جامعة الاقصى
AL-AQSA UNIVERSITY - GAZA
AL-AQSA UNIVERSITY - GAZA
Analysis
Lebesgue measure: outer measure, measurable sets and functions, Egoroff's theorem, Lusin's theorem, convergence in measure, the Lebesgue integral: the integral of a bounded function over a set of finite measure, the integral of a nonnegative function, the general Lebesgue integral, Riemann and Lebesgue integrals, differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolute continuity, Lp classes: the Holder and Minkowski inequalities, completeness of Lp classes, the duals of Lp classes, Banach spaces: linear operators, the Hahn-Banach theorem and other basic results, Hilbert spaces.
Functional Analysis
Hilbert spaces: the geometry of Hilbert space, the Riesz representation theorem, orthonormal bases, isomorphic Hilbert spaces, operators on Hilbert space: basic properties and examples, adjoints, projections, invariant and reducing subspaces, positive operators and the polar decomposition, self-adjoint operators, normal operators, isometric and unitary operators, the spectrum and the numerical range of an operator, operator inequalities, compact operators, Banach spaces: basic properties and examples, convex sets, subspaces and quotient spaces, linear functionals and the dual spaces, the Hahn-Banach theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.
Complex Analysis
Analytic functions: power series, Laurent series, analytic functions as mappings, Mobius transformations, linear fractional transformations, conformal mappings, cross ratio, complex integration: zeros of analytic functions, Cauchy's theorem and formula, the argument principle, the open mapping theorem, the maximum modulus principle, Schwartz lemma, singularities: classification of singularities, residues, residue theorem, evaluation of real definite and improper integrals, normal families: Riemann mapping theorem, Schwartz reflection principle, Schwartz-Christofell formulas, harmonic functions: Dirichlet problem, Poisson’s formula, mean value property
Topology
Topological spaces, neighborhoods, bases and subbases, continuous functions, product spaces, weak topologies, quotient spaces, filters, separation axioms, regular and completely regular spaces, normal and perfectly normal spaces, Lindelof, separable spaces and second countable spaces, compact spaces, locally compact spaces, sequentially and countably compact spaces, one point compactification, paracompact spaces, connected spaces. Locally compact and k-spaces, metric and metrizable spaces, complete metric spaces and the completion theorem, Baire spaces and Baire category theorem.
Differential Equations
Basic definitions – some models of partial differential equations – classifications of P.D.E – separation of variables method – series method – laplace transform method– Fourier transform method – perturbation method.
Fluid Mechanics
Introduction to tensor analysis – basic definitions- Newtonian fluids – Equation of continuity – Nervier – Stokes equations. The motion between two plates – The motion through cylinder – Non Newtonian fluids – Equations of motion for non-Newtonian fluids – Applications Basic definitions – some models of partial differential equations – classifications of P.D.E – separation of variables method – series method – lap lace transform method – Fourier transform method – perturbation method.
Analytical Dynamics
Survey of the elementary principles- Variational principles and lagrange's equations- The Rigid Body Equations of motion- The Hamilton Equations of motion- Canonical Transformations- Hamilton – Jacobin theory- Constrained lograngians as fidd systems.
Algebra I
Groups, definitions and examples , groups of permutations , direct product, homomorphism and isomorphism, fundamental theorem of finite abelian groups, simple groups, Sylow theorems, introduction to free groups.
Ring Theory, integral domains, ideals and factor rings, prime ideals, maximal ideals, ring homomorphism, polynomial rings, factorization, divisibility in integral domains, fields, extension fields, algebraic extensions, finite fields .
Algebra II
General noncommutative ring theory, chain conditions, sum and direct sum of rings, Wedderbern theorem, modules, module decomposition, modules over finite rings.
Advanced Number Theory
Divisibility, primes, congruence, primitive roots and quadratic reciprocity, quadratic residues, Arithmetic functions, introduction to algebraic number theory, quadratic number fields, cyclotomic fields.
Coding Theory
Basic concepts for linear codes : linear codes, generators and parity check matrix, dual codes, weights and distances, Hamming codes, bounds on the size of codes, cyclic codes, zeros of a cyclic codes, minimum distance of cyclic codes, self dual codes, codes over Z4
Cryptography
Mathematical Basics: Divisibility, Primes, An Introduction to Congruences ,
Euler, Fermat ,Complexity.
Cryptographic Basics: Definitions and Illustrations ,Classic Ciphers , Stream Ciphers , LFSRs, Modes of Operation , Attacks, DES and AES,
Public-Key Cryptography: RSA , ElGamal, DSA — The DSS
Factoring: Classical Factorization Methods ,The Continued Fraction, Algorithm, Pollard’s Algorithms ,The Quadratic Sieve, The Elliptic Curve Method (ECM).
Statistics
Counting, conditional probability and independence, random variables, distribution functions, density and mass functions. distributions of functions of a random variable, expected values, moments and moment generating functions. Discrete distributions.Continuous distributions, exponential families, location and scale Families, inequalities and identities. Joint and marginal distributions, conditional distributions and independence, bivariate transformations. Hierarchical models and mixture distributions, covariance and correlation, multivariate distributions. Sums of random variables from a random sample, sampling from the normal distribution, properties of the sample mean and variance.Student's t and Snedecor's F.Order Statistics.
Advanced Statistics
Principles of Data Reduction: the sufficiency principle, the likelihood principle, the likelihood function.Methods of finding estimators; method of moments, maximum likelihood estimators, bayes estimators.Methods of evaluating estimators.Best unbiased estimators, sufficiency and unbiasedness.Hypothesis testing: likelihood ratio tests,Bayesian tests, union-intersection and intersection-union Tests, error probabilities and the power function, most powerful tests.Methods of evaluating interval Estimators.Point Estimation; consistency, efficiency.The mean and the median.Approximate maximum likelihood intervals. Analysis of variance. Linear regression. Models and distribution assumptions, estimation and testing with normal errors.
