PHD theses : Statistics
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Item The Qeometry Of Space - time(Al-Neelain University, 2007) Mohammed Hassen ElzubairIn this research we utilized twister theory to describe the geometry of space-time. The twistors are derived from Spinors which are also used to write zero rest- mass fields equations. We mainly used the properties of twistor function to generate zero rest- mass fields, where this function is formulated from both the geometry and topology of Minkowski spaceItem STOCHASTIC CONTROL PROBLEM WITH PARTIAL OBSERVATIONS(Al-Neelain University, 2006-09) HOYAM TAG ELDINThe problem of stochastic control with observations had been discussed. First we start with the problem of linear filtering to get the best estimate for the system according to the given observations. To obtained the solution we solve Kalman filter equation. In the case of nonlinear filtering the problem reduces to complete study of Zakai equation. The optimal stochastic control dynamic system had been discussed in the case of partial infonnation which yields Kalman filter. But in the case of complete observation we solve the problem using Hamilton-Jacobi equation. Finally we introduce and discuss some applications of the problem of stochastic control.Item Ridge Regression and the Multicollinearity Problem(Al Neelain University, 2016-02) Tahani Ali Esmail AdamThis research primarily aims at evaluating the performance of the ridge regression estimators as remedial techniques to the multicollinearity problem. The study is based on Monte Carlo experiments in which the performance of ridge estimators is investigated under different levels of multicollinearity, population variance & sample size. A new method suggested by the author and based on using a variable biasing constant with values proportional to the variances of the estimates of the regression coefficients led to great improvement in the performance of the ridge estimate. This weighted estimator resulted not only in increased precision of the regression estimate compared to the unweighted estimate, but also worked as a controlling factor to the mean squares of the regression estimates when these explode at large values of biasing constants. The thesis also reviews in detail the performance of the weighted & unweighted ridge regression estimators under varying levels of sample size, population variance & degree of linear correlation. It also examined the effect of linear correlation under different levels of variance & sample size on the ordinary regression estimates. A value of the biasing constant of 0.1 appeared to be a dividing line for the mean square of the ridge regression estimates as it explodes greatly after it if weighing is not used. Hence the author recommended that weighted estimates should be used for biasing factor greater than 0.1 as well as when the independent variables differ greatly in their variances.Item Geometrical Reformulation of some Equations of Fluid Mechanics(Neelain University, 2015) Mansour Hassan MansourAbstract The aim of this thesis is to explain some of the connections between fluid mechanics and differential geometry and to shed light on formulation of classical fluid mechanics in a differential geometric language. The thesis presents a reformulation of some of the most basic entities and equations of fluid mechanics, the continuity equation and the momentum equation of motion, in a modern differential geometric language using calculus of exterior differential forms on manifold (exterior calculus). Also, the study investigates the integrability of some fluid problems from geometrical perspective, with particular attention to the Eulefequations of motion.Item The Algebraic Treatment of Symmetric Spaces A thesis submitted(Neelain University, 2008) Nemaat Hamed TalebABSTRACT We study manifolds spaces which lead to Lie group, then by definition of transitive action of a Lie group in a manifold we get homogeneous spaces. Also we study symmetric spaces, which are particular homogeneous spaces. We classify simple Lie algebra over C according to Dykin diagram. Also by means of real form we classify simple Lie algebra over DR. We show that every symmetric space gives rise to an orthogonal symmetric Lie algebra. Finally we classify Riemannian symmetric spaces of types I, II, III and IV according to the classification of the irreducible orthogonal symmetric Lie algebra of types I, II, III, and IV.Item STOCHASTIC CONTROL PROBLEM WITH PARTIAL OBSERVATIONS(Neelain University, 2006) HOYAM TAG ELDIN AHMEDAbstract The problem of stdchastic control with observations had been discussed. First we start with the problem of linear filtering to get the best estimate for the system according to the given observations. To obtained the solution we solve Kalman filter equation. In the case of nonlinear filtering the problem reduces to complete study of Zakai equation. The optimal stochastic control dynamic system had been discussed in the case of partial infonnation which yields Kalman filter. But in the case of complete observation we solve the problem using Hamilton-Jacobi equation. Finally we introduce and discuss some applications of the problem of stochastic control.Item The Formulation Of Equivalence Problems And Its Application(Neelain University, 2014) Runda Abdalhafiz Abdelrhim . BashirAbstract In this research we considered Cartans equivalence problem . We defined the concept of equivalence and illustrated that by several cases . We introduced the equivalence problem for Coframs and then used Cartans equivalence method to determine whether two differential operators are equivalent , we also handle the classification problem of Lie algebras, ln particular we treated the case of semi-simple Lie algebras.Item Boundary Element Method For Porous Media Flow(Al Neelain University, 2005-07) Azhari Ahmed AbdallaThe focus point of this study is to develop BEM formulation to overcome the difflculties caused by nonlinearity and heterogeneity in the solution of partial differential equations governing the porous media flow .This dissertation consist of two major parts ,theory of BEM , and applications of BEM to two important fields of porous media ,water flow in aquifers and oil flow in reservoirs. The contribution done in the theory sections is mainly a mathematical derivation of the standard boundary element method for Lapiace's equation and the step by step formulation to the BEM , starting from its correspondence differential equation ,beside the developed form of BEM based on GEM and DRBEM that has presented. to handle both heterogeneous and nonlinearity. In the applications section a novel boundary integral solution was applied for determining : (1) Water table elevation in an unconfined homogeneous aquifer subjected to recharge and dewatering from a stream as well as fluctuations induced by constant and continuous recharge in a two stream unconfined-aquifer system. (2) Changes in water table exposed to a transient boundary condition and space- dependent recharge. This technique was compared with the closed form solution obtained in [111] and excellent results were obtained. (3) Characteristics of the flow through heterogeneous unsaturated porous aquifer . (4) Solution of reservoir engineering problems. This work adapted the most recent developments in boundary element methods to reservoir engineering problems. The transient pressure (diffusion) and convection- diffusion equations were solved in heterogeneous media using the Dual Reciprocity Boundary Element Method (DRBEM) and the Green Element Method (GEM). Numerical experiments showed that DRBEM is more accurate than a standard finite difference method. However like finite difference methods, DRBEM is subject to spurious oscillation at high Peclet numbers. DRBEM also requires the solution of a dense system of equations. GEM, which is a hybrid boundary elementlfinite element method, overcomes these disadvantages. The method was found to produce very accurate solutions to convection-diffusion problems and only shows small oscillations in the solution at very high Peclet numbers. A further important advantage is the sparse nature of the matrix system. GEM is also amenable to solving transient nonlinear problems, which makes it the basis for a new technique for multiphase flow simulation. This work explores the advantages of a hybrid boundary element method known as the Green element method for modeling pressure transient tests. Boundary element methods are a natural choice for the problem because they are based on Green's functions, which are an established part of well test analysis. The classical boundary element method is limited to single phase flow in homogeneous media. This works presents formulations which give computationally efflcient means to handle heterogeneity. Comparisons of the proposed Green element approach to standard finite difference simulation show that both methods are able to model the pressure change in the well over time. When pressure derivative is considered however the finite difference method produces very poor results which would give misleading interpretations. The Green element method in conjunction with singularity programming reproduces the derivative curve very accurately. Boundary element method was applied for solving Stokes flow equations on multi particle system. Also, the method is modified for estimating flow parameters for a specified porous media. A new method for the so/ut/‘an of the unsteady incompressible Navier-Sta/res equations was presentedItem Generalization on LP - Contractivity of Semigroups Commutators and C 0-Semigroups of Resolvent Estimates(Neelain University, 2008) Ria Hassan MohamrnedWe derive a pointwise estimate on the absolute difference between two corresponding diffusion kernels of two diffusion semigroups , as well as an L” —operator norm bound. We show that linear partial differential operators of order higher than two can not generate contraction semigroups on the Lebesgue space except for some fourth order operators in a restricted compact interval . We consider a comparison between two semigroups , a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions . We give a sufficient condition for the criterion in the setting of square field operator. We also consider the essential self a djointness of a perturbed semigroup . We discuss the existence and the continuity of the boundary values problem on the Lebesgue space of the resolvent of a self — adjoint operator of the conjugate operator method . we allow the conjugate operator to be the generator of a Co —semigroup and that first commutator is not comparable to the self —adjoint operator . Strong application include the spectral theory of zero mass quantum field models are considered .Item Symmetric Spaces and Their Applications(Neelain University, 2007) Mohamed Alamin Abdalla HamidAbstract Syrnrnet:n'c spaces is a special topic in Riemannian geometry . These spaces werefirst studied and classified by Elie Cartan . _ In this research we study in a logical ordering their snucture through manifolds , Lie groups , Lie algebras and basics of Riemannian geometry. _ The study covers locally and globally symmetric spaces , endowed with some examples for them such as Euclidean spaces , spheres hyperbolic spaces and some applications on locally syrrunetric spaces in the field of arithmetic and algebraic groups , including quadratic and modular fonns , lattices , the realization of discrete series representations of groups , Poincare and linear symmetric spaces . Their classification is also discussed by introducing compact and noncompact symmetric spaces besides the types ( l , ll , Ill , IV ) . This classification is carried through Lie algebras , root systems and their Dynkin diagrams . The aim of the research is to display in a simplified manner the connection between symmetric spaces and the differential geometric temis such as manifolds , Lie groups , Lie algebras and basics of Riemannian geometry with some of the applications of symmetric spaces.