Mathematics

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https://hdl.handle.net/20.500.11888/3206

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The S-Property and Best Approximation
(2000) Sawsan Azmi Sabri Al-Dwaik; Dr. Abdallah A. Hakawati; Dr. Waleed Deeb
The problem of best approximation is the problem of finding , for a given point x and a given set G in a normed space (X,||.||) , a point go in G which should be nearest to x among all points of the set G . However , in our study , we shall mainly take as X not an arbitrary normed space but Orlicz space , we shall denote by P(x,G) , the set of all elements of best approximants of x in G. i.e P(x,G)= { gₒ є G llx- g= inf{||x-g||: g є G } The problem of best approximation began , in 1853 , with P. L. Chebyshev who considered the problem in the space of all real valued continuous function defined on [a,b] , a closed real interval in R . My theses consist of four chapters. Each chapter is divided into sections. A number like 2.1.3 indicates item (definition, theorem, corollary or lemma) number 3 in section 1 of chapter 2. Each chapter begins with a clear statement of the pertinent definitions and theorems together with illustrative and descriptive material. At the end of this thesis we present a collection of references. In chapter (1) we introduce the basic results and definitions which shall be needed in the following chapters. The topics include projection, normed space, compactness , Hilbert space and measure theory . This chapter is absolutely fundamental. The results have been stated without proofs, for theory may be looked up in any standard text book in Functional Analysis. A reader who is familiar with these topics may skip this chapter and refer to it only when necessary. Chapter (2) will be devoted to give an introduction to fundamental ideas of Best Approximation in Normed Space. We will start by introducing the definition of best approximants of x є X in a closed subspace G of X . We denote the set of all best approximation of x in G by P(x,G) . In section (2) we study the properties of P(x,G) . In section (3) we define proximinal set and Chebyshev subspace , and we mention some conditions that can assure that G is proximinal in X . Finally , we define Lᵖ- summand and give a simpler proof for the fact that “every a closed subspace of a Hilbert space is proximinal ". Chapter (3) has two purposes .First, we review the properties of Orlicz spaces. Second, we introduce some ofthe basic theory ofproximinality in Orlicz space . This material was designed to meet the needs of chapter (4). W. Deeb and R. Khalil proved the following results. (1) If G is 1-complemented in X, then G is proximinal in X. [1, p.529] . (2) If Lᶲ(μ,G) is proximinal in Lᶲ(μ,X) ,then G is proximinal in X. [3 , p.8] , [2 ,p.297] , [4 , p.37] (3) If L¹(μ,G) is proximinal in L¹(μ,X), then L∞(μ,G) is proximinal in L∞(μ,X) . [1 , p.528] Some questions about proximinality in Lᶲ (μ,G now suggest themselves. (1) Let X be a Banach space and let G be proximinal in X. Under what conditions can it be asserted that G is l-complemented in X? (2)If G is proximinal in X, Under what conditions can it be concluded that Lᶲ (μ,G) is proximinal in Lᶲ(μ,X)? In particular, is the proximinatily of G in X a sufficient condition? (3)If L∞(μ,G) is proximinal in L∞(μ,X). Under what condition can be asserted that L¹(μ,G) is proximinal in L¹(μ,X). These questions are addressed in the section (1) of chapter (4). The answer depends on the S-property. Some interesting results have been achieved. Among of which it is shown that if G has the S-property then L∞ (μ,G) has the S-property . lt is also proved that if G has the S-property then Lᶲ (μ P⁻¹ɢ (0))= P⁻¹Lᶲ (μ,G) (0) I ask our God to be our assistant to continue our efforts so as to achieve the hopes and desires of all scholars in mathematics.
On Cyclic Zpm-Codes
(2000) Ali Saleh Hussein Shaqlaih; Dr. Ali Abdel-Mohsen; Dr. Mohammad A. Saleh
In this thesis, cyclic codes, their generators, their idempotent, and their dual have been studied. Also, coding and decoding of cyclic codes and Algorithm for decoding linear cyclic codes were under focus. Moreover, cyclic Z,. codes, their generators, their dual and their idempotent have been deeply discussed.
A Study On Projective Modules and Some Weak forms of Projectivity
(2000) Iyad Khalil Yousef Al-Hrabat; Dr. Ali Abdel-Mohsen
In this thesis we consider projective modules and some weak forms of projectivity and we try to study the most important known results concerning these modules. In chapter one we summarize some of the essential and basic concepts in and n. This chapter consists of three sections; section l presents definitions and basic properties of modules. In section 2, we study the radical of modules and rings which plays an important role in our study. In section 3 we study simple and semi simple modules and rings. In chapter Two, which is the main body of our thesis, we study the main characterizations and properties of projective modules. Moreover we study radicals and endomorphism rings of projective modules. Finally, "projective covers" was studied in this chapter. In chapter three we study semi-perfect and perfect rings as an a applications to projective modules, those over which all finitely generated modules and, respectively, all modules have projective covers. 5 2 9 51 9 In chapter four, we study some weak forms of projective modules such as, weakly projective modules, ideal projectivity , Jacobson radical projectivity, and simple projectivity.
On A Mathematical Design System: Maximum Reliability, Minimum Cost
(2001) Saleh Sadeq Afaneh; Dr. Mohammad Najib Assa'd
In this thesis, reliability concepts, measures of reliability and static models have been studied. Also, comparisons between exponential and logistic distributions have been discussed. We used two methods; dynamic programming and heuristic approach to maximize the reliability of an electronic device systems, where the optimum structure of components and units of the assumed system have been determined. Examples are given to show the optimum structure ofthe system with the maximum reliability and minimum cost. Comparison between dynamic programming and heuristic approach shows that the dynamic programming results are better than the results obtained by the heuristic approach. Finally, our objective is to characterize marginal cost and minimize cost capacity plans for a typical service delivery system. Results indicate that marginal costs are convex with respect to reliability of service, while changes in the demand distribution’s variability may impact optimal capacity by either increasing or decreasing required capacity. Two demand distributions are assumed; uniform and logistic distributions. The results show that the logistic demand distribution gives an optimum criterion which are more realistic. Also, the optimum capacity using logistic is greater under the condition b/B> 0.5.
Best Approximation in General Normed Spaces
(2001) Mu'tas Hasan Mahmoud Al-Sayed; Dr. Abdallah A. Hakawati
Let X=(X,||.||) be a normed space and suppose that any given x in X is to be approximated by an element Y in Y, where Y is a fixed subspace of X. We let d denoted the distance from x to Y. By definition, d = d(x, Y) = infyєY ||x – y|| .Clearly, d depends on both x and Y, which we keep, fixed, so that the simple notation d is in order. If there exists a YoєY such that ||x - Yo|| =d. then Yo is called a best approximation of Y to x or a best approximant of x in Y. We see that a best approximation Yo is an element of minimum distance from the given x. Such a YoєY mayor may not exist; this raises the problem of existence. The problem of uniqueness is of practical interest, too, since for a given x and Y there may be more than one best approximation. My thesis consists of three chapters. In chapter one we summarize some of the essential and basic concepts which shall be needed in the following chapters, this chapter consists of two sections; in the first one we present metric, normed, Banach spaces, and the last one we present inner product, and Hilbert spaces. This chapter is absolutely fundamental. In chapter two, we define best approximations in section one. In section two we study some properties of the set of all best approximations P(x,Y). In section three we study some properties of the proximinaI set and show that compact subspace and finite-dimensional subspace are proximal. In section four we consider the problem of uniqueness of best approximation. In section five we review the properties of OrIicz spaces in which we introduce some of the basic theory of proximinality In chapter three, which is the main body of our thesis, we, in section one, study the main characterizations and properties of best approximations and some consequences of the characterization in arbitrary normed linear spaces. In sections two and three we gives some application in several spaces like L ¹(T,v), C(K) and CR(K).
Convexity, Fixed Point Theorems and Walrasian Equilibrium
(2002) Abdul-Rahim Omar Amin Nur; Dr. Abdallah A. Hakawati; Dr. Rimon Abduh' Y. Jadoun
In this thesis, I will deal with an application of fixed-point theorem of set valued map [Let X and Y be two subsets of Rⁿ: A set-valued map F from X to Y, is a map that associates with any x є X a subset F(x) of Y, A fixed point x for F exists if x є F(x) ], and convexity to prove the existence of Walrasian Equilibrium under sufficient conditions for both pure exchange economy and private ownership economy. Then I will show how to modify these theorems in more general cases under uncertainty and externalities.
The Effect Of Using The Computer As An Educational Teaching Aid In The Achievement Of Fifth Grade Students In A Unit On Areas
(2003) Wael Abdel Lateef Abdellah Afaneh; Dr. Salah Alden Yasin
This study aimed at investigating the effect of using the computer as an educational teaching aid in the achievement of the fifth grade students in a unit on Areas , compared with two ways the traditional method with work-sheets, and the method of traditional teaching .This study fried to answer the following questions: 1- Are there any significant statistical differences at ( α = 0.05 ) in the achievement of the elementary fifth grade students in mathematics between the first controlling group and the experimental group the computer. 2- Are there any significant statistical differences at (α = 0.05 ) in the achievement of the elementary fifth grade students in mathematics between the second controlling group work sheets and the experimental group the computer. 3- Are there any significant statistical differences at α = 0.05 ) in the achievement of the elementary fifth grade students due to educational method. 4- Are there any significant statistical differences at (α = 0.05 ) in the achievement of the elementary fifth grade students in mathematics by the use of computer due to sex .To answer all these questions, the sample of study consisted of (86) students from the students of fifth grade from Ramallah private schools. Those students were randomly chosen with (3) schools. Students were distributed into three group two controlling groups and the third experimental. Where male and female students study in different classes, in two groups, the first controlling and the experimental, while the second controlling group consisted of female only. In this study an educational programmer prepared by the research was used within the program (Power Point), in which the material was presented as it is in the authorized book for fifth elementary grade. (8) lessens out of ( 10) in the area unit were explained in average (5) lessons every week , in real ( 12 ) lessons .All the group studied the same number of lessons and the same subject. The results of experimental shows the following: 1-The existence of significant statistical difference at (α = 0.05 ) in the achievement of the elementary fifth grade students in Mathematics in the first controlling group the traditional and second controlling group work sheets and experimental group the computer the points were in favor of experimental group the computer . 2-There are some significant statistical differences at (α = 0.05) in the achievement of the elementary fifth grade students in mathematics by using the computer due to sex , and this in favor of females . Because of the results the researcher found , the researcher advised of looking for especial computer laboratory that helps in the educational process in all subjects , and this can be used to present educational material or helping the Teacher in making any experiment or any educational method during explanation and the research advised in making additional studies on The educational methods in Mathematics and especially using(Power Point) as an Educational method in presenting the educational material.
Eigenvalues of The Matrix of The Distances Reciprocals for The Complete Bipartite and Cycle Graphs
(2004) Riad Kamel Hasan Zaidan; Dr. Subhi Ruzieh
This work deals with the spectra and eigenspaces of some matrices related to the distance matrix of some connected graphs. In particular, we investigate the (n x n) matrix Bn whose nonzero entries are the reciprocals of the corresponding nonzero entries in the distance matrix. We derive formulas for the eigenvalues and eigenvectors related to the complete bipartite graph K(r, n-r), and the cycle graphs Cn, for any positive integer n. In the beginning, we state some needed facts in graph and matrix theory. In chapter three, we present some known results about the distance matrix and related topics. In chapter four, the discussion was focused on the matrix Bn related to the graphs K(r, n-r), and K(2, n-2). In chapter tive we deal with the matrix Bn related to the cycles Cn for any positive integer n. New Accomplishments In chapter two, we state and prove a theorem for computing the eigenvalues and eigenvectors of circulant matrices, and relate them to the permutation matrices. ln chapter four: We state and prove a theorem for computing the eigenvalues and eigenvectors of a matrix, that appears in our main study. Also, we state for the first time, a theorem for computing the eigenvalues and eigenvectors for the matrix Bn,whose nonzero entries are the reciprocals of the corresponding nonzero entries of the distance matrix of the graph K(r, n-r), and as a special case the graph K(2. n—2). We will construct a table, which contains numerical values of the spectral radius ( 1, and those of some complete bipartite graphs. obtained by direct calculation and by the resulting formulas. Also, we will construct a table which contains numerical values of for K(2,n—2), K(3,n-3), K(4,n-4) as n goes bigger and bigger, then we state and prove a lemma for calculating the limit of as n approaches . In chapter five, we will find the eigenvalues and eigenvectors for Bn related to the cycle graphs Cn, and we state and prove a theorem which shows that the eigenvalues of Bn are real for any positive integer n. We will also calculate the eigenvalues and eigenvectors of` Bn related to the cycle graph Cn, and those of Cn. We will present some graphs, then we will compute the eigenvector that corresponds to the spectral radius, and will note that vertices with greater eigenvector entries are with smaller eccentricities, and tend to be in the center of the graph.
Cantor Set in Measure Theory
(2005) Alaa Jamal Moustafa Yaseen; Dr. Abdallah A. Hakwati - Supervisor; Dr. Jasser H. Sarsour - Co-Supervisor
This thesis is a survey for the using of Cantor sets and in measure theory. It is proved that and are measurable and have zero measure. Following that it is shown that the measure of is positive and the measure of is zero. Also it is shown that there exists a subset of such that is non-measurable. At the end of this thesis it is shown that there is no subset such that is Bernstein in.
On Best Approximation Problems In Normed Spaces With S-property
(2008) Ghadeer Ghanem Fayez Qwadreh; Dr. Abdallah Hakawati
The problem of best approximation is the problem of finding, for a given point xX and a given set G in a normed linear space ( X, ), a point g G which should be nearest to x among all points of the set G.This thesis contains properties of best approximations in spaces with the S-property. We provide original results about Orlicz subspaces, and about subspaces with the S-property. As a major result we prove that: if G is a closed subspace of X and has the S-property. Then the following are equivalent:1.G is a Chebyshev subspace of X.2.L (m,G) is a Chebyshev subspace of L (m,X).3.L (m,G) is a Chebyshev subspace of L (m,X), 1pound
On Singular Value Decomposition of Rectangular Matrices
(2009) Sheren Najeh Issa Odeh; Dr. "Mohammad Othman" Omran
The singular value decomposition of matrices stands as one of the most important concepts in mathematics, because of its variety of applications in mathematics, statistics, biology and many other areas of science. In this thesis, we present the singular value decomposition and its relation to the spectral decomposition . We also investigate the singular value decomposition of a matrix together with some of its applications. Some of these applications include the Moore-Penrose psuedoinverse, the effective rank of matrices and image compression.
Error-Detecting and Error-Correcting Using Hamming and Cyclic Codes
(2009) Ne'am Hashem Ibraheem Ibraheem; Dr. Mohammad Omran
In this thesis we provide an overview of two types of linear block codes: Hamming and cyclic codes. We study the generation, encoding and decoding of these codes as well as studying schemes and/or algorithms of error-detecting and error-correcting of these codes.
Mathematical Theory of Wavelets
(2009) Bothina Mohammad Hussein Gannam; Dr. Anwar Saleh
Wavelets are functions that satisfy certain requirements and are used in representing and processing functions and signals, as well as, in compression of data and images, in many fields such as: mathematics, physics, computer science, engineering, and study of wavelet transform had been motivated by the need to the medicine. The overcome some weak points in representing functions and signals by the classical Fourier transform such as Gibbs phenomenon. In addition, wavelet transform have showed superiority over the classical Fourier transform. They converge faster than Fourier transform, leading to more efficient processing of signals and data. In this thesis, we overview the theory of wavelet transform, as well as, the theory of Fourier transform and make a comparative theoretical study between the tow major transforms proving the superiority of wavelet transform over the Fourier transforms in the speed of convergence and the accuracy or many functions
Multigrid Methods for Elliptic Partial Differential Equations
(2010) Rania Taleb Mohammad Wannan; Dr. Anwar Saleh
Partial differential equations appear in mathematical models that describe natural phenomena. Various methods can be used for solving such equations. In this thesis, an overview of classical iterative methods, as well as, the most recent multigrain methods is given. The classical iterative methods used are; the Jacobi, the Gauss-Seidel, and the SOR methods. Jacobi and Gauss-Seidel methods are efficient in smoothing the error but not in reducing it. The smoothing property of some classical methods motivated the work done on multigrain methods. Poisson's problem in one and two dimensions has been used as model problem in the study of multigrain methods. The study shows that the rate of convergence of multigrain methods does not depend on the mesh size, a feature that makes multigrain methods good accelerator of classical methods like Gauss-Seidel.
Decoding Turbo Codes with Linear Programming
(2013) Hisham Hamed Abdel-Raouf Salahat; Dr. Mohammad Assa`d; Dr. Mohammad Omran
In this thesis we investigate the application of Linear Programming LP relaxation to the problem of decoding an error-correcting code. LP relaxation is a standard technique in approximation algorithms and operation research, and it is used to find good suboptimal solution to very difficult optimization problems. The method of a posteriori probability and iterative decoding algorithm is used to decode product codes (a special type of turbo codes). We investigate a program using Matlab to make computations to our algorithm. The logistic distribution with variance one is used. We compare the results of our computations to those of other authors; we find that our results are the best all over the others. The LP method has its place in the generic turbo code, which is made up of asset of simpler" trellis-based" codes, we formulate the LP for a single trellis-based code as a min-cost flow problem, using the trellis as a directed flow. We extend this formulation to any turbo codes by applying constraints between the LP variables used in each component code. One of the most advantages for LP decoding is that whenever the decoder output a result it is guaranteed to be the optimal solution, the most likely (ML) information sent over the channel, we refer to this property as the ML certificate property.
Fuzzy Fredholm Integral Equation of the Second Kind
(2014) Muna Shaher Yousef Amawi; Prof. Dr. Naji Qatanani
Fuzzy Fredholm integral equations of the second kind have received considerable attention due to the importance of these types of equations in studies associated with applications in mathematical physics and fuzzy financial and economic systems. After addressing the basic concepts of fuzzy integral equations, we have investigated the analytical and the numerical aspects of the fuzzy Fredholm integral equation of the second kind. The analytical methods include: Fuzzy Laplace transform method, Homotopy analysis method (HAM), Adomain decomposition method (ADM) and Fuzzy differential transformation method (FDTM). For the numerical treatment of the fuzzy integral equation of the second kind, we have employed the Taylor expansion method and the trapezoidal method. Some numerical test cases are included. A comparison between the analytical and the numerical methods has been presented. Numerical results have shown to be in a closed agreement with the analytical ones.
Extending Topological Properties to Fuzzy Topological Space
(2014) Ruba Mohammad Abdul-Fattah Adarbeh; Dr. Fawwaz Abudiak
In this thesis the topological properties of fuzzy topological spaces were investigated and have been associated with their duals in classical topological spaces. Fuzzy sets, fuzzy functions and fuzzy relations were presented along with their properties. Different types of fuzzy topological spaces (FTS) were introduced in Chang’s and Lowen’s sense as well as intuitionistic (FTS). Many topological properties were proved to be extensions to those in non fuzzy setting, while examples were presented for those non extension properties. For instance, the closure of the product is not equal to the product of the closures. Also different approaches of separation axioms were investigated using Q-neighborhoods and fuzzy points, it turns out that most of them are not extension of classical separation axioms. Fuzzy topological properties are considered, for instance, we studied fuzzy connectedness and fuzzy compactness. It is found that the product of an infinite number of fuzzy compact spaces may not be compact. Finally, fuzzy continuity, fuzzy almost continuity and fuzzy ??-continuity were introduced with a theorem proved the way they are related.
An Analytic and Dynamic Programming Treatment for Solow and Ramsey Models
(2014) Ahmad Yasir Amer Thabaineh; Dr. Mohammad Assa`d
In this thesis, we studied two of the most important exogenous economic growth models; Solow and Ramsey models and their effects in microeconomics by using dynamic programming techniques. Dynamic programming (DP) is a general approach to solve economic growth problems. The main differences between Solow and Ramsey models are discussed in details. Bellman value function for the growth models is applied to the two models and an analytic formula are derived. Concerning the models under study, we then discussed the steady states for the model and derived a closed formula for the capital. This formula was checked by computer using Python codes where a new concave assumed value function is given; , to be compared with a value function given by other . These two initial functions have the same properties of being monotone and concave up. The comparison shows the excellence and advantages of our assumption. We reached the true value function faster.
On The Theory of Convergence Spaces
(2015) Raed Juma Hassan Shqair; Dr. Mohammad Abu Eideh
In this thesis we investigate some information about convergence space concepts such as closure and interior of sets , open sets, closed set, cluster point of a filter , closed adherences of convergence spaces , separation axioms, continuity, homeomorphism, compactness, connectedness spaces and obtain some results about the aforesaid concepts and provide basic ideas of convergence theory, which would enable One to tackle convergence -theoretic without much effort . In this thesis some results on the cluster set of functions in convergence spaces are obtained.
On Fuzzy Metric Spaces and their Applications in Fuzzy Environment
(2015) Sondos Abdelrahim Mohammad Eshtaya; Dr. Mohammad Al-Amleh
In this thesis, the fuzzy metric spaces were investigated using different definitions and point of views. Some were applied on regular sets while others were applied on sets of fuzzy points. The concept of complement of fuzzy metric spaces using fuzzy scalars were studied and parallel results in classical analysis were found under fuzzy Setting . And finally , Fuzzy fixed point theorems on fuzzy metric space were proved .

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