Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Tree‐Properties for Ordered Sets

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree‐property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ₀‐arborescent.     

Set theory for fuzzy sets of higher level

We extend the usual notion of fuzzy set in such a way that the elements of fuzzy sets again can be fuzzy sets. For such fuzzy sets of higher level the fuzzy set theoretic operations are generalized up to the notion of a fuzzy mapping. In our presentation of the results we use a suitable many valued logic, indicating in this way the close formal connections between fuzzy and classical set theory.     

A convex sets in general position

The notion of convex cones in general position has turned out to be useful in convex programming theory. In this paper we extend the notion to convex sets and give some characterizations which yield a better insight into this concept. We also consider the case of convex sets in S-general position.     

复广义权的拟拓扑和可权集

本文讨论了Hausdorff空间中Borel集、细Borel集和拟开集之间的关系;引入复广义可权集的概念并讨论其性质.     

On a Family of Covering Extended Building Sets

Recently Davis and Jedwab introduced the notion of covering extended building sets to construct abelian difference sets. In this paper we consider a family of covering extended building sets similar to the ones corresponding to Hadamard difference sets and Spence difference sets and derive some numerical restrictions on the parameters.     

Between *-closed sets and ℐ-<Emphasis Type="Italic">g</Emphasis>-closed sets in ideal topological spaces

Dontchev et al. [4] introduced the notion of ℐ-g-closed sets. In [16], the further properties of ℐ-g-closed sets are investigated. In this paper, we introduce the notion of ℐ-mg-closed sets and obtain the unified characterizations for certain families of subsets between *-closed sets and ℐ-g-closed sets in an ideal topological space.     

Some Supports of Fourier Transforms of Singular Measures are not Rajchman

The notion of Riesz sets tells us that a support of Fourier transform of a measure with non-trivial singular part has to be large. The notion of Rajchman sets tells us that if the Fourier transform tends to zero at infinity outside a small set, then it tends to zero even on the small set. Here we present a new angle of an old question: Whether every Rajchman set should be Riesz.     

Well-posedness and convexity in vector optimization

We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers.     

Normal cone approximation and offset shape isotopy

This work addresses the problem of the approximation of the normals of the offsets of general compact sets in Euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the μ-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets that is stable under perturbation.     

Differential games with fixed terminal time and estimation of the instability degree of sets in these games

This paper contributes to the theory of differential games. A game problem of bringing a conflict-controlled system to a compact target set is analyzed. Sets in the position space that terminate on the target set and are not stable bridges are considered. The notion of stability defect of these sets is examined. It is demonstrated how the notion of stability defect can be used to construct sets with relatively good geometry that are at the same time convenient for the first player to play the game successfully.     

模糊粗糙集及粗糙模糊集的模糊度

1965年,Zadeh提出了Fuzzy集理论,1982年,Z.Pawlak提出Rough集理论。将二者结合而形成的模糊粗糙集（FR集）及粗糙模糊集（RF集）近年来越来越受到国际学术界的关注。本文所研究的FR集及RF集的模糊度,是对FR集及RF集模糊程度的一种度量,进而引进了相应的明可夫斯基距离,明可夫斯基模糊度和Shannon模糊度。     

自由半群的估计熵和△-弱混合集

本文对自由半群作用的动力系统引入了估计熵和△-弱混合集的概念,得到一些性质.通过引入△-熊混沌集,给出了△-弱混合集的一个等价刻画.     

VIP systems in partial semigroups

A VIP system is a polynomial-type generalization of the notion of an IP system, i.e., a set of finite sums. We extend the notion of VIP system to commutative partial semigroups and obtain an analogue of Furstenberg's central sets theorem for these systems which extends the polynomial Hales–Jewett Theorem of Bergelson and Leibman. Several Ramsey theoretic consequences, including the central sets theorem itself, are then derived from these results.     

L-可定义集与L-粗糙概念格

对基于剩余格L的L-广形式背景引入了L-可定义集概念,研究L-可定义集的性质及其与L-粗糙概念之间的关系.培出L-广可定义集的刻画;得到常值L-案均为L-可定义集的若干等价条件;证明全体L-可定义集在L-集包含序下是相应L-幂集格及L-粗糙概念格的子完备格;给出全体L-可定义集恰为全体L-粗糙概念外延的一个充要条件.     

Twofold fuzzy sets and rough sets—Some issues in knowledge representation

This paper deals with the representation of sets where the membership of some elements may be ill-known rather than just a matter of degree as in a fuzzy set. The notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete. A twofold fuzzy set is made of a nested pair of fuzzy sets: the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong. Twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by Pawlak. Set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined. Twofold fuzzy relations are also introduced. Finally, various applications of twofold fuzzy sets in knowledge representation are briefly discussed.     

Approximate decidability in euclidean spaces

We study concepts of decidability (recursivity) for subsets of Euclidean spaces ?^k within the framework of approximate computability (type two theory of effectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.     

Hausdorff dimension of homogeneous perfect sets

Summary We introduce the notion of homogeneous perfect sets as a generalization of Cantor type sets and determine their exact dimension based on the length of their fundamental intervals and the gaps between them. Some earlier results regarding the dimension of Cantor type sets are shown to be special cases of our main theorem.     

A characterization of smooth normed linear spaces

In this paper a new characterization of smooth normed linear spaces is discussed using the notion of proximal points of a pair of convex sets. It is proved that a normed linear space is smooth if and only if for each pair of convex sets, points which are mutually nearest to each other from the respective sets are proximal.