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# PHD-Department of Mathematics

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Item Quasiaffine Inverse and Moore-Penrose Inverse of Operators in Hilbert Spaces(Kenyatta University, 2024-05) Mwanzia, Joel MutukuShow more The study of inverses of operators by the concept of the Moore-Penrose Inverse and the quasiaffine inverse started in 1920 and 1985 respectively. Precisely, Moore (1920) and Penrose (1955) independently gave conditions satisfied by the MPI. Hongke and Chuan (1985) and Khalagai (1996) studied invertibility of normal (sub-normal) operators and one-sided invertible operators respectively, by the concept of the quasiaffine inverse. Since then, several researchers have contributed to these areas. Particularly, it has been shown that the Moore-Penrose inverse of an operator A with closed range satisfies the following conditions:AA^+ A=A,〖 A〗^+ AA^+=〖A 〗^+, 〖(AA〗^+ )^*=AA^+ and 〖(A〗^+ A)^*=A^+ A. If A is a quasiaffine inverse of B then both A and B are quasiaffinities and if A is an EP operator then 〖Ran(A)=Ran(A〗^*). It is also known that the Fuglede-Putnam Theorems and Fuglede-Putnam type commutativity theorems hold for normal operators and EP operators under some conditions. On quasiaffine inverses, this thesis establishes the uniqueness of the quasiaffine inverse of A given AXB=X and BYA=Y as well as establishing that B=A^(-1) under given conditions. The invertibility of quasinormal partial isometry with dense range as well as results on invertibility of operators A and B satisfying the equations AX=XB or BY=YA or both is shown under some given conditions. On MPI, the invertibility of an EP operator in terms of its Moore-Penrose inverse is established. In particular, the case where the Moore-Penrose inverse of an EP operator turns to be its usual inverse under some given conditions is shown. The Moore-Penrose inverse of a perturbed operator A+B with closed range, where A is expressible as a product of two operatorsP,Q∊B_C (H) with closed ranges and B a bounded operator satisfying some given conditions is exhibited as well as the relation between the ranges and null spaces of these operators. Moreover, this thesis establishes that Fuglede-Putnam-type results hold for EP operators, injective operators and operators with dense range satisfying some commutativity conditions involving operators 〖 AA〗^*, A^* A, 〖A^* A〗^+, BB^*,B^* B and 〖B^* B〗^+. The study of inverses of operators by the concept of the Moore-Penrose Inverse and the quasiaffine inverse started in 1920 and 1985 respectively. Precisely, Moore (1920) and Penrose (1955) independently gave conditions satisfied by the MPI. Hongke and Chuan (1985) and Khalagai (1996) studied invertibility of normal (sub-normal) operators and one-sided invertible operators respectively, by the concept of the quasiaffine inverse. Since then, several researchers have contributed to these areas. Particularly, it has been shown that the Moore-Penrose inverse of an operator A with closed range satisfies the following conditions:AA^+ A=A,〖 A〗^+ AA^+=〖A 〗^+, 〖(AA〗^+ )^*=AA^+ and 〖(A〗^+ A)^*=A^+ A. If A is a quasiaffine inverse of B then both A and B are quasiaffinities and if A is an EP operator then 〖Ran(A)=Ran(A〗^*). It is also known that the Fuglede-Putnam Theorems and Fuglede-Putnam type commutativity theorems hold for normal operators and EP operators under some conditions. On quasiaffine inverses, this thesis establishes the uniqueness of the quasiaffine inverse of A given AXB=X and BYA=Y as well as establishing that B=A^(-1) under given conditions. The invertibility of quasinormal partial isometry with dense range as well as results on invertibility of operators A and B satisfying the equations AX=XB or BY=YA or both is shown under some given conditions. On MPI, the invertibility of an EP operator in terms of its Moore-Penrose inverse is established. In particular, the case where the Moore-Penrose inverse of an EP operator turns to be its usual inverse under some given conditions is shown. The Moore-Penrose inverse of a perturbed operator A+B with closed range, where A is expressible as a product of two operatorsP,Q∊B_C (H) with closed ranges and B a bounded operator satisfying some given conditions is exhibited as well as the relation between the ranges and null spaces of these operators. Moreover, this thesis establishes that Fuglede-Putnam-type results hold for EP operators, injective operators and operators with dense range satisfying some commutativity conditions involving operators 〖 AA〗^*, A^* A, 〖A^* A〗^+, BB^*,B^* B and 〖B^* B〗^+.Show more Item Modeling HIV Aids Dynamics with Funding along the Northern Corridor Highway in Kenya(Kenyatta university, 2023) Kimulu, Ancent Makau; Winifred N. Mutuku; Samuel M. Mwalili; David MalonzaShow more For the past three and a half decades, HIV/AIDS has been a worldwide health problem. Because of its severe repercussions, it necessitates a significant financial commitment to stem its spread and prevent death-related disorders. In 2019, 18.6 million dollars were spent globally to finance HIV response. UNAIDS estimated that the HIV response required $26.2 billion USD by 2020. This emerged as a result of increased infection and fatality rates between 2015 and 2020. East and Southern Africa are the most affected by HIV/AIDS in Africa. However, owing to financial commitments to combat the pandemic, Kenya and South Africa have seen enhanced prevention and treatment facilities, with the largest difficulty being reliance on donors for funding HIV response. Kenya has a record of more than 1.5 million cases of people living with HIV in 2019 with a prevalence of 4.8% among adults in the same year. This ranked Kenya as the seventh-largest HIV population in the world. A recent study among 3,805 truckers along the Northern corridor highway in Kenya found that 55.9% had commercial sex in the past 6 months and 46.6% had regular sex partners along their trucking route besides their wife or girlfriend at home. HIV is mostly spread sexually in Kenya, with a high rate of infection among critical demographics such as sex workers and truckers, among others. Our study was based on the Northern Corridor highway which runs from Mombasa to Busia-Malaba on the Kenya-Uganda border, passing through Nairobi, Mai Mahiu, and Salgaa. The major mechanism of HIV/AIDS transmission along Kenya's Northern Corridor route is transactional sex, which involves financial transfers. The sexual network of truckers are complex and can be a conduit for the widespread of HIV between truckers and FSWs along the corridor. This prompted the need to better understand the dynamics of transmission of HIV/AIDS between truckers and female sex workers. In this study, a model was formulated for HIV/AIDS dynamics along the Northern corridor highway in Kenya which included circumcision and funding of treatment as control measures. The reproduction number, disease-free equilibrium and endemic equilibrium points were determined and their stabilities were also determined using the next-generation matrix method. The disease-free equilibrium is stable when R_0u<1, R_0c<1 and R_0f<1 while the endemic equilibrium point is stable when R_0u>1, R_0c>1 and R_0f>1. It was found that circumcision can be used as an intervention to minimize the infection of HIV among truckers and female sex workers. In addition, a model was developed to investigate the impact of funding on HIV transmission between truckers and female sex workers. According to the model's findings, boosting circumcision and funding decreases the rate of transition from the Susceptible to the Infected classes. Additionally, boosting funding for any class of treatment raises the Treatment class and lowers the total number of AIDS-related deaths. This also indicates that the Kenyan government should increase internal funding for HIV/AIDS due to the dwindling donor funding since 2018. This is because funding is a key in increasing treatment which decreases the AIDS cases.Show more Item Classical and Bayesian Approaches For the Zero-Inflated Dynamic Categorical Panel Ordered Probit Model(Kenyatta University, 2023) Wanjiru, John Kung’u; Leo Odiwour Odongo; Ananda Omutokoh KubeShow more The Zero inflated ordered categorical data with time series structure are often a characteristic of behavioral research attributed to non-participation decision and zero consumption of substances such as drugs. The existing Semi-parametric zero inflated dynamic panel probit model with selectivity have exhibited biasness and inconsistency in estimators as a result of poor treatment of initial condition and exclusion of selectivity in the unobserved individual effects respectively. The model assumes that the cut points are known to address heaping in the data and therefore cannot be used when the cut points are unknown. The Simulated maximum likelihood was applied to evaluate the double integrals in the Semi-parametric zero inflated dynamic panel probit model. This procedure could be very time-consuming even with fast modern computer and imprecise even with the use of modern simulator like Halton simulators. The aim of this research was to develop the Zero inflated dynamic panel ordered probit models with independent and correlated error terms to address the above challenges. Interpretation of the coefficients in the proposed models were extra difficult than in the normal regression scheme because a shift in one of the variables in the equation is conditioned on other variables and their parameters. Average partial effects that gave the effects on the particular probabilities per unit change in the covariates was proposed to address the above challenge. The integrals were evaluated using Two step Gauss Hermite quadrature that is five times faster than the Simulated maximum likelihood. Since the solutions are not of closed form, maximum likelihood estimation based on Newton Raphson algorithm and Bayesian approach were used to estimate the parameters of the proposed models. Monte Carlo simulations were conducted to investigate the theoretical properties of the estimators of the developed models. Using National Longitudinal Survey of Youth (1997) dataset sponsored by the Bureau of labour Statistics of the U.S. Department of labour with zero inflation, the study investigated the determinants of smoking tobacco among the youths. The study found that the proposed models produced consistent estimators and their estimates were more accurate than the Dynamic panel ordered probit model estimates. The proposed models fitted the data better than dynamic panel ordered probit model in both classical and Bayesian approaches in the simulated data. The study found positive associations between the initial period participation decision and consumption levels observations and unobserved latent participation decision and consumption levels. Therefore, this indicated that it is essential to control for participation decision and consumption levels at the initial period. The models showed a strong and significant positive state dependence in both participation decision and at various consumption levels. The unobserved individual effects accounted for 49.90% of the unexplained variation in decision to participate in smoking and 47.65% of the unexplained variation at all levels of consumption. The main causes of persistence in smoking decision were the state dependence, unobserved heterogeneity and race while the main causes of persistence at consumption level were state dependence, unobserved heterogeneity, gender and age. The study is significant to policy analyst in identifying the socioeconomic and demographic factors associated with drug abuse and providing useful information to facilitate well-targeted public health policies.Show more Item Simulating Pressure Distribution of a Horizontal Well in an oil Reservoir Subject to Single Edged and Bottom Constant Pressure(Kenyatta University, 2022) Mutisya, Mutili Peter; Kennedy Otieno Awuor; Stephen Ezizanami Adewole; Daniel Okang’a OyooShow more In this study the pressure distribution in an oil reservoir with a horizontal well is investigated. A horizontal well with single-edged and constant bottom pressure is outlined. A reservoir bounded with two constant pressure boundaries, like an edge and bottom water, requires that the production engineer should adhere diligently to a production schedule, developed by a reservoir engineer, for clean oil production to be possible. This means that arbitrary production practices through selection of production rates could lead to production of these external fluids. This can mar the economics of the project. Production schedules or plans show acceptable rates, well design and production time that can guarantee only clean oil production. In this study, pressure behaviour of a horizontal well drilled and completed in a reservoir subject to with simultaneous single-edged and bottom water drives is investigated in detail. All possible flow periods or patterns that can be exhibited by the well are determined. Fluid flow in oil reservoirs in real time is governed by a heterogenous diffusivity equation, which describes reservoir pressure as a function of reservoir, fluid and wellbore properties. To solve this unsteady state problem, Green’s functions were deployed to represent the boundaries of the reservoir selected for study. The Green’s functions selected are for flow from start of transient to late time, when all the external boundaries are felt. Newman product rule was used to derive a dimensionless pressure expression for the reservoir system oil flow. The source of pressure transient was production throughout. All the resulting integrals were performed numerically. MATLAB programming was used to plot the curves by applying spline functions interpolation. Influence of reservoir, fluid and wellbore properties on reservoir pressure was investigated in real time. To assist interpretation, dimensionless pressure derivatives were also computed. Near wellbore problems, like skin and wellbore storage, which affect well performance only at very early time, were not considered in the study. From the results, 𝑃𝐷 and 𝑃𝐷′ vary directly with ℎ𝐷 and inversely as 𝐿𝐷. The 𝑃𝐷′ gradually reduces to zero when 𝑃𝐷 begins to exhibit a constant trend. 𝑃𝐷′ vary inversely with ℎ𝐷 and 𝑦𝑒𝐷 at all flow times. The number of flow periods varies with reservoir size, well length and production time. The time at which the 𝑃𝐷′ starts to exhibit a downward trend is the external fluid breakthrough time. The breakthrough time is affected by well design. Longer wells exhibit delayed breakthrough time because of lower pressure drawdown associated with increased well length. If production rate is sustained for any particular well design, the well will completely water-out. Finally, infinite conductivity 𝑥𝐷=0.732 and uniform flux condition do not really affect 𝑃𝐷 and 𝑃𝐷′ at early time.Show more Item Phytochemical Analysis, Anti-Inflammatory Activity, Antioxidant Activity and Toxic Effects of Aqueous Root Extract of Launaea CORNUTA (Hochst. Ex Oliv. & Hiern.) Evans Kapanat Akimat (Bs.C)(Kenyatta University, 2022) Akimat, Evans Kapanat; George Isanda OmwengaShow more Anti-inflammatory, anti-nociceptive and antioxidant drugs have alleviated the agony of millions of people, especially in the developing countries where management of inflammation remains a big challenge. Nonetheless, despite being beneficial, these drugs are now seriously jeopardized by the adverse side effects associated with these synthetic compounds which include heart attack, stomach ulcers, liver and kidney diseases. In most African countries, the anti-inflammatory agents are limited and expensive. Therefore, scientists are tasked to generate new ideas of alternative and novel drugs. The root extracts of Launaea cornuta have been locally used in traditional medicine for decades to manage inflammatory conditions and other oxidative-stress-related syndromes; however, their pharmacologic efficacy has not been scientifically investigated and validated. Hence, we investigated the in vitro antioxidant activity, anti-inflammatory (in vitro, ex vivo and in vivo) efficacy, acute oral toxicity, and qualitative phytochemical composition of the aqueous root extract of L. cornuta. The ferric-reducing antioxidant power (FRAP) and the 2,2-diphenyl-2-pycrylhydrazyl (DPPH) methods were used to determine the studied extract’s antioxidant activity. Besides, the anti-inflammatory efficacy of the studied plant extract was investigated using in vitro (anti-proteinase and protein denaturation), ex vivo (membrane stabilization), and in vivo (carrageenan-induced paw oedema in Swiss albino mice) methods. OECD guidelines were used to conduct acute oral toxicity test using mice model. The studied plant extract demonstrated significant in vitro antioxidant effects, that were evidenced by higher DPPH radical scavenging (53.30% to 86.84%) and FRAP activities (0.56 to 0.74 absorbance), which were in a concentration-dependent manner (p<0.05). Generally, the studied plant extract exhibited significant in vitro, ex vivo, and in vivo anti-inflammatory efficacy, respectively, and in a concentration/dose-dependent manner, compared with respective controls (p<0.05). For instance, the extract dose of 250 mg/kgbw had higher potency than the standard drug Dexamethasone in dose and time dependent manner in all mice used. Moreover, the studied plant extract did not cause any observable signs of acute oral toxicity even at the highest dose of 2000 mg/Kg BW (LD50 >2000 mg/Kg BW). Additionally, the qualitative phytochemistry revealed the presence of tannins, cardiac glycosides, anthraquinones, alkaloids, steroids, terpenoids, phenols and flavonoids. Some of these phytochemicals like flavonoids are antioxidant- and anti-inflammatory-associated phytochemicals and were deemed responsible for the reported pharmacologic efficacy. Therefore, aqueous root extract of L. cornuta has in vitro, ex vivo and in vivo anti-inflammatory, and in vitro antioxidant activity thus further studies to characterise bioactive molecules and their mode(s) of pharmacologic efficacy are encouraged.Show more Item Numerical Analysis of the Thermophysical Properties of Hybrid Nanofluids for Industrial Use(Kenyatta University, 2022) Okello John AcholaShow more Hybrid nanofluids engineered using two or more different types of nanoparticles suspended in the base fluid have numerous industrial and engineering applications. The applications range from heat transfer (coolant) fluids in industrial thermal processes, automotive industry, electronic devices, to being used in the biomedical field in areas such as nanocryosurgery, nano-drug delivery, magnetic fluid hyperthermia, etc. The current study examines the thermophysical properties of hybrid nanofluids for application as industrial coolants and lubricant additives (nanolubricants). The hybrid nanofluid (hybrid nanocoolant) consists of ethyleneglycol as the base fluid with Copper-Alumina (Cu-Al2O3), Copper-Titania (Cu-TiO2), and Titania-Alumina (TiO2-Al2O3) as the hybrid nanoparticles. The partial differential equations governing the flow of the hybrid nanofluid are formulated and transformed into a system of coupled non-linear ordinary differential equations using suitable similarity transformation variables. The shooting technique together with the fourthorder Runge-Kutta-Fehlberg integration scheme was used to obtain the numerical solutions to the coupled non-linear ordinary differential equations. The numerical analysis and simulation is achieved using MATLAB and the graphical results are depicted for the various pertinent parameters involved in the flow. The presence of nanoparticles makes the fluid susceptible to the effects of magnetic field. Increasing magnetic field intensity (𝐻𝑎) applied to the flow retards the flow of the fluid and enhances the fluid’s thermal boundary layer thickness maximizing the cooling effect of the hybrid nanofluid. The (TiO2-Al2O3/EG) hybrid nanofluid maintains a low temperature profile thus emerging as the optimal industrial coolant. For the nanolubricant study, the fluid’s velocity profiles, temperature profiles, local Nusselt number and skin friction coefficient were investigated for different pertinent parameters namely; Eckert number (𝐸𝑐), suction/injection parameter (𝑓𝑤), magnetic field intensity (𝐻𝑎), slip parameter (𝛽), nanoparticle volume fraction (𝜙), and Grashof number (𝐺𝑟). The study considered MHD incompressible boundary layer flow of engine oil-based ((MWCNTs-Cu), (MWCNTs-Al2O3), and (MWCNTs- TiO2)) conducting hybrid nanofluids past a convectively heated vertical porous plate with Navier slip boundary conditions. The study revealed increment in fluids velocity and a decrease in local skin friction with increasing values of the slip parameter (𝛽). The (MWCNTs-TiO2/engine oil) hybrid nanofluid registered the least coefficient of skin friction thus emerging as the most suitable nanolubricant. The (TiO2-Al2O3/EG) hybrid nanofluid that emerged as the best nanocoolant can be utilized in the cooling of electronic devices, transformers, in the automobile radiators, cooling of drilling equipment and sensors used in the extraction of geothermal energy, cooling of nuclear reactors etc. The best nanolubricant (MWCNTs-TiO2/engine oil) can be used as a lubricant in high temperature applications (lubrication).Show more Item Well Test Analysis of a Horizontal Well in a Completely Bounded Reservoir(Kenyatta University, 2022) Kitungu, Nzomo Timothy; Kennedy Awuor; Stephen Ezizanami; Daniel Okang'a OyooShow more Well test analysis as an important part of reservoir engineering has gone through tremendous improvement through the years since the discovery of oil. This is in terms of the tools used, the technology involved, and the mathematical modeling involved. Since most oil reservoirs are underground and in some cases thousands of feet from the surface, it’s impossible to physically observe them and see how they behave or determine their character. Mathematical models play an important role in reservoir system characterization by predicting the well and reservoir behaviour and properties. Over the years horizontal wells have proved that they are more productive compared to vertical wells. In this dissertation, possible mathematical models that can be applied in well test analysis for horizontal wells in a completely bounded oil reservoir are developed. In developing the models, source and Green’s functions are used. Using these functions, dimensionless pressure and dimensionless pressure derivative distributions in real time are derived. Mathematical analyses of the models developed and how they can be applied to characterize a completely bounded oil reservoir penetrated with a horizontal well are presented. All possible flow periods and the effects of reservoir, fluid properties and well design on horizontal well performance are investigated and presented. The effects of reservoir anisotropy on well performance are also investigated. The results of this study show that assuming isotropic cases might reduce the accuracy and reliability of the results obtained and thus recommend consideration of anisotropy in computations. This should be in all the three directions. Further, the results of this study show that well design, directional permeability and reservoir geometry will affect the horizontal well performance differently at early flow time. This applies when the infinite-acting flow is considered as compared to the pseudosteady state flow at late time. It is also noted that the number of flow periods can be many; four full and at least three transitional flow periods from inception of early transient, when the well is infinite-acting, to late transient, when all the external reservoir boundaries are felt. The results also suggest that oil wells that are in the same reservoir and closely spaced will experience pressure communication (interference) faster and vice-versa. Thus, well pressure interference affects well performance. The results obtained in this study can be used for complete reservoir system characterization and to investigate the best well design for optimum oil recovery in a bounded well reservoir penetrated with a horizontal well.Show more Item Campaigns and Treatment on Alcohol Abuse in Kenya(Kenyatta University, 2021) Muthuri, Grace Gakii; David M Malonza,; Farai NyabadzaShow more Alcoholism is a serious problem in Kenya today and many adults are addicted to alcohol. The harmful use of alcohol causes a large burden concerning diseases, social and economic problems to the society. Mass media campaigns against alcohol act as sources of information to halt alcohol abuse and its potentially harmful effects. In this research, we developed deterministic models for alcohol abuse driven by the light and heavy drinkers taking into consideration the influence of pre exposure to mass media campaigns. Two models were developed, one with perfect pre exposure campaigns where the campaigns were successful and the other with imperfect pre exposure campaigns where the campaigns were not successful. The two models were analyzed through the determination of the model’s steady states and their respective stabilities analysis in terms of the alcohol abuse reproduction numbers R0. The analysis shows that alcohol-free equilibrium (AFE) is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Numerical simulations were carried out and the sensitivity analysis of the model parameters was done to determine where the campaigns should be targeted for effective control of the abuse. The results from the simulations illustrated that increasing the rate of treatment reduces the number of alcohol addicts in the community. The results also showed that mass media campaign against alcohol consumption reduces alcohol abuse. From the study we conclude that if alcohol treatment is emphasized and mass media campaign regulated then alcohol addiction will be reduced from the community.Show more Item Study of Projective Curvature Tensor, 𝑾𝒋𝒌𝒉𝒊(𝒙,𝒙̇) in Bi-Recurrent Finsler Space, 𝟐𝑹−𝑭𝒏(Kenyatta University, 2021) Opondo, Mary Atieno; Surendra P. Singh; James GatotoShow more The Weyl (W) projective curvature tensor, 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) has properties that have a wide range of applications in various fields but are still not easy to understand in a fundamental way since 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is a function of both position and direction. In this study three selected properties of the projective curvature tensor, 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) namely inheritance symmetry, collineation symmetry and decomposition are investigated in bi-recurrent Finsler space for purposes of new applications. Many authors have studied inheritance, collineation and decomposition properties for 𝐻,𝐾,𝑁,𝑅 and 𝑈 curvature tensors in recurrent Finsler spaces and in this thesis we have extended these studies to 𝑊 curvature tensor in bi-recurrent Finsler space which is still relatively under explored. The Finsler space is viewed as regular metric space and the three properties are described from the modern geometry view point using the relevant geometric and symbolic computation tools such as Lie derivatives, transvection, commutation and covariant differentiation in the sense of Berwald. The W-Curvature inheritance is defined by a Lie derivative (𝐿𝑣) proportional to the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) while the W-Curvature collineation is defined by vanishing Lie derivative (𝐿𝑣) of 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇). The results of the study show that every motion admitted in a bi-recurrent Finsler space is also a W-curvature inheritance if the space is isotropic otherwise it is a W-curvature collineation. The contra field and concurrent field are considered as special cases. The study reveals that both fields do not admit a W- curvature inheritance however they both admit a W- curvature collineation when the vector field (𝑉𝑖) of the infinitesimal transformation is orthogonal to the recurrence vector (𝐾𝑙). The geometrical properties of both inheritance and collineation symmetries have physical significance which make them useful in spacetime applications. The decomposition property of the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is also investigated for specified decomposition tensors denoted by Ψ(𝑥,𝑥̇) using different symbolic tensor computation algorithms. The study has established that tensor decomposition for the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇) is not unique but specific conditions discussed in this study introduce uniqueness into the decomposition algorithm. The study has also established that in both recurrent and bi-recurrent Finsler spaces the decomposition tensors have some properties similar to those of the original tensor and therefore decomposition can be used to compress tensors for further applications. The tensorial computation algorithms are presented in form of step by step equations and the principal results obtained have helped to identify some hidden components of the projective curvature tensor 𝑊𝑗𝑘ℎ𝑖(𝑥,𝑥̇). The results of the study are summarized in form of theorems which have already been verified and can be used in various fields for theoretical investigations and practical applications of tensors.Show more Item Modelling Jiggers Infestation and Interventions in Humans: A Case Study of Murang'a County, Kenya(Kenyatta University, 2020-06) Matendechere, Nancy ImbusiShow more Tungiasis is a parasitic skin disease caused by jiggers, also known as sandeas. The disease predominantly a ects impoverished populations living in Sub-Saharan Africa, the Caribbean and South America.In areas especially with limited or no interventions, jiggers infestationremains a problem. Mathematical models have been used for decades to inform public health policies and have been useful for the evaluation of control strategies and interventions. While some studies may have been done on jiggers, majority of them focused on social aspects of the disease. Very few mathematical models have been done on jiggers infestation. Considering the results and the interventions that come from models on vector borne diseases, a model on jiggers infestation and interventions in humans would be useful for the policy makers and government to intervene and come up with a solution to this menace. In this thesis, we present two deterministic mathematical models. First, we present a deterministic model with four compartments that represent the dynamics of the human population and an age-structured model for flea. Second, we incorporated media campaigns in the first model with the aim of investigating the potential role of awareness through media campaigns on jiggers infestation dynamics. We introduce a class of those that are aware in which the awareness does not completely protect individuals from jiggers. The model equilibria are computed and stability analyses carried out based on the reproduction number R0. Sensitivity analysis is performed on the model parameters and the results suggest that the executive infestation contact rate, as well as the rate at which the larvae develop into adult fleas are the main parameters that fuel jiggers infestation. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the parameter that accounts for the proportion of larvae that develop into adult female fleas involved in jiggers transmission is included. The model points to control of the flea through treatment of infested humans and enhancing efficacy of media campaigns.Show more Item Suborbital graphs of the groups Cn ,Dn and UP and graphs whose automorphism groups contain the permutation groups Cn and Dn(Kenyatta University, 2018-05) Olum, Fredrick OdondoShow more Item Implementation of a modified procesi algorithm to compute covariants of binary forms of up to degree five and their relations(Kenyatta University, 2016-11) Kariuki, Njau LawrenceShow more In his book (Procesi,C., 2007), Claudio Procesi suggested a new algorithm for computing covariants of binary forms under the action of SL(2;C), based on an iterative computations of covariants of the simpler group U+. In Procesi book the computation was carried out only for binary forms of degree 3 and 4, but the rst signi cant test for the algorithm would be the computation for degree 5. In 2010 summer school in Algebra organized by ICTP in Kenya, Procesi suggested the implementation of his algorithm as a project. In this thesis we implement a modi cation of the original Procesi algorithm on the computer algebra system CoCoA, study its general properties and test it with the complete description of generators and relations of the algebra of covariants of binary forms of degree 5. The modi ed form of Procesi algorithm computes covariants iteratively with respect to the degree of a covariant. The implementation was tested in the computation of covariants of binary forms of degree 5, which produces 23 covariants of degree up to 18. The algorithm produces the explicit list of covariants and rheir relations. As far as we know this is the most explicit description of the complete list of relations which is made available so far.Show more Item A numerical investigation of turbulent natural convection in a 3-d enclosure using k-w SST model and piso method(Kenyatta University, 2016-10) Kimunguyi, Kiliswa JosephsShow more Item Cycle Indices, Subdegrees and Suborbital Graphs of PSL(2,Q) Acting on the Cosets of Some of its Subgroups(Kenyatta University, 2016-01) Rotich, Stanley KiplagatShow more Item Simulating the Effects of Dam Breakage on the Downstream Topography: Morphological Evolution of Mounds and a Furrow(Kenyatta University, 2015-12) Magua, Amos Ng'ang'aShow more In this work we apply a nite volume discretization technique based on a relaxation scheme to simulate the morphological evolution of the topography as a result of a dam break, that causes ooding downstream of the breach location. The considered mathematical model comprise of shallow water equations coupled with the bed updating equation which is modi ed to account for sediment entrainment process. Thus the model comprises a set of highly nonlinear hyperbolic partial di erential equations written in compact conservation form. In order to ensure that the resulting ux matrices are non-singular and are in compact conservation form, C-formulation was used. This formulation is an unsteady approach where the water ow and bed update are discretised simultaneously. The resulting Jacobian matrices could not be diagnolised easily and the eigenvalues were determined using the formulae for cubic functions as given by Spiegiel and Liu (1999). The non-linear partial di erential equations written in C formulation were rst relaxed into a set of linear hyperbolic system using the relaxation variables ~V = (V1; V2; V3; V4) ; ~W = (W1;W2;W3;W4). The relaxed equations were then discretized spatially (semi-discretization) using the Vanleer's MUSCL scheme which is total variation diminishing, and nally the time discretization (full discretization) was done using implicit-explicit Runge kutta scheme. The numerical model developed was used to simulate dam break ows and sediment transport on topographical surfaces with a deep narrow furrow and a topographic surface with two mounds located downstream of the breach location. Results on simulations showed that the entrainment and bed load transport signi cantly a ected topography containing the furrow. The furrow widened and became shallower. Secondly the entrainment and the bed load sediment transport signi cantly a ected the topography containing the two mounds. The mounds were eroded and there was high depositions of the sediments in the vicinity of the mounds and thirdly the dam break scenario with entrainment had a higher morphological evolution than the dam break scenario without the entrainment. The results thus obtained showed that the model is conservative, accurate, stable, robust, capable of resolving shocks and can handle even more complex geometries including simulations of real life dam break scenarios.Show more Item Ranks and subdegrees of the cyclic group, the dihedral group and the affine group and associated suborbital graphs(Kenyatta University, 2015-11) Kangogo, Moses RuttoShow more Item Cycle Indices, Subdegrees and Suborbital Graphs of PSL(2,Q) Acting on the Cosets of Some of its Subgroups(Kenyatta University, 2015-12) Magero, Bunyasi FideliusShow more Item Heat and mass transfer past a semi - infinite vertical porous Plate in MHD flows in turbulent boundary layer(2019) Ngesa, Joel OcholaShow more Turbulent flows in electrically conducting media (MHD) remains one of the last unresolved problems in engineering industry and classical physics, but has general importance for the evolution of astro and geophysical plasmas. Turbulence in plasmas, i.e. ionized gases, also offers valuable insights into the not yet fully understood nonlinear dynamics of spectral cascades and structure formation due to the presence or generation of magnetic fields. These allow additional diagnostic access to the underlying nonlinear interaction of turbulent fluctuations. In experimental devices for thermonuclear fusion the magnetically confined hot plasma is basically collisionless and requires kinetic treatment. Exceptions are the thin and comparably cool edge layer near the vessel boundaries and plasmas in reversed-field pinch configurations. Turbulent plasmas in or beyond the earth often allow a fluid description due to the immense size of the dynamical regions and associated timescales of interest compared to the effective mean-free-path and the frequencies related to the plasma particles. Since plasma turbulence is a fully nonlinear problem comprising the dynamics of many interacting degrees of freedom, the relatively simple single fluid description of magnetohydrodynamics (MHD) represents a sensible starting point for theoretical and numerical investigations. The interesting properties of MHD turbulence lies mainly in its potential universality, that is to say the inherent properties of turbulence might well be important for the dynamics of systems involving gravity, radiation, rotation, or convection. Many authors have studied the theory of magnetohydrdynamics (MHD)flow problems as well as to various methods of solving these problems though mostly addressed heat and mass transfer with Hall and ion-slip currents in laminar boundary layer and rotating turbulent system past a semi-infinite vertical porous plate. In this research work we address the problem of heat and mass transfer of unsteady free convection incompressible fluid flow past a semi-infinite vertical porous plate in (MHD) flow in turbulent boundary layer, in the presence of a strong magnetic field inclined at an angle to the plate with Hall and Ion-Slip currents. The determination of the concentration, temperature and velocity profiles’ distribution for fluid flow, the rate of heat transfer, the skin friction, rate of mass transfer and effects of various flow parameters on the turbulent boundary layer fluid flow field are carried out. An explicit finite difference approximation method is used to analyze the partial differential equations governing the flow for a heat generating fluid with Hall and ion-slip effects. The computation of skin friction, rate of heat and mass transfers at the plate is achieved by Newton’s interpolation approximation over the first five points. In both cases when Gr < 0 (in the presence of heating of the plate by free convection currents) and Gr> 0 (in the presence of cooling of the plate by the free convection currents) have been discussed extensively. The effects of various flow parameters on the convectively cooled or convectively heated plate restricted to turbulent boundary layer is considered. The results demonstrate that, Hall current, Schmidt number, Modified Grashof number, Heat source parameter, Suction velocity, Time, Angle of inclination, Ion-Slip current on the convectively cooled or convectively heated plate affect the velocity, temperature and concentration profiles. Increases in Hall current parameter cause a decrease in both primary and secondary velocity profiles while increase in Ion Slip current, decreases primary velocity profiles but increases secondary velocity profiles. As a result, skin friction, rate of heat and mass transfers are altered by their variations.Show more Item Quantile Autoregression and its application to Financial Risk Management and Portfolio Optimization(2014-09-01) Ananda, Omutokoh KubeShow more Increasing globalization, complexity of capital markets and the expanding range of exotic financial instruments have made financial risk management difficult to evaluate. As a consequence, a rise in use of more sophisticated risk management systems has not led to better results. Most financial data exhibits time varying volatility and heavy tails therefore an appropriate risk measure should capture these features. Volatility patterns reflect different characteristics in different stock markets. The main aim of this study is to improve on volatility estimation by use of Quantile Autoregression frameworks. To avoid strong assumptions about the form of innovations, an initial proxy of volatility estimator is proposed. The estimator is assumed to capture the intraday volatility based on the conditional Interquantile Autoregressive Range. A class of a-mixing time series models based on quantile regression are used and direct estimation of coefficients as introduced by Koenker and Bassett (1978) are adopted. We study the estimation of scale function in the Quantile Autoregressive models discussed. Similar methods used to show the asymptotic properties of conditional autoregressive coefficient estimators are applied to the conditional Interquantile Autoregressive Range (IQAR) estimator and show that under some mild regularity conditions, it is consistent and asymptotically normal. A Monte Carlo study is carried out to verify theoretical properties derived for the estimator which confirms the estimator is consistent. The estimator is fitted to simulated data to show how to perform Risk management and Portfolio Optimization. An application to real data is included to illustrate Financial Risk Management and Portfolio Optimization.Show more Item An analysis of international tourism demand for Kenya(2014-08-18) Ngugi, Kamau W. L.Show more Tourism sector in Kenya plays an important role in the national economy and has been identified as one of the six priority sectors in vision 2030 meant to drive the economy to attain 10 percent economic growth. International tourism demand for Kenya lags behind other African countries like Egypt, Morocco, Tunisia and South Africa. Furthermore, the number of tourists‟ arrivals to Kenya from different world regions does not increase constantly but have experienced cyclical fluctuations over the years. Moreover, the Kenya tourism product offered is becoming increasingly noncompetitive. There is need therefore, for Kenya to offer demand driven tourism products that ensure visitors come to Kenya and stays longer. GOK, tourism planners and marketers therefore need to clearly understand which important factors influence international tourists‟ decision to visit Kenya as their destination. Motivated by this need, the study sought to investigate the determinants of international tourism demand for Kenya. Specifically the study established the effect of economic factors, tourist socio-demographic characteristics, political factors and destination characteristics on international tourism demand for Kenya. The study used both longitudinal and cross sectional research designs. The study used panel data for economic variables from eleven countries for the period 1991 to 2011 collected from the World Bank database, United Nations database, International Monetary Fund database and Kenya National Bureau of Statistics. Survey data were collected from individual tourists leaving the country through Jomo Kenyatta International Airport using questionnaires. The study used a dynamic panel regression model to determine the effect of economic factors on international tourism demand for Kenya and a count data regression model to determine the effect of socio-demographic characteristics, political factors and destination characteristics on international tourism demand. The study results indicated that tourism price, travelling cost, trade openness and word of mouth effect were the main economic factors influencing international tourism demand for Kenya. The dummy variable representing the 2008 post election clashes and the lagged dependent variable representing the word of mouth effect were also significant. In addition, the tourist‟s socio-demographic characteristics such as annual household income, age and occupational status were found to significantly influence international tourism demand. The political factors composite index and destination characteristics composite index were also important determinants of international tourism demand. Taking into consideration of all these factors affecting tourism demand, the government and all the tourism stakeholders should work towards making Kenya tourism product competitive by ensuring that the prices remain competitive, diversifying tourism by creating tourism products which meets the needs of specific groups. The existing tourism products should also be improved in order to remain competitive, the tourism infrastructure and services should be well established and of good quality. The government should continue to engage in bilateral trade. Therefore, all the tourism stakeholders should work towards making Kenya a destination of choice.Show more