In this article we first correct a recent misconception about a topology that was suggested by Zeeman as a possible alternative to his Fine topology. This misconception appeared while trying to establish the causality in the ambient boundary-ambient space cosmological model. We then show that this topology is actually the intersection topology (in the sense of G.M. Reed) between the Euclidean topology on R 4 and the order topology whose order, namely horismos, is defined on the light cone and we show that the order topology from horismos belongs to the class of Zeeman topologies.These results accelerate the need for a deeper and more systematic study of the global topological properties of spacetime manifolds.
AcknowledgementThe Authors would like to thank the Reviewers for the creative comments and remarks, as well as for their corrections, which have improved the text significantly.
Topological indices describe mathematical invariants of molecules in mathematical chemistry. M-polynomials of chemical graph theory have freedom about the nature of molecular graphs and they play a role as another topological invariant. Social networks can be both cyclic and acyclic in nature. We develop a novel application of M-polynomials, the ( m , n , r ) -agent recruitment graph where n > 1 , to study the relationship between the Dunbar graphs of social networks and the small-world phenomenon. We show that the small-world effects are only possible if everyone uses the full range of their network when selecting steps in the small-world chain. Topological indices may provide valuable insights into the structure and dynamics of social network graphs because they incorporate an important element of the dynamical transitivity of such graphs.
It is well known that statistics deals with quantitative analysis. Thus, there is a lack of approach to do quantitative analysis in the presence of qualitative attributes. Soft set theory has the freedom to deal with attributes, [0, 1], etc. along with quantity. Thus, we introduce some fundamental ideas of soft statistics.Here, soft mean, soft standard deviation, soft coe±cient of variation, soft correlation coefcient are introduced and some theorems are proved with respect to utility. Utility theory provides an analysis of choice behavior. As an application of our notions and results, we¯nd soft correlation coe±cients between vulnerability and government responses of various regions across the world. The data from \The Global Slavery Index 2016" are considered for application purposes.
In this article we show that a contra second countable bitopological space is a p1-Lindelöf space, but the converse part is not necessarily true in general. We provide suitable example with the help of concepts of nest and interlocking from other areas related to bitopology. The relation between pairwise regular spaces and p 1 -normal spaces has been investigated. Finally, we propose some open problems which may enrich various concepts related to Lindelöfness in a bitopological space and other areas of mathematical ideas.
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