On b-locally closed sets and related topics

The aim of this paper is to introduce the class of b-locally closed sets and two different notions of generalizations of continuous maps namely b-LC-continuous and b-LC-irresoluteness in a topological space. We discuss some of their properties. Several examples are provided to illustrate the behaviour of these new types of sets and functions.     

几种绝对连续函数的等价性

本文讨论了取值于Banach空间中函数的三种绝对连续相互等价性,得到强绝对 连续与绝对连续等价当且仅当空间是有限维的;绝对连续与弱绝对连续等价当且仅当空 间不含c0.     

Choquet like sets in function spaces

In convex analysis when studying function spaces of continuous affine functions, notions of a geometrical character like faces, split and parallel faces, exposed or Archimedean faces were investigated in detail by many authors. In this paper we transfer these notions to a more general setting of Choquet theory of abstract function spaces. We prefer a direct functional analytic approach to the treatment of problems instead of using a transfer of a function space to its state space. Methods invoked are based mainly on a measure theory and basic tools of functional analysis and are different from ones using a geometric visualization.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC(X, Y) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y. Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC(X, Y) and study some of its basic properties.     

A decomposition of continuity in ideal topological spaces

We introduce the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces and investigate some of their properties. Additionally, we obtain a decomposition of continuous functions.     

The notion of exhaustiveness and Ascoli-type theorems

In this paper we introduce the notion of exhaustiveness which applies for both families and nets of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and -convergence (continuous convergence). Using these results we obtain some Ascoli-type theorems dealing with exhaustiveness instead of equicontinuity. Also we deal with the corresponding notions of separate exhaustiveness and separate -convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function.     

Paley Type Inequalities for Orthogonal Series with Vector-Valued Coefficients

We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients with respect to a general orthonormal system Φ. This leads us to introduce the notions of Paley Φ-type and Φ-cotype for a Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.     

Computable Real-Valued Functions on Recursive Open and Closed Subsets of Euclidean Space

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.     

Integrals for functions with values in a partially ordered vector space

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.     

Completions of partial metrics into value lattices

In this paper we investigate some notions of completion of partial metric spaces, including the bicompletion, the Smyth completion, and a new “spherical completion”. Given an auxiliary relation, we show that it arises from a totally bounded partial metric space, and the spherical completion of such a space is its round ideal completion. We also give an example of a totally bounded partial metric space whose bicompletion and Smyth completion are not continuous posets. Finally, we present an example of a totally bounded partial metric giving rise to the Scott and lower topologies of a continuous poset, but whose spherical completion is not a continuous poset.     

Vector-valued interval functions and the Dedekind completion of <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">E</Emphasis>)

We consider interval valued functions with values in a Banach lattice E. Certain notions of continuity introduced earlier for real interval valued functions are generalised to the more general case considered here. As an application, we characterise the Dedekind completion of the space of continuous, E-valued functions on a paracompact \(T_{1}\)-space, extending a result of Anguelov.     

General,direct and self-implementation of social choice functions via protective equilibria

This paper investigates the possibility of implementing social choice functions via protective equilibria — a refinement of maximin behavior. We describe a necessary condition, as well as a set of sufficient conditions for a social choice function to be implementable. The set of implementable functions is certainly not empty, and it contains interesting procedures, allowing for some agents to have some veto power. However, positional or Condorcet-type methods cannot be implemented via protective equilibrium. The connections and the differences between different notions of implementation are also discussed.     

Some bitopological concepts based on the alternative effects of closure and interior operator

In this paper we give an approach for constructing classes of near open and near closed sets which have unusual implication relations. These new classes of subsets are based on the alternative effect of closure and interior operators with respect to two topologies. Also these classes of subsets are applied for constructing several classes of near continuous functions and some types of separation axioms called mildly binormal, almost ij-normal, almost ij-regular, quasi ij-regular and strongly S-ij-regular. Using the introduced functions, we generalize several preservation theorems of normality and regularity to bitopological spaces. Implications between notions are given and counter examples for some reverse directions are obtained. It should be noted that considering the space time as the product of two topologies, the topology of space and that of the space time will open the way for new line of research in the field of quantum gravity initiated by Witten and El-Naschie and many others (cf. [Chaos, Solitons & Fractals 17 (2003) 989; Chaos, Solitons & Fractals 7 (1996) 499; Int. J. Theor. Phys. 37 (1998) 2935; Phys. Today (1996) 24]).     

Complete observability of differential-algebraic systems with delays

We study the statement and solvability of complete observability problems for linear stationary differential-algebraic dynamical systems with delays (DAD systems. Since in the general case, the state space of such systems is infinite-dimensional and is not necessarily “minimal,” we consider various statements of problems depending on what states are observed. Our attention is focused on the simplest DAD system in symmetric form. We obtain efficient parametric criteria and analyze relationships between various notions of complete observability for DAD systems. In the case of DAD systems with scalar coefficients, we obtain a complete classification of notions of complete observability in the class of continuous initial functions with the continuous matching condition. We analyze the problem of computing the minimum number of outputs of a spectrally observable DAD system.     

广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.     

STRONGLY SEQUENTIALLY CONTINUOUS FUNCTIONS

Abstract

Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).     

Intrinsic metrics and Lipschitz functions

We study the notions of measurable metric and Lipschitz function which were introduced by N. Weaver ([12]), in the framework of Dirichlet spaces. To this respect, we bring some precisions and complements to [15], notably concerning links with the notion of intrinsic metric ([2]). In the particular case of an abstract Wiener space, we establish the relationship between these notions and that of H-metric ([5]) and μ-a.e. H-Lipschitz continuous function ([4]).     

Information widths

We introduce the notions of adaptive information widths of a set in a metric space and consider the problem of comparing them with nonadaptive widths. Exact results are obtained for one class of continuous functions that is not centrally symmetric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1506–1518, November, 1995.This work was partially supported by the International Science Foundation, Grant UB 1000.     

Discrete unbounded sets in a finite dimensional space and beyond

Discrete infinite sets in a finite dimensional space, i.e., infinite sets without finite limit points appear in various branches of analysis (zero and pole sets of meromorphic functions, various models in the mathematical theory of quasicrystals, and so on). Here we introduce some notions and present some new theorems connected with such sets.