PHD-Department of Mathematics

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Model-based non parametric regression estimation of the finite population total under two-stage cluster sampling
(2011-07-21) Karoki, Githinji Samuel
A model-based nonparametric regression estimator for the finite population total under two-stage sampling is proposed. In stage one, a sample of clusters is obtained and in stage two, sub samples of elements within each selected cluster are obtained. It is assumed that auxiliary information is available for all the clusters in the population and that a non parametric model describes the relationship between the cluster total and the auxiliary variable. Under these conditions, the estimator for the finite population total due to Dorfman (1992) is adapted. The condition mean squared error of the proposed estimator given the auxiliary information is derived. Asymptotic properties of the estimator are studied by deriving its Asymptotic Mean Integrated Squared Error (AMISE). In particular the applicability to the choice of bandwidth which minimize AMISE is explored. It is shown that plug- in methods for estimating bandwidth based on the AMISE will be ineffective. In choosing the bandwidth, a method akin to the one used i n Dorfman(1992) for the direct element sampling situation is suggested. The results of an empirical study are used to compare the performance of the proposed estimator with that of the standard estimators of the total in current use. It is observed that the estimator performs better (in terms of efficiency) than the linear with the other non parametric estimators in most situations. It is also noted that the proposed estimator is easier to understand and implement than the local polynomial estimator which, in its present form, requires derivations of expressions of the inclusion probabilities which can be difficult to obtain especially when a complex design is involved.
Some investigations on singular cauchy problems
(2011-11-17) Iyaya, Wanjala; Nyandwi, C.; Mutio, J.
The purpose of our study is to get a solution to the Cauchy problem of (i) The wave equation in n-dimension space Rn which is effectively a good example of regular Cauchy problems (ii) The Euler Poisson Darboux equation which we call singular Cauchy problem by use of Riemann's method. The Riemann-Green function for each case is calculated, which enables us to evaluate any solution at a point by the Cauchy data on a non-characteristic curve. In case (i) the Riemann-Green function is in terms of Legendre polynomial and the solution obtained is shown to solve the wave equation as well. In case (ii) the Riemann-Green function written in terms of the Appell's hyper geometric function of two variables is arrived at, this is of interest and may be a good model for a more general theory. A discussion of the generalized singular Cauchy problem of Euler-Poisson-Darboux equation is included and found to have solution that is continuous and analytic over the interval that contains the singular point.
Some Aspects of MHD Flows with Hall and Ion-Slip Currents
(2012-01-05) Singh, Amarjit
The purpose of this thesis is to give the basic assumption and the formulation of the theory of the flow problems of megnetohydrodynamics (MHD), as well as to the various methods of solving these problems. ''An explicit finite- difference method is used to analyse megnetohydrodynamics (MHD) stokes problem for a heat generating fluid with Hall effect'' And '' Perturbation method is also applied to study the effects of both Hall and Ion-slip currents on convective flow in a rotating fluid with wall temperature oscillations.'' Velocity and temperature profiles have been interpreted graphically before the conclusions are drawn as a result of applying the above stated methods to solve the highly non- linear differential equations. The present thesis consists of four chapters followed by bibliography. Before the beginning of Chapter 1 the nomenclature of the various symbols used in the thesis is given for easy reference. Chapter 1 contains the general introduction the basic assumptions and the formulation of the theory of the flow problems. The important non-dimensional parameters for the flow problems in magnetohynamics have been given. A brief description of the survey of the applications is also given. In chapter 11 the work done by the previous workers in the field of MHD has been reviewed. The last two chapters (III and IV) specifically deal with the study of the effects of Hall and ion- slip current on convective flow in the presence of strong magnetic field. Analytical expression for the velocity and temperature fields are obtained. The velocity and temperature profiles have been shown on graphs and the results are discussed in terms of the parameters me or be (the Hall parameter), mi or bi (their ion-slip parameter) and Grashof number Gr. Both the cases when GrO (in the presence of heating of the plate by free convection currents) have been discussed extensively. One of the most important applications of such problems are in the studies of coronal plasma flows in the configuration of plasma sheet formation in the active region of the sun or in the magnetic tail region. Further trends are discussed at the end of Chapter IV. The validity for the results and conclusions of the work done together with brief details of computational fluid dynamics codes for computerization is also added towards the end of Chapter IV. Bibliography of the research papers and books, concerning the subject matter of the present work is given at the end of the thesis arranged in the alphabetical order by names of the authors.
Ranks and subdegrees of the symmetric groups sn acting on ordered r-element subsets
(2012-04-13) Rimberia, Jane Kagwiria
The action of the symmetric group S n on ordered subsets from the set X = { 1, 2, ... , n} is an aspect that seems to have received little attention for a long time. Most studies have focused on the action of S; on unordered subsets leaving many properties about its action on ordered subsets unknown. This research is set to determine the rank and subdegrees of S; acting on X[r], the set of all ordered r-element subsets from X. Particular cases when r = 2, 3 and 4 have been considered first and then a generalization has been made for any value of rand n. In the action of Sn on Xl2], X(3] and Xl4] , the rank is shown to be 7, 34 and 209 respectively. By generalizing these results, we have come up with the formulas for the rank and subdegrees of S; acting on Xl']. This study shows that if n ~ 2r, then for a fixed value of r, the rank of S; on Xl'] is a constant while the subdegrees vary with n. The action of S; on Xl'] has been shown to be both transitive and imprimitive. We have also formulated the conditions for a suborbit of S; corresponding to this action to be either self-paired or paired with another. A formula for computing the number of self-paired suborbits has also been derived using a theorem from character theory. Finally, the suborbital graphs corresponding to this action have been constructed and their theoretic properties studied. The results show that all these graphs are disconnected. We have also come up with the formulas for computing the number of connected components in these graphs. The girth sizes of the suborbital graphs corresponding to suborbits of S; containing exactly r elements and no element from A = {l, 2, ... , r} have also been determined. For the suborbital graphs corresponding to self-paired suborbits of S; with exactly r elements from A, the girth is shown to be zero while that of paired suborbits with precisely r elements from A is shown to be three. This study also reveals that the girth of the suborbital graph corresponding to the suborbit of S; with no element fromA is three provided n ~ 3r. The results obtained have been summarized in form of theorems while others are displayed in tables.
On the spectrum of the cesaro operator
(2012-05-24) Okutoyi, J. I.
This thesis consists of four chapters. In chapter one, we determine the most general continuous linear functional f X, where X is any semiconservative BK-space with +) as its  Schauder basis. We also determine the necessary and sufficient conditions for(Ak)°bv°(X), bv(X), where Ak B(X, Y), X and Y are any Banach spaces. In chapter two, we study the general FH-spaces with H = S, where S is the space of all double sequences in which coordinates are continuous. We then specialize and obtain the relation between c(c) , c(c), c(c) and RCN. We also prove that c(c) = RC. In chapter three, we determine the spectrum of the Cesaro operator C1 = (C, 1) on c, bv, bv, wp (0) and wp (1p < ). In chapter four, we study 4-dimensional matrices and then go on to determine the spectrum of the Cesaro operator C11 = (C, 1,1) on c(c) by a direct method. A method, which consists of both classical and analytical techniques.
A study of junior secondary school pupils competence in selected processes of science
(2012-05-30) Otieno, Alego
Junior secondary school pupils were tested in four processes of science. The test subjects consisted of a total of 516 students. There were 113 male second form students, 143 female second formers, 131 male third form students and 129 female third formers. The objective of the study was to compare these four groups on the basis of sex and grade level, on how they performed in the tests which were administered to them. The process skills in which these subjects were tested were: Observation, Prediction, Generalization and in Controlling Variables. The attitude of Open-mindedness was also investigated using an appropriately designed test. The competence of the subjects in the process skills was also analyzed using a laid down criterion of performance of 0.5 (or 50) to decide on group competence. The observation test required the subjects to utilize all their senses in order to identify and communicate in writing all the differences that they could observe. These were sixteen (including one item for demonstration), which could be written down. In the prediction tests, the subjects watched a demonstration, then on the basis of the demonstration they were required to reason in order to make prediction about the outcomes of some tasks which were presented to them. The prediction levels were varied depending on the number of variables a subject was to take into consideration during the test. The test on generalization required the subjects to make conclusion on the basis of their earlier observation about the liquids and balls used in the test. The items testing subjects' ability to control variables required them to identify which variables to keep constant and which ones to vary in order to set up appropriate experiment. The test to measure the attitude of Open-mindness required the subject to identify situations in which the response "It is not possible to tell" were applicable and to give some basis for stating so. The mean performance was calculated for each group in each of the tests administered. A test of significance of difference between two means was performed for the group pairs (a total of 6 group pairs) to find out whether or not the groups deferred in performance. The Chi-Square Technique was also used to test for significance of differences between frequencies of sample pairs which satisfied the criterion which had been set. Analysis of the results revealed that there were significant differences between the group means in the tests administered. Of particular interest, the results of the analysis revealed that in some processes such as in prediction and controlling variables, the dimension of sex contributed to differential in performance. In observation and generalization, sex as a dimension did not seem to feature. The difference of one year in school attendance did not produce any difference in performance between the subjects. The Chi-Square Test analysis also revealed some significant results for some groups in some tasks. Form Two male students did not significantly differ from their female counterparts in nearly all tests except in prediction tests. Male Form Two students did not differ from male Form Three students in all the tests given. Form Two girls differed from Form Three girls in only one test, namely generalization. There were four tests in which Form Three boys significantly differed from their female counterparts. These were in prediction (Level Two and Three), Open-mindedness and in Controlling Variables. The competence of the subjects in the skills tested was found to be very low. Pupils’ competence was particularly low in Observation; Prediction (Level Three) in Open-mindedness where in all the cases the percentages of competent subjects were all less than 30.It was suggested that the observed low competence in the skills should be of great concern for science teachers. It was argued that pupils who are incompetent in science skills are not likely to manifest high-level ability in science tasks. To function effectively in the scientific field requires a practitioner to be competent in both content and process. It was hence suggested that because of observed low competence in the skills, Kenya' secondary school science curriculum should strive to improve pupils' competence in the processes of science particularly that of observation. The call for incorporation of these activities would not require additional funding nor would the number of science lessons have to be increased. The development of competence in the skills can be enhanced by using the same number of lessons using available materials without necessarily incurring additional costs. The long-term benefits in teaching of future scientists was bound to be enormous should attempts to implement these recommendation succeed.
Robust variance estimation for finite population sampling
(2012-06-08) Otieno, Romanus Odhiambo
After a sample has been obtained the statistic of interest can be computed. The next step (and a ore formidable one) is the assessment of the accuracy (precision) of the resulting statistic. The most commonly used measure of accuracy in the model based survery is the variance of the prediction error associted with teh considered statistic. In general variance are not known and must be estimated using the sampled data. In this thesis we have jproposed new methods for estimating the error variance for finite population sampling. in particular we have considered fixed bandwidth kernel smoothing of the sqared residual and bootstrap technique based on resampling of the rresiduals. Analytical and empirical performances of the new variance estimmators are studied vis avis the robust estimators favoured in teh current practice. On average the proposed estimators have better robustness properties than estimators favored in the current practice. Further more the new estimators have the desired properties of non negativity, simplicity and extend even to cases where some of the current etstimators can not be applied.
Investigating Sedenion Extension Loops and Frames of General Hypercomplex Numbers
(2013-03-21) Njuguna, Lydia Nyambura; Kivunge, B.; Kamuti, I.N.
A sequence of algebras over the field of real numbers can be constructed, each with twice the dimension of the previous one. The algebra constructed by doubling complex numbers is the 22 - dimensional quaternions. Next we have the 23 - dimensional octonions, constructed by doubling the quaternions.The algebra constructed-by doubling octonions is the 24 -dimensional sedenions. The oldest method of constructing these algebras is the Cayley-Dickson formula. Since they extendthe complex numbers, they are called hypercomplex numbers in general for dimensions greater than 24 . Our main emphasis is on the general Z"-ons. Split extensions of 2n -ons are studied from the point of view of Loop Theory. Multiplication of basis elements of complex, quaternion, octonion and sedenion split extensions using the Jonathan Sooth doubling formula is done. It is shown thatNim addition gives a way of determining the subscripts for the products of the basis elements for the split extensions. This result is extended to split extensions of general2n - ons. Subloops of sedenions are also investigated, and it is shown that they do not necessarily reduce to either subloops of octonions or sedenion split extensions. This is done by showing the existence of two-sided subloops of sedenions that are neither subloops of octonions or sedenion split extensions. It is well known that when L, the multiplicative sub loop of octonions is abelian, the sedenion split extension L.x SO formed is a group. The structure of Lx SO when L is non-abelian is studied: IJ? 'particular, it is shown that Lx SO fails to satisfy group properties. When the satisfaction of standard Loop theoretical properties is investigated, L x SO is seen to satisfy the Jordan identity and flexible properties. It however fails to satisfy the left alternative, right alternative and anti-automorphism properties. Finally, multiplication of the basis elements of complex numbers, quatemions, octonions and sedenions is done using the Jonathan Smith doubling formula. In each case, it is shown that the formula gives a way of constructing Hadamard matrices, as did the Sooth- Conway formula. This result is generalized for genera12n - ons .
An analysis of international tourism demand for Kenya
(2014-08-18) Ngugi, Kamau W. L.
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.
Quantile Autoregression and its application to Financial Risk Management and Portfolio Optimization
(2014-09-01) Ananda, Omutokoh Kube
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.
Ranks and subdegrees of the cyclic group, the dihedral group and the affine group and associated suborbital graphs
(Kenyatta University, 2015-11) Kangogo, Moses Rutto
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 Fidelius
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'a
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.
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 Kiplagat
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 Josephs
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 Lawrence
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.
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 Odondo
Heat and mass transfer past a semi - infinite vertical porous Plate in MHD flows in turbulent boundary layer
(2019) Ngesa, Joel Ochola
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.
Modelling Jiggers Infestation and Interventions in Humans: A Case Study of Murang'a County, Kenya
(Kenyatta University, 2020-06) Matendechere, Nancy Imbusi
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.
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 Gatoto
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.

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