PHD theses : Statistics

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The Geometric Structure 0f Group Invariant Solutions Of Diﬁerential Operators
(Neelain University, 2005) Abdelgadir Ahmed Hammdan Omer
Abstract Group invariant solutions have been used to a great effect in the description of the asymptotic behaviour of many general solution to systems of partial differential equations. The rigorous foundation of the general method for constructing group — invariant solution requires advanced formalism of differential geometry. From our point of view the Fibre bundle over a quotient manifold approach seems very promising in analyzing the mechanism of group-invariant solutions and classifying different types of behaviour of a system. In this research we shall explore the bundle over a quotient space approach. Our applications will are on some partial differential equations
The Geometric Structure 0f Group Invariant Solutions Of Diﬁerential Operators
(Neelain University, 2005) Abdelgadir Ahmed Hammdan Omer
Abstract Group invariant solutions have been used to a great effect in the description of the asymptotic behaviour of many general solution to systems of partial differential equations. The rigorous foundation of the general method for constructing group — invariant solution requires advanced formalism of differential geometry. From our point of view the Fibre bundle over a quotient manifold approach seems very promising in analyzing the mechanism of group-invariant solutions and classifying different types of behaviour of a system. In this research we shall explore the bundle over a quotient space approach. Our applications will are on some partial differential equations.
Boundary Element Method For Porous Media Flow
(Al Neelain University, 2005-07) Azhari Ahmed Abdalla
The 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 presented
SOME STOCHASTIC MODELS IN MATHEMATICAL BIOLOGY
(Neelain University, 2006) ABDALSALAM B. H. ALDAIKH
Abstract Many recent scientiﬁc works have addressed the need for a better understanding of the underlying theory of modeling in biology. However, much more attention has been paid to the area of deterministic modeling in biology than to stochastic modeling, although it is more realistic to consider biological processes as stochastic rather than deterministic. The goal in writing this thesis is to ~ introduce a contribution to ﬁll the arising gap, ~ provide some of deterministic and stochastic biological models, and to 0 compare between such models. To achieve this, we present the thesis in the following chapters: Chapter One: Measure Theorv & Basic Modern Probability The underlying mathematical theory of stochastic modeling is stochastic processes, and the theory of stochastic processes is based on probability theory. The axiomatic development of probability theory was initiated by Kolmogorov in the early l930’s. Modem probability theory is technically a branch of measure theory, but it has developed characteristics and methods of its own, so any systematic exposition of the subject must begin with some basic measure-theoretic facts. The fundamental concept in this approach to probability theory is the probability space. In this Chapter, topics from measure theory and probability theory are reviewed which are particularly relevant to stochastic processes. Chapter Two: Stochastic Process In this chapter some basic concepts from the theory of stochastic processes are presented, in particular those are needed for an exploration of stochastic differential equations. ln the ﬁrst section, after deﬁning the stochastic process, the concept of Brownian motion is introduced as one ofthe most important examples of such process. The Markov process is a stochastic process exhibits speciﬁc property, called Markov dependence, is presented in the second section. Another related concepts such as: Random Walks, Martingales, Ito Integration and Ito Formula are introduced in the remain sections. Chapter Three: Stochastic Differential Equations One ofthe main problems in the stochastic modeling is how to solve the analog stochastic differential equation explicitly or at least numerically, so this chapter is devoted to many related concepts to the solutions of stochastic differential equations: the ﬁrst part of this chapter is devoted to Analytical Solutions of SDEs, many related concepts are introduced such as: Interpretation of Stochastic Differential Equations, an Existence and Uniqueness Solutions, Strong and Weak Solutions, Martingale Problem,.... However, except in simple cases, it is generally not possible to obtain explicit solutions to SDEs, so the second part is Numerical Methods for Solving SDEs, to reach the last part, which is the main stone in the remain chapters, that is the Diffusion Processes and SDEs. Chapter Four: Deterministic Models in Biology Numerous d6I8tTl1lItlSIlC models from biology for single and two interacting species are presented, such as: Exponential Growth Model for Single population, Logistic Growth Model, Competition Model, Predator-Prey Model and Harvesting Problem. For these models, the behavior of deterministic model is discussed, and will be developed to the corresponding stochastic model, in the next chapter. Chapter Five: Stochastic Models in Biology Both deterministic and stochastic models have important roles to play and should therefore be considered together. We start with an introduction to show the importance of considering stochastic models, and then introduce a stochastic analogue to each one of the previous deterministic models. The explicit solutions of some few models are presented. For the majority, however, it is impossible to get such solutions. Ito SDEs for Interacting Populations offer much help to solve many population models numerically, therefore, we devote a section to discuss these equations, and then use them to solve many models in the remain sections. In biology we are often asked to infer the nature of population development from a single data set, yet different realizations of the same process can vary enormously. Since even stochastic solutions are only of limited help here, we shall construct simple computer simulation procedures which provide much needed insight into the underlying generating mechanisms. Indeed, such model-based simulations can highlight hitherto unforeseen features of a process and thereby suggest further proﬁtable lines of biological investigation. All MAT HEMATICA-4.1 and MATLAB-6.2 programs used to generate the graphs are provided in the Appendix for easy referencing and developing to further studies. Chapter six: Conclusion and General Remarks Although full conditions for agreement between deterministic and expected stochastic solutions are at present unknown, it could be, however, compared them. In this chapter, a comparison between the Cl6l6I'I1'lll'llSliC & stochastic models is given to show the feature and the better use of each one.
STOCHASTIC CONTROL PROBLEM WITH PARTIAL OBSERVATIONS
(Neelain University, 2006) HOYAM TAG ELDIN AHMED
Abstract 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.
the geometry of space-time
(Neelain University, 2006) Mohammed Hassen Elzubair
Abstract In 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 ﬁlnction to generate zero rest- mass ﬁelds, where this function is formulated from both the geometry and topology of Minkowski space
An Appraisal of the Educational Statistical System of the Sudan
(Neelain University, 2006) Abbas Ali AAMIR
Abstract The Thesis is about improvement of the current educational statistical system. It concentrates on the deﬁciencies and shortcomings inherent in the present system and on ways and means of their alleviation and improving their contents. A good statistical database will help education authorities to assess the well-being of the system and to pinpoint areas of weaknesses. Thus, it would help to layout policies for possible remedies and improvements. This can better be done through calculation of comparative educational indicators and by building of models that single out variables or factors that have signiﬁcant effects on the progress of the educational system as a whole. Impoltant indicators are deﬁned and ways of calculating them are shown with illustrative examples that are based on the currently existing data. Limitations of the currently available educational data rendered it unattainable to calculate other important indicators. However, the purpose of an indicator is to characterize the components of a system: how they are related and how they change overtime. Indicators are descriptive tools that do not have the ability to measure the signiﬁcance of the changes that they reﬂect. Indicator systems, however, unlike a single indicator measure distinct components of the system and provide information about how the individual components work together to produce the overall effect. Models are more efﬁcient means of testing real variations. While expectations from social indicators are generally modest and, hence, can not substitute for a well designed, in-depth evaluations of social programs, models do the job. They give scientiﬁcally veriﬁable information about the different variables that are contained in them. Data collected should cover the complex variables that compose and interfere with the educational process and that are essential elements to be included in the planned models. There are different methods of data collection that can be used to suit the planned goals. Besides full coverage which, like all censuses, is limited in scope of coverage, other means of data collection like sample surveys, experimentation and case study evaluations can be applied wherever suitable. The World Conference on Education For All of 1990 and its assessment of Sub-Saharan Africa has urged the necessity of the establishment of National Educational Statistical Information Systems NESIS in all countries of Sub-Saharan Africa. Details of the ﬁndings of that conference with-respect-to NESIS program are included in the thesis with an illustration of the experience of Zambia in applying that program. Our educational system has gone through a multiplicity of changes in size, curriculum, policies and in economic, political and social conditions. The educational statistical system should be able to supply a wide variety of data that enables researchers to study the effect of those changes on the educational process. Examination results tell a lot about the quality of education students get and about the attainment of students from that education. Examinations usually comprise a number of components which may vary in the importance of their contribution to the ﬁnal assessment of the students’ academic achievement. A weighting system is thus sometimes introduced on the examination results to give more weight to some components that are considered more effective in infonning about the educational attaimnent of a student. It is also sometimes introduced for reasons the examiners think are justified by the ﬁndings of those particular examination results. Certain weighting methods are provided and discussed in the thesis and the formulae for a scientiﬁcally veriﬁable weighting system is discussed in detail. But the practice in the ﬁnal assessment of the Secondary School Certiﬁcate’s examination results of Sudan seems to differ from what has been originally meant by the introduction of the aforementioned weighting procedures. Those results were analyzed in detail throughout the years 1993-2001 for two components, Mathematics and English, being normally hypothesized to be the weakest areas of the students’ educational attaimnent. This comparative statistical analysis reﬂected that, rather than an introduction of a veriﬁable weighting system, the actual marks attained by the candidates were in fact seemingly adjusted to cover up the apparently weak student attainment. The true examinations results are one mean for the education system to hold itself accountable for whatever weaknesses that occur and remedy them in the best way it can afford putting in mind the known limitations of the economic conditions of the country.
A CONTRIBUTION TO RISK AND DECISION ANALYSIS OF PETROLEUM EXPLORATION
(Neelain University, 2006) Hala Abbas Bedawi
Abstract In this study the problem specifications are as follows: 1) Assessment of the factors of risk in petroleum exploration. 2) Evaluation of the geological risk. 3) Estimating prospect reserves. 4) Utilization of the probability of geological risk and the results of estimating prospect reserves in the decision making for drilling. ln petroleum exploration the basic concept in assigning risk for the geologic factors is contraverse; therefore, the subjective probability is used to assess the factors which are independent thus; the multiplication rule of probability is used to evaluate the geological risk. The prospect reserves distribution being a lognormal distribution, is estimated by finding out first the mean and variance of all the reserve parameters in the prospect reserves. In our study two methods were explained for finding the mean and variance of the reserve parameters: 1) Swanson's 30-40-30 rule. 2) Three-point method Since the prospect drilling is too expensive, and to avoid the loss if a dry hole is resulted, one can use the decision tree initially to decide whether to drill or not. The calculation of prospect reserve will be illustrated by the use of worksheet from one of the Sudanese Petroleum Companies.
APPLICATION OF - " THE VIRIAL METHOD TO THE SOLUTION OF SOME PROBLEMS < ' IN FLUID DYNAMICS
(Alneelain University, 2006) Mohamed Saad E1-Din Abdel Gafoor Abdel Magid
The problem of the charged spheroidal bubble is studied using the virial method. The conditions that are necessary for equilibrium are deduced and the oscillation of the bubble is studied . The frequencies belonging to the second order harmonics are found . Further extension of the virial theorem is made by studying the viscous ﬂuid sphere in an incompressible viscous ﬂuid giving the different and necessary virial equations of motion for both the exterior and the interior media and then the equilibrium state is studied.
solation of some stochatic paril differentionl equations
(Alneelain University, 2006) Sana Hussein F adl Allah
STOCHASTIC CONTROL PROBLEM WITH PARTIAL OBSERVATIONS
(Al-Neelain University, 2006-09) HOYAM TAG ELDIN
The 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.
The Qeometry Of Space - time
(Al-Neelain University, 2007) Mohammed Hassen Elzubair
In 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 space
Thethory of vanianalsymmetry Qroup and Conservationlawsa
(Neelain University, 2007)
Abstract In this research we talk about symmetry of D.E"‘ via symmetry of variation and we write the variation by modern methods 0lD.E"‘. So we utilize methods of global analysis such as differential geometry . ln this research we introduce the theory of ﬁber bundles .We give application ol" ﬁber bundles : Gauge theory and variational principles .Als0 we formulate Euler — Lagrange equations by three methods ( without coordinate ) our treatment is global since we employed ﬁber bundle formulation .
Symmetric Spaces and Their Applications
(Neelain University, 2007) Mohamed Alamin Abdalla Hamid
Abstract 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.
OPTIIVIAL CONSUMPTION PROBLEMS IN INCOMPLETE MARKETS
(Neelain University, 2007) WAFAA AHMED MOHAMMED SAID
the theory of varitional symmetry group and conservation laivs
(Neelain University, 2007) Sami Higazi Mustafa
I would like to express my deepest sincere gratitude and honest appreciation to my supervisor professor Mohammed Ali Bashir for helping me with enthusiasm and ideas .Also express my gratefulness to Al Neelain University for providing us with facilities to work together. Also special thanks are due to my friends who push me all the time to ﬁnish this work .Finally , I wish to express my heartfelt thanks to my family and special thanks to my mother .
THE GEOMETRICAL QANTIZATION OF PHYSICAL FIELDS
(Neelain University, 2007) Khalid Masoud Makin Mohamed Ali
ABSTRACT We considered the problem of geometric quantization. We ﬁrst started with the classical symplectic geometry and then'we used the complex line bundle to describe prequantization and quantization. Geometrically the Hilbert space of quantum states is constructed from the sections of the complex line bundle over the phase space. We then used the invariance group approach to the geometric quantization.
OPTIMAL CONSUMPTION PROBLEMS IN INCOMPLETE MARKETS
(Al Neelain University, 2007-01) WAFAA AHMED MOHAMMED SAID
We solve the problem of an agent who seeks to maximize expected utility from consumption plus expected utility from terminal wealth and give asummary of optimal consumption and portfolio method , the market model is quite general , allowing the coefﬁcient processes to be stochastic process. We study hedging strategies in incomplete market by compute the upper hedging price hup (K) . Finally , we get an optimal portifolio problem with logarithmic utility in markets with insider where the deriving process is a l'evy process .
Symmetric Spaces and Their Applications
(Al Neelain University, 2007-02) Mohamed Alamin Abdalla Hamid
Symmetric spaces is a special topic in Riemannian geometry . These spaces were ﬁrst studied and classiﬁed by Elie Cartan . In this research we study in a logical ordering their stnicture 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 symmetric spaces in the ﬁeld of arithmetic and algebraic groups , including quadratic and modular forms , 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 . lll . l\‘ ) . This classiﬁcation is carried tluough Lie algebras . root systems and their Dynkin diagrams t The aim ofthe research is to display in a simpliﬁed manner the connection between symmetric spaces and the differential geometric terms such as manifolds . Lie groups , Lie algebras and basics of Riemannian geometry with some of the applications of symmetric spaces.
THE GEOMETRICAL QANTIZATION OF PHYSICAL FIELDS
(Alneelain University, 2007-10) Khalid Masoud Makin Mohamed Ali
ABSTRACT We considered the problem of geometric quantization. We ﬁrst started with the classical symplectic geometry and then'we used the complex line bundle to describe prequantization and quantization. Geometrically the Hilbert space of quantum states is constructed from the sections of the complex line bundle over the phase space. We then used the invariance group approach to the geometric quantization.