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The intention of this article is to introduce and characterise the concept of preopen sets and prelocally closed sets in Generalised Topology and Minimal structure spaces.

This research paper aims to introduce the concept of lacunary ideal Cauchy sequences of fuzzy variables in a credibility space. We establish the interrelationships between this notion with lacunary ideal convergent sequences in the same structure from several aspects of credibility. Furthermore, we explore the concepts of strongly lacunary Cauchy, strongly ℐ -lacunary Cauchy, and strongly ℐ ∗ -lacunary Cauchy sequences of fuzzy variables within the context of credibility. We also examine the interconnections between these concepts and analyze their relationships. [ABSTRACT FROM AUTHOR]

In this article we introduce the notion of double sequences of complex uncertain variables associated with multiplier sequences. We study their different algebraic and topological properties like solidity, symmetricity, completeness etc. [ABSTRACT FROM AUTHOR]

The increasing pollution of river systems poses a significant challenge to water quality management and public health. This study was conducted to evaluate the water quality of the Gomati River in India, focusing on the impact of anthropogenic activities on pollution levels across different seasons. The primary objective was to develop a comprehensive assessment model using an effective weighted fuzzy soft expert system (WFSES) to derive a weighted water pollution score (WWP-score) for rating water pollution levels based on various water quality parameters. To achieve this, we employed a fuzzy soft set (FSS) methodology integrated with a weighted fuzzy soft set (WFSS) model to analyze water quality indices at six sampling stations along the Gomati River, the largest river in Tripura, India. Originating from the Raima and Sarma streams, the river flows westward through key regions, providing 249.39 million cubic meters of water annually, essential for drinking, agriculture, fishing, and transportation. The study utilized crisp data summaries of key water quality parameters, including pH, total dissolved solids, total suspended solids, electrical conductivity, biochemical oxygen demand, dissolved oxygen, total hardness, chloride, total alkalinity, and total coliform, across the pre-monsoon, monsoon, and post-monsoon seasons from May 2022 to April 2023, guided by the distribution of waste discharge points. Fuzzy membership functions were defined for these parameters, and the root mean square (RMSQR) operation was used to combine multiple FSSs. The results revealed that the WWP-scores, a measure of water quality, ranged from 0.744 to 0.872 across the sampling stations, with Srimantapur exhibiting the highest level of pollution and Udaipur showing comparatively better quality. These scores indicate that water quality at all sites falls into the poor category, with Srimantapur experiencing the most severe contamination due to high levels of coliform bacteria and organic pollutants. The analysis highlights the impact of domestic wastewater and agricultural runoff on the river's health. This research offers a novel application of FSS and WFSS theory in environmental management, providing a robust framework for evaluating and ranking water quality across multiple parameters. The findings are valuable to the scientific community as they offer detailed insights into the pollution dynamics of the Gomati River and propose a methodological approach for assessing water quality in other river systems facing similar challenges. The study's results underscore the need for targeted pollution control measures and continuous monitoring to safeguard water resources and public health. [ABSTRACT FROM AUTHOR]

The conception of single-valued quadripartitioned neutrosophic d-ideal (SVQN-d-I) of single-valued quadripartitioned neutrosophic d-algebra (SVQN-d-A) as an expansion of neutrosophic d-Ideal and neutrosophic d-Algebra has been attempted to be introduced in this article. Additionally, we identify various characteristics of them. Additionally, SVQN-d-I and SVQN-d-A examples have been provided. [ABSTRACT FROM AUTHOR]

In this article, we introduced the idea of lacunary weak convergence of sequences defined by the class of Orlicz function. Various topological and algebraic properties, such as solidity, symmetry, and monotonicity, have been studied. To validate the failure instances, we gave appropriate examples and detailed discussions. Additionally, we have established some inclusion relationships between these spaces. [ABSTRACT FROM AUTHOR]

Convex functions play an important role in finding the inequalities, those help in finding the solutions of different types of equations and equations involving functions. In this article, we have considered convex functions in a normed linear space. We have established some results on composite convex sets and composite convex functions. We have considered quasi-arithmetic mean, that unifies efficiently all types of power means. On applying the principles of composite convex functions, we have established a Hermite-Hadamard like inequality. The functions considered are composite convex functions with respect to a strictly monotonic continuous composite function. The composite convex functions serve as a comprehensive generalization of composite convex functions. As an application, we have established some inequalities on integrable composite convex functions. The results are deferred mean type inequalities. The results on inequalities can be applied for further investigations as well as for application in finding the solutions in different areas of research. [ABSTRACT FROM AUTHOR]

In this paper we investigate the concept of ℐ {\mathcal{I}} -monotonic convergence of sequences of bi-complex numbers. We also discuss some partial order relations ≼ i 2 {\preccurlyeq_{i_{2}}} and ≼ i id {\preccurlyeq_{i_{\rm id}}} between two bi-complex numbers and ℐ {\mathcal{I}} -convergence, ℐ {\mathcal{I}} -monotonic sequences with respect to different partial order relations, ℐ {\mathcal{I}} -Cauchy sequences, ℐ {\mathcal{I}} -bounded monotonic sequences of bi-complex numbers. The analysis indicates that ℐ {\mathcal{I}} -monotonicity with respect to ≼ i 2 {\preccurlyeq_{i_{2}}} is not comparable with ℐ {\mathcal{I}} -monotonicity with respect to ≼ i id {\preccurlyeq_{i_{\rm id}}} . [ABSTRACT FROM AUTHOR]

The notion of relative uniform convergence of a sequence of functions was introduced and investigated by Moore, followed by E. W. Chittenden in the twentieth century. The concept attracted the researcher at the beginning, of the twenty-first century. Jackson F. H. initiated the concept of the notion of the quantum difference operator. We introduced the quantum difference sequence space m(ɸ, ru,∇p), p ≥ 0 of W.L.C. Sargent type. We examine its various properties such as solidity, convergence-free, completeness, etc. Also, some induction results have been established. [ABSTRACT FROM AUTHOR]

We say that a regular graph G of order n and degree r ≥ 1 (which is not the complete graph) is strongly regular if there exist non-negative integers τ and θ such that |Si ∩ Sj | = τ for any two adjacent vertices i and j, and |Si ∩ Sj | = θ for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let λ1 = r, λ2 and λ3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, λ2 and λ3, respectively. We here describe the parameters n, r, τ and θ for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 7/2 , 7/3 , 7/4 , 7/5 , 7/6 .

In this research article, we introduce ℐ {\mathcal{I}} -statistically pre-Cauchy sequences of complex uncertain variables in five different aspects of uncertainty, namely: in mean, in measure, in distribution, in almost sure, and in uniformly almost sure. We also explore the connection between ℐ {\mathcal{I}} -statistically pre-Cauchy sequences and ℐ {\mathcal{I}} -statistically convergent sequences using complex uncertain variables. Additionally, we initiate the study of ℐ {\mathcal{I}} -statistically pre-Cauchy sequences of complex uncertain variables through Orlicz functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of I3-convergent,I⋆3-convergent and I3-Cauchy triple sequences in a fuzzy normed linear space. Besides, we establish some basic results related to those notions. Furthermore, we define the concepts of I⋆3-Cauchy sequence in fuzzy normed space and some relations between I3-Cauchy sequence in a fuzzy normed space are shown. [ABSTRACT FROM AUTHOR]

The article reveals the investigation of properties of multiplication operation of neutrosophic fuzzy matrices. We study commutative property, associative property, distributive property of them. We show with suitable example that neutrosophic fuzzy matrices do not obey commutative property with respect to multiplication operation. We prove that neutrosophic fuzzy matrices hold associative property property with respect to multiplication operation. We also prove that neutrosophic fuzzy matrices hold distributive property with respect to multiplication operation over addition. These results are further justified by providing suitable numerical examples. The important aspects of the article is that the investigation of commutative, associative and distributive properties of neutrosophic fuzzy matrices with respect to multiplication operation will fill up the gaps in the existing literature. [ABSTRACT FROM AUTHOR]