Open Access Original Research Article

Kennedy Mwila, Darius Mangwatu, Enala S. Lufungulo, Alex Mugala, Maimbolwa Namuchana, Onwubuya Gift Chinemerem, Manuel Siampule

This study sought to investigate the effectiveness of the mathematics laboratory method in enhancing learner achievements in mathematics of Grade 12 pupils. The specific objectives of this study were to; investigate the extent to which the Mathematics Laboratory Method enhances Grade 12 academic performance in Mathematics, and to establish the challenges faced by Secondary Schools in establishing Mathematical laboratories. The study employed a mixed method approach, a concurrent triangulation design was used, with a sample size of 120 pupils and 12 teachers from the 3 selected secondary schools. Simple random sampling was used on the pupils and purposive sampling teaching was used to select the teachers. The control group and the experimental group were subjected to a pre-test where an independent sample T-test was used to test the hypothesis. The mean pre- test scores were 52% and 54% for the experimental and control group respectively. Further, statistical analysis revealed that there was no statistically significant difference in the pre-test results between the two groups. On the other hand, results for the post test, which was administered after the control group was taught using the traditional method of teaching and experimental group using the mathematics laboratory method showed a statistically significant difference. The mean score for the post-test rose to 63.97%% and 68.13% for the control and experimental group respectively. Pupils who were taught using the Mathematics Laboratory Method performed better than those who were taught using the Traditional Method of teaching. The study recommends the adoption of Mathematics Laboratory Method in Secondary Schools so as to enhance learner achievements in Mathematics.

Open Access Original Research Article

Alwanyi Kevin Shilaviga, Achiles Nyongesa Simiyu, Olege Fanuel

It is possible to express all geometric notions connected with closed linear subspaces in terms of algebraic properties of the orthoprojectors onto these linear spaces. In this paper, sufficient conditions for the calculus of a family of orthoprojectors in *B(H)* have been given with meaningful consideration of the sum, the product and difference of orthoprojectors to be a projector. This has been done by giving the algebraic formulations of orthogonality for the sum, product and difference. From the paper, it is observed that there is a natural one-to-one correspondence between the set of all closed linear subspaces of a Hilbert space *H* and the set of all orthoprojectors on *H*. This paper will help in the study of vector space with many diverse applications such as orthogonal polynomials, *QR* decomposition of projectors and Gram-Schmidt orthogonalization.

Open Access Original Research Article

Darwin P. Mangubat, Isagani S. Cabahug, Jr.

Let G be a nontrivial, undirected, simple graph. Let S be a subset of V (G). S is a restrained cost effective set of G if for each vertex v in S, degS(v) \(\leq\) degV (G)rS(v) and the subgraph induced by the vertex set, V (G) r S has no isolated vertex. The maximum cardinality of a restrained cost effective set is the restrained cost effective number, CEr(G). In this paper, the restrained cost effective sets of paths, cycles, complete graphs, complete product of graphs and graphs resulting from line graph of graphs with maximum degree of 2 were characterized. As a direct consequence, the bounds or exact values for the restrained cost effective number were determined as well.

Open Access Original Research Article

Wekesa Waswa Simon, Winifred Nduku Mutuku

**Aim: **This study focuses on the numerical analysis of Jeffrey nano-fluid bound with magnetic field in presence of convectively heated boundary.

**Study Design: **Abstract, introduction, Equations formulation, numerical analysis and conclusion

**Place and Duration of Study: **Department of Mathematics and Actuarial Science, Kenyatta University, between 2021-2022

**Methodology: **This paper discusses the imposed magnetic field on Jeffrey fluid suspended with nanometre-sized particles moving over a vertical sheet with a convectively heated boundary. The partial differential equations are formulated by considering assumptions and the boundary conditions to describe the continuity, momentum, energy and concentration of the fluid. The similarity transformation technique was applied to convert the partial differential equations into first-order linear differential equations which were simulated in Matlab by invoking the Adam’s-Moulton predictor-corrector scheme in ode113.

The graphs have been analysed with the effects of Deborah, Dufour-Lewis, Hartman, and Prandtl numbers respectively, solutal stratification, diffusion, thermophoresis, temperature Grashof, mass Grashof, relaxation-retardation parameters on the flow velocity, concentration, temperature, skin friction, heat and mass transfers looked into.

**Results: **While Deborah number increased velocity, it reduced concentration, skin friction and thermal boundary layer at lower numbers hence improved mass and heat transfer. Solutal stratification, Retardation-relaxation parameter and diffusion raised temperature thus heat transfer.

**Conclusion: **Deborah number, solutal stratification, retardation-relaxation parameter and diffusion improves heat and mass transfer.

Open Access Original Research Article

Amonyela Hillary Isabu, Achiles Nyongesa Simiyu, Aldrin Wekesa Wanambisi

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

Open Access Original Research Article

Ngutor Nyor, Muhammad Izhar Muazu, Samuel Abu Somma

The business of Sachet water (popularly called pure water) in Nigeria is often competitive due to the high demand for Sachet water by the populace. This is so because sachet water is the most affordable form of pure drinking water in Nigeria. As such, Sachet Water Firms that want to succeed in an ever increasing competitive market need to have the knowledge of Game Theory to identify which strategy will yield better profit independent of the strategy adopted by other competitors. This paper is aimed to investigate and determine the equilibrium point for three Sachet Water Firms using the Nash Equilibrium Method as it provides a systematic approach for deciding the best strategy in competitive situation. The result showed two Nash Equilibriums (promo, promo) and (stay-put, stay-put) with their respective payoffs of (82; 82; 82) and (147; 147; 147).