Abstract

In this paper, the concept of a ternary relation (named as C-relation) is introduced. It is observed that every closure operator can be used to define a C-relation and conversely, any C-relation induces a closure operator. Thus, topological concepts can be studied in terms of relations.

Keywords: Relation; uniform space; proximity space; connectedness; separability.

MSC: 54A05, 06A15

Abstract

The distance between two edges in a simple connected graph can be described as the distance between the corresponding vertices in its line graph. In this paper, we determine the distance between edges in strong double graphs and apply our results to compute some edge distance-related invariants for this family of graphs.

Keywords: Distance between edges in graph; graph invariants; strong double graphs.

MSC: 05C12, 05C76

Abstract

We characterize those $\alpha\in \mathbb{R}$ and $\mu$ positive Borel measure on $(0,1]$ for which generalized Hausdorff operator acts on Hardy spaces of the unit disk. Further, certain conditions on $\mu,$ we prove the operator is bounded linear on $H^p(\mathbb{D}),$ for different cases of $p.$ For $\alpha=0,$ we determine the characterization of the operator on weighted spaces of integrable functions.

Keywords: Hausdorff matrix; Cesàro operator; Hardy spaces; weighted spaces.

MSC: 47B38, 46E15

Abstract

Using maximum instead of sum, nonlinear Favard-Szász-Mirakjan operator of maximum product kind was introduced. The present paper deals with the approximation processes for this operator. Especially in a previous study, it was indicated that the order of approximation of this operator to the function $f$ under the modulus is $\sqrt{x/n}$ and it could not be improved except for some subclasses of functions. Contrary to this claim, under some special conditions, we will show that a better order of approximation can be obtained with the help of classical and weighted modulus of continuities.

Keywords: Nonlinear Favard-Szász-Mirakjan operator; modulus of continuity; weighted modulus of continuity.

MSC: 41A36, 41A10, 41A25

Abstract

A sequence $\{x_n\}$ is $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergent to $\xi$, if there exists a "big enough" subsequence $\{x_{n_k}\}$ which $\mathcal{K}$-converges to $\xi$ via semi-open sets. In this paper, we introduce the concept of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes $\mathcal{S}$-$\mathcal{I}$-convergence and discuss some properties, as well as its relation with compact sets. For two given ideals $\mathcal{I}$ and $\mathcal{K}$, we justify the existence of an ideal such that $\mathcal{I}^\mathcal{K}$-convergence and convergence with the third ideal coincides for semi-open sets. Moreover, the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of a sequence is defined and studied here. We characterize the collection of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster points of a sequence as semi-closed subsets of the space.

Keywords: $\mathcal{I}^\mathcal{K}$-convergence; $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence; $\mathcal{S}$-$\mathcal{I}^{\mathcal{K}}$-cluster points; semi-open sets; semi-compactness; semi-dense set.

MSC: 40A35, 40A05, 54A05, 54A20

Abstract

The aim of this paper is to introduce some separation notions of graded ditopological texture spaces by means of spectrum idea and investigate some of their properties. Also, the relations between separation spectrums of graded ditopological texture spaces and separation axioms in ditopological structure are studied. Further, the hierarchy of the separation spectrums and a categorical aspect corresponding to the separation spectrums are given.

Keywords: Separation spectrums; graded ditopology; texture.

MSC: 54A05, 54A40, 54D10

Abstract

In this paper, we consider all the non-metabelian groups of order $162$ and characterize the structure of the unit group of the corresponding group algebras. Overall, there are $55$ non-isomorphic groups having order $162$ and $11$ among them are non-metabelian. We study the unit group of the semisimple group algebras over any finite field whose characteristic does not divide the order of these eleven groups.

Keywords: Finite field; group algebra; unit group; non-metabelian groups.

MSC: 16U60, 20C05