On fuzzy regular volterra spaces

The aim of this paper is to introduce the concepts of regular Gδ-sets, regular Fσ-sets and regular Volterra spaces in fuzzy setting are introduced and studied. Several characterizations offuzzy regular Volterra spaces in terms of fuzzy regular Fσ-sets, fuzzy first category sets, fuzzy residual sets and fuzzy σ-nowhere dense sets are also established in this paper.

Classifications of Pairwise Fuzzy Volterra Spaces

The main focus of this paper is to introduce the new types of pairwise fuzzy Volterra spaces such as by introducing pairwise fuzzy residual sets in the place of pairwise fuzzy Gδ -sets in the definition of pairwise fuzzy Volterra space, a new kind of fuzzy bitopological space namely, pairwise fuzzy εr -Volterra spaces has been introduced and studied and also by introducing pairwise fuzzy pre-open sets in the place of pairwise fuzzy dense sets in the definition of pairwise fuzzy Volterra space, another kind of fuzzy bitopological space namely, pairwise fuzzy εr - Volterra spaces has been introduced and studied. Some of their characterizations and relationships with the other fuzzy bitopological spaces have been investigated and studied.

An Extension of Hypercyclicity forN-Linear Operators

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.

Preimages of Residual Sets of Continuous Functions under Operation of Superposition

Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

The sobolev embeddings are usually sharp

Letx0∈ℝd; we study the Hölder regularity atx0of a generic function of the Sobolev spaceLp,s(ℝd)and of the Besov spaceBps,q(ℝd)fors−d/p>0. The setting for genericity is supplied here by HP-residual sets.