Mathematics

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    On orthogonality and micro transitivity characterization of Hilbert spaces
    (Chuka University, 2024) Mugure Damaris Njeri
    Characterization of a transitive separable Banach spaces as Hilbert spaces has been an open area of research. It has been shown that separable Banach spaces which are transitive, almost transitive, convex transitive and micro transitive together with isometries of various characteristics such as unitary, reflection, differentiable properties are Hilbert spaces. It has also been shown that a separable real Banach space which is almost transitive with vector orthogonalities of dimension greater than three is a Hilbert space. However, properties such as micro transitive together with vector orthogonalities for n-dimension have essential property that can be utilized to characterize Banach spaces as Hilbert spaces. Additionaly, by this characterization, properties of matrix numerical range and numerical radius can also be determined. Therefore, by utilizing micro transitivity and Isosceles vector (I-vector), Pythagorean vector (P-vector) and Isosceles Pythagorean vector (IP-vector) in the unit sphere of separable Banach space this research determined that an n-dimension separable Banach spaces is a Hilbert space. In addition, by the use of properties of numerical range in general Banach space the study also established properties of matrix numerical range and numerical radius in separable transitive Banach space. The findings of this study will find use in algebra and differential operators for the purpose of calculation of wave functions and formulation of theory. In addition, the findings of the study will find use in spectral analysis of functions for the study of vibrations and interfacial waves stability analysis.
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    APPLICATION OF QUEUING THEORY FOR OPTIMAL CUSTOMER CENTRICITY TO THE BANKING SECTOR IN KENYA
    (Chuka University, 2023-10) JUMA SAMWEL KISIANG’ANI
    Long queues and waiting times are common in banks, resulting in customer dissatisfaction and low customer retention. The study applied a descriptive research design to investigate queuing dynamics in a banking hall at a commercial bank in Kenya. A single server system (M/M/1) queuing model was used to estimate the average waiting time, system intensity, service time, and optimal number of staff during peak and off-peak periods (July). The study used secondary data on daily waiting times, service times, the number of customers, and servers for May and July 2019, 2020, and 2021 during working hours between 8.30 a.m. and 4 p.m. on Monday to Friday and 8:30 a.m. and 12 p.m. on Saturdays. Data analysis was done using R and Excel. The research findings indicated that the peak periods (May) recorded an average waiting time (AWT) of 13 minutes, 35 seconds in 2019, 10 minutes, 14 seconds in 2020, and 8 minutes, 36 seconds in May 2021. In the off-peak periods (July), an AWT of 3 minutes, 46 seconds, was registered in 2019, 5 minutes, 12 seconds in 2020, and 7 minutes, 42 seconds in 2021. An average service time (AST) of 1 minute 52 seconds in May 2019, 2 minutes 34 seconds in May 2020, and 2 minutes 27 seconds in May 2021. In the off-peak periods (July), an AST of 3 11 seconds was registered in 2019, 3 4 seconds in July 2020, and 2 43 seconds in July 2021. Overall, the system intensities are low to moderate, with the COVID-19 pandemic severely impacting the peak period more than the off-peak. In the peak periods, the service rates averaged 33, 24, and 25 persons per hour in May 2019, May 2020, and May 2021. The respective system intensities were 0.534, 0.360, and 0.492. In the off-peak periods, the average service rates were 19, 20, and 23 persons per hour in July 2029, July 2020, and July 2021. The respective associated system intensities of 0.535, 0.461, and 0.487. From the pooled data for 2019 and 2021, the study recommends that banks operate with an AWT of 6 minutes, 24 seconds, and an AST of 3 minutes. Further, the study established that a bank could work with an optimal four servers with an AST of 2 minutes, 35 seconds (a service rate of 20 people per hour), and achieve a moderate average service intensity of 0.552.
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    ON CHARACTERIZATION OF PERMUTATION GRAPHS
    (Chuka University, 2020-12) Nyabate, Nyabuto Fancy
    Graph theory is an area in discrete mathematics with numerous theoretical developments and many applications to practical problems in computer science, chemistry, biology and operational research. As a result, it has attracted much attention to researchers in many dimensions including graph labeling, graph coloring, combinatorics, graph isomorphism, matroid theory and graph representations among others. Many researchers in this area have also worked on permutation graphs paying attention to the properties of cyclic permutation graphs, including crossing numbers and isomorphism. So far isomorphism between two cyclic permutation graphs has been determined by positive and negative natural isomorphism. However, construction of other classes of permutations graphs and establishing an alternative approach for determining isomorphism between permutation graphs as well as finding some properties of permutation graphs would be of significance. The aim of this study was to develop a class of permutations, determine algebraic properties of the permutations, construct permutation graphs and establish some properties of the constructed graphs including isomorphism. A class of [nxk - permutations was first obtained by coming up with a bijection on a finite set, which resulted into permutations. Some algebraic properties were established, in particular, the permutations resulted in an abelian group as well as it formed a subgroup. Graphs were then constructed from these permutations and some properties including symmetry, unique, connectedness, distance and isomorphism determined by enumeration. The results of this research are of significance in other practical areas of application of graphs, including computer science, chemistry, biology, operational research and combinatorics.