وصف مساقات الرياضيات (دكتوراه)

تحليل عددي(Math 9372) :

Error analysis. Solutions of linear systems: LU factorization and Gaussian elimination, QR factorization, condition numbers and numerical stability, computational cost. Least squares problems: the singular value decomposition (SVD), QR algorithm, numerical stability. Eigenvalue problems: Jordan canonical form and conditioning, Schur factorization, the power method, QR algorithm for eigenvalues. Iterative Methods: construction of Krylov subspace, the conjugate gradient and GMRES methods for linear systems, the Arnoldi and Lanczos method for eigenvalue problems.

Prerequisite: Consent of the department.

طرق الرياضيات التطبيقية  :(Math 9361)

Calculus of variations, necessary conditions for extrema, isoperimetric problems. Partial differential equations. Cauchy Kowalevsky theorem, linear, quasi-linear, half-linear equations. Initial and boundary value problem for the wave and heat equations, Riemann method. Laplace equation, minimum and maximum principles. Integral transforms. Stability and bifurcation. Inverse scattering transform.

Prerequisite: MATH 3303 or equivalent.

جبر متقدم  :(Math 9312)

Noetherian and Artinian modules and rings. Hilbert basis theorem. Wedderburn-Artin Theorem. Uniform modules, primary modules, compactly packed modules, primary compactly packed modules, Prime Avoidance theorem for modules, Primary Avoidance theorem for modules, and Noether-Lasker theorem. Smith normal form over a principal ideal domain and rank. Fundamental structure theorem for finitely generated modules over a Principal ideal domain and its application to finitely generated Abelian groups.

Prerequisite: MATHA6341 or equivalent.

تحليل اقتراني متقدم  :(MATH 9351)

Compact operators, compact operators in Hilbert spaces, Banach algebras. The spectral theorem for normal operators, unbounded operators between Hilbert spaces. The spectral theorem for unbounded self-adjoint operators, self adjoint extensions.

Prerequisite: MATHA6310 or equivalent.

تبولوجى متقدم  :(MATH 9331)

Compactications, Stone-Cechcompactication. Paracompact spaces, countably paracompact spaces. Metrizablity, complete metric spaces, Baire's Theorem. Local connectedness, totally disconnected spaces. Uniform spaces, complete uniform spaces, proximity spaces. Function spaces.

Prerequisite: MATH 6360 or equivalent.

نظرية الاحتمالات (1) (MATH 9343):

Axiomatic definition of probability, properties of probability measure, conditional probability. Stochastic independence. Random variables, distribution functions, expectation and moments of random variables.  Convergence of sequences of random variables. Law of large numbers, characteristic function, central limit theorems.

نظرية الاحتمالات (2) (MATH 9344):

Stochastic process, random walk, Markov chains, martingales, Brownian motion.

Prerequisite: MATH 9343.

جبر خطى متقدم  :(Math 9317)

Basic properties of vector spaces and linear transformations, algebra of polynomials, characteristic value and diagonalizable operators, invariant subspaces and triangulable operators. The primary decomposition theorem, cyclic decompositions and generalized Cayley-Hamilton theorem. Rational and Jordan forms, inner product spaces. The spectral theorem, bilinear forms, symmetric and skew symmetric bilinear forms.

نظرية الدوال في متغير مركب :(MATH 9321)

Analytic functions. Singular points and zeros. The argument principle. Conformal mappings. Riemann mapping theorem. Mittag-Lefler theorem. Infinite products. Canonical products. Analytical continuation. Elementary Riemann surfaces.

مقدمة في الدوال في عدة متغيرات مركبة (MATH 9322):

Holomorphic functions, comparison of one and several variables, domains of holomorphy, subharmonicity, pseudoconvexity.  Invariant metrics, holomorphic maps, Stein and CR-manifolds, integral formulas equation. 

Prerequisite:  Math 9321 

التبولوجيا الجبرية (Math 9332):

Fundamental group, Van Kampen's Theorem, covering spaces. Singular homology, Homotopy invariance, homology long exact sequence, Mayer-Vietoris sequence, excision. Cellular homology. Homology with coefficients. Simplicial homology and the equivalence of simplicial and singular homology. Axioms of homology. Homology and fundamental groups. Simplicial approximation. Applications of homology.

 Prerequisite: Consent of the department.

فضاءات المتجهات التبولوجية (Math 9333):

Vector space topologies. Metrizability. Locally convex topological vector spaces, Hahn-Banach theorem, projective and inductive topologies, barreled and bornological

spaces. Linear mappings, Banach’s homomorphism theorem, uniformboundedness and the Banach-Steinhaus theorem.Duality, dual systems and weak topologies, strong dual, bi-dual, and reflexive spaces. Theorems of Grothendieck, weak compactness, open mappings, and closed graph theorems. Linear manifoldsand applications.

Prerequisite: Consent of the department 

التحليل الاقتراني التطبيقي :(Math 9376)

Distributions, Review of Banach and Hilbert Spaces. Sobolev spaces. Semigroups, Some techniques from nonlinear analysis.  

Prerequisite: Consent of the department. 

معادلات تفاضلية جزئية متقدمة (Math 9383):

Sobolev spaces: Weak derivatives, Approximation by smooth functions, Extentions, Traces, Sobolev Inequalities, The Space H-1. Second Order Elliptic Equations: Weak solutions, Lax-Milgram Theorem, Energy Estimates, and Fredholm Alternative. Regularity.  Maximum principles.  Eigenvalues and Eigenfunctions. Linear Evolution Equations: Second Order Parabolic Equations, (Weak solutions, regularity, Maximum Principle), Second Order Hyperbolic Equations, (Weak solutions, Regularity, Propagation of disturbances), Hyperbolic Systems of First Order Equations, Semigroup Theory.  

Prerequisite: MATH 9361 or equivalent.

طرق متقدمة في الرياضيات التطبيقية  :(Math 9362)

Initial and initial-boundary value problems for the first order nonlinear PDEs. Continuous solutions. Conservation laws and weak solutions. Burgers' equation. Quasi-linear hyperbolic systems. Riemann invariants. Nonlinear waves in gases and deformable solids. One parameter group transformation and similarity solutions of nonlinear problems for PDEs. Nonlinear waves in strings under transverse impact.  

Prerequisite: MATH 9361 or equivalent.

معادلات تفاضلية عادية متقدمة :(Math 9382)

Nonlinear Periodic Systems: Limit Sets, Poincare-Bendixon Theorem. Linearization near Periodic Orbits. Orbital stability. Bifurcation: Bifurcation of Fixed Points, the Saddle-Node Bifurcation, the Transcritical Bifurcation, the Pitchfork Bifurcation, Hopf Bifurcation. Boundary Value Problems:  linear Differential Operators, Boundary Conditions, existence of Solutions of BVPs, Adjoint Problems, Eigen values and Eigen functions for Linear Differential Operators, Green's Function of a Linear Differential Operator. 

Prerequisite: MATH 6302 or equivalent.

  

مواضيع خاصة في الرياضيات :(Math 9350) 

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor. The course introduce the students to research method and scientific research ethics and contains topics vary from year to year according to interest of students and instructor in charge.

نظرية الاعداد الجبرية :(Math 9316)

The arithmetic of algebraic number fields, the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field.

مقدمة في الجبر الغير إبدالي :(Math 9313)

Noncommutative rings and their ideals and modules, concentrating on two fundamental results, the Wedderburn structure theorem and Maschke’s theorem. Further insight into the structure of rings will be provided by the Jacobson, prime and primary radicals.

نظرية الترميز (Math 9318):

This course is a continuation of the course has been taken at master level in Mathematics department, it includes the following topics: BCH and Reed Solomon codes, Duadic codes, Quadratic residue codes. Weight distribution, Macwilliams equations, designs.  Self- dual codes, convolutional codes, codes over Z4 and some topics suggested by the instructor.

نظرية اللاتس (Math 9314):

- First concepts: Definitions of lattices, how to describe lattices, some algebraic concepts, polynomial identities and inequalities, Free lattices, special elements characterization theorems and representation theorems and congruence relation and topological representation.

- Distributive lattices with pseudo complementation, Introduction and Stone algebra, Heyting algebra, Post algebra, identities and congruences, Representation theorems.

الجبر البولى  :(Math 9315)

- Finite joins and meets: Definition of Boolean algebras, some consequences of the axioms. Ideals and filters, subalgebras, homomorphisms and isomorphosims, maximal ideals and filters. Reduced and perfect field of sets, a fundamental representation Theorem. Atoms, quotient algebras. Induced homomorphisms between fields of sets, theorems on extending to homomorphisms, independent subalgebras. Products.  Free Boolean algebras, induced homomorphisms between quotient algebras, direct unions, connection with algebraic rings.

- Boolean spaces, the representation theorem, duality for ideals, duality for homomorphisms, completion.

- Relation to other algebras:

1. Application to mathematical logic.

2. Application to Topology.

شبه الزمر التبولوجية (Math 9334):

Semigroups and their Ideals. Right Topological Semigroups, Ultrafilters, the Topological space βD, Stone-Cechcompactification, uniform limits vaiultrafilters. The Semigroup βS, Ramsey Theory, ideals and commutativity in βS, groups in βS, cancellation, idempotent and homeomorphisms.

 مواضيع متقدمة في التحليل الرياضي :(Math 9320)

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor Contents vary from year to year according to interest of students and instructor in charge.

مواضيع متقدمة في الجبر:(Math 9310) 

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor Contents vary from year to year according to interest of students and instructor in charge.

مواضيع متقدمة في الاحصاء(Math 9390) :

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor Contents vary from year to year according to interest of students and instructor in charge.

مواضيع متقدمة فى الرياضيات التطبيقية  :(Math 9360)

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor Contents vary from year to year according to interest of students and instructor in charge.

مواضيع متقدمة في نظرية الاحتمالات (Math 9340):

Students choose and study a topic under the guidance of a faculty member, normally his/her advisor Contents vary from year to year according to interest of students and instructor in charge.