MAA 5210. Topics in Advanced Calculus
4(4,0). PR: MAA 4226 or equivalent, graduate status or senior standing, or C.I. Topics in multivariable calculus, including limits, continuity, integration, differentiation, Taylor's theorem, inverse and implicit function theorems
Fall.
COS-Mathematics |
MAA 6238. Measure and Probability
3(3,0). PR: MAA 5210 or C.I. Measure and integration, probability measures, random variables, distribution and characteristic functions. Convergence in LP, probability, distribution and with probability one.
Even Fall.
COS-Mathematics |
MAA 6306. Real Analysis
3(3,0). PR: MAA 5210. Sets, function spaces, Lebesque measure, Lebesque-Stieltjes measure, measurable functions, convergence notions, general measure and integration, Radon-Nikodym theorem.
Occasional.
COS-Mathematics |
MAA 6404. Complex Analysis
3(3,0). PR: MAA 6405, MAP 4307, MAA 4226, or C.I. Review of complex variable theory; advanced topics chosen from conformal mapping and its applications, boundary behavior, numerical techniques; singular integrals.
Occasional.
COS-Mathematics |
MAA 6405. Complex Variables
3(3,0). PR: MAA 4226, MAP 4307, and graduate standing or C.I. Analytic functions. Harmonic functions. Integration in the complex plane. Laurent series. Residue calculus. Inversion of Laplace transformations. Maximum Modulus Principle. Conformal mappings.
Occasional.
COS-Mathematics |
MAA 6416. Topology
3(3,0). PR: MAA 4226, MTG 4302. Topological spaces and continuous functions, connectedness and compactness, separation axioms, metrization theorems, Baire spaces and dimension theory, the fundamental group and homotopy paths.
Even Spring.
COS-Mathematics |
MAA 6506. Functional Analysis
3(3,0). PR: MAA 4226 or C.I. Normed vector spaces, linear operators, Baire Category theorem, Banach fixed point theorem, Hahn-Banach theorem and applications, open mapping and closed graph theorem with applications, Hilbert space, Gateaux and Frechet.
Even Spring.
COS-Mathematics |
MAA 6508. Hilbert Spaces with Applications
3(3,0). PR: MAP 2302, MAS 3106, or C.I. Normed and inner product spaces; Hilbert spaces; orthonormal systems; linear operators and spectral decomposition; applications to differential and integral equations.
Occasional.
COS-Mathematics |
MAA 6531. Analysis of Manifolds
3(3,0). PR: Matrix or Linear Algegra, MAA 4226 or MAA 5210, or C.I. Derivatives as linear transformations, inverse function theorem, manifolds and integration of real-valued functions on manifolds, wedge products, differential forms, vector analysis as a specific case.
Occasional.
COS-Mathematics |
MAA 7239. Asymptotic Methods in Mathematical Statistics
3(3,0). PR: MAP 6111 or C.I. Large sample theory, martingale sequences, probability measures on metric spaces, absolute continuity and singularity, Hellinger distance, functions of statistics, asymptotic theory of estimation and applications.
Occasional.
COS-Mathematics |
MAD 5205. Combinatorics and Graph Theory II
3(3,0). PR: MAD 4203, graduate status or senior standing, or C.I. Polya?s theory of counting; Latin squares and rectangles; block designs; coding theory; probabilistic methods; hypergraphs; applications.
Odd Spring.
COS-Mathematics |
MAD 6309. Advanced Graph Theory I
3(3,0). A seminar devoted mainly to reading papers and presenting their content. Advanced areas of graph theory will be covered. Primarily for PhD students in Mathematics and Computer Science.
Occasional.
COS-Mathematics |
MAD 6608. Finite Fields and Coding Theory
3(3,0). PR: MAS 5311 or C.I. General theory of fields, existence, construction and implementation of finite fields, polynomials over GF(pn), solving equations: emphasizing fields of characteristic 2.
Occasional.
COS-Mathematics |
MAE 5935. Post-Secondary Mathematics
3(3,0). PR: Graduate status or senior standing or C.I. The course will focus on issues which are faced by teachers of collegiate mathematics. Topics will be selected from teaching issues, program issues, and other issues.
Even Fall.
COS-Mathematics |
MAP 5106. Introduction to Quantitative Aspects of Modeling and Simulation
3(3,0). PR: MAC 2253, graduate status or senior standing, or C.I. An introduction to calculus, matrix algebra, probability and statistics, and high level programming languages. A student who has mastered this content does not have to take this course.
Occasional.
COS-Mathematics |
MAP 5117. Mathematical Modeling
3(3,0). PR: STA 4321, MAP 4363, graduate status or senior standing, or C.I. Introduction to modeling in industrial and scientific applications; techniques for studying statistical and deterministic models.
Fall.
COS-Mathematics |
MAP 5336. Ordinary Differential Equations and Applications
3(3,0). PR: MAP 2302, and graduate status or senior standing or C.I. Existence and uniqueness of solutions of differential equations, systems of ordinary differential equations, autonomous systems, phase plane analysis, stability, bifurcations.
Spring.
COS-Mathematics |
MAP 5396. Splines and Data Fitting
3(3,0). PR: MAS 3106, MAS 3105, MAP 2302, and graduate status or senior standing or C.I. Univariate splines and their application to data fitting. Applications to regression analysis, differential and integral equations. Algorithms to use different types of splines in computation.
Occasional.
COS-Mathematics |
MAP 5404. Mathematical Foundations for Industrial Engineering and Operations
3(3,0). PR: MAP 2302, ESI 5219 or equivalent, ESI 4312, and graduate status or senior standing or C.I. Methods of proof, set theory; basic elements of topology, real analysis, graph theory, and matrix analysis.
COS-Mathematics |
MAP 5426. Special Functions
3(3,0). PR: MAP 2302, and graduate status or senior standing or C.I. Series and integral representations, generating functions, recurrence relations and orthogonality properties of the special functions. Emphasis on Bessel, Legendre and hypergeometric functions.
Occasional.
COS-Mathematics |
MAP 5435. Advanced Mathematics for Engineers
3(3,0). PR: MAP 2302, and graduate status or senior standing or C.I. Linear Algebra and matrix methods, ordinary differential equations, Fourier series, partial differential equations, numerical methods for differential equations, and applications to engineering.
Occasional.
COS-Mathematics |
MAP 5514. Linear and Nonlinear Waves I
3(3,0). PR: MAP 2302, MAP 4363, and graduate status or senior standing, or C.I. Equations of motion in inviscous and viscous fluids, energy equation and energy flux, linear theory of gravity and capillary-gravity waves, variational principles for water waves.
Occasional.
COS-Mathematics |
MAP 5931. Research Seminar
1(1,0). PR: Graduate status or senior standing or C.I. Four instructors will introduce the students to a research area by presenting necessary background and presenting current investigations. Different branches of mathematics will be presented for a sense of diversity.
COS-Mathematics |
MAP 6111. Mathematical Statistics
3(3,0). PR: MAA 6238 Measure and Probability or C.I. Strong laws of large numbers, consistency and asymptotic normality, complete and sufficient statistics, maximum likelihood and least squares, optimal estimators, hypothesis testing.
Spring.
COS-Mathematics |
MAP 6118. Introduction to Nonlinear Dynamics
3(3,0). PR: MAP 5336, PHY 2048 or equivalent, or C.I. Nonlinear differential equations; bifurcation theory; Hamiltonian dynamics; integrable systems and breakdown of integrability; chaos in conservative and dissipative systems.
Occasional.
COS-Mathematics |
MAP 6207. Optimization Theory
3(3,0). PR: MAA 4226 or C.I. Lagrangian function and duality, Kuhn-Tucker' theorem, quadratic programming and Wolfe?s theorem, Griffith and Stewar?s method, search methods for unconstrained optimization.
Occasional.
COS-Mathematics |
MAP 6356. Partial Differential Equations
3(3,0). PR: MAP 4364 or MAP 5435 or equivalent. First and second order linear equations; classification; analytical methods including Green?s functions and integral representations; introduction to nonlinear equations; applications.
Odd Spring.
COS-Mathematics |
MAP 6383. Mathematical Methods for Image Analysis
3(3,0). PR: MAP 2302, MAS 3106, MAT 5711 or COT 4500, or C.I. Linear spaces, eigenvalue problems, linear and nonlinear optimization methods, calculus of variations and numerical; solutions of partial differential equations, compressive sampling, diffusion maps, graphical models.
Odd Fall.
COS-Mathematics |
MAP 6385. Applied Numerical Mathematics
3(3,0). PR: MAP 5117, MAA 5210, graduate standing, or C.I. Numerical linear algebra, Gaussian elimination, Gauss-Seidel iteration, optimization procedures, numerical solution of ordinary differential equations.
Occasional.
COS-Mathematics |
MAP 6398. Multivariate Splines and Surface Fitting
3(3,0). PR: MAP 5396 or C.I. Approximation of functions of several variables, tensor product splines, theory of multivariate splines, box splines, surface fitting, applications to statistics, computer graphics.
COS-Mathematics |
MAP 6407. Applied Mathematics I
3(3,0). PR: MAP 2302, MAS 3105, MAA 4226, MAP 4307, MAP 4363, or C.I. Linear functionals, calculus of variations, Hamilton's principle, Rayleigh-Ritz method, Sturm-Liouville theory, Green's functions for ordinary differential equations, Volterra and Fredholm integral equations.
Occasional.
COS-Mathematics |
MAP 6408. Applied Mathematics II
3(3,0). PR: MAP 2302 and MAA 6405 or equivalent. Asymptotic series, asymptotic expansion of integrals, regular and singular perturbation expansions, boundary layer, multiple scales, WKB theory.
Spring.
COS-Mathematics |
MAP 6419. Advanced Transform Methods
3(3,0). PR: MAP 6424 or C.I. Fourier analysis and sliding-window Fourier transform, sampling theory and its applications in signal analysis and optics, Radon transforms, the technique of back projection.
Occasional.
COS-Mathematics |
MAP 6420. Generalized Functions
3(3,0). PR: MAA 6506 or C.I. Spaces of test functions and their duals, calculus of distributions, convolution and tempered distributions, Fourier transforms of distributions, and applications to PDEs.
COS-Mathematics |
MAP 6421. Integral Equations
3(3,0). PR: MAA 6405 or C.I. Successive approximations, Volterra equations, Fredholm theory, Hilbert-Schmidt theory, Newmann series, singular integral equations, the Riemann-Hilbert problem.
Occasional.
COS-Mathematics |
MAP 6424. Transform Methods
3(3,0). PR: MAA 6405 or C.I. Laplace, Fourier, Hankel, and other integral transforms, inversion theorems; the Z transform; applications to physical problems.
Occasional.
COS-Mathematics |
MAP 6438. Mathematical Fluid-Flow Theory I
3(3,0). PR: MAP 2302, MAP 4363, MAP 4307, PHY 3220 or equivalent, or CI. Mathematical theory of incompressible fluid flows along with analytical methods in solving the equations of fluid dynamics in various situations.
Even Fall.
COS-Mathematics |
MAP 6445. Approximation Techniques
3(3,0). PR: MAA 4227, MAA 5210 or C.I. Normed linear spaces; Weierstrass approximation theorem; Tchebycheff approximation by polynomials; trigonometric approximation; orthogonal expansions and least squares approximations.
Occasional.
COS-Mathematics |
MAP 6463. Doubly Stochastic Measures
3(3,0). PR: MAA 6506, MAA 6416, MAP 6111, MAA 6238, or C.I. Doubly stochastic matrices, Birkhoff?s theorem, double stochastic measures, Douglas-Linden-Strauss theorem, copulas, Frechet bounds, dependence of random variables, Markov operators.
Occasional.
COS-Mathematics |
MAP 6465. Wavelets and Their Applications
3(3,0). PR: MAP 4364, MAA 6508, or C.I. Continuous wavelet transforms, discrete wavelet transforms, frams, Zak transform, multi-resolution analysis, orthonormal bases of compactly supported wavelets, spline wavelets.
Occasional.
COS-Mathematics |
MAP 6507. Wave Propagation through Random Media
3(3,0). PR: MAP 2302, EEL 5542, or C.I. Development of mathematical models for laser communications and laser radar in atmospheric turbulence. Free-space propagation of Gaussian beams and classical theories of propagation.
Odd Spring.
COS-Mathematics |
MAP 7119. Advanced Nonlinear Dynamics
3(3,0). PR: MAP 6118 or C.I. Solitons, inverse scattering transform, breakdown or integrability, analytic structure of dynamical systems, fractal aspects of turbulence.
Occasional.
COS-Mathematics |
MAP 7357. Advanced Topics in Partial Differential Equations
3(3,0). PR: MAP 6356 or C.I. Variational techniques, perturbation and asymptotic methods, hyperbolic systems, Lie group methods, parabolic, elliptic, or free boundary value problems, spectral analysis.
Occasional.
COS-Mathematics |
MAP 7386. Numerical Solutions of PDE
3(3,0). PR: MAP 6356, MAP 6385 or C.I. Numerical solution of linear and nonlinear partial differential equations of parabolic, elliptic and hyperbolic type. Solution of PDE using finite difference and spectral methods.
Occasional.
COS-Mathematics |
MAP 7439. Mathematical Fluid-Flow Theory II
3(3,0). PR: MAP 6438 or C.I. Mathematical theory of compressible potential flow, nonlinear acoustics, exact solutions to equations of viscous fluid flow, viscous fluid flows at low or high Reynolds numbers.
Odd Fall.
COS-Mathematics |
MAS 5145. Advanced Linear Algebra and Matrix Theory
3(3,0). PR: MAS 3105, and graduate status or senior standing or C.I. LU and LDU decompositions, linear spaces, inner product spaces, systems of linear equations, eigenvalues and canonical forms, variational principles and applications.
Occasional.
COS-Mathematics |
MAS 5311. Abstract Algebra with Applications
3(3,0). PR: MAS 4301 or undergraduate abstract algebra, and graduate status or senior standing or C.I. Group actions, the class equation, Sylow Theorems, polynomial rings, Euclidian domains, principal ideal domains, field extensions, modules, and semi-simple rings.
Fall.
COS-Mathematics |
MAS 6147. Multilinear Algebra
3(3,0). PR: MAS 5145 or C.I. Algebraic theory of tensor and exterior products of finite and infinite dimensional vector spaces and linear transformations. Some category theory will be discussed. Applications to other areas of algebra will be presented.
Occasional.
COS-Mathematics |
MAS 7919. Doctoral Research
var. May be repeated for credit.
Occasional.
COS-Mathematics |
MAS 7980. Doctoral Dissertation
var. May be repeated for credit.
COS-Mathematics |
MAT 5711. Scientific Computing
3(3,0). PR: MAC 2313, MAP 2302, graduate status or senior standing, or C.I. Basic programming skills using Mathematica, Maple, Matlah, or Java in solving basic scientific computing problems; preparing students for advanced computational methods and algorithms.
Fall.
COS-Mathematics |
MTG 5256. Differential Geometry
3(3,0). PR: MAA 4227, graduate status or senior standing, or C.I. Differentiable manifolds, tangent space and tangent bundle, flows and vector fields, Lie derivatives, cotangent space and cotangent bundles, Riemann metrics, connections and geodesics, applications in classical mechanics.
Occasional.
COS-Mathematics |
MTG 6348. Topological K-Theory
3(3,0). PR: C.I. or MTG 4302. Chain and cochain complexes; general cohomology theories; exact couplas and spectral sequences; Ariyah-Hirzebruch spectral sequence; topological K-theory; Chein character; applications.
Occasional.
COS-Mathematics |