The main aim of this work is to give further studies for the multi-dimensional integral equations. This work includes the following aspects: Discuss the existence of the unique solution for special types of the multi-dimensional integral equations, namely the multi-dimensional Volterra non-linear integral equations of the second kind via Picard iterative procedure and Banach fixed point theorem. Devote the multi-dimensional differential transform for functions of multiple independent variables with some of its properties. Solve special types of the multi-dimensional Volterra non-linear integral equations of the second kind via the multi-dimensional differential transform method. Apply the homotopy perturbation method for solving special types of the multi-dimensional Volterra non-linear integral equations of the second kind.

The main purpose of this work is to study integral and integro-differential equations that contain two different types of integral operators, these equations named as Volterra-Fredholm integral and integro-differential equations. This study includes the following aspects: (1) Discuss the existence and the uniquness theorem of solution of solution of the non-linear Volterra- Fredholm integral and integro-differential equations of the second kind via Banach fixed point theorem. (2) Gives some approximation methods for solving linear Volterra-Fredholm integral and integro-differential equations of the second kind, namely the expansion methods via Bernestain polynomials, Neumann series and the power series method. (3) Uses the power series method to solve special types of non-linear Volterra-Fredholm integral and integro-differential equations of the second kind.

The Periodicity, Linear Complexity, Randomness and Correlation Immunity used as basic criterions to measure Key Generator Efficiency. The key generator depends basically on LFSR which is considered as one of the basic units, beside the combining function, of stream cipher systems. Through this thesis, these basic criterions can be calculated theoretically for any key generator before it be implemented or constructed (software or hardware). The implementation of the basic efficiency criterions on keygenerator reduces the time and the cost since the implementation of these tests applied before the final practical construction of the keygenerator. To approach efficient keygenerator, some conditions, which must be available in any keygenerator, are introduced. The Linear, Product and Brüer cryptosystems are chosen to be our study cases. This work introduces the mathematical proof of the good efficiency of the Linear and Brüer keygenerator deterministically. In the same time, we introduce the fail of the Product keygenerator to pass the basic efficiency criterions tests. Lastly this thesis introduce the theoretical and practical evolution of the three study cases and the evolution results appear that Brüer keygenerator is the best among the other keygenerator.

The main aim of this thesis is to study and develop some mathematical properties of some classes of non-linear dynamical control problems in infinite dimensional space with bounded perturbation and unbounded linear control operator using the concept of one-parameter family of strongly continuous semigroups generated by some unbounded linear generators. The necessary background for this approach has also been presented and supported by some useful concluding remarks. The mild principles, spectral conditions for stability and unstability have also been discussed and developed for certain class of non-linear dynamical control system in the presence of input in some infinite dimensional space, by using the principle of composite perturbation strongly continuous semigroup generated by some infinitesimal unbounded generator. Finally, -optimal control problem in infinite dimensional spaces are also presented. Some optimal control problems and solvability, controllability, null controllability, perturbed Recatti operator equation as well as some of its properties are discussed and developed via evolution perturbed strongly continuous semigroup generated by the linear unbounded perturbed generators.

The main objective of this thesis is to study the stability of switched non-linear systems and to find the conditions for stability under arbitrary switching. Also, this work consist of analyzing stability for switched systems which are composed of both continuous and discrete time subsystems by considering a Lie algebra generated by all subsystems of matrices, we showed that if all subsystems are Hurwitz-Schure stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. If some subsystems are unstable while the same Lie algebra is solvable, then we had prove that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme.

The first objective of this thesis, is oriented towards function approximation using special type of spline functions, which is called the generalized spline functions including its basic theory. General formula for each type of generalized spline is derived. The second objective of this work is the generalization of generalized spline functions for two-dimensional spaces by tensor product methods. The properties of tensor product are used in this generalization to get the two dimensional approximation function. Finally, applications of the generalized spline functions method in solving ODE and PDE problems are given.

The purpose of this thesis is to construct surfaces and complete arcs in the projective 3 – space PG(3,q) over Galois fields GF(q), q = 2, 3 and 5. A(k,n) – arc in PG(3,q) is a set of k points, no n + 1 of them are coplanar. A(k,n) – arc is complete if it is not contained in a (k + 1,n) – arc. In this work the (k,r) – caps and (k, ) – spans are constructed in PG(3,2) and PG(3,3) and it is found that the maximum (k,2) – cap, which is called an ovaloid, exists in PG(3,2) when k = 5 and also exists in PG(3,3) when k = 8. Moreover, the maximum (k, ) – span, which is called a spread, is found to exist in PG(3,2) when k = 5 and exists in PG(3,3) when k = 10.

الاهتمام الاساسي في هذه الاطروحة ينصب على تأسيس خلفية نظرية لتعريف ريمان-ليوفيل Riemann-Liouville) ) للاشتقاق والتكامل الكسري لدالة لمتغيرين وكذلك حل نوع معين من معادلات فريدهولم(Freadholm) الخطية الجزئية التكاملية التفاضلية ومعادلات فريد هولم الخطية الجزئية التكاملية التفاضلية ذات الرتب الكسرية بأستخدام بعض الطرق التقريبية، حيث تم أستخدام نظرية النقطة الثابتة (Banach Fixed Point Theorem) لبرهان وجود ووحدانية الحل لهذه النوعية من المعادلات. تم استخدام طريقتين من طرق البواقي الموزون (Weighted Residual Method) وهما طريقة التجميع(Collocation) وطريقة كاليركين(Galerkin) لمعالجة هذه المعادلات بصورة تقريبية وايضا" تم استحداث طريقة الفروق المنتهية الصريحة وتم استخدامها لتقريب الحل لكلتا المعادلتين(PIDE) و(FPIDE) . أضافة الى ذلك فقد تم مناقشة تقارب واستقرارية الطرق التقريبية الثلاث. وفي النهاية تم كتابة برنامج لكل واحدة من الطرق بأستخدام Matlab(v.7) .

Let R be a ring not necessarily with an identity element. A wellknown result proved by I.N.Herstein concering derivations in prime rings have been extensively studied by many authors like, M.Bresar, N.M.Shammu and M.Ashraf and N.Rehman. Also, C. Haetinger and M.Ferro extended this result to higher derivations. The main purpose of this work is i. Extend N.M.Shamm's theorem to higher N-derivations by giving the concept of higher N-derivation. A higher Nderivation of a ring R is defined as a family of additive mappings of R into itself such that d (ur su) d (u)d (r) d (s)d (u) i j i j n n + = ∑ i j + + = , for all u ∈U , r,s∈R , n ∈ N,where U is a Jordan ideal of R. ii. We answer the question of C.Haetinger and W.Cortes whether the result of C.Haetinger and M.Ferro is also true for Jordan generalized triple higher derivations. iii. We introduce the concept of (U,R) derivations and generalized (U,R) derivations. Then we extend Awatar's theorem and we extend this result to higher (U,R) derivations and generalized higher (U,R) derivations by giving corresponding definitions. A well-known result of I.N.Herstein concerning Jordan homomorphism and Jordan triple homomorphism has been extensively extended by M. Bresar. Also R.C. Shaheen extended these results to generalized Jordan homomorphism and generalized Jordan triple homomorphism iv. We introduce the concepts of higher homomorphism, Jordan homomorphism and generalized Jordan triple homomorphism and their generalization and we extend the above results and study these concepts onto 2-torsion free prime ring.

The aim behind this thesis is to find out the approximate solutions of the singular integral equation with Cauchy kernel. These equations embrace singular points. To do away with these singular points, special types of approximation must be found owing to the fact that direct methods cannot be applied. The approximate solutions for these equations have been arrived at by the use of the following methods:- Collocation method: In this method the solution is represented as a power series with unknown coefficients multiplied by a weighted function and then Chebyshev zeros are used as collocation points to find unknown coefficients. Gaussian-quadrature method: In this method a new approximation technique has been used in which the required unknown function is represented as a result of multiplying two orthogonal functions including (Chebysheve polynomials of first and second kinds and Jacobi polynomials). The properties of this function have been employed to eliminate the effect of the singular points. Trigonometric polynomials method: by the use of the same methods, a special type of approximation, then trigonometric functions and orthogonality properties, to find a system of linear algebraic equations which can be solved numerically. All these employed methods have yielded reliable and stable results even in the case of using a large-scale node. Algorithms and programs have been written in Fortran language for all the above-mentioned methods.

Let R be a commutative ring with identity and let M be a unitary left R-module. A proper submodule N of M is said to be Prime if re ∈ N for r ∈ R, e ∈ M implies that either e ∈ N or r ∈ [N : M]. prime submodules have been studied extensively by many authors, like S.A.Saymeh, C.P.Lu, R.L.McCasland, M.E.Moore, P.F.Smith, K.Divaani-Aazar and M.A.Esmkhani. primary submodules are generalizations of prime submodules. The main goal of this thesis is to look for generalizations of results on prime submodules to primary submodules. A proper submodule N of an R-module M is said to be Primary if re ∈ N for r ∈ R and e ∈ M implies that either e ∈ N or r ∈ p [N : M]. One of our goals of this work is to give some useful properties and characterizations of primary submodules. Furthermore we generalize the concept of the prime radical of a submodule to the primary radical. The primary radical of a submodule N , pradM(N ) is defined as the intersection of all primary submodules of M which contain N , and if no such submodule exists, we put pradM(N ) = M. And we give a characterization of the primary radical of a submodule N as: pradM(N ) = \ K(N ,P n )| P is a prime ideal of R, n ∈ Z + . Then we describe the elements of pradM(N ) for a finitely generated submodule of a free module. One of our important results is the Primary Avoidance Theorem, as a generalization of the Prime Avoidance Theorem. We introduce the concepts of p-compactly packed , strongly p-compactly packed, coprimarily packed and strongly coprimarily packed submodule. We investigate their vii properties, and discuss the relations between them. Finally, we give the definitions of associated and supported primary ideals of a module M and a characterization for each of them and study some of their fundamental properties. Also we define the associated and supported primary submodules of M and give some results about them.

الهدف الرئيسي لهذه الاطروحة هو دراسة وتطوير بعض الطرق التحليلية والتقريبية مع عرض بعض الخوارزميات الجديدة لمعالجة منظومة معادلات فولتيرا التكاملية اللاخطية من النوع الثاني. بالنسبة للمعالجات التحليلية تم تطوير احدى الطرق لحل معادلة فولتيرا اللاخطية وهي طريقة التقريبات المتتالية والتي تسمى أيضا (طريقة بيكرد) والتي أستخدمت لحل منظومة أعلاه. كذلك تم تعميم نظريات الوجود والوحدانية الخاصة بالمعادلة تكاملية اللاخطية الى منظومة معادلات تكاملية اللاخطية. أضافة الى ذلك , تم أستخدام ثلاث أنواع مختلفة من دوال الثلمة (التقليدية, التوصيلية وكاتمول-رام) لحل المنظومة أعلاه عدديا. ثلاث أنواع مختلفة من دوال الثلمة التقليدية(الخطية, التربيعية, التكعيبية) طورت وطبقت لاول مرة لمعالجة المنظومة أعلاه. خمس رتب مختلفة من دوال الثلمة التوصيلية (الأول, الثاني, الثالث, الرابع, الخامس) طورت وطبقت لاول مرة لمعالجة المنظومة أعلاه. نوعيين مختلفين لنوع جديد من دوال الثلمة وهي كاتمول-رام (ألاول و الثاني) طورت وطبقت لاول مرة لمعالجة المنظومة أعلاه. الصيغة العامة لمشتقة الصيغة الجديدة لدوال الثلمة التوصيلية أشتقت وطبقت لمعالجة المنظومة أعلاه عدديا. تم مناقشة أستقرارية وأقتراب ألانواع المختلفة لدوال الثلمة. أخيرا, تم كتابة الخوارزميات والبرمجيات بلغة (MATLAB\R14 ) لكل من الطرق الأنفة الذكر.