCP function enjoys favorable properties, such as equivalence between a stationary point of the minimization problem and the nonlinear complementarity problem, strict complementarity at a solution of the minimization problem, and boundedness of the level sets of the objective function. We examine these properties for three restricted NCP functions and show that the merit function based on the restricted NCP function proposed in this paper enjoys favorable properties compared with those based on the other restricted NCP functions.     

广义凸集的支撑函数

本文讨论了笔者在[1]中提出的伪凸集,拟凸集的支撑函数与障碍锥的性质,并通过这些性质得出了二个闭性准则。     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

Optimal and perfect difference systems of sets

Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented.     

Plateaued functions,partial geometric difference sets,and partial geometric designs

Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ${\mathbb{Z}}_2$‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.