Catalog Course Inventory

601. Higher Mathematics for Engineers and Physicists. (4-0). Credit 4.

602. Higher Mathematics for Engineers and Physicists. (4-0). Credit 4.

603. Operator Theory and Partial Differential Equations. (4-0). Credit 4.

605. Mathematical Fluid Dynamics. (3-0). Credit 3.

606. Theory of Probability I. (3-0). Credit 3.

607. Real Variables I. (3-0). Credit 3.

608. Real Variables II. (3-0). Credit 3.

609. Numerical Analysis. (3-3). Credit 4.

610. Numerical Methods in Partial Differential Equations. (3-3). Credit 4.

611. Ordinary Diffential Equations. (3-0). Credit 3.

612. Partial Diffential Equations. (3-0). Credit 3.

613. Graph Theory. (3-0). Credit 3.

614. Dynamical Systems and Chaos. (3-0). Credit 3.

617. Theory of Functions of a Complex Variable I. (3-0). Credit 3.

618. Theory of Functions of a Complex Variable II. (3-0). Credit 3.

621. Mathematical Logic. (3-0). Credit 3.

622. Differential Geometry of Curves. (3-0). Credit 3.

623. Riemannian Geometry. (3-0). Credit 3.

626. Theory of Probability II. (3-0). Credit 3.

627. Theory of Numbers. (3-0). Credit 3.

631. Ring Theory. (3-0). Credit 3.

633. Group Theory. (3-0). Credit 3.

636. Topology I. (3-0). Credit 3.

637. Topology II. (3-0). Credit 3.

639. Iterative Techniques. (3-3). Credit 4.

640. Linear Algebra for Applications. (4-0). Credit 4.

641. Analysis for Applications I. (3-0). Credit 3.

642. Analysis for Applications II. (3-0). Credit 3.

643. Algebraic Topology I. (3-0). Credit 3.

644. Algebraic Topology II. (3-0). Credit 3.

651. Optimization I. (3-0). Credit 3.

652. Optimization II. (3-0). Credit 3.

653. Algebra I. (3-0). Credit 3.

654. Algebra II. (3-0). Credit 3.

655. Functional Analysis I. (3-0). Credit 3.

656. Functional Analysis II. (3-0). Credit 3.

657. Spline Approximation I. (3-0). Credit 3.

658. Spline Approximation II. (3-0). Credit 3.

660. Computational Linear Algebra. (3-0). Credit 3.

661. Finite Element Methods. (3-0). Credit 3.

662. Seminar in Algebra. (3-0). Credit 3.

663. Seminar in Analysis. (3-0). Credit 3.

664. Seminar in Applied Mathematics. (3-0). Credit 3.

665. Seminar in Topology. (3-0). Credit 3.

666. Seminar in Geometry. (3-0). Credit 3.

667. Approximation Theory. (3-0). Credit 3.

668. Wavelet Analysis. (3-0). Credit 3.

670. Applied Mathematics I. (3-0). Credit 3.

671. Applied Mathematics II. (3-0). Credit 3.

673. Information, Secrecy, and Authentication I. (3-0). Credit 3.

674. Information, Secrecy, and Authentication II. (3-0). Credit 3.

685. Problems. Credit 1 to 6 each semester.

689. Special Topics in... Credit 1 to 4.

691. Research. Credit 1 or more each semester.

695. Frontiers in Mathematical Research. (3-0). Credit 3.

697. Seminar in the Teaching of Calculus. (1-0). Credit 1.

Course Descriptions

Methods of linear algebra, vector analysis and complex variables. Prerequisite: MATH 308 or equivalent.

602. Higher Mathematics for Engineers and Physicists.

Classification of linear partial differential equations; special functions, Sturm-Liouville theory, application to boundary value problems' introduction to Green's functions, finite Fourier integrals. Prerequisites: MATH 601 or 308, 405, and 407.

603.Operator Theory and Partial Differential Equations.

Theory of operators in partial differential equations and boundary value problems: Laplace and Fourier transforms, adjoint operator, self adjoint and differential operators. Prerequisites: MATH 601 or 308, 312, and 407.

605.Mathematical Fluid Dynamics.

Derivation of basic equations of motion; Navier-Stokes equations; potential equations; some exact solutions in two and three dimensions; equations of boundary layer theory; vorticity-stream function formulation and vortex dynamics; introduction to hydrodynamic stability; introduction to equations of turbulence. Prerequisites: MATH 411 and 447 or approval of instructor.

606. Theory of Probability I.

Measure and integration, convergence concepts, random variables, independence and conditional expectation, laws of large numbers, central limit theorems, applications. Prerequisites: MATH 411 and 447 or approval of instructor.

607. Real Variables I.

Lebesgue measure and integration theory, differentiation, Lp-spaces, abstract integration, signed measures; Radon-Nikodym theorem, Riesz representation theorem, integration on product spaces. Prerequisite: MATH 447 or equivalent.

608. Real Variables II.

Banach spaces, theorems of Hahn-Banach and Banach-Steinhaus, the closed graph and open mapping theorems, Hilbert spaces, topological vector spaces and weak topologies. Prerequisite: MATH 607.

609. Numerical Analysis.

Interpolation, numerical evaluation of definite integrals and solution of ordinary differential equations; stability and convergence of methods and error estimates. Prerequisite: Knowledge of computer programming (C or FORTRAN.)

610. Numerical Methods in Partial Differential Equations.

Introduction to finite difference and finite element methods for solving partial differential equations; stability and convergence of methods and error bounds. Prerequisite: MATH 417 or 609 or their equivalent.

611. Ordinary Differential Equations.

General methods for first order equations, singular solutions, applications, special methods, linear equations of second order, method of succesive approximations, systems of ordinary equations. Prerequisite: MATH 601 or equivalent.

612. Partial Differential Equations.

General solution of first order equations, second order equations from physics and mechanics. Prerequisite: MATH 611 or equivalent.

One or more broad areas of graph theory or network theory, such as planarity, connectivity, Hamiltonian graphs, colorings of graphs, automorphisms of graphs, or network theory. Prerequisite: MATH 431 or equivalent, or approval of instructor.

614. Dynamical Systems and Chaos.

Discrete maps; continuous flows; dynamical systems; Poincare maps; symbolic dynamics; chaos, strange attractors; fractals; computer simulation of dynamical systems. Prerequisites: MATH 308 and MATH 601 or equivalent.

Holomorphic functions, complex integral theorems, Runge's theorem, residue theorem, Laurent series, conformal mapping, harmonic functions. Prerequisite: MATH 410.

Infinte products, Weierstrass factorization theorem, Mittag-Leffler's theorem, normal families, Riemann mapping theorem, analytic continuation, Picard's theorems and selected topics. Prerequisite: MATH 617.

Axiomatic formal theories and their models; model theory in propositional logic; model logic and its philosophical bases; metatheorems and the Lowenhiem-Skolem theorem. Prerequisites: PHIL 341 or approval of instructor. Cross-listed with PHIL 642.

Local and global theory of parameterized curves; regular surgaces, local coordinates, first fundamental form, orientation, area; Gauss map, second fundamental form; topics chosen from special surfaces, intrinsic geometry of surfaces, global differential geometry of curves and surfaces. Prerequisites: MATH 311 or equivalent; approval of instructor.

Smooth manifolds and mappings; tensors; curvature equations, geodesics, completeness; special manifolds and constructions. Prerequisites: MATH 311 or equivalent; approval of instructor.

Topics chosen from weak convergence of probability measures, Brownian motion and invariance principals, Gaussian processes, empirical processes, martingales, Martin processes. Prerequisite: MATH 606 or approval of instructor.

Quadratic residues; the Legendre, Jacobi and Kroneckor symbols; quadratic reciprocity; residue characters; character sums; sums of squares; diophantine equations. Prerequisite: Approval of instructor.

Rings and ideals, chain conditions, radicals, simplicity and semisimplicity, modules, homology. Prerequisite: MATH 653 or approval of instructor.

Abelian groups, Sylow theorems, group actions, Jordan-Holder theorem, solvable and nilpotent groups, additional topics. Prerequisite: MATH 653 or approval of instructor.

Set theory, topological spaces, generalized convergence, compactness, metrization, connectedness, uniform spaces, function spaces. Prerequisite: MATH 436 or approval of instructor.

Continuation of MATH 636. Prerequisite: MATH 636 or approval of instructor.

Numerical methods for solving linear and nonlinear equations and systems of equations; eigenvalue problems. Prerequisites: Elementary linear algebra and knowledge of computer programming (C or FORTRAN).

Review of linear algebra; spectral theory in inner product spaces; decomposition theorems; duality theory and multilinear algebra; tensor products; applications. May be taken concurrently with MATH 641. Prerequisites: MATH 304 or equivalent.

Review of preliminary concepts; sequence and function spaces; normed linear spaces, inner product spaces; spectral theory for compact operators; fixed point theorems; applications to integral equations and the calculus of variations. Prerequisites: MATH 447, 640, or approval of instructor.

Distributions and differential operators; transform theory; spectral theory for unbounded self-adjoint operators; applications to partial differential equations; asymptotics and perturbation theory. Prerequisite: MATH 641.

Fundamental ideas of algebraic topology, homotopy and fundamental group, covering spaces, polyhedra. Prerequisite: MATH 436 or approval of instructor.

Homology and cohomology theory. Prerequisite: Math 643.

Fundamentals of mathematical analysis underlying theory of constrained optimizations for a finite number of variables, necessary and sufficient conditions for constrained extrema of equality constraint problems, sufficient conditions for fulfillment of constraint qualification, computational methods for concave programming problems and applications. Prerequisite: MATH 410 or approval of instructor.

Necessary conditions of calculus of variations, elementary theory of games, formulation of basic control problem, Hestenes' necessar conditions for optimal control, transformations, methods of computation and applications. Prerequisite: MATH 651.

Survey or groups, rings, ideals. Prerequisite: MATH 415 or approval of instructor.

Survey of modules, field extensions, Galois theory. Prerequisite: MATH 653 or approval of instructor.

Normed linear spaces, duality theory, reflexivity, operator theory, Banach algebras, spectral theory, representation theory. Prerequisite: MATH 608.

Topological linear spaces, locally convex spaces, duality in locally convex spaces, ordered topological vector spaces, distribution theory, applications to analysis. Prerequisite: MATH 655.

Review of fundamental concepts of approximation, polynomials and other tools; basic univariate spline theory including bases, computational algorithms and approximation power; applications to interpolation, discrete approximation and data fitting. Prerequisite: MATH 304 and 417 or equivalents.

Tensor-product methods using polynomials and B-splines; computation and application of tensor methods to interpolation and approximation; triangle-based methods; dimension problems, local bases and approximation power; application to scattered data fitting, computer-aided design and finite element analysis. Prerequisite: MATH 657.

Techniques in matrix computation; elimination methods, matrix decomposition, generalized inverses, orthogonalization and least-squares, eigenvalue problems and singular value decomposition, iterative methods and error analysis. Prerequisites: MATH 417 or equivalent or CPSC 442 or equivalent. Cross-listed with CPSC 660.

Introduction to difference equations, finite element analysis and splines. Prerequisite: Approval of Instructor.

Problems, methods and recent developments in algebra. This course may be taken three times for credit as content varies. Prerequisite: Approval of Instructor.

Problems, methods and recent developments in analysis. This course may be taken three times for credit as content varies. Prerequisite: Approval of Instructor.

Problems, methods and recent developments in applied mathematics. This course may be taken three times for credit as content varies. Prerequisite: Approval of Instructor.

Problems, methods and recent developments in topology. This course may be taken three times for credit as content varies. Prerequisite: Approval of Instructor.

Problems, methods and recent developments in geometry. This course may be taken three times for credit as content varies. Prerequisite: Approval of Instructor.

Existence, uniqueness and characterization of best approximations; polynomial and rational approximants; inequalities; order of approximation; interpolation, algorithms; n-widths; saturation theorems; approximation in Hankel norm. Prerequisites: MATH 407 and 409.

Time-frequency analysis, integral wavelet transform, multiresolutional analysis, dyadic wavelets and inversions, frames, classification of wavelets, dual basis and a duality principle, wavelet decompositions and reconstructions, spline wavelets, zero-crossings of spline-wavelets, wavelet packets, multivariate wavelets. Prerequisites: MATH 304, MATH 409, MATH 414 or equivalents.

Mathematical tools of applied mathematics; Fredholm alternative; integral operators; Green's functions; unbounded operators; Stone's theorem; distributions; convolutions; Fourier transforms; applications. Prerequisites: MATH 642 or equivalent.

Mathematical tools of applied mathematics; Sobolev spaces; convexity; variational inequalities; variational methods for partial differential equations; maximum principles; elements of nonlinear analysis; compact operators; fixed point theorems; applications. Prerequisites: MATH 670 or equilavent.

Preliminaries; probability, information, entropy, signals, channels; group-theoretic view of messages; contemporary secrecy and digital signature systems; one-time pads, DES, RSA, DSS, wheels, LFSR-based systems; analog scramblers; key exchange, key management, secret sharing, access structures; measures of security. Prerequisites: Graduate classification and approval of instructor.

Classical and recent attacks; login, compression, error control and genetic codes; finite and infinite codes; matrices, graphs, duals, groups, morphisms, composites, products, rates and classification of codes; the confusion/diffusion/arithmetic/calculus extension of Shannon's two design primitives. Prerequisites: Graduate Classification and MATH 673 or approval of instructor.

Offered to enable students to undertake and complete, with credit, limited investigations not within their thesis research and not covered by any other courses in the curriculum. Prerequisites: Approval of instructor.

Selected topics in an identified area of mathematics. May be repeated for credit. Prerequisites: Approval of instructor.

Research for thesis or dissertation.

This course is designed to acquaint the graduate student with the present status of investigative work in a variety of mathematical fields. Content will depend on the availability of visiting lecturers who will be selected because of distinguished international recognition in their field of research. May be taken two times for credit. Prerequisite: Graduate classification.

Theorems, applications and concepts of calculus, methods and mechanics of teaching calculus and college mathematics, discussion of computer assisted instruction. May not be repeated for credit. Prerequisite: Teaching assistant in the Mathematics Department.