Description of Courses (Master)
‫ بخصوص الوصف بالعربي يتم تغير االسم بالعربي ويبقى الوصف باالنجليزي لمواد الماجستير والدكتوراه‬:‫ملحوظة‬
1. Algebra (1) (MATH A 6341)
-Structure of groups, including: isomorphism, automorphisms theorems, conjugacy, group
action, Sylow theorems and fundamental theorem of finite abelian groups. Solvable groups,
simple groups, nilpotent groups and semidirect products. Rings, . Chainese reminder theorem,
integral domains, ideals, PID UFD, polynomial rings and irreducibility tests. Modules,
product and sums of modules, exact sequences , simple and semisimple R- modules, the ring
of endomorphisms of R- modules and free modules.
2. Algebra (2) (MATH B 6341)
Projective and injective modules, the radical of an R-modules, tensor products
over
PID, flat modules, chain conditions, composition series, Jordan-Holder theorem, primery
decomposition and Noetherian and Artinian R-modules.
-Fields, including: algebraic elements, algebraic extension, splitting fields, finite fields,
irreducible polynomials over finite fields, quadratic reciprocity.
-Galois Theory, including: the Galois group of a polynomial, root multiplicity, fundamental
theorem of galois theory, solvability by radicals, cyclotomic polynomials, Galois groups.-
3. Analysis (1) (MATH A 6310)
Lebesgue Measure, including: Algebras and - algebras, monotone families,
Outer measures and measurable sets, measurable functions, Lebesgue measure.
-Lebesgue integral, including: Riemmann integral of bounded functions over
a set of finite measure, integral of nonnegative functions, the general
Lebesgue integral, limit theorems.
-Differentiation, including: differentiation of monotone functions, functions
Of bounded variation, Differentiation of an integral(the fundamental theorem of calculus)
-Absolute continuity, including: convex functions and Jensen's inequality.
4. Analysis (2) (MATH B 6310)
- L p spaces. including: Riesz-Fischer theorem and Riesz representation theorem.
-General measure and integration, including: measure spaces and measurable functions,
integration and general convergence theorems, signed and complex measures including total
variation, absolute continuity, singularity, Lebesgue decomposition, the Radon-Nikodym
theorem and the duals of the L p spaces.
-Measures, outer measures and product measures, including:
outer measures and
measurability, the extension theorem, measurability on Cartesian products, Fubini and
Tonnelli's theorems.
-The Daniell Integral, including: the Riesz representation theorem, Regular and Borel
measures.
5. Complex Analysis (MATH 6312)
Complex calculus (Review).
Evaluation of improper integral by the method of residues, conformal mapping, Riemann
mapping theorem, singular points and zeros, the argument principle, Rouche's theorem, power
series expansions, Weierstress theorem, Hurwitz theorem, partial fractions and factorizations,
Mittag Leffer's theorem.
6. General Topology (MATH 6360)
Set Theory. Relations. Ordinal and cardinal numbers. Zorn's Lemma and the
Axiom of Choice. Topological spaces, product and quotient spaces. Connected,
locally connected, locally pathwise connected, and totally connected spaces.
Compact and locally compact spaces. Compactifications. Product topology and
the Tychonoff theorem. Baire spaces and the Baire category theorem.
Separation and Countability Axioms. T0, T1, T2 spaces. Regular and completely
regular spaces. Normal and completely normal spaces. The Urysohn's lemma
and the Tietze's extension theorems. Metrizability and Paracompactness. The
Urysohn's metrizationtheorem. Paracompactness
7. Functional Analysis (MATH 6324)
Metric spaces, continuous function in metric spaces, linear spaces, subspaces, bases, Normed
spaces, convergence, completeness, Banach spaces, Banach algebra, spectral theory, inner
product and Hillbert space, orthonormal basis, Hahn_Banach theorem, uniform boundedness
theorem, open mapping theorem, closed graph theorem, and Baire-category theorem.
8. Mathematical Statistics(1) (MATH A6330)
Probability theory, conditional probability and independence, random variables, distribution
functions, density functions, distributions of functions of a random variables, moments and
moment generating functions, exponential families, location and scale families, joint and
marginal distributions, conditional distributions, bivariat transformations, covariance and
correlation, multivariate distributions, inequalities and identities.
9. Mathematical Statistics(2) (MATH B6330)
Basic concepts of random samples, sums of random variables from a random sample,
convergence concepts, sampling from the normal distribution, order statistics, the
sufficient principle, the likelihood principle, the invariance principle, methods of
finding estimators, methods of moments, maximum likelihood estimators, bayes
estimators, byes estimators, properties of estimators, hypothesis testing, methods of
finding tests, methods of evaluating tests, interval estimation.
10. Applied Mathematics (MATH 6306)
The existence and uniqueness theorem, Nonlinear differential equations, Power series
solutions, Equations of hypergeometric type, Linear system of differential equations.
11. Probability (MATH 6331)
Probability spaces, random variables, distribution functions, conditional probability, laws of
large numbers, and central limit theorems.
12. Advanced course in Statistics(Math 6339)
This course covers advanced topics in different statistics subjects. The contents of the
course are dedermined by the department according to the students’ interests
12. Advanced course in Mathematics (MATH6399)
This course covers advanced topics in different mathematics subjects. The contents of
the course are dedermined by the department according to the students’ interests.
13.Ordinary differential equations(Math 6302)
Existence and uniqueness theorem, Frobenius method, Equations of hypergeometric
type, Linear systems of differential equations, Dynamical systems and stability.
Partial differential equations(Math 6303) 14.
Cauchy Kowalevsky theorem, Linear and quasi-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, Green's functions.
15. Numerical Analysis(Math 6313)
Error analysis, Solutions of linear systems, LU factorization and Gaussian
elimination, QR factorization, condition numbers and numerical stability,
computational cost, Interpolation, Numerical evaluation of definite integrals, Solution
of ordinary differential equations, Stability and convergence of methods and error
estimates.
16. Special topics (1) (MATH A 6200)
The content of this course is compatible with the interest of the student’s advisor.
17. Special topics (2) MATH B 6100
Continuation of special topics (1).
18. Thesis MATH 6600