A decomposition of α-continuity and semi-continuity

The following two decomposition theorems are obtained. (1) A function f is α-continuous if and only if f is pre-continuous and αα-continuous, (2) A function f is semi-continuous if and only if f is spr-continuous and αLC-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

On decomposition of continuity

Summary The main purpose of this paper is to introduce the concepts of η-sets, ηζ-sets,η-continuity and ηζ-continuity and to obtain a decomposition of continuity.     

Oscillation for almost continuity

Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function a _f : X → [0, ∞] such that f is almost continuous at x if and only if a _f (x) = 0. Some properties of this function are investigated. Supported by grant VEGA 2/6087/26 and APVT-51-006904.     

A note on δ-I-sets

We characterize δ-I-sets, discuss the relation between δ-sets, δ-I-sets and I _δ-sets and generalize some of the results.     

A Decomposition of a Weaker Form of Continuity

The aim of this paper is to give a decomposition of a weaker form of continuity, namely gα^**-continuity by providing the concepts of K-sets and K-continuity. This revised version was published online in June 2006 with corrections to the Cover Date.     

A unified theory of R-continuity

We introduce and investigate R-M-continuous functions defined between sets satisfying some minimal conditions. The functions enable us to formulate a unified theory of modifications of R-continuity [22]: R-irresoluteness [6], R-preirresoluteness [7].      

On decompositions of continuity via idealization

We have introduced α-I-open, semi-I-open and β-I-open sets via idealization and using these sets obtained new decompositions of continuity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Local approximation properties of certain class of linear positive operators via <Emphasis Type="Italic">I</Emphasis>—convergence

In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I—convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I—convergence sense.      

A ∞-modules over A ∞-algebras and Hochschild cohomology for modules over algebras

For each left graded module M over a graded algebra A, a Hochschild cochain complex S*(A, M) whose homology is responsible for the existence of nontrivial structures of A _∞-modules over A _∞-algebras on the given module is constructed.     

On Λ-generalized closed sets in topological spaces

We introduce new classes of sets called Λ _g -closed sets and Λ _g -open sets in topological spaces. We also investigate several properties of such sets. It turns out that Λ _g -closed sets and Λ _g -open sets are weaker forms of closed sets and open sets, respectively and stronger forms of g-closed sets and g-open sets, respectively. Dedicated to Professor Maximilian Ganster on the occasion of his 50th birthday     

A ∞-simplicial objects andA ∞-topological groups

The notion ofA _∞-topological group is introduced. It is proved that, if a space is obtained by deformation retraction of a topological groupG, then it has the structure of anA _∞-topological group, and theA _∞-homotopy equivalence holds. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 913–919, December, 1999.     

‖A −1‖∞ and equidiagonal-dominance

In this paper, the matrix of equidiagonal-dominance is defined and several theorems about ‖A ⁻¹‖_∞ and its evaluation are established. Many interesting numerical examples are given. This work is supported by the National Natural Science Foundation of China and the Science Foundation of Academy of Engineering Physics of China     

A note on W-I-continuous functions

In [1], Aç?kgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.     

Optimality of mappings and some separation axioms

Optimal mappings and optimally irresolute mappings are introduced. Fundamental properties of them are obtained. Preservation of well-known separation axioms (semi-ℐ₂, ℐ₂, semi-normality, normality, s-regularity, regularity) under such types of mappings (with some additional conditions) is studied.      

On fuzzy covering properties

The purpose of this paper is to introduce properties of the notion of α-compactness for fuzzy topological spaces. Moreover, α _c-compact spaces are introduced and properties of them are also discussed for fuzzy topological spaces.      

On Weak Generalized Stability and (c,d)-Pseudostable Random Variables via Functional Equations

In this paper we give a first attempt to define and study stable distributions with respect to the weak generalized convolution, focusing our attention on the symmetric weakly stable distribution. As in the case of the classical convolution, characterization of distributions stable in the sense of weak generalized convolution depends on solving some functional equations in the class of characteristic functions. This paper was partially written while the second author was a visiting professor of Delft Institute of Applied Mathematics, Delft University of Technology, Holland.     

A note on strong matrix summability via ideals

Following the line of recent works of Das et al. (2011) [15] and Savas and Das (2011) [19] we apply the notion of ideals to strong matrix summability, and in this note, we introduce the notion of strong A^I-summability with respect to an Orlicz function. We make certain investigations on the classes of sequences arising out of this new summability method.     

On elongations of totally projective groups and <Emphasis Type="Italic">α</Emphasis>-<Emphasis Type="Italic">Σ</Emphasis>-groups

We prove that a special α-elongation of a totally projective group is an α-Σ-group if and only if it is a totally projective group. This parallels our recent result in [5].     

Properties of Distortion Risk Measures

The current literature does not reach a consensus on which risk measures should be used in practice. Our objective is to give at least a partial solution to this problem. We study properties that a risk measure must satisfy to avoid inadequate portfolio selections. The properties that we propose for risk measures can help avoid the problems observed with popular measures, like Value at Risk (VaR _α ) or Conditional VaR _α (CVaR _α ). This leads to the definition of two new families: complete and adapted risk measures. Our focus is on risk measures generated by distortion functions. Two new properties are put forward for these: completeness, ensuring that the distortion risk measure uses all the information of the loss distribution, and adaptability, forcing the measure to use this information adequately. This research was partially funded by ^1,3 Welzia Management, SGIIC SA, RD Sistemas SA, Comunidad Autónoma de Madrid Grant s-0505/tic/000230, and MEyC Grant BEC2000-1388-C04-03 and by ² the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 36860-06.