On Countable Products of Finite Hausdorff Spaces

We investigate in ZF (i.e., Zermelo‐Fraenke set theory without the axiom of choice) conditions that are necessary and sufficient for countable products ∏_m∈ℕX_m of (a) finite Hausdorff spaces X_m resp. (b) Hausdorff spaces X_m with at most n points to be compact resp. Baire. Typica results: (i) Countable products of finite Hausdorff spaces are compact (resp. Baire) if and only if countable products of non‐empty finite sets are non‐empty. (ii) Countable products of discrete spaces with at most n + 1 points are compact (resp. Baire) if and only if countable products of non‐empty sets with at most n points are non‐empty.     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

Local completeness, drop theorem and Ekeland's variational principle

By using a very general drop theorem in locally convex spaces we obtain some extended versions of Ekeland's variational principle, which only need assume local completeness of some related sets and improve Hamel's recent results. From this, we derive some new versions of Caristi's fixed points theorems. In the framework of locally convex spaces, we prove that Daneš' drop theorem, Ekeland's variational principle, Caristi's fixed points theorem and Phelps lemma are equivalent to each other.     

Minimum Distance to the Complement of a Convex Set: Duality Result

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.     

Quantitative (p,q) theorems in combinatorial geometry

We show quantitative versions of classical results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.     

Properties of Solutions to Second Order Evolution Control Systems. I

We consider a control system described by a nonlinear second order evolution equation defined on an evolution triple of Banach spaces (Gelfand triple) with a mixed multivalued control constraint whose values are nonconvex closed sets. Alongside the original system we consider a system with the following control constraints: a constraint whose values are the closed convex hull of the values of the original constraint and a constraint whose values are extreme points of the constraint which belong simultaneously to the original constraint. By a solution to the system we mean an admissible trajectory-control pair. In this part of the article we study existence questions for solutions to the control system with various constraints and density of the solution set with nonconvex constraints in the solution set with convexified constraints.     

On a theorem of E. Helly

E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

     

A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

知识规律与规律的属性扰动

By employing the knowledge(R-element equivalence class)in one direction Srough sets and dual of one direction S-rough sets,the concept of knowledge law is given;the generation theorem of knowledge law,the excursion theorem of knowledge law,and the attribute disturbance discernible theorem of knowledge law are proposed.Knowledge law is a new characteristic of S-rough sets.     

可列集合套

[1]中给出模糊集的分解定理与表现定理，指出可利用集合套H＝{H（λ）|λ∈[0,1]}刻化模糊集。本文定义可列集合套H＝{H（a）|a∈Q}，Q是（0，1）的可列稠密子集。相应地给出新的分解定理与表现定理。指出可利用可列集合套刻化的模糊集。     

Probabilities of incidence between lines and a plane curve over finite fields

We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.     

简论局部指标定理的证明

本文描述一个形式算法（即Diana算法）来证明Dirac算子的局部指标定理,并指出算法的魔力需附上一个逻辑的证明,在这种理解下,可看出Getzler的证明有一些缺陷．     

Function S-rough Sets and Their Law Characteristic

Function S-rough sets(Function Singular rough sets) are defined by R-function equivalence class which has dynamic characteristic, and a function is s law, function S-rough sets have law characteristic. Function S-rough sets has these forms: function one direction S-rough sets, function two direction S-rough sets and dual of function one direction S-rough sets. This paper presents the law characteristic of function one direction S-rough sets and puts forward the theorems of law-chain-attribute and law-belt. Function S-rough sets is s new research direction of the rough sets theory.     

P-集合与双信息规律生成

P-集合(pacdet sets)是由内P-集合X~F(internal packet set X~F)与外P-集合X~F(outer packet set X~F)构成的集合对.P-集合是把动态特性引入到普通集合中,利用普通集合(cantor set)被提出的.在一定的条件下,P-集合能够回到普通集合的原点.利用P-集合,给出双信息规律生成概念,给出双信息规律依赖的属性特征,提出双信息规律生成定理,辨识定理,给出应用.P-集合是研究动态信息系统的一个新的数学理论与数学方法.     

随机集的极限性质

本文研究了Banach空间随机紧集的极限性质.利用嵌入定理,得到了Banach中独立同分布随机紧集的Minkowski和的中偏差原理.     

Attracting Sets for Increasing Co‐Radiant and Topical Operators

A generalization of the Perron‐Frobenius theorem to increasing positively homogeneous of degree one operators is extended to increasing co‐radiant and topical operators, which are of interest in mathematical economics. In particular, small attracting sets containing the limit points of all sequences generated by iteration of such operators are determined.     

Ned sets on a hyperplane

Under study are the sets in ℝ ⁿ (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.     

知识的属性扰动与属性扰动定理

By using the dynamic characteristic of one direction S-rough sets(one direction singular rough sets) and dual of one direction S-rough sets(dual of one direction singular rough sets),the concepts of attribute disturbance of knowledge,the attribute disturbance degree of knowledge,and the disturbance coefficient of knowledge are given.By employing these concepts,the cardinal order theorem of the attribute disturbance knowledge,the unit circle theorem of the attribute disturbance knowledge,and the discernib...     

Medians of Discrete Sets according to a Linear Distance

In this paper we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q . We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q . We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions. Received April 20, 1999, and in revised form November 16, 1999.