Course Description - Master in of Mathematics Comprehensive exam& Thesis Tracks

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1 Course Description - Master in of Mathematics Comprehensive exam& Thesis Tracks Theory of ordinary differential equations Review of ODEs, existence and uniqueness of solutions for ODEs, existence and uniqueness of solutions for systems. Sturm-Louiville's theory and orthogonal functions. Non-linear ODEs and their stability Partial differential equations PDEs in Mathematical Physics, separation of variables, Transform Methods, Eigen function expansions, Green s Function. Elliptic PDEs. Parabolic and Hyperbolic PDEs Methods of Applied mathematics-1 Integral Transforms, Green s Function, Integral Equations. Voltera- Fredholm Theorem Methods of Applied mathematics-2 Prerequisite: Variational methods. Perturbation methods. Approximation methods Dynamical Systems and Chaos Prerequisite: One-dimensional maps. Graphical representation of an orbit. Stability of fixed points. Periodic points. Notion of dynamical systems. Examples of dynamical systems(discrete and continuous). Existence and uniqueness of solutions. Linear systems. Asymptotic behavior. Invariant sets. Attractors. Jordan normal form of a matrix and solutions for general linear systems. Stability for general linear systems. Nonlinear systems: Iterations of the logistic map. Chaos and Fractals. Properties of chaos: Lyapunov exponents for discrete-time 1-d systems and for higher dimensional systems. Symbolic analysis. Bifurcations.

2 Real Analysis (Measure Theory & Integration) 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, Fato's Lemma, Lebesgue dominated and convergence theorem. The general Lebesgue integral, Riemann and Lebesgue integrals. Differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolute continuity. Lp spaces: the Holder and Minkowski inequalities, completeness of Lp spaces, the duals of Lp spaces, Banach spaces: linear operators, the Hahn-Banach theorem and other basic results. Hilbert spaces Functional Analysis Prerequisite:( ) 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 Advanced Matrix Theory Similarity and canonical forms, diagonalization and simultaneous diagondization of matrices, location of eigenvalues, special classes of matrices, unitary equivalence of matrices, Schur s theorem and spectral theorem, singular value decomposition and polar

3 decomposition, generalized inverses, least-squares solutions to linear systems, determinant and trace inequalities, the min-max principle, singular value inequalities, perturbation inequalities, vector and matrix norms, the spectral radius and the numerical radius, unitarily invariant norms, norm inequalities Mathematical Statistics (3 Cr. Hr.) Univariate and multivariate distribution Theory, sufficiency and related theorems, Completeness, Rio Blackwell theorem, methods of point estimation and properties of point estimators, Bayesian and Minimax estimation methods, Cramer's raw inequality, Confidence Intervals, Testing Hypotheses, Neyman-Pearson Lemma, Randomized Tests, most powerful and Uniformly most powerful, LR test, Sequential tests, Unbiased tests Probability Theory (3 Cr. Hr.) Kolmogorov's axioms, Random variables, Distributions and their functions, truncated Distributions, Expectations, Chebyshev's inequality, Conditional probability, Independence, Borel- Cantelli Lemma, Convergence concepts, Characteristic functions, Central limit theorem, strong and weak laws of large numbers Stochastic Processes (3 Cr. Hr.) Distributions and general properties of processes, continuity, Bernoulli and Poisson Processes, Markov chains, Markov pure jump, Continuous Markov and elements of second order processes, Gausian processes, wiener diffusion and Feller processes Linear Statistical Models (3 Cr. Hr.) Least squares estimation, solutio0n of normal equations by generalized matrix inverse, multivariate normal distribution, distributions of quadratic forms, noncentral chi-square distribution, independence of several quadratic forms, full rank and less than full rank linear models, analysis of variance for linear models, variance components and mixed models, Gauss-Markov and BLUE estimators.

4 Nonparametric Statistics (3 Cr. Hr.) Distribution free Statistics, Counting and ranking statistics, goodness of fit test, wilcoxon test, wald- wolfowitz test, Kolmogorov- Smirnov- test, Mann- Whitney test, Median, Mood, Siegel-Tukey test, Kloz test, Sukatme and Kruskal- Wallis test, Asymptotic relative efficiency. rdering of Hermitian matrices, Hadamard product of matrices, applications Modern Algebra-1 Isomorphism theorems of groups, group automorphism, finite direct products, finitely generated groups, groups actions, Sylow theorems, rings and ideals, prime and maximal ideals, polynomial rings and irreducibity tests, unique factorization domains, Euclidean domains Modern Algebra-2 Prerequisite: ( ) Review of basics of rings and ideals, fields and extension of fields. Galois theory Analytic Number Theory Review of some analytic concepts and techniques used in number theory. The Prime number theorem. The Riemann zeta function and Dirichlet character L-functions. Some explicit formulas related to primes in arithmetic progressions. Functional equations for the Riemann zeta function and Dirichlet $L$-functions. Modular forms, automorphic forms and higher level modular forms General Topology-1 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, separable spaces and second countable spaces, compact spaces, locally compact spaces, Lindelof spaces, sequentially and countably compact spaces, one point compactification, paracompact spaces, connected spaces Topology-2 Prerequisite:( ) Locally compact and k-spaces, Čech complete spaces, metric and metrizable spaces, complete metric spaces and the completion theorem, Baire spaces and Baire category theorem, uniform and proximity spaces.

5 Algebraic topology Prerequisite:( ) Contractible spaces, retract and strong deformation retract. Manifolds, special Euler function. The fundamental groups. The fundamental group of S. Brouwer fixed point theorem. The fundamental group of the surfaces, covering spaces. Singular homology groups. Mayer-Vietories sequences. The jordan separation theorem. CW-Complexes, The homology of CW-complexes. The Ellenberg-Steen rod axioms. poincare duality. Applications on the fixed point theorem. The Lebschetz fixed point theorem Graph Theory Oriented Linear Graphs, Non-Oriented linear Graphs, Incidence set and cut-set, Static Maximal Flows.