Lower Porosity on ℝ2

In this paper, we study the properties of a lower porosity of a set in R2. It turns out that the properties of the lower and upper porosity are symmetrical, except that the main tools for testing the lower porosity are not balls but cones. New families of topologies on R2 generated by the lower porosity are defined. Furthermore, by applying the notion of the lower porosity, we introduce the definition of generalized continuity. Using defined topologies, we study properties of this continuity. We show that the properties of topologies generated by the lower and (upper) porosity are symmetrical.

Topological aspects of continuity via generalized limit

This article aims to introduce the concept of generalized continuity of an function with respect to another function and to analyze the topological aspects of this concept Initially, the article presents the concept and the properties of generalized limit ofan function with respect to another function, highlighting that the Riemann integral is a particular case of generalized limit. Then, the definition of generalized continuity is presented, evidencing that this concept does not coincide with the classical continuity, being the last one a particular case of the first. Finally, some topological aspects associated with the concept of generalized continuity are approached in order to present proofs about the preservation of topological invariants via generalized continuity, such as preservation of compactness and connectedness.

$$\mathbf {O}(D,D)$$ completion of the Friedmann equations

In string theory the closed-string massless NS-NS sector forms a multiplet of $$\mathbf {O}(D,D)$$ O ( D , D ) symmetry. This suggests a specific modification to General Relativity in which the entire NS-NS sector is promoted to stringy graviton fields. Imposing off-shell $$\mathbf {O}(D,D)$$ O ( D , D ) symmetry fixes the correct couplings to other matter fields and the Einstein field equations are enriched to comprise $$D^{2}+1$$ D 2 + 1 components, dubbed recently as the Einstein Double Field Equations. Here we explore the cosmological implications of this framework. We derive the most general homogeneous and isotropic ansatzes for both stringy graviton fields and the $$\mathbf {O}(D,D)$$ O ( D , D ) -covariant energy-momentum tensor. Crucially, the former admits space-filling magnetic H-flux. Substituting them into the Einstein Double Field Equations, we obtain the $$\mathbf {O}(D,D)$$ O ( D , D ) completion of the Friedmann equations along with a generalized continuity equation. We discuss how solutions in this framework may be characterized by two equation-of-state parameters, w and $$\lambda $$ λ , where the latter characterizes the relative intensities of scalar and tensor forces. When $$\lambda +3w=1$$ λ + 3 w = 1 , the dilaton remains constant throughout the cosmological evolution, and one recovers the standard Friedmann equations for generic matter content (i.e. for any w). We further point out that, in contrast to General Relativity, neither an $$\mathbf {O}(D,D)$$ O ( D , D ) -symmetric cosmological constant nor a scalar field with positive energy density gives rise to a de Sitter solution.

More on decomposition of generalized continuity

In this paper a new class of sets termed as ω∗μ-open sets has been introduced and studied. Using these concept, a unified theory for decomposition of (μ, λ)-continuity has been given.

Generalized Derivative and Generalized Continuity

The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.

Некоторые оценки для обобщенного преобразования Фурье, ассоциированного с оператором Чередника - Опдама

In the classical theory of approximation of functions on $\mathbb{R}^+$, the modulus of smoothness are basically built by means of the translation operators $f \to f(x+y)$. As the notion of translation operators was extended to various contexts (see [2] and [3]), many generalized modulus of smoothness have been discovered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [4] and [5]). In [1], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we also discuss this subject. More specifically, we prove some estimates (similar to those proved in [1]) in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated with the differential-difference operator $T^{(\alpha,\beta)}$ in $L^{2}_{\alpha,\beta}(\mathbb{R})$. For this purpose, we use a generalized translation operator.

On Lembda Gamma- Set in Fuzzy Bitopological Spaces

The aim of this paper is to introduce the concept of lambda operator of a fuzzy set in a fuzzy bitopological space. Then we study (i, j)-fuzzy Lembda Gamma- set and its properties. Moreover we define (i, j)-fuzzy Lembda-closed set, (i, j)-fuzzy Lembda Gamma-closed set and (i, j)-fuzzy generalized closed set in fuzzy bitopological space. The concepts (i, j)-fuzzy Lembda-closed set and (i, j)-fuzzy generalized closed set are independent to each other but jointly they gives the taui-fuzzy closed set. To this end as the application of (i, j)-fuzzy Lembda Gamma-closed set we shall study (i, j)-fuzzy Lembda Gamma continuity and (i, j)-fuzzy Lembda Gamma-generalized continuity and their properties.