Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

Relay Synchronization in a Weighted Triplex Network

Relay synchronization in multi-layer networks implies inter-layer synchronization between two indirectly connected layers through a relay layer. In this work, we study the relay synchronization in a three-layer multiplex network by introducing degree-based weighting mechanisms. The mechanism of within-layer connectivity may be hubs-repelling or hubs-attracting whenever low-degree or high-degree nodes receive strong influence. We adjust the remote layers to hubs-attracting coupling, whereas the relay layer may be unweighted, hubs-repelling, or hubs-attracting network. We establish that relay synchronization is improved when the relay layer is hubs-repelling compared to the other cases. We determine analytically necessary stability conditions of relay synchronization state using the master stability function approach. Finally, we explore the relation between synchronization and the topological property of the relay layer. We find that a higher clustering coefficient hinders synchronizability, and vice versa. We also look into the intra-layer synchronization in the proposed weighted triplex network and establish that intra-layer synchronization occurs in a wider range when relay layer is hubs-attracting.

Quantum exceptional chamber induced by large nondipole effect of a quantum dot coupled to a nano-plasmonic resonator

Exceptional points (EPs) are the singularities of a non-Hermitian system where the eigenenergies and eigenstates simultaneously coalesce, a topological property that gives rise to a plethora of exotic phenomena. Probing the EPs and associated effects requires the system to go through the EPs. However, the ultrahigh sensitivity of an isolated EP to the external disturbances makes accessing the EPs difficult. To overcome this limit, many approaches have been presented to form the exceptional line/ring and surface. Here, we demonstrate that a quantum exceptional chamber, which is a three-dimensional collection of the EPs, can be constructed in the coupled plasmon-quantum dot (QD) systems by the nondipole effect of the QD. For an asymmetric QD adjacent to a plasmonic nanoparticle, it is found that the contributions of multipole transitions to the coupling strength can be larger than that of dipole transition. The orientation-dependent quantum interference between the dipole and multipole transitions can lead to controllable switch between the weak and strong coupling, and provides an extra degree of freedom to form a high-dimension EP space. Our approach provides a robust platform for accessing the quantum EPs and related applications.

Some New Variants of Relative Regularity via Regularly Closed Sets

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

Spatial Configurations and Walkability Potentials. Measuring Urban Compactness with Space Syntax

This contribution demonstrates how space syntax methods on various scale levels can be used to identify and describe the spatial features of a compact city. Firstly, the term urban compactness is discussed. A short discussion of some writings on the compact city are elaborated. As it transpired, urban compactness can best be approached from a spatial topological point of view, since compactness is a topological property. Secondly, urban compactness will be reconsidered in spatial configurative terms through the use of space syntax and urban micro scale tools. Examples from car-, pedestrian-, and public transport-based centres in Oslo and Bergen will be used throughout this contribution. Discussions of the examples in this contribution are discussed with references to other space syntax research results. As the case studies show, enhancing compact neighbourhoods with good walkability potential from a spatial perspective relies on spatial interaccessibility on all scale levels. Accessibility depends on spatial configurative compactness. Seemingly, it depends on the following complex set of sufficient conditions: a spatially integrated street network on all scale levels, short urban blocks and streets with building entrances with windows and doors on the ground floor level.

Some new separation axioms in fuzzy soft topological spaces

In this paper, a new form of separation axioms called r-fuzzy soft Ti;(i = 0,1,2,3,4), r-fuzzy soft regular and r-fuzzy soft normal axioms are introduced in a fuzzy soft topological space based on the paper Ayg?no?lu et al. [7]. Also, the relations of these axioms with each other are investigated with the help of examples. Furthermore, some fuzzy soft invariance properties, namely fuzzy soft topological property and hereditary property are specified.

Separable Topological Space of Hereditary

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.