American scientific publishing group

Many studies have employed a variety of techniques to further investigate topological space, particularly the notion of bitopological spaces, due to the significance of topological space in data processing as well as certain implementations. Numerous extended topological structures have been laid out subsequently. Of those abstractions, functions in topology was one of which was most noteworthy. In order to assist in this trend, we focused our research on the idea of open and closed sets, which is one of the strongest techniques available to present scientists for the study of computer graphics and digital topology.New functions, pairwise ω−closed functions, which are strictly weaker than pairwise closed functions, will be introduced in this study. By applying the P˝−space definition, whose is a P−space modification. Additionally, we establish different projection and product theories pertaining to pairwise Lindelöf of and pairwise paracompact spaces utilizing P˝−spaces. We analyze images and inverse images that have been chosen topological attributes for every one of these functions. In the final analysis, we explore several counterexamples that correspond to the offered definitions and theorems.

This paper introduces and explores the concept of Neutrosophic MR-Metric Spaces (NMR-MS), which generalize classical metric spaces by incorporating neutrosophic logic to handle uncertainty, indeterminacy, and truth-membership in complex systems. We define the structure of NMR-MS, including MR-metrics, neutrosophic functions, and associated algebraic operations. Key contributions include the introduction of the Neutrosophic Graph MR-Metric Space, where graph-theoretic concepts such as shortest paths and betweenness centrality are integrated with neutrosophic degrees. We establish several fundamental results, including contraction lemmas, robustness under targeted attacks, and the existence of fixed points for neutrosophic centrality mappings. Furthermore, we define Neutrosophic MR-Homotopy and the associated Neutrosophic MR-Fundamental Groupoid, proving that it forms an equivalence relation and possesses a well-defined algebraic structure. The theoretical framework is applied to diverse real-world networks, including social, neural, and internet topologies, demonstrating its utility in network analysis, robustness testing, and path planning. Computational algorithms and performance metrics are provided, showcasing the practical applicability of the proposed framework.

In this paper, we study ideals defined with respect to arbitrary multiplicatively closed subsets S⊆R of a commutative ring R. An ideal I⊆R is called an S-ideal if for all a,b∈R, the condition ab∈I and a∈S implies b∈I. This is equivalent to the identity I=S⁻¹I∩R, where S⁻¹I is the extension of I in the ring of fractions S⁻¹R. The concept of S-ideals provides a unified framework encompassing several classical ideal types. For instance, r-ideals arise when S=reg(R), the set of regular elements. If S=R∖P for a prime ideal P, then the S-ideals coincide with P-primary ideals. Ideals that admit primary decomposition correspond to S-ideals for which S is the complement of a finite union of prime ideals. Moreover, z₀-ideals are S-ideals when S is the complement of a union of minimal prime ideals of R. We generalize several results known for r-ideals to this broader setting and investigate structural and closure properties of S-ideals in various contexts. As an application, we give a characterization of the von Neumann regularity of the localization S⁻¹R in terms of S-ideals. We also study the behavior of S-ideals in polynomial rings, idealizations, and amalgamated constructions with respect to different choices of S.

Recent developments in complex fuzzy (CF) sets have extended the classical fuzzy framework from the unit interval [0,1] to the unit disk in the complex plane C, allowing for the modeling of uncertainties in both magnitude and phase. Building upon this foundation, this study introduces-for the first time-the use of complex intuitionistic fuzzy (CIF) numbers to solve partial differential equations, specifically focusing on the CIF heat equation. The CIF framework integrates both membership and non-membership functions with a complex-valued representation, enabling a more expressive treatment of uncertainty, including hesitation. An explicit finite difference method, namely the Forward Time Central Space (FTCS) scheme, is employed to discretize and solve the CIF heat equation. The model considers fuzziness in the initial and boundary conditions, where uncertainty impacts amplitude and phase terms. To represent this uncertainty, triangular fuzzy numbers are used for the real and imaginary parts within the complex unit disk. The proposed approach demonstrates numerical stability and achieves second-order spatial and first-order temporal accuracy, validating its reliability and effectiveness. A numerical example confirms the feasibility of the method, showing strong alignment with theoretical predictions. This work generalizes existing CF heat equation models and provides a foundation for solving more complex systems involving higher-order and bipolar uncertainties in future studies.

In this careful study , through the concept possibility interval valued neutrosophic hyper soft set (abbreviated as piv-NHSS) which is combined from the hypersoft set (HSS) and Interval-valued neutrosophic set under the posobolity degree and each iv-NHSS is assigned a possibility degree in the interval [0, 1]. Based on this concept, we present a more flexible, expanded method for a previous concept named possibility interval valued neutrosophic hyper soft matrix (piv-NHSM) as a new generalization of piv-NHSS. In this work, we also present nseveral algebraic operations and also all the mathematical properties associated with this model. In addition to the above, we have presented a clear algorithm based on the matrix properties of this model, which has been used to solve one of the multi-property decision-making problems. Finally, the correlation coefficient for this concept was defined and explained in detail according to an approved mechanism, with a numerical example provided to illustrate the mechanism of use. Moreover, we develop a new algorithm for solving the decision-making issue based on the proposed correlation coefficient for piv-NHSS .