https://repository.maseno.ac.ke/handle/123456789/103 2026-05-15T12:08:10Z https://repository.maseno.ac.ke/handle/123456789/6278 Enumeration of plane and d-ary Tree-like structures ONYANGO, Christopher Amolo Trees are connected graphs which do not have loops, multiple edges and cycles. A variety of trees such as binary trees, ordered trees, d-ary trees, Cayley trees and noncrossing trees have been studied at length. Tree-like structures such as cacti and Husimi graphs have the properties of trees where we consider blocks of the structures instead of vertices. Plane Husimi graphs, plane cacti and plane oriented cacti have been enumerated with regards to leaves, number of blocks and block types. However, there is no literature on the study of plane tree-like structures according to root degree and degree sequence. Moreover, d-ary tree-like structures have not been enumerated at all . In this work, we have enumerated plane Husimi graphs, plane cacti and plane oriented cacti according to the degree of the root and outdegree sequences. We have also enumerated bicoloured plane tree-like structures with regards to number of vertices, blocks and block types. Finally, we have introduced and enumerated d-ary Husimi graphs, cacti and oriented cacti with given indegree sequence, number of leaves, blocks and block types. To obtain our results we have used symbolic method to obtain generating functions for tree-like structures, used Lagrange Inversion formula and Lagrange Burmann to extract the coefficients of the variables in the generating¨ functions and in some instance, we constructed a bijection. The results of this study will add to literature in this area of study. Master's Thesis 2024-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5986 Mathematical modeling of Dual protection and art Adherence for a high risk HIV Population ORIEDO, Indayi Samson The spread of HIV/AIDS remains a major concern to public health enthusiasts world over. In spite of interventions such as medical male circumcision, condom use, treatment using Antiretroviral Therapy (ART), as well as use of Pre-Exposure Prophylaxis, the number of new HIV/AIDS infections in Sub-Sahara Africa remains high. This may be attributed to factors such as PrEP failure and inconsistency in condom use especially among the high risk group. The e ectiveness of condoms depends on quality and proper use, while the success of ART largely depends on adherence. Mathematical models for these interventions exist in literature. However the challenges associated with the use of a single approach consequently necessitate the use of dual protection for better outcome against infection especially for the high risk population. In this study, a mathematical model for dual protection, incorporating PrEP and Condom use, and ART adherence is formulated, based on a system of ordinary di erential equations and analyzed. The results obtained from stability analysis indicate that provided the basic reproductive number (R0) is less than unity, the disease free equilibrium point is both locally and globally asymptotically stable, while provided that R0 is greater than unity, the endemic equilibrium point is locally asymptotically stable. Sensitivity analysis showed that the most sensitive parameter is 1, the mean contact rate with undiagnosed infectives. Numerical simulation results revealed that dual protection and ART adherence are key in the ght against the spread of HIV among the high risk population. These ndings will help in reducing the number of new HIV infections as well as lower the infectivity of those who are already infected. Master's Thesis 2023-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5261 Operation of mutation on Polar quivers OYENGO, Michael Obiero Michael Obiero In recent times, there has been a lot of interest in the study of quivers, both by mathematicians and theoretical physicists. We introduce a new concept of polar quivers and their mutation. The idea of polar quivers arises from the concept of anomaly free R-charges in theoretical physics. Mutation of polar quivers is build on mutation quivers with potential, which was defined by Derksen, Weyman and Zelevinsky. An R-charge assigns angles to the arrows of a quiver. In a polar quiver we assign angles and positive non-zero integers to vertices and impose conditions equivalent to the anomaly conditions for R-charges. We then establish that mutation of a polar quiver will give a polar quiver if and only if a simple additional condition is satisfied. We use families of quivers linked by mutation, from the work of Stern, as our source of examples. The results of this study have applications in geometry and theoretical physics. 2009-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5260 Norms of tensor products and elementary Operators ODERO, Beatrice Adhiamb In this thesis, we determine the norm of a two-sided symmetric operator in an algebra. More precisely, .we investigate the lower bound of the operator using the injective tensor norm. Further, we determine the norm of the inner derivation on irreducible C*-algebra and confirm Stampfli's result for these algebras. 2009-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5257 On norms of elementary operators NYAARE , Benard Okelo The study of elementary operators has been of great interest to many mathematicians for the past two decades. Of special interest has been to determine the norms of these operators. The norm problem for elementary operators involves finding a formula which describes the norm of an elementary operator in terms of its coefficients. The upper estimates of these norms are easy to find but approximating these norms from below has proved to be difficult in generaL Several mathematicians have produced known results for special cases on the lower estimates, for example, Mathieu found that for prime C*-algebras, the coefficient is ~, Stacho and Zalar obtained 2( v'2-1) for standard operator algebras on Hilbert spaces, Cabrera and Rodriguez obtained 20!I2 for JB* -algebras while Timoney came up with a formula involving the tracial geometric mean to calculate the norm of elementary operators. An operator T: A ~ A is called an elementary operator if T can be expressed in the formZ'[z) = L~=Iaixbi, \j x E A where A is an algebra and tu, b; fixed in A. The norm of an operator T is defined by IITII= sup{IITxll : x E H, Ilxll = I} where H is a Hilbert space. The purpose of this study therefore, has been t,o determine the lower estimate of the norm of the basic elementary operator on a' C*- algebra through tensor products. To do this we needed to have a good background knowledge on functional analysis, general topology, operator theory and C*-algebras by understanding the existing theorems and relevant examples especially on tensor product norms. We used the approach of tensor products in solving our particular problem. We hope that the results obtained shall be useful to applied mathematicians and physicists especially in quantum mechariics. 2009-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5251 Mathematics of Pesticide Adsorption in a Porous Medium: Convective-Dispersive Transport with steady state water flow In two Dimension WETOYl, A.Seth Harrisson The transport of solutes through porous media where chemicals undergo adsorption or change process on the surface of the porous materials has been a subject of research over years. Usage of pesticides has resulted in production of diverse quantity and quality for the market and disposal of excess. material has also become an acute problem. The concept of adsorption is essential in determining the movement pattern of pesticides in soil in order to asses the effect of migrating chemical, from their disposal sites, on the quality of ground water. In the study of movement of pesticides in the soil, the mathematical models so far developed only consider axial movement. The contribution of radial movement to the overall location of solutes in the porous media seems to have been disregarded by researchers in this field. The objective of this study is to close this gap by developing a mathematical model to determine the combine radial and axial movement of pesticides due to Convective - Dispersive transport of pesticides with steady - state water flow in a porous media. The methodology will involve determining the comprehensive dispersion equation accounting for both axial and radial movement of solutes in the porous media and finding the solution of the governing equation using finite difference methods. The solution of this equation will be applied to the data from experiments carried out on adsorption and movement of selected pesticides at hi~h concentration by soil department, University of Florida, Gainesville U.S.A. We will confme our study to single - Region Flow and Transport. 2007-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5250 Numerical solution of Korteweg-de vries equation ONAM, Joel Otieno The Kotteweg-de Vrr-es(KdV)is a mathematical model of waves on shallow water surfaces. The mathematical theory behind the KdV equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. The equation is named after Diederik Korteweg and Gustav de Vries, It has long been known that conservative discretization schemes for the KdV and other nonlinear equations tend to become numericrtlly unstable. Although finite difference approximations have been used, there are always instabilities of the solutions obtained, In this work we solved the Korteweg-ds Vries (KdV) equation using an explicit finite difference method, subject. to various boundery conditions which are travelling wave solutions to the KdV equation. The methodology involved carefully designing conservative finite difference discretization that can remain stable and deliver sharp solution profiles fora long time. We then determined the accuracy of the finite diffurence scheme by comparing the graphical outputs of the numerical results. 2008-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5249 On a generalized q-numerical Range Musundi, Sammy Wabomba We 'consider numerical ranges of a bounded linear operator on complex Hilbert spaces. Many properties of the classical numerical range are known. We investigate the properties of the q-numerical range in relation to those of the classical numerical range. We also establish the relationship between the q-numerical range and the algebra q-numerical range. Furthermore, we extend the results of the classical numerical range and q-numerical range to the C-numerical range and investigate how the C-numerical range is an explicit generalization of both the classical numerical range and q-numerical range. 2008-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5230 Mathematical modelling of flood Wave: a case study of Budalang'i Flood plain basin in Busia county, Kenya MUSINDAYI, Stephen Miheso Flooding is a worldwide problem with more adverse e ects in developing countries. In Kenya, severe ooding is experienced on the lower tributaries of Lake Victoria, mainly Budalang'i area. This is indicated in the historical oods of 2003, 2007, 2017 and 2019, leading to mass displacement of people and property destruction. This has attracted attention of researchers worldwide and application of di erent measures to curb ood in the study regions. Mathematical modeling of ood wave has however not been adopted in Budalang'i ood plain. Therefore this study formulated, analyzed and simulated the 2D ood wave model with incorporation of a sink to the Budalangi ood plain. Formulation was applied on existing Navier Stokes equations with the addition of a sink term on continuity equation. Analy- sis of the shallow water model entailed transforming the equations using Jacobian transformation and assessing the nature of ow using Froude number. For simula- tions of the 2D shallow water model, the study adopted a nite di erence scheme to make approximations which solved the system of equations and displayed in the gures . It is realized that in the formulation of the 2D shallow equations, appro- priate model for Budalang'i ood plain is easily derived from the 3D Navier Stokes equations under ood plain assumptions and addition of a sink term is necessary for modelling in the ood plain. Assessment of the properties reveals that super- critical ows are dominant. Addition of a sink term ensures steady state velocity thus reducing higher frequency and turbulence as well as over bank ows while incorporating coriolis term has signi cant e ect on the turbulence. The study concludes that addition of a sink term to the 2D shallow water model will enable control of the oods in the area of study. The ndings will aide disaster manage- ment stakeholder to come up with a more reliable ood prevention technique and new knowledge on how source terms can help reduce ood risk. 2022-01-01T00:00:00Z https://repository.maseno.ac.ke/handle/123456789/5205 Norms of elementary operators. RUTO, Peter Kiptoo The norm of an elementary operator has been investigated over long period by several mathematician under various special circumstances. Timoney working on algebra of bounded linear operators on Hilbert spaces, established the lower bound of norms of eleementary operators on Calkin algebra. Similary, mathieu studied norm properties of elementary operators on Calkin algebra and established a result whose key basis is the Haagerup tensor norm. We joined results from these eminent mathematicians to establish norms of elementary operators, particularly determine the lower bounds of elementary operators. 2010-01-01T00:00:00Z