Department of Mathematics, Statistics and Actuarial Sciences
Permanent URI for this collectionhttp://localhost:4000/handle/20.500.12092/1962
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Item Onaquotient group 74:(3X2S7) of a 7-local subgroup of the Monster M(Karatina University, 2023-01) Musyoka, David Mwanzia; Njuguna, Lydia Nyambura; Prins, Abraham Love; Chikamai, LucyItem Bayesian Estimation of Parameters of Weibull Distribution Using Linex Error Loss Function(Canadian Center of Science and Education, 2020-02) Kinyanjui, Josphat Kamau; Kori, Betty ChemutaiThis paper develops a Bayesian analysis of the scale parameter in the Weibull distribution with a scale parameter θ and shape parameter β (known). For the prior distribution of the parameter involved, inverted Gamma distribution has been examined. Bayes estimates of the scale parameter, θ , relative to LINEX loss function are obtained. Comparisons in terms of risk functions of those under LINEX loss and squared error loss functions with their respective alternate estimators, viz: Uniformly Minimum Variance Unbiased Estimator (U.M.V.U.E) and Bayes estimators relative to squared error loss function are made. It is found that Bayes estimators relative to squared error loss function dominate the alternative estimators in terms of risk function.Item ON THE MAXIMAL NUMERICAL RANGEOF ELEMENTARY OPERATORS(2017) Runji, Flora Mati; Agure, John Ogonji; Nyamwala, Fredrick OluochThe notion of the numerical range has been generalized in different directions. One such direction, is the maximal numerical range introduced by Stampfli (1970) to derive an identity for the norm of a derivation on L(H).Unlike the other generalizations, the maximal numerical range has not been largely explored by researchers as many only refer to it in their quest to determine the norm of operators. In this paper we establish how the algebraic maximal numerical range of elementary operators is related to the closed convex hull of the maximal numerical range of the implementing operators A = (A1, A2,...,A), B = (B 1 ,B 2,...,B ), on the algebra of bounded linear operators on a Hilbert space H. The results obtained are an extension of the work done by Seddik [2] and Fong [9]Item D-optimal Rotatable Central Composite Designs Constructed through Resolutions(Mathematical Theory and Modeling, 2016) Kinyua, Margaret; Koskei, Joseph; Kinyanjui, JosphatResponse surface methodology is widely used for developing, improving, and optimizing processes in various fields. A design is of resolution 𝑅 if no 𝑝 factors effect is confounded with any other effect containing less than 𝑅 −𝑝 factors. In this study, a method for constructing second order rotatable designs based on resolution R, in particular resolution III and IV for three and four factors respectively, argumented with star points is presented. Attention is given to the moment matrices and the related information surfaces based on the parameter subsystem of interest on the second-order Kronecker model and their corresponding rotatable Central Composite Designs (CCDs). Weighted Central Composite Designs (WCCDs) are derived by assigning different weights to two portions of the CCD namely the cube and star portion. The derived designs achieve the property of rotatability and high efficiency and are shown to be D-optimal. Experimental runs are reduced hence economical and the resulting designs are improved in terms of optimality and estimation efficiency. The results show that the cube portion is of great importance in D-optimal resolution III design while the two portions are of equall importance in resolution IV design.Item D-Optimal Designs for Third-Degree Kronecker Model Mixture Experiments with an Application to Artificial Sweetener Experiment(IOSR Journal of Mathematics, 2014) Kinyanyui, Josphat; Kungu, Peter; Ronoh, Benard; Korir, Betty; rutto, Mike; Koske, Joseph; Kerich, GregoryThis study investigates some optimal designs in the third degree Kronecker model mixture experiments for non-maximal subsystem of parameters, where Kiefer’s functions serve as optimality criteria. Based on the completeness result, the considerations are restricted to weighted centroid designs. First, the coefficient matrix and the associated parameter subsystem of interest using the unit vectors and a characterization of the feasible weighted centroid design for a maximal parameter subsystem is obtained. Once the coefficient matrix is obtained, the information matrices associated with the parameter subsystem of interest are generated for the corresponding factors. We apply the optimality criteria to evaluate the designs.Item An Introduction to Differential Geometry: The Theory of Surfaces(Science Publishing Group, 2017) Gikonyo, Kuria Joseph; Kinyua, Kande DicksonFrom a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.Item Multinomial Logit Modeling of Factors Associated With Multiple Sexual Partners from the Kenya Aids Indicator Survey 2007(American Journal of Theoretical and Applied Statistics, 2015-05) Kinyanjui, Josphat; Mwalili, Samuel; Ang’ir, BerylThe number of lifetime sex partners of an individual has an important effect on Human Immunodeficiency Virus (HIV) status of an individual; hence modeling multiple sexual partnerships is an essential component of any analysis of HIV outcome. Multiple sexual partnerships are associated with greater risk of HIV, Sexually Transmitted infections (STIs) and intimate partner violence. This research project presents a general approach for modeling logit of clustered (correlated) ordinal and nominal responses using polytomous data from the Kenya AIDS Indicator Survey 2007 (NASCOP 2010). We review multinomial logit models as generalized linear models. The model is applied to HIV prevalence data among men and women in Kenya, derived from the Kenya AIDS Indicator Survey 2007 (KAIS). We generalize logistic regression to handle multinomial response variables, with separate models for nominal and ordinal cases. When modeling a nominal response variable we are interested in finding if certain predictors have an effect on the probabilities. The baseline category logit model, models the odds of being in one category relative to being in a designated category (last category), for all pairs of categories. It is used for nominal responses. A maximum likelihood estimation (MLE) approach is used for baseline category logit model. To model an ordinal response variable one models the cumulative response probabilities or cumulative odds. The cumulative logit model is used when the response of an individual unit is restricted to one of a finite number of ordinal values. This study shows the practicality of multinomial logit model in analyzing epidemiological data. Other studies have found education to be associated with multiple sexual partners. In this study, we observed that multiple sexual partners is not related to education. Other covariates like Gender, Place of residence, sexually active individuals for the past 12 months and marital status were found to be associated with multiple sexual partners. Individuals that are sexually active for the past 12 months were found to be ten times more likely to have multiple sexual partners compared to those that are not. After controlling for all other factors, the odds of male to female having multiple sexual partners doubled to 7.6 meaning male are almost 8 times likely to have multiple sexual partners compared to female. Partner testing or couples testing is a main strategy of national testing initiatives in Kenya. Respondents are encouraged to learn their test results with their partner.Item Differential Geometry: An Introduction to the Theory of Curves(Science publishing group, 2017) Gikonyo, Kuria Joseph; Kinyua, Kande DicksonDifferential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.Item Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments(Science publishing group, 2017) Gikonyo, Kuria Joseph; Mwaniki, Wambua Alex; Elizabeth, Njoroge; Joseph, Koske; Mutiso, John; Gitunga, Muriungi Robert; Kipkoech, CheruiyotThe aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker modelbn for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.Item Language practices involving two languages among trilingual undergraduate students of Mathematics(IMHOTEP, 2015-08) Njurai, E.W.