Remarks on enlargements of generalized topologies

The concept of enlargement of a generalized topology was introduced by á. Császár [4]. He also introduced the concepts of (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity on enlargements. In this paper, we characterize the (κ,λ)-continuity and introduce the concept of strong (κ,λ)-continuity on enlargements. In particular, we study characterizations for the strong (κ,λ)-continuity and the relationships among (μ,ν)-continuity, (κ,λ)-continuity, strong (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity.     

Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ? 2 ^X with τ ? C we have f ^?1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ? C* ? 2 ^X , C*-continuity is a dual to C-continuity if C?C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.     

A note on θ(g, g′)-continuity in generalized topological spaces

The concept of θ(g, g′)-continuity was introduced by Császár [1]. In this paper, we investigate characterizations for θ(g, g′)-continuous functions and introduce the concept of weak θ(g, g′)-continuity, and study characterizations for weak θ(g, g′)-continuity and the relationships among θ(g, g′)-continuity, weak (g, g′)-continuity and weak θ(g, g′)-continuity.     

On weak neighborhood systems and spaces

We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s. This work was supported by a grant from Research Institute for Basic Science at Kangwon National University.     

A unified theory of weak continuity for functions

In this paper we introduce a new notion of weaklyM-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [27], almosts-continuity [43],p(θ)-continuity [10] andp-continuity [59].     

On weakly (τ, m)-continuous functions

In this paper we introduce a new notion of weakly (τ, m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [20], almosts-continuity [33],p(θ)-continuity [10] andp-continuity [41].     

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.     

Decompositions of continuity and complete continuity

We introduce the notions of δ-t-sets, δ_β-t-sets, δ-B-continuity and δ_β -B-continuity and obtain decompositions of continuity and complete continuity.     

Another weak forms of faint continuity

A function f:X→Y is said to be faintly continuous [Kyungpook Math. J. 22 (1982) 7] if f⁻¹(V) is open in X for every θ-open set V of Y. In this paper, we introduce and investigate two weaken forms of faint continuity which are called faint α-continuity and faint γ-continuity. We obtain their characterizations, their basic properties and their relationships with other types of functions between topological spaces also, some results in (A.A. El-Atik, A study of some types of mappings on topological spaces, M.Sc. Thesis, Tanta University, Egypt, 1997) are improved. We speculate that weak-faint continuity may be relevant to the physics of fractal and Cantorian spacetime.     

Stability of penalty finite-element methods for nonconforming problems

Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth-order problems in which C¹-continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C¹-continuity constraint appears promising. In this study we demonstrate equivalence of the finite-element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.     

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].      

Strongly-$\mathcal{I}$-locally closed sets and decompositions of ∗-continuity

We introduce the notion of strongly-I\mathcal{I}-locally closed sets to obtain decompositions of ∗-continuity.     

Weak continuity on generalized topological spaces

We introduce the notions of weak (g, g′)-continuity and weak (ψ, ψ′)-continuity on GTS’s and GNS’s, respectively, and investigate characterizations for such functions and relationships between weak (g, g′)-continuity and weak (ψ, ψ′)-continuity.     

On moduli of p-continuity

Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<??).     

Fuzzy (Θ, S)-continuity and fuzzy (Θ, S)-closed graphs

The concept of (,s)-continuity [6] is considered and studied in fuzzy setting. It is seen that althought it is independent with each of the concepts of fuzzy continuity [2], fuzzy -continuity [10], fuzzy almost continuity [1] and fuzzy semicontinuity [1]; it implies fuzzy weak continuity [1], but the converse may not be true. The image of a compact fts [2] under a fuzzy (,s)-continuous surjective function isS-closed [5]. Finally the concepts of fuzzy (,s)-closed graphs, fuzzy (,s)-T ₂ spaces and fuzzy Urysohn spaces are introduced and mainly their connections with fuzzy (,s)-continuity are studied.     

A Transference Result of the L p Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms

In this paper we develop a transference method to obtain the Gaussian-Riesz transform’s L ^p -continuity and the Laguerre-Riesz transform’s L ^p -continuity from the L ^p -continuity of the Jacobi-Riesz transform, in dimension one, using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials.     

Superposition operator on the space of sequences almost converging to zero

We study the superposition operator f on on the space ac ₀ of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac ₀ with a = f 0, b ∈ D(ac ₀), g a superposition operator from ℓ_∞ into I(ac ₀), D(ac ₀) = {z: zx ∈ ac ₀ for all x ∈ ac ₀}, and I(ac ₀) the maximal ideal in ac ₀. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f acting on ac ₀ and the conditions for which there exists a sequence y ∈ ac ₀ such that y − f y ∈ ac ₀. In terms of f, criteria for the boundedness, continuity, and sequential σ(ac ₀ℓ₁)-continuity of f on ac ₀ are given. It is shown that the continuity of f is equivalent to the weak sequential continuity. Finally, properties of spaces D(ac ₀) and I(ac ₀) are studied, and in particular it is established that the inclusion I(ac ₀) ⊕ {λe: λ ∈ ℝ} ⊂ D(ac ₀) is proper, where e = (1, 1, …). By means of D(ac ₀), a number of Banach-Mazur limit properties are derived.     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.     

On general perturbations of symmetric Markov processes

Let X be a symmetric right process, and let be a multiplicative functional of X that is the product of a Girsanov transform, a Girsanov transform under time-reversal and a continuous Feynman–Kac transform. In this paper we derive necessary and sufficient conditions for the strong L²-continuity of the semigroup given by T_tf(x)=E_x[Z_tf(X_t)], expressed in terms of the quadratic form obtained by perturbing the Dirichlet form of X in the appropriate way. The transformations induced by such Z include all those treated previously in the literature, such as Girsanov transforms, continuous and discontinuous Feynman–Kac transforms, and generalized Feynman–Kac transforms.     

Lp-Bounded Pseudodifferential Operators and Regularity for Multi-quasi-elliptic Equations

Using a classical result of Marcinkiewicz and Lizorkin about the L^p-continuity for Fourier multipliers, the authors study the action of a class of pseudodifferential operators with weighted smooth symbol on a family of weighted Sobolev spaces. Results about L^p-regularity for multi-quasi-elliptic pseudodifferential operators are also given.