# Mathematics

## Course Description

**41701 ****Ordinary Differential Equations**

Perturbation theory and Poincare-Bendixson theorem, the asymptotic analysis of differential equations. Theory and theoretical examples are complemented by computational, model driven examples from biological and physical sciences.

**41711 ****Real Analysis (I)**

Concept of integration, generalized Riemann integral, Borel sets, Bair functions, outer measures, measurable sets, Lebesgue measure, Lebesgue density theorem, Hausdorff dimension, measurable functions, Lusins and Egorovstheorm, limit theorems for lebesgue integral, spaces of integrable functions, signed measure.

**4171****4** **Complex Analysis**

- Fundamentals of complex numbers: basic mathematical operations and basic properties of complex numbers.
- Powers and roots of a complex number.
- Continuity, differentiability, analyticity of a function of a complex variable using limits. Determine where a function is differentiable, analytic by using the Cauchy-Riemann Equations.
- exponential, trigonometric and hyperbolic functions of a complex number. Equations involving exponential, trigonometric and hyperbolic functions, logarithms and complex powers.
- Contour integral using the parametrization of the contour. Contour integral with an integrand having singularities inside or outside the simple closed contour, apply the Cauchy’s integral formula and the generalizedCauchy’s integral formula for computing contour integral.
- The circle of convergence of a power series. Taylor series of a given function. Laurent series of a rational function. The annulus of convergence of a Laurent series.
- The residue of a function at a point. Using the residue theorem to evaluate a contour integral.

**41721 ****Numerical Analysis (I)**

Solution of linear systems, least squares problems, and eigenvalue problems via matrix factorizations, the singular value decomposition (SVD) and basic sensitivity analysis. Polynomial interpolation, numerical integration, numerical solution of ordinary differential equations by single and multi-step methods, Runge-Kutta, Predictor-Corrector; numerical solution of boundary value problems for ordinary differential equations by shooting methods, finite differences and spectral methods.

**41741 ****Algebra (I)**

Isomorphism theorems of groups, auto morphism ,finite direct sums, Finitely generated groups, group action, Sylow theorems, rings and ideals, maximal and prime ideals, polynomial rings ,reducible and irreducible polynomials, unique factorization domains and Euclidean domains.

**41761 ****Topology (I)**

Topological spaces, bases , and sub bases ,neighborhoods, continuous function, Homeomorphisms, subspaces, product spaces, quotient spaces, convergence, nets and filters, separation Axioms, count ability Axioms, compact spaces, connected spaces. Regular spaces, completely regular spaces, normal spaces, linderof spaces

The student should know and be able to define toplogical spaces and related topics, cont. functions, and homeomorphism, identify the top logical properties. Also, to give the separation and countabety Axioms. Moreover, to deal with different spaces like compact, connected, normal, regular, etc….

**41702 ****Applied Mathematics**

The course provides a systematic approach to modeling and analysis of physical processes. For specific applications, relevant differential equations are derived from basic principles, for example from conservation laws and constitutive equations. Dimensional analysis and scaling are introduced to prepare a model for analysis. Analytic solution techniques, such as integral transforms and similarity variable techniques, or approximate methods, such as asymptotic and perturbation methods, are presented and applied to the models. A broad range of applications from areas such as physics, engineering, biology, and chemistry are studied

**41703 ****Partial Differential Equations**

Basic model equations describing wave propagation diffusion and potential functions; characteristics, Fourier transform, Green function, and eigenfunction expansions; elementary theory of partial differential equations; Sobolev spaces; linear elliptic equations; energy methods; semigroup methods; applications to partial differential equations from engineering and science.

**41712 ****Real Analysis (II)**

General convergence theorem, decomposition of measures, radon-Nikodym theorem, Caratheodery extension theorem, product measure, Fubinistheorm, Riesz representation theorm, elements of spectral theory, Frechet space

**4171****3** **Functional Analysis**

Review of metric spaces, normed linear spaces, Banach spaces, dual spaces, Hahn_Banach theorem, bidual and reflexivity, Bairestheorm, dual maps, projections, Hilbert spaces, the spaces L(X,Y), L_{p}(X), C(X), locally convex vector spaces.

**41722 ****Numerical Analysis (II)**

The course introduces numerical methods especially the finite difference method for solving different types of partial differential equations. The main numerical issues such as convergence and stability will be discussed. It also includes an introduction to the finite volume method, finite element method and spectral method.

**41742 ****Algebra (II)**

A review of rings and ideals, zerodivisors modules, simple and semisimple modules, small and large submodules projective and nijective modules, regular rings, radical and socle of amodule and Noetherian and Artinian modules.

**41743 ****Finite Fields**

Algebraic foundations: Group theory, ring theory, field theory. Structure of finite fields: Characterization of finite fields, roots of irreducible polynomials, traces, norms, and bases, roots of unity and cyclotomic polynomials, representations of elements of finite fields.

**41732 ****Applied Statistics**

Intensive statistics course with application to the sciences. Topics include continuous and discrete random variables, covariance and correlation, simple, multiple and logistic regression. Sampling methods including simple random sample, stratified, systematic, and cluster samples. Truncated and censored data. A major statistical software package will be utilized.

**41744 Advanced Linear Algebra**

Matrix Algebra (review), transpose and conjugate transpose, submatrices and partitions of a matrix, determinants, Laplace’s theorem, Binet Cauchy Formula, Systems of Linear equations and matrices, the LU decomposition. Linear, Euclidean and Unitary Spaces. Linear transformations and matrices image and kernel of a linear transformation, invertible transformations. Eigenvalues and eigenvectors of linear transformations, the adjoint of a linear transformation and dual spaces.

**41745 ****Coding Theory**

Introduction to error-correcting codes, with a focus on the theoretical and algorithmic aspects arising in the context of the "channel coding" problem: We want to transmit data over a noisy communication channel so that the receiver can recover the correct data despite the adverse effects of the channel.

**41746 ****Matrix Analysis**

Schur's Triangularization Theorem, the Jordan Canonical Form, Spectral Theorems for Hermitian and Normal Matrices, The Courant-Fischer Theorem, Interlacing Eigenvalues Theorem, Singular Value Theorem, Perron-Frobenius Theorem.

**41762 ****Topology (II)**

Metric spaces , Metric topologies, metrization of topological spaces, uniform spaces, topological groups, Function spaces, CoveringSpaces