Department of Mathematics, Statistics and Actuarial Sciences
Permanent URI for this collectionhttp://localhost:4000/handle/20.500.12092/1962
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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 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.