MAT 401 Computer Programming For Data Management

Definition of database, structure of databases, Microsoft Access program.

MAT 402 Advanced Computer Programming

An Introduction to Visual Basic, Working with Forms, Controls, Events, Sub Procedures, Sub Functions, Variables in Visual Basic and Properties, File Processing, Database and Visual Basic.

MAT 403 System for Doing Mathematics by Computer

Running Mathematica, Numerical Calculations, Building up Calculations, Using Mathematica System, Algebraic Calculations, Symbolic Mathematics, Numerical Mathematics, Functions and Programs, Lists, Graphics and Sound, Files and External Operations.

MAT 408 Perturbation techniques

Gauge functions, order symbols, asymptotic series, convergency of asymptotic series.  Root finding for quadratic, cubic, higher-order equations and transcendental equations. Expansion of integrands, integration by parts, Laplace’s method. Perturbed second order differential equations, dimensional analysis, the straightforward approximation, the Lindstedt-Poincare technique, the method of renormalization,   the method of multiple scales, the method of averaging.

MAT 409 Operational Calculus

Fourier series, Fourier transform and inverse Fourier transform. The Laplace transform. The inversion integral for the Laplace transform. Applications of Laplace transform to linear ordinary, partial and integral equations. The Z-transform. The inversion integral for the Z-transform. Applications of Z-transform to difference equations and linear networks.

MAT 410 Modules and Rings

Categories, functors, product and coproduct. Rings, ideals and subrings. Modules, submodules and factor modules. Homomorphism of modules and rings. Generators and cogenerators. Jordan-Hölder-Schreirer theorem. Endomorphism ring of a module. Dual modules.  Direct sums and direct product of modules. Free modules. Pushout and pullback. Split exact sequences. Projective modules. Injective modules. Injective hulls and projective covers. Baer’s criterion. Length of a module. Artinian and Noetherian modules. Artinian and Noetherian rings. Local rings. Local endomorphism rings. Krull-Remak-Schmidt theorem. Simple and semisimple modules. Simple and semisimple rings. The density theorem. Radicals of modules and rings. The socle of modules. Finitely generated modules over principal ideal domains. Rings of linear transformations of vector spaces. Von Neumann regular rings. The tensor product. Flat Modules. Semiperfect modules. Perfect rings.

MAT 411 Numerical Solution of Ordinary Differential Equations

Theory of ordinary differential equations, One-step methods, Euler’s method, Higher order Taylor methods, Runge-Kutta methods, truncation errors of one-step methods, convergence of one-step methods, Multistep methods, methods based on numerical integration, explicit methods, implicit methods, predictor and corrector methods, extrapolation method, Error analysis, Stiff differential equations, stability and convergence, boundary value problems, shooting method, finite difference methods for linear and nonlinear problems, global errors.

MAT 412 Elementary Topology and Geometry

Introduction to Algebraic Topology and Geometry. Point set topology, Naive set theory, topological spaces, connected and compact spaces. Homotopy Theory, Homotopy , fundamental group, covering spaces, homotopy of maps, Homology Theory, Homology groups, Homological algebra, The Alternating Algebra, de Rham Cohomology, Chain complexes and their cohomology, The Mayer-Vietrois Sequence, Applications of de Rham cohomology, Brouwer’s fixed point theorem, Jordan-Brouwer separation theorem, Smooth manifolds, differential forms on smooth manifolds.

MAT 413 Applied Mathematics I

Main equations and systems of applied mathematics. Notions of well-posed and ill-posed problems. Solutions of problems by Fourier series expansion. Solution of the problems by the Laplace and Fourier transform techniques. Initial boundary value problems for hyperbolic systems and their solving by the method of characteristics.

MAT 414 Applied Mathematics II

Multi-dimensional integral equations with a continuous and polar kernels. Iterated kernels. Resolvents. Volterra integral equation of the first and second kinds. Fredholm’s theorems for   multi-dimensional integral equations with a continuous and polar  kernels. Multi-dimensional integral equations with an Hermitian continuous and polar  kernels. Statement of the problems of the calculus of variations. Fundamental lemma of the calculus of variations. Euler’s equations.. Examples of the problems of the calculus of variations.

MAT 416 Riemannian Geometry

Vector fields, Covector fields and Mappings, Symmetric Forms and Skew-symmetric forms, Immersion, Bilinear forms and Riemannian metric, Symmetric, positive definite bilinear forms, Riemannian manifolds as metric spaces, The arclength for the Riemannian metric in local coordinates, Riemannian curvature tensor.

MAT 417 Infinite Abelian Groups

Abelian groups. Maps and diagrams. Commuative diagrams. The most important types of groups. Categories of abelian groups. Functorial subgroups and quotient groups. Topologies in groups. Direct sums and products. Direct summands. Pullback and pushout diagrams. Direct limits. Inverse limits. Completeness and completions. Direct sums of cyclic groups. Free abelian groups. Finitely generated groups. Linear independence and rank. Direct sums of cyclic p-groups. Subgroups of direct sums of cyclic p-groups. Countable free groups. Divisible groups. Injective groups. Systems of equations. The structure of divisible groups. The divisible hull. Finitely cogenerated groups. Pure subgroups. Bounded pure subgroups. Quotient groups modulo pure subgroups. Basic subgroups. p-basic subgroups. Basic subgroups of p-groups. The Ulm sequence.

MAT 418 Special Topics in Abelian groups

Essential subgroups of abelian groups, uniform groups, small Subgroups, complements, supplements, pure-exact sequences, pure subgroups

MAT 419 Asymptotic Analysis

The basic of asymptotics: order relations, asymptotic equivalence, asymptotic sequences and asymptotic series, elementary operations on asymptotic series. The Laplace method for integrals: a general case, maximum at the boundary, asymptotic expansions, multiple integrals. The saddle point method:  steepest descent,  range of a saddle point, small perturbations. Applications of the saddle point method: a modified gamma function, the entire function. Differential equations: a Riccati equation, an unstable case, application to a linear second order equation, oscillatory cases. Linear equations with variable coefficients: first order scalar equations, second order equations, solutions near regular singular points, singularity at infinity, solutions near an irregular singular point.

MAT 420 Project

Will be announced.

MAT 424 Numerical Solution of Partial Differential Equations

Finite difference approximations to derivatives, notations for functions of several variables, an explicit finite difference approximation to parabolic equations, Crank-Nicolson implicit method, solution of the implicit equations by Gauss elimination method, the stability of the elimination method, derivative boundary conditions, miscellaneous methods for improving accuracy, convergence, local truncation errors and consistency, stability, stability analysis by the matrix method, the eigenvalues of a common tridiagonal matrix, stability criteria for derivative boundary conditions, von neumann’s method, lax’s equivalence theorem,finite difference methods to hyperbolic equations.

MAT 427 Commutative Algebra I

Commutative rings and Subrings, ideals, prime ideals and maximal ideals, primary decomposition, rings of fractions.

MAT 428 Commutative Algebra II

Modules, chain conditions on modules, commutative noetherian rings, more module theory.     

MAT 429 Theory of Manifolds

Euclidean n-Space, Linear n-Space Affine n-Space, Differentiable functions and mappings on linear n-Space, The tensor product of vector spaces, The tensor product of linear transformations, Symmetry and Alternation, Differentiable varieties in , The implicit functions theorems. Construction of differentiable manifolds by identification, Germs of differentiable functions, the Tangent and Cotangent spaces at a point, Differentiable mappings and their induced linear transformations.

MAT 430 Difference equations

The difference calculus, linear difference equations, stability theory, asymptotic methods, the self-adjoint second order linear equation, Sturm-liouville problem, discrete calculus of variations, boundary value problems for nonlinear equations.

MAT 431 General Topolgy

Topolgical spaces, Open sets and closed sets. Closure, Interior, boundary, limit points and isolated points of a set, Dense sets, Nowheredense sets, Neighborhoods, Basis and subbases for a topology, Coarser and Finer topologies, Subspaces and Relative topologies, Continuous functions, Homeomorphisms, The weak topology generated by a collection maps, Metric spaces, Sequences and Failure of Sequence, Nets and Filters, Product topology, Quotient topology, The Countability Axioms, Lindelöf space, seperability, Seperation Axioms: T₀, T₁, T₂, T₃ and T₄ spaces, Regular, completely regular, normal, completely normal spaces, Compactness, Covers and Compact sets, Connectedness, pathwise connectedness, local connectedness and locally pathwise connected, Totally disconnectedness.Topology of cell structures.

MAT 432 Boundary value problems and integral equations

Linear ordinary differential equations. Qualitative analysis of the difference between Cauchy problem and Initial boundary value problem.

Ordinary differential equation and its self adjoint  form. Solvability conditions of non-homogeneous initial boundary value problem: Fredholm’s theorems. Green’s functions. The construction of the Green’s functions according to Fredholm’s theorems. Solving  boundary value problems by Green’s functions. Linear Fredholm integral equations with degenerate and continuous kernels. Fredholm theorems for integral equations. Linear and non-linear Volterra integral equations. Method of successive approximations. Existence and uniqueness theorems. Applications.

MAT 433 Nonlinear Differential Equations and Dynamical Systems

Existence and Uniquness Theorems, Lineer and Nonlineer Systems, Periodic Solutions, Stability, Stability by Linearization and by direct methods, Bifurcation, Applications.

MAT 434 Nonlinear Partial Differential Equations

Nonlinear partial differential equations: basic concepts and definitions. Nonlinear model equations and variational principles: some nonlinear model equations, variational principles and Euler-Lagrange equations, the variational principle for nonlinear Klein-Gordon equations and for nonlinear water waves. Nonlinear first order partial differential equations: generalized method of characteristics, characteristic directions, characteristic strip. Complete and singular solutions: special procedures for finding solutions, transformations, Charpit’s method, examples of aplication to nonlinear(geometrical) optics and to analytical dynamics.  Cauchy problem: Cauchy problem in two and n independent variables, examples of aplication to nonlinear (geometrical)  optics and analytical dynamics. Principles of higher order equations: the Cauchy problem, normal form, power series and the Cauchy-Kowalewski theorem.

MAT 435 Mathematical Methods in Computer Aided Geometric Design

A symbolic language   software (Maple), expressions, operations, plots and programming. Vector spaces, affine spaces and affine maps. Bezier curves, de Casteljau algorithm, subdivision, derivatives, blossoms and degree elevation techniques. Interpolation techniques Lagrange and divided difference form, Aitken’s algorithm, Hermite form, and least square. Spline curves, de Boor algorithm, B-splines and cubic B-splines.  Bezier surfaces, tensor and triangular patches.

MAT 438 Principles of Economics

Introductio to Economics. Demand and Supply.Production, Cost, and Supply. Monopoly, Oligopoly and Imperfect Competition. Introduction to Macroeconomics. The Financial System. Money and Prices.

MAT 439 History and Philosophy of Mathematics I

What is mathematics? The epistemology of mathematics. The mathematical meaning. Historical aspects and development of mathematics. Logicism. Set theory. Intutionism. Formalism. Finitism. Historical examples to all topics.

MAT 440 History and Philosophy of Mathematics II

The philosophy of mathematics education. Historical and philosophical contributions in mathematics teaching. Mathematics as a language. Semiotical analysis of mathematical language. Discussion of basic articles in the philosophy of mathematics. The incompleteness theorems. The concept of infinity. Historical examples to all topics.