iosr-Jm   Volume-22 ~ Issue-5 ~ Sep. – Oct. 2026

Abstract : Let 𝑅 be a non reduced commutative ring and let 𝑁(𝑅) be the set of nil- elements of 𝑅 of index two. The nil graph of 𝑅, denoted by 𝛤𝑁(𝑅) is a graph whose vertex set is the set 𝒵𝑁(𝑅)∗={𝑥∈𝑅 \ {0}∶𝑥𝑦 ∈ 𝑁(𝑅) 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑦∈ 𝑅 \ {0}} and two vertices 𝑥 and 𝑦 are adjacent if and only if their product is a nil element of index two. The chromatic number of a graph is the assignment of minimum colors to the vertices of the graph so that no two adjacent vertices have assigned......

Word: Nil Graph; Graph Coloring; Clique; Topological Indices; Zagreb Index. 2020 AMS Subject Classification: 05C25, 05C09, 05C15

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[3]. Akbari, S., Mohammadian, A., & Shahriari, M. (2004). On the zero-divisor graphs of a commutative ring. Journal of Algebra, 276(2), 683–691.
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Abstract : Elementary operators have been widely studied based on properties such as numerical range, spectrum, compactness, norm, and rank. Previous studies have considered elementary operators on bounded operators on Hilbert spaces, matrix algebras, C*-algebras, von Neumann algebras, and JB*-algebras. This study focused on the multiplication elementary......

Keywords: Elementary operators; C*-algebras; Operator norm; Multiplication operators.

[1]. Kawira, E., King’ang’i, D., and Musundi, S.(2018). on the norm of an elementary operator of finite length in a C ∗ algebra.
[2]. Mathieu, M.(2001). The norm problem for elementary operators. In North-Holland Mathematics Studies (Vol. 189, pp. 363-368). North-Holland.
[3]. Magajna, B.(2004). The norm of a symmetric elementary operator. Proceedings of the American Mathematical Society, 132(6), 1747-1754.
[4]. Muiruri, P., King’ang’i, D., and Musundi, S.(2019). On the Norm of Basic Elementary Operator in a Tensor Product.
[5]. Muiruri, P. G.(2024). On The Norm of Finite Length Elementary Operator in Tensor Product of C ∗ -Algebras.

Abstract : This paper presents a unified application of the Homotopy Analysis Method (HAM) to two- and three-dimensional linear and nonlinear partial differential wave equations. A common solution framework is formulated for both dimensions, while the nonlinear terms are incorporated into the successive deformation equations. The method constructs a series solution without requiring a small perturbation parameter. An auxiliary convergence-control parameter provides.....

Keywords: Homotopy Analysis Method; wave equations; partial differential equations; two-dimensional models; three-dimensional models; nonlinear equations; series solutions; convergence analysis.

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Abstract : Differential equations are fundamental tools for describing physical and engineering phenomena, yet their teaching is often dominated by symbolic procedures that can obscure the connection between a mathematical model and an observable process. This paper presents a didactic application of a first-order linear differential equation to model solar-light attenuation with water depth in an outdoor swimming pool. Experimental information was obtained in a private swimming pool in Mexico, using a digital lux meter with a remote sensor and bar-graph display. Measurements were taken at depths from 0 to 2.0 m in 0.25 m increments. Each reported experimental value is the.....

Keywords: Differential equations; mathematical modeling; light attenuation; swimming pool; illuminance; educational technology; engineering mathematics

[1] E. Lozada, C. Guerrero-Ortiz, A. Coronel, and R. Medina, "Proposal of a mathematical modelling activity to facilitate students’ learning of ordinary differential equation concepts," Sustainability, vol. 15, no. 16, Art. no. 12483, 2023, doi: 10.3390/su151612483.
[2] K. Spooner, "Using mathematical modelling to provide students with a contextual learning experience of differential equations," Int. J. Math. Educ. Sci. Technol., vol. 55, no. 2, pp. 565-573, 2024, doi: 10.1080/0020739X.2023.2244472.
[3] R. Durandt, W. Blum, and A. Lindl, "Fostering mathematical modelling competency of South African engineering students: Which influence does the teaching design have?," Educ. Stud. Math., vol. 109, pp. 361-381, 2022, doi: 10.1007/s10649-021-10068-7.
[4] R. Durandt, W. Blum, and A. Lindl, "A mathematical modelling unit for first-year engineering students," Modelling Sci. Educ. Learn., vol. 15, no. 1, pp. 77-92, 2022, doi: 10.4995/msel.2022.16646.
[5] M. F. Merck, M. A. Gallagher, E. Habib, and D. Tarboton, "Engineering students’ perceptions of mathematical modeling in a learning module centered on a hydrologic design case study," Int. J. Res. Undergrad. Math. Educ., vol. 7, pp. 351-377, 2021, doi: 10.1007/s40753-020-00131-8.

Abstract : This paper employ a Pareto Multi-Objective Optimization (POO) to reconciles the reported sectoral credit allocation, non-performing-loan (NPL) and electronic-payment statistics into a single, internally consistent panel of seventeen credit sectors, and formulates the allocation problem as a tri-objective constrained optimization , maximize portfolio growth, minimize portfolio NPL-risk exposure, and maximize a digital-banking-impact index, subject to a fixed lending envelope and sector-level absorptive-capacity bounds. The resulting problem is a multi-objective linear program.....

Keywords: Pareto multi-objective optimization; sectoral credit allocation; non-performing loans; digital banking; and linear programming

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Abstract : Rwanda's hilly terrain and increasingly erratic rainfall expose its districts to recurrent weather-related disasters, including floods, landslides, and windstorms. National agencies report disaster effects annually but rarely test whether observed differences among districts are jointly statistically significant across multiple impact indicators. This study evaluates whether weather disaster-impact profiles differ significantly among three districts representing.....

Keywords: Multivariate Analysis of Variance (MANOVA); disaster risk management; Rwanda; climate variability;

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[4]. Rwanda Red Cross Society / IFRC. Rwanda: Floods and Landslides DREF Final Report (MDRRW022). ReliefWeb; 2023.
[5]. Ministry in Charge of Emergency Management [MINEMA]. Disaster Effects Situation Report 2023. Government of Rwanda, Kigali; 2023..