Description of Courses (Doctorate)
Numerical Analysis (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.
Methods of applied mathematics (Math 9361):
Calculus of variations, necessary conditions for extrema, isoperimetric problems.
Partial differential equations. Cauchy Kowalevsky theorem, linear, quasi-linear, halflinear 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.
Advanced Algebra (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.
Advanced Functional Analysis (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.
Topology (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.
Probability Theory 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.
Probability Theory 2 (MATH 9344):
Stochastic process, random walk, Markov chains, martingales, Brownian motion.
Prerequisite: MATH 9343.
Advanced linear algebra (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.
Theory of Functions of a Complex Variable (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.
Introduction to Functions of Several Complex Variables (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
Algebraic Topology (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.
Topological Vector Spaces (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 andweak 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
Applied Functional Analysis (Math 9376):
Distributions, Review of Banach and Hilbert Spaces. Sobolev spaces. Semigroups,
Some techniques from nonlinear analysis.
Prerequisite: Consent of the department.
Advanced Partial Differential Equations (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.
Advanced methods for Applied Mathematics (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.
Advanced Ordinary Differential Equations (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.
Special Topics in Mathematics (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.
Algebraic Number Theory (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.
Introduction to Non-commutative Algebra (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.
Coding Theory (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.
Lattice Theory (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.
Boolean algebra ( 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.
Topological Semigroups (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.
Advanced Topics in Analysis (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.
Advanced Topics in Algebra (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.
Advanced Topics in Statistics (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.
Advanced Topics in Applied Mathematics (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.
Advanced Topics in probability (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.