## Graduate Program in Mathematics - Course Descriptions

Degree Plan | Requirements​  | Course Descriptions

​​MATH 502 General Relativity                                                                                 (3-0-3)

Minkowski space. Tensor analysis on differentiable manifolds. The Einstein field equations. Exact solutions; the Schwarzschild and Reissner-Nordstrom solutions. The three classical tests of general relativity. Energy momentum tensor for perfect fluids and the electromagnetic field. The interior Schwarzschild solution. Black holes and analytic extensions. Robertson-Walker and other cosmological models of the universe. Distance measurements in cosmology.

Prerequisite: MATH 301. (Credit may not be obtained for both MATH 502 and PHYS 575)

MATH 505 Mathematical Theory of Elastodynamics                                           (3-0-3)

An introduction to Cartesian tensors. Stress and strain tensors. Conservation of mass, energy and momentum. Hooke’s law and constitutive equations. Isotropic solids and some exact solutions of elasticity. Elastodynamic equations. Elastic waves in an unbounded medium. Plane waves in a half space. Reflection and refraction at an interface. Surface waves.

Prerequisite: MATH 301 or equivalent

MATH 513 Mathematical Methods for Engineers                                                (3-0-3)

Laplace transforms including the convolution theorem, error and gamma functions. The method of Frobenius for series solutions to differential equations. Fourier series, Fourier-Bessel series and boundary value problems, Sturm-Liouville theory. Partial differential equations: separation of variables and Laplace transforms and Fourier integrals methods. The heat equation. Laplace equation, and wave equation. Eigenvalue problems for matrices, diagonalization.

Prerequisite: Math 202. (Not open to mathematics majors. Students cannot receive credit for both MATH 301 and MATH 513)

MATH 514 Advanced Mathematical Methods                                                      (3-0-3)

Integral transforms: Fourier, Laplace, Hankel and Mellin transforms and their applications. Singular integral equations. Wiener-Hopf techniques. Applications of conformal mapping. Introduction to asymptotic expansion

Prerequisite: MATH 301or MATH 430 or MATH 513

MATH 521 General Topology I                                                                              (3-0-3)

Basic set theory (countable and uncountable sets, Cartesian products). Topological spaces (basis for a topology, product topology, functions, homeomorphisms, standard examples). Connected spaces, path connectedness. Compact spaces, compactness in metrizable  paces. Countability axioms, first countable and second countable spaces. Separation axioms, Urysohn’s Lemma, Urysohn’s metrization theory. Complete metric spaces.

Prerequisite: MATH 421

MATH 523 Algebraic Topology                                                                              (3-0-3)

Concept of categories and functors. Simplicial complexes, subdivision and simplicial approximations. Homotopy, fundamental group and covering spaces. Fundamental group of polyhedron. Chain complexes, homology groups and their topological invariance.

Prerequisite: MATH 421. (MATH 521 is recommended)

MATH 525 Graph Theory                                                                                       (3-0-3)

Review of basic concepts of graph theory. Connectivity, matching, factorization and covering of graphs, embeddings, edge and vertex coloring. Line graphs. Reconstruction  of graphs. Networks and algorithms.

Prerequisite: MATH 425

MATH 527 Differential Geometry                                                                          (3-0-3)

Curves in Euclidean spaces: arclength, tangent, normal and binormal vectors, curvature and torsion. Frenet formulas. Isoperimetric inequality. Differential geometry and local theory of surfaces, the first and second fundamental forms. Local isometries. Geodesics. Gaussian and mean curvature of surfaces. The Gauss- Bonnet theorem. Manifolds and differential forms. Introduction to Riemannian geometry.

Prerequisite: MATH 421

MATH 531 Real Analysis                                                                                         (3-0-3)

Lebesgue measure and outer measure. Measurable functions. The Lebesgue integral. Lebesgue convergence theorem. Differentiation and integration. Lp spaces. Riesz representation theorem. Introduction to Banach and Hilbert spaces.

Prerequisite: MATH 411

MATH 533 Complex Variables I                                                                             (3-0-3)

Analytic functions. Cauchy’s theorem and consequences. Singularities and expansion theorems. Maximum modulus principle. Residue theorem and its application. Compactness and convergence in space of analytic and meromorphic functions. Elementary conformal mappings.

Prerequisites: MATH 411; MATH 430 or consent of the instructor

MATH 535 Functional Analysis I                                                                            (3-0-3)

Normed linear spaces, Banach spaces, Hilbert spaces, Banach Algebras (definitions, examples, geometric properties), bounded linear operators, convex sets, linear functionals, duality, reflexive spaces, weak topology and weak convergence, Banach fixed point theorem, Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, representation of functionals on Hilbert spaces (Riesz Representation Theorem).

Prerequisite: MATH 411

MATH 536 Functional Analysis II                                                                          (3-0-3)

Algebra of bounded operators, self-adjoint operators in Hilbert Spaces, Normal operators, compact operators, projections, spectral theory of linear operators in normed spaces and Hilbert spaces, spectral mapping theorem, Banach-Alaoglu theorem.

Prerequisite: MATH 535

MATH 537 Topological Vector Spaces                                                                   (3-0-3)

Topological vector spaces, locally convex spaces, Krein-Milman theorem, duality in locally convex spaces, separation theorem for compact convex sets, topological tensor products, nuclear mappings and spaces.

Prerequisite: MATH 535

MATH 538 Applied Functional Analysis                                                                (3-0-3)

A quick review of basic properties of topological, metric, Banach and Hilbert spaces. Introduction of Hausdorff metric and iterated function system. Fixed point theorems and their applications. Introduction to infinite dimension calculus – Frechet and Gateaux derivatives, Bochner integral. Introduction to weak and w*-topologies. Algorithmic optimization including complementarity problems and variational inequalities.

Prerequisite: MATH 411

MATH 540 Harmonic Analysis                                                                               (3-0-3)

Fourier series on the circle group (Fourier coefficients, Fourier series of square summable functions, absolutely convergent Fourier series, Fourier coefficients of linear functionals), The convergence of Fourier series, Fourier transforms on L1(R), Fourier transforms on Lp(R), The Payley-Wienner theorems. Fourier analysis on locally compact groups (locally compact groups, the Haar measure, characteristic and the dual group, Fourier transforms, almost periodic
functions and the Bohr compactification).

Prerequisite: MATH 411

MATH 545    Algorithms and Complexity                                                              (3-0-3)

Polynomial time algorithms and intractable problems; relationship between the
classes P, NP, and NP-complete; Cook’s theorem and the basic NP-complete problems. Techniques for proving NP-completeness; NP-hardness. Hierarchy of complexity classes.

Prerequisite: Consent of the Instructor

MATH 550    Linear Algebra                                                                                  (3-0-3)

Basic properties of vector spaces and linear transformations, algebra of polynomials, characteristic values and diagonalizable operators, invariant subspaces and triangulable operators. The primary decomposition theorem, cyclic decompositions and the generalized Cayley-Hamilton theorem. Rational and Jordan forms, inner product spaces, The spectral theorem, bilinear forms, symmetric and skew symmetric bilinear forms.

Prerequisite: MATH 280

MATH 551    Abstract Algebra                                                                               (3-0-3)

Basic definitions of rings and modules, homomorphisms, sums and products, exactness, Hom and tensor, adjoint isomorphism, free, projective and injective
modules. Chain conditions, primary decomposition, Noetherian rings and modules, Artinian rings, structure theorem.

Prerequisite: MATH 345. (MATH 450 is recommended)

MATH 552    Fields and Galois Theory                                                                  (3-0-3)

Field extensions, the fundamental theorem. Splitting fields and algebraic closure, finite fields, separability, cyclic, cyclotomic, and radical extensions. Structure of fields: transcendence bases.

Prerequisite: MATH 345. (MATH 450 is recommended)

MATH553    Homological Algebra                                                                          (3-0-3)

Review of free, projective, and injective modules, direct limits. Watt’s theorems. Flat modules. Localization. Noetherian, semisimple, Von Neumann regular, hereditary, and semi-hereditary rings. Homology, homology functors, derived functors. Ext. and Tor. homological dimensions, Hilbert Syzygy theorem.

Prerequisite: MATH 551

MATH 554    Rings and Categories of Modules                                                    (3-0-3)

Classical ring structure theorems, functors between module categories, equivalence and duality for module categories. Decomposition properties of injective and projective modules. Specific Artinian rings.

Prerequisite: MATH 551

MATH 555    Commutative Algebra                                                                       (3-0-3)

Basics of rings and ideals. Rings of fractions, integral dependence, valuation rings, discrete valuation rings, Dedekind domains, fractional ideals. Topologies and completions, filtrations, graded rings and modules. Dimension theory.

Prerequisite: MATH 551

MATH 560    Applied Regression and Experimental Design                                (3-0-3)

Simple linear regression. Testing of intercept and slope. Multiple linear regression. Estimation parameters and testing of regression coefficients. Prediction and correlation analysis. Analysis of variance technique. Completely randomized and randomized block designs. Latin square design. Incomplete block design. Factorial design, 2k factorial design and blocking and confounding in 2k factorial design.

Prerequisite: STAT 201, STAT 319, or Instructor’s Consent. (Students cannot receive credit for both MATH 560 and STAT 430 or SE 535)

MATH 561    Mathematical Statistics                                                                      (3-0-3)

Axioms and foundations of probability. Conditional probability and Bayes’ theorem. Independence. Random variables and distribution functions and moments. Characteristic functions, Laplace transforms and moment generating functions. Function of random variables. Random vectors and their distributions. Convergence of sequences of random variables. Laws of large numbers and the central limit theorem. Random samples, sample moments and their distributions. Order statistics and their distributions.

Prerequisite: STAT 302 or Consent of the Instructor

MATH 563    Probability Theory                                                                            (3-0-3)

Foundations of probability theory. Measure-theoretic approach to definitions of
probability space, random variables and distribution functions. Modes of convergence and relations between the various modes. Independence, Kolmogorov type inequalities. Tail events and the Kolomogorov 0-1 law. Borel-Cantelli lemma. Convergence of random series and laws of large numbers. Convergence in distribution. Characteristic functions. The central limit theorem. Weak convergence of probability measures. Conditional expectations and martingales.

Prerequisite: STAT 301

MATH 565    Advanced Ordinary Differential Equations I                                 (3-0-3)

Existence, uniqueness and continuity of solutions. Linear systems, solution space, linear systems with constant and periodic coefficients. Phase space, classification of critical points, Poincare’-Bendixson theory. Stability theory of linear and almost linear systems. Stability of periodic solutions. Laypunov’s direct method and applications.

Prerequisite: MATH 465

MATH 566    Fractional Differential Equations                                                     (3-0-3)

Special functions (Gamma, Mittag-Leffler, and Wright), Riemann fractional integral, Riemann-Liouville and Caputo fractional derivatives, composition rules, embeddings, equivalence with integral equations, well-posedness for Cauchy type problems, Successive approximations method, Laplace and Mellin transform methods.

MATH 568    Advanced Partial Differential Equations I                                       (3-0-3)

First order linear and nonlinear equations. Classification of Second order equations. The wave equation, heat equation and Laplace’s equation. Green’s functions, conformal mapping. Separation of variables, Sturm-Liouville theory. Maximum principles and regularity theorems.

Prerequisite: MATH 470

MATH 569    Linear Elliptic Partial Differential Equations                                 (3-0-3)

Sobolev spaces, Mollifiers, Dual spaces, Poincare’s inequality, Lax-Milgram Theorem, linear elliptic problems, Weak formulation, weak derivatives, Weak solutions, Existence uniqueness and regularity, maximum principle.

Prerequisite: MATH 531

MATH 571     Numerical Analysis of Ordinary Differential Equations              (3-0-3)

Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations. One-step, linear multi-step, Runge-Kutta, and extrapolation methods; convergence, stability, error estimates, and practical implementation, Study and analysis of shooting, finite difference and projection methods for boundary value problems for ordinary differential equations.

Prerequisite: MATH 471 or Consent of the Instructor

MATH 572    Numerical Analysis of Partial Differential Equations                    (3-0-3)

Theory and implementation of numerical methods for boundary value problems
in partial differential equations (elliptic, parabolic, and hyperbolic). Finite difference and finite element methods: convergence, stability, and error estimates. Projection methods and fundamentals of variational methods. Ritz-Galerkin and weighted residual methods.

Prerequisite: MATH 471 or Consent of the Instructor

MATH 573    Matrix Computations and Optimization Algorithms                     (3-0-3)

Survey of practical techniques of numerical analysis for engineering and graduate students. Topics include computational and theoretical aspects of direct and iterative methods for linear systems, iterative solutions of nonlinear systems (successive approximations, relaxation, conjugate gradient, and quasi-Newton methods), sparse materials, least-squares problems (both linear and nonlinear), eigenvalue problems, and optimization problems. Problems include case studies in various disciplines.

Prerequisites: MATH 280; MATH 321 or SE 301. (Not Open to Mathematics Majors)

MATH 574     Numerical Methods of Partial Differential Equations                   (3-0-3)

Concepts of consistency, stability, and convergence of numerical schemes. Initial and boundary value problems for ordinary differential equations. Various finite difference and finite element methods and their applications to fundamental partial differential equations in engineering and applied sciences. Case studies selected from computational fluid mechanics, solid mechanics, structural analysis, and plasma dynamics.

Prerequisite: MATH 321, SE 301, or Consent of the Instructor. (Not Open to  mathematics Majors)

MATH 575     Introduction to Approximation Theory                                          (3-0-3)

Best approximation in normed linear spaces: basic concepts. Lagrange and Hermite interpolation. Approximate solution of over-determined system of linear equations. Linear approximation of continuous functions in Chebyshev and least squares norms. Rational approximation. Piecewise polynomial approximation. Cubic and B-splines.

Prerequisite: Consent of the Instructor.

MATH 577     Introduction to Industrial Mathematics                                          (3-0-3)

Why and how industrial mathematics? The description of air bag sensor. How to judge the quality of a non-woven fabric? Damage estimation in a machine (fatigue life time). Mathematics to solve the above mentioned problems.

Prerequisite: MATH 202, MATH 280, or Consent of the Instructor

MATH 578: Applied Numerical Methods II: (3-0-3)

This course introduces finite element, finite difference, and finite volume methods. Applications of these methods to steady-state, diffusion and wave models. Stability and convergence. Homogenization, upscale and multiscale methods. Implementations and computer labs.

MATH 579     Wavelets and Fractals                                                                       (3-0-3)

The continuous wavelet transform, the discrete wavelet transform, advantages of using wavelet transforms over the classical Fourier transform. Applications of wavelets in solution of differential and partial differential equations. Iterated function system and deterministic fractals.

Prerequisite: MATH 202

MATH 580     Convex Analysis                                                                                (3-0-3)

Convex sets and convex functions; epigraphs, level sets. Inf-convolution; continuity and semi-continuity. Separation theorems and the Hahn-Banach theorem. Representation theorems, Caratheodory theorem. Polyhedra. Farkas lemma. Fenchel’s theorem. Applications to linear systems. The weak duality theorem. Convex systems. Differentiability. Subdifferentials and subgradients, generalized gradients. Inf-compactness. Applications to Math programming and control theory. Cones of tangent. Constraint qualifications and optimality conditions for non-smooth minimization problems.

Prerequisite: MATH 411, or Consent of the Instructor

MATH 581     Advanced Linear Programming                                                      (3-0-3)

A rigorous and self-contained development of the theory and main algorithms of linear programming. Formulation of linear programs. Theory of linear programming (linear inequalities, convex polyhedral duality). Main LP algorithms (simplex, revised simplex, dual, and ellipsoidal algorithms). Geometry and theory of the simplex, dual, and ellipsoidal algorithms. Geometry and theory of the simplex method. Sensitivity analysis. Related topics (games, integer programming, parametric programming, stochastic programming). Representative applications in Economics, Engineering, Operations Research, and Mathematics. Familiarity with computer implementation of LP methods will be acquired by working on individual (or small group) projects of applying LP to student’s chosen areas.

Prerequisite: MATH 321, MATH 573, or Consent of the Instructor. (Credit cannot be given to both MATH 581 and SE 503)

MATH 582     Nonlinear Programming                                                                  (3-0-3)

An advanced introduction to theory of nonlinear programming, with emphasis on convex programs. First and second order optimality conditions, constraint qualifications, Lagrangian convexity and duality. Penalty function methods. Theory and algorithms of main computational methods of nonlinear programming. Representative applications of nonlinear programming in Economics, Operations Research and Mathematics.

Prerequisite: MATH 412

MATH 585: Computational Inverse Problem (3-0-3)

This course introduces students to fundamental concepts in linear and nonlinear inverse problems. Emphasis is placed on describing how to integrate various information sources from measured data and prior knowledge about the inverted model. Subjects studied will include topics and tools such as: Regression, Least squares, Maximum likelihood estimation, Rank deﬁciency, Ill-conditioning, Generalized and Truncated SVD solutions, regularizations (Tikohonov, spectral filtering), proximal and primal-dual iterative schemes, Nonlinear inverse (gradient-based and global optimization methods), OCCAM method. Computer lab sessions will be organized to combine classroom learning with hands-on applications.

MATH 586     Design and Analysis of Experiment                                                (3-0-3)

Concepts of statistical designs and linear models. Basic designs: Completely randomized design. Randomized block design. Latin square designs (computer aided selection) models: Fixed, random and mixed models, estimation of parameter using Gauss-Markov theorem. Expectation of mean squares with and without use of matrix theory. Incomplete block designs. Factorial experiment, 2p confounding, fractional replicate and orthogonal designs. 3p confounding, fractional replicate and orthogonal designs. P q N confounding; fractional replicate and orthogonal designs. Tagouchi method as applied to design of experiments for engineering, industrial and agricultural data analysis. Extensive use of computer packages and computer aided designs.

Prerequisites: Graduate Standing, Consent of the Instructor

MATH 587     Advanced Applied Regression                                                         (3-0-3)

Least square method and properties. Simple and multiple linear regression with
matrix approach. Development of liner models. Residual analysis. Polynomial models. Use of dummy variables in multiple linear regression. Analysis of variance approach. Selection of ‘best’ regression equation. Concepts of mathematical model building. Non-linear regression and estimation. Extensive use of computer packages.

Prerequisites: Graduate Standing, Consent of the Instructor

MATH 590     Special Topics in Mathematics (Variable Credit 1-3) Variable Contents.

MATH 591     Introduction to the Mathematical Literature                                 (0-1-0)

Research and expository survey journals in mathematical sciences, review journals, citation journals, journal abbreviations and literature citations. Classification  of mathematical subjects. Library search: books, bound journals, current periodicals, microfilms. Searching for publications on a specific subject or by a certain author. Structure and organization of a research paper in mathematics. Methods of dissemination of mathematical results: abstracts, conferences, research papers, books and monographs. Major mathematical societies and publishers and their publication programs. The course will consist of one lecture a week and «workshop» sessions at the KFUPM Library supervised by the instructor.

MATH 595     Reading and Research I (Variable Credit 1-3) Variable Contents

MATH 596     Reading and Research II (Variable Credit 1-3) Variable Contents

MATH 599     Seminar                                                                                              (1-0-0)

MATH 601 Stochastic Differential Equations and Applications                          (3-0-3)

Probability spaces, characteristic functions, stochastic processes, martingales, Markov Chains, Brownian motion,  Itô calculus, Itô formula, stochastic differential equations, applications of stochastic differential equations.

Prerequisite: Math 531 or Instructor’s Consent

MATH 602 Topics in Fluid Dynamics                                                                    (3-0-3)

Kinematics and dynamics. Potential flow. Navier-Stokes equations. Some exact solutions. Laminar boundary layers. Stokes and Oseen flows. Sound waves.  Topics in gas dynamics. Surface waves. Flow in porous media. Darcy’s law and equation of diffusivity.

Prerequisite: MATH 505 or equivalent

MATH 605 Asymptotic Expansions and Perturbation Methods                          (3-0-3)

Asymptotic sequences and series. Asymptotic expansions of integrals. Solutions of differential equations at regular and irregular singular points. Nonlinear differential equations. Perturbation methods. Regular and singular perturbations. Matched asymptotic expansions and boundary layer theory. Multiple scales. WKB theory.

Prerequisites: MATH 430; MATH 301 or MATH 513

MATH 607 Inverse and Ill-Posed Problems                                                           (3-0-3)

Mathematical and numerical analysis of linear inverse and/or ill-posed problems for partial differential, integral and operator equations. Tikhonov regularization. Constraints and a priori bounds. Methodologies for achieving «optimal» compromise between accuracy and stability. Applications to practical problems in remote sensing, profile inversion, geophysics, inverse scattering and tomography.

Prerequisite: MATH 513, MATH 573, or Consent of the Instructor.

MATH 610 MSC Thesis                                                                                           (0-0-6)

MATH 611 Hilbert Space Methods in Applied Mathematics I                             (3-0-3)

Review of normed and product spaces. Theory of distributions, weak solution.
Complete orthonormal sets and generalized Fourier expansions. Green’s functions and boundary-value problems, modified Green’s functions. Operator theory, invertibility, adjoint operators, solvability conditions. Fredholm alternative. Spectrum of an operator. Extremal principles for eigenvalues and perturbation of eigenvalue problems. Applications.

Prerequisite: MATH 535

MATH 612 Hilbert Space Methods in Applied Mathematics II                           (3-0-3)

Integral equations; Fredholm integral equation, spectrum of a self-adjoint compact operator, inhomogeneous equation. Variational principles and related approximation methods. Spectral theory of second-order differential operator, Weyl’s classification of singular problems. Continuous spectrum. Applications. Introduction to nonlinear problems. Perturbation theory. Techniques for nonlinear problems.

Prerequisite: MATH 611

MATH 619: Project for either MX (Computational Analytics) or MX (Data Science)

A graduate student will arrange with a faculty member to conduct an industrial research project related to the Data Science field. Subsequently the students shall acquire skills and gain experiences in developing and running actual industry-based project. This project culminates in the writing of a technical report, and an oral technical presentation in front of a board of professors and industry experts.

MATH 621 General Topology II                                                                             (3-0-3)

The Tychonoff theorem, one-point compactification, the Stone-Cech compactification. Paracompactness, Lindelof spaces, Stone’s theorem. Metrizability, the Nagata-Smirnov metrization theorem. Homotopy paths, fundamental group, simply-connected spaces, retracts and deformation retracts; the fundamental groups of the circle, the punctured plane and the n-sphere; Van Kampen’s theorem.

Prerequisite: MATH 521

MATH 627 Differentiable Manifolds and Global Analysis                                  (3-0-3)

Calculus on manifolds. Differentiable manifolds, mappings, and embeddings.
Implicit functions theorem, exterior differential forms, and affine connections. Tangent bundles. Stoke’s theorem. Critical points. Sard’s theorem. Whitney’s embedding theorem. Introduction to Lie groups and Lie algebras. Applications.

Prerequisite: MATH 527

MATH 631 Advanced Topics in Real and Abstract Analysis                               (3-0-3)

Topics to be chosen from Measure and Integration, Measurable Selections, Locally Convex Spaces, Topological Groups, Harmonic Analysis, Banach Algebras.

Prerequisite: MATH 531

MATH 633 Complex Variables II                                                                           (3-0-3)

Harmonic functions. The Riemann mapping theorem. Conformal mappings for multi-connected domains. Elliptic functions and Picard’s theorem. Analytic continuation. Entire functions. Range of an analytic function. Topics in univalent functions and geometric function theory.

Prerequisite: MATH 533

MATH 637 Non-linear Functional Analysis and Applications                             (3-0-3)

Fixed points methods. Nonexpansive mappings. Differential and integral calculus in Banach spaces. Implicit and inverse function theorems. Potential operators and variational methods for linear and nonlinear operator equations. Extrema of functionals. Monotone operators and monotonicity methods for nonlinear operator equations. Applications to differential and integral equations and physical problems.

Prerequisite: MATH 535

MATH 640 Calculus of Variations                                                                          (3-0-3)

Gateaux and Fréchet differentials. Classical calculus of variations. Necessary conditions. Sufficient conditions for extrema. Jacobi and Legendre conditions. Natural boundary conditions. Broken extrema, Erdmann-Weierstrass condition. Multiple integral problems. Constrained extrema. Hamilton principle with applications to mechanics and theory of small oscillations. Problems of optimal control. Direct methods including the Galerkin and the Ritz-Kantorovich methods. Variational methods for eigenvalue problems.

Prerequisite: MATH 411, or Consent of the Instructor

MATH 641 Topics in Calculus of Variations                                                         (3-0-3)

Selected topics from the following: Variational inequalities, weak lower semicontinuity and extremal problems in abstract spaces, theory of optimal control, stochastic control, distributed parameter systems, optimization problems over infinite horizons, algorithmic and penalty methods in optimization.

Prerequisite: MATH 640

MATH 642 Control and Stability of Linear Systems                                            (3-0-3)

Review of systems of linear differential equations to include existence and uniqueness, contraction mappings, fixed points, transition matrix, matrix exponentials, the Laplace transform and stability. Linear control systems. Controllability, observability and duality. Weighting patterns and minimal realizations. Feedback. Linear regulator problem and matrix Riccati equations. Fixed-end point problems. Minimum cost and final-value problems in control theory. Stability of linear systems. Uniform stability. Exponential stability.

Prerequisites: MATH 465; MATH 460 or MATH 550

MATH 645 Combinatorics and Graph Theory                                                     (3-0-3)

Enumerative analysis, generating functions. Sorting and searching. Theory of codes. Block design. Computational combinatorics. Methods of transforming combinatorial ideas into efficient algorithms. Algorithms on graphs, network flow.

Prerequisite: MATH 425

MATH 651 Universal Algebra                                                                                 (3-0-3)

Lattices: basic properties, distributive and modular lattices, complete lattices, equivalence relations and algebraic lattices; Algebras: definition and examples, isomorphisms, subalgebras congruences and quotient algebras, homomorphism theorems, direct products, subdirect products, simple algebras, class operators and varieties, terms and term algebras, free algebras, Birkhoff’s theorem, equational logic, Boolean algebras: Boolean algebras and Boolean rings, filters and ideals, Stone duality, connections with model theory: First-order languages and structures, reduced products and ultraproducts.

Prerequisite: MATH 551

MATH 652 Advanced Topics in Group Theory                                                    (3-0-3)

Advanced theory of solvable and nilpotent groups. General free groups. Krull- Schmidt theorem. Extensions. The general linear group. Group rings and group algebras. Representation theory of groups.

Prerequisite: MATH 450. (MATH 551 is recommended)

MATH 653 Advanced Topics in Commutative Algebra                                       (3-0-3)

Selected topics from: prime spectra and dimension theory; class groups; ideal systems and star operations; multiplicative ideal theory; generator Property; homological aspects of commutative rings; pullbacks of commutative rings.

Prerequisite: MATH 555. (MATH 552 and MATH 553 are recommended)

MATH 654 Advanced Topics in Algebra                                                               (3-0-3)

Selected topics from: groups, rings, modules, and general algebraic systems.

Prerequisites: Graduate Standing, Consent of the Instructor

MATH 655 Applied & Computational Algebra                                                     (3-0-3)

Contents vary. Concepts and methods in algebra which have wide applications in mathematics as well as in computer science, systems theory, information theory, physical sciences, and other areas. Topics may be chosen from fields of advanced matrix theory; algebraic coding theory; group theory; Grِbner bases; or other topics of computational and applied algebra.

Prerequisites: Graduate Standing, Consent of the Instructor

MATH 661 Mathematical Statistics                                                                         (3-0-3)

Theory of point estimation, Properties of estimators. Unbiased estimation and lower bounds for the variance of an estimator. Methods of moments and maximum likelihood. Bayes’ and minimax estimation. Minimal sufficient statistics. Neymann-Pearson theory of testing of hypotheses. Unbiased and invariant tests. Confidence estimation. Confidence intervals (shortest length, unbiased and Bayes’). The general linear hypothesis and regression. Analysis of variance. Nonparametric statistical inference.

Prerequisite: MATH 561

Measurable functions and integration. Radon-Nikodym theorem. Probability space. Random vectors and their distributions. Independent and conditional probabilities. Expectation. Strong laws of large numbers. The weak compactness theorem. Basic concepts of martingales. Invariance principles. The Law of the Iterated Logarithm. Stable distributions and infinitely divisible distributions.

Prerequisites: MATH 531, MATH 563

MATH 665 Advanced Ordinary Differential Equations II                                  (3-0-3)

Self-adjoint boundary-value problems, Sturm-Liouville theory. Oscillation and comparison theorems. Asymptotic behavior of solutions. Singular Sturm-Liouville problems and non self-adjoint problems. Hypergeometric functions and related special functions. Bifurcation phenomena.

Prerequisite: MATH 565

MATH 667 Advanced Partial Differential Equations II                                       (3-0-3)

Classification of first order systems. Hyperbolic systems, method of characteristics. Applications to gas dynamics. Dispersive waves; application to water waves. Potential theory, single and double layers, existence theory for Dirichlet and Neumann problems.

Prerequisite: MATH 568

MATH 668    Evolution Equations                                                                          (3-0-3)

Maximum Monotone Operators, Bounded and unbounded operators, Pseudo monotone operators, Self-adjoint, Evolution Equations in Hilbert and Banach spaces,  Hille-Yosida Theorem, application to linear heat and wave Equations, Nonlinear Evolution equations, The Galerkin Method

Prerequisite: MATH 569

MATH 669 Integral Equations                                                                                (3-0-3)

Review of the Fredholm and Hilbert-Schmidt theories for Fredholm integral equations of the second kind. Kernels with weak and logarithmic singularities. Singular integral equations of the first and second kind (Abel, Carleman, and Wiener-Hopf equations). Nonlinear integral equations (Volterra and Hammerstein equations). Application of the Schauder fixed point theorem. Nonlinear eigenvalue problems and integral equation methods for nonlinear boundary-value problems. Nonlinear singular integral equations. Applications to engineering and physics (the nonlinear oscillator, the airfoil equation, nonlinear integral equations arising the radiation transfer, hydrodynamics, water waves, heat conduction, elasticity, and communication theory).

Prerequisite: MATH 535

MATH 673 Numerical Solution of Integral Equations                             (3-0-3)

Numerical methods and approximate solutions of Fredholm integral equations of the second kind (both linear and nonlinear). Approximation of integral operators and quadrature methods. Nystrom method. Method of degenerate kernels. Collectively compact operator approximations. Numerical methods for Volterra integral equations. Methods of collocation, Galerkin, moments, and spline approximations for integral equations. Iterative methods for linear and nonlinear integral equations. Eigenvalue problems.

Prerequisite: MATH 471 or Consent of the Instructor

MATH 674 Numerical Functional Analysis                                                           (3-0-3)

Theoretical topics in numerical analysis based on functional analysis methods. Operator approximation theory. Iterative and projection methods for linear and nonlinear operator equations. Methods of steepest descent, conjugate gradient, averaged successive approximations, and splittings. Stability and convergence. Abstract variational methods and theoretical aspects of spline and finite element analysis. Minimization of functionals. Vector space methods of optimization. Newton and quasi-Newton methods for operator equations and minimization.

Prerequisite: MATH 535 or MATH 611

MATH 680 Dynamic Programming                                                                        (3-0-3)

Development of the dynamic programming algorithm. Optimality principle and characterizations of optimal policies based on dynamic programming. Shortest route problems and maximum flow problems. Adaptive process. One-dimensional allocation processes. Reduction of dimensionality. Additional topics include imperfect state information models, the relation of dynamic programming to the calculus of variations, and network programming. Computational experience will be acquired by working on individual projects of applying dynamic programming to case study problems.

Prerequisite: MATH 640

MATH 681 Topics in Mathematical Programming                                               (3-0-3)

Contents vary. Topics selected from: Nonconvex optimization, geometric programming, Lagrangian algorithms, sensitivity analysis, large-scale programming, nonsmooth optimization problems and optimality conditions in infinite-dimensional spaces, combinatorial optimization, computation of fixed points, complementarity problems, multiple-criteria optimization, and semi-infinite programming.

Prerequisite: MATH 582, or Consent of the Instructor

MATH 690 Special Topics in Mathematics (Variable Credit 1-3) Variable Contents

MATH 695 Reading and Research I (Variable Credit 1-3) Variable Contents