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.     

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.     

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 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].      

Note on generalized open sets

Characterizations of γ-open sets and locally γ-regular sets are given. We generalize some already established results and answer an open question by giving a characterization to γ-quasi-open sets.      

Theorems on unions ofU-sets

The properties of families of subsets of topological spaces that make it possible to separate out the common core of the proofs of theorems on unions of sets of uniqueness for series in various systems of functions are considered. As a consequence of the results obtained, it is proved, in particular, that the countable union of closed sets of uniqueness for the trigonometric series is a set of uniqueness for the Abel-Poisson summation method. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 778–787, May, 2000.     

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.     

On pre-Λ-Sets and pre-v-Sets

We introduce the notions of a pre--set and a pre-V-set in a topological space. We study the fundamental properties of pre--sets and pre-V-sets and investigate the topologies defined by these families of sets.     

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.     

Plateau of α-function and c-minimal homoclinic orbits

We consider the construction of the plateau of the α-function in a hyperbolic and positive definite Lagrangian system, and link the boundries of the α-function’s plateau with the distribution of c-minimal homoclinic orbits to Aubry sets.     

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.      

A New View on Fuzzy Hypermodules

We describe the relationship between the fuzzy sets and the algebraic hyperstructures. In fact, this paper is a continuation of the ideas presented by Davvaz in （Fuzzy Sets Syst., 117： 477- 484, 2001） and Bhakat and Das in （Fuzzy Sets Syst., 80： 359-368, 1996）. The concept of the quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set is introduced and this is a natural generalization of the quasi-coincidence of a fuzzy point in fuzzy sets. By using this new idea, the concept of interval-valued （α,β）-fuzzy sub-hypermodules of a hypermodule is defined. This newly defined interval-valued （α,β）-fuzzy sub-hypermodule is a We shall study such fuzzy sub-hypermodules and sub-hypermodules of a hypermodule. generalization of the usual fuzzy sub-hypermodule. consider the implication-based interval-valued fuzzy     

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.     

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.      

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.     

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].     

Some constructions of two-associate class PBIB designs

Summary Two-associate class PBIB designs, having association schemes of GD orL ₂ types, are constructed by using patterned matrices and by methods of taking unions of sets of blocks.     

Redefined fuzzy B-algebras

By two relations belonging to and quasi-coincidence (q) between fuzzy points and fuzzy sets, we define the concept of (α, β)-fuzzy subalgebras where α, β are any two of with . We state and prove some theorems in (α, β)-fuzzy B-algebras.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.