Descriptions of Courses:

Note: Topic courses (1504-1509-1511-1513-1515-1518-1519-1520-1521-1522-1540-1570) are to be decided by instructors.

(1) compulsory courses:

Functional Analysis

Theory of Diff. Equations

Rings and Modules

Advanced Mathematical Methods

Topology

(2) Optional  courses:  

1501

Definition and Basic Properties of Rings, Ideals, Quotient Rings Homomorphisms of Rings Prime Ideal Domains, Unique Factorization Domains, Rings of Polynomials, Simple Rings, Definition and Basic Properties of Modules (Left and Right, R-Modules), submodules, Bases Homomorphisms of Modules Exact Sequences of Modules, Diagram Chasing, Direct Product of Modules, External Direct Sum of Modules, Free Modules, Projective and Injective Modules, Finitely Generated Modules, Noetherian and Artinian Modules, Tensor Product of Modules.

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Existence Theorems: Picard Lindelof, Peano, Caratheodory and Contraction Mapping Theorems, Uniqueness Theorems, Extension Solution, Maximum and Minimum Solutions, Stability and Asymptotic Behaviour of Solution-Laplace, Liapunov and Poincare Stabilities, Liapunov Stability by Direct Methods, Cheyshev Instabilitiy Theorem, Stability Based on First Approximation, Perron Theorem.

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Complete Metric Spaces, Normed Linear Spaces, Banach Spaces, Equivalent Norms in Banach Spaces, Convergence of Series, Bounded Linear Operators and Functionals, Dual Spaces, Open Mapping, Closed Graph and Hanhn-Banach Extension Theorems, Strong and weak Convergence, Hilbert Spaces, Spectral Theory of Self-Adjoint Bounded Linear Operators in Hilbert Spaces.

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Connectednes, path-Connectednes, Local Connectedness and Components, Compactness, Local Compactness and Compactifications, Function Spaces, Topological Groups and Orbit Spaces. Homotopy and Fundamental Groups.

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Laplace and Fourier transforms; theory and applications. Self-adjoint differential equations. Sturm-Louiville form. Eigenfunctions and eigenvalues Green functions. Orthogonal functions.

Gram-schmidt orthogonalization. Orthogonal polynomials generated by Gram-schmidt.

General Integral transform. Sturm-Louiville transforms. Finite Hankel transforms.

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1500

Classical Variational Problems, Spaces of Functions, Derivatives in Normed Spaces (Weak, Frechet, Strong), Linear and Nonlinear Functionals, First and Second Variations of Functionals, Necessary Conditions for Relative Maximum (Minimum), Weak and Strong Relative Extreme Values, Euler-Lagrange Equation, Discussion of Solutions, Boundary Conditions, Applications, Some Generalizations.

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1503

Classification of Second Partial Differential equations, Hyperbolic, Parabolic and Elliptic, Canonical Form of 2nd Order P.D.E., Eave Equation, Heat Equation, Cauchy Problems, Cauchy-Kowalewski Theorem for 2nd Order P.D.E.

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Are to be decided by instructors.

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Numerical Solution of Ordinary Differential Equations: Taylor Series Method, Runge-Kutta Method, Multistep Method, Convergence Criteria, Errors, Systems of Equations and Higher Order Equations, Boundary-Value Problems, Numerical Solution of Elliptic Partial Differential Equations: Representation as Difference Equation, Laplaces Equation on Rectangular Region, Poisson Equation, Derivative Boundary Conditions, Parabolic Partial Differential Equations: Explicit Method, Crank-Nicolson Method, Stability and Convergence Criteria, Hyperbolic Partial Differential Equations: Wave Equation, Stability of Numerical Method, Method of Characteristic.

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Classical Logic: Statement Calculus and Predicate Calculus, Non-Classical Logic: Many valued-Logic, Fuzzy Logic, Modal Logic, and Intuiotionistic Logic, Consistency, Completeness, and Decidability, Gödel’s Theorems.

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Are to be decided by instructors.

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Homotopy, Fundamental Groups, Higher Homotopy Groups, Singular Homology, Singular Cohomology, and Eilenberg Steenrod Axioms, CW-Complexes, Products.

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Are to be decided by instructors.

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Are to be decided by instructors.

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Sylows Theorems, P-groups, Free Groups, Group Presentations, Orbit Space, Burnside Theorem, Groups of orders £ 17.

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Are to be decided by instructors.

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Rings and Fields of sets, Measurable Spaces, Measure Spaces, Properties of Measure, Existence of Measure, Product Measure, Simple Functions, Measurable Functions and their Properties, Convergence a.e and in Measure, Lebesgue Integral, Properties, Limit Theorems, Fubini Theorem, Spaces of Integrable Functions.

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- Field Theory :

Rings, Ideals, and Quotient Rings, Maximal and Prime Ideals Polynomial Rings, Field Extensions, Splitting Fields, Finite Fields, Galois Theory, Insolvability of Quintic.

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Are to be decided by instructors.

   الانتقال إلى جدول المقررات الدراسية 

Are to be decided by instructors.

   الانتقال إلى جدول المقررات الدراسية

Are to be decided by instructors.

   الانتقال إلى جدول المقررات الدراسية 

Are to be decided by instructors.

   الانتقال إلى جدول المقررات الدراسية 

Are to be decided by instructors.

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Are to be decided by instructors.

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Poset, Lattice, Distributive Lattice, Algebraic Lattice, Closure Operators, Boolean Algebras, Universal Algebras, Subalgebras, Isomorphisms, Quotient Algebras, Homomorphisms, Direct Products, and Free Algebras.

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Are to be decided by instructors.

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