Topics in Advanced Calculus (MAA 5210)
- AS 4 (4,0)
- PR: MAC 3313 or C.I. Selected topics in multivariable calculus including limits, continuity, Euler's theorem,the Jacobian, and double series; extension of single variable concepts including uniform convergence and improperintegrals.
Techniques of Complex Variables (MAA 5405)
- AS 3 (3,0)
- PR: MAC 2313 or C.I. Analytic functions; integration in the complex plane; Laurent series and residue calculus,inversion of Laplace transforms; conformal mappings; application in engineering and the physical sciences.
Foundations of Analysis (MAA 5416)
- AS 3 (3,0)
- PR: MAA 4226. Topological spaces, compactness results, connectedness, analytical and differentiable manifolds,topological groups, Lie groups, representation theory for classical groups, Green, Stokes and Gauss' theorems.
Hilbert Spaces with Applications (MAA 6508)
- AS 3 (3,0)
- PR: MAP 2302, MAS 3106 or C.I. Normed and inner product spaces; Hilbert spaces; orthonormal systems; linearoperators and spectral decomposition; applications to differential and integral equations.
Real Analysis (MAA 6306)
- AS 3 (3,0)
- PR: MAA 5210. Sets, function spaces, Lebesque measure, Lebesque-Stieltjes measure, measurable functions, convergencenotions, integrations, general measure and integration, Radon-Nikodym theorem.
Complex Analysis (MAA 6404)
- AS 3 (3,0)
- PR: MAA 5405, Advanced Calculus I; or C.I.. Analytic and harmonic functions, Cauchy's integral theorem, Cauchy'sintegral formula, Taylor and Laurent series, singularities and residue theory, conformal mapping.
Combinatorics and Graph Theory II (MAD 5205)
- AS 3 (3,0)
- PR: MAD 4203 or C.I. Polya's theory of counting; Latin squares and rectangles; block designs; coding theory;probabilistic methods; hypergraphs; applications.
Advanced Graph Theory I (MAD 6309)
- AS 3 (3,0)
- A seminar devoted mainly to reading papers and presenting their content. Advanced areas of graph theory willbe covered. Primarily for Ph.D. students in Mathematics and Computer Science.
Finite Fields and Coding Theory (MAD 6608)
- AS 3 (3,0)
- PR: MAS 5311 - Abstract Algebra with Applications or C.I. General theory of fields, existence, constructionand implementation of finite fields, polynomials over GF (pn), solving equations: emphasizing fields of characteristic2.
Applied Numerical Mathematics (MAP 5385)
- AS 3 (3,0)
- PR: MAP 2302 or C.I. Classical topics or numerical analysis and their applications, Romberg integration, Richardsonextrapolation, Gaussian quadrature schemes.
Splines and Data Fitting (MAP 5396)
- AS 3 (3,0)
- PR: MAP 2302, MAS 3105 or 3106, or C.I. Topics on univariant splines and data fitting; applications to regressionanalysis, differential and integral equations; algorithms for different types of splines.
Ordinary Differential Equations and Applications (MAP 5336)
- AS 3 (3,0)
- PR: MAP 2302, Differential Equations; or consent of the instructor. Existence and uniqueness of solutions ofdifferential equations, systems of ordinary differential equations, autonomous systems, phase plane analysis, stability,bifurcations.
Applied Mathematics I (MAP 5407)
- AS 3(3,0)
- PR: MAP 2302 or CI.
- Calculus of variations, Hamilton’s principle, Rayleigh-Ritz method, Sturm-Liouville theory, Green’s functionsfor ordinary differential equations, introduction to integral equations.
Special Functions (MAP 5426)
- AS 3(3,0)
- PR: MAP 2302 or C.I
- Series and integral representations, generating functions, recurrence relations and orthogonality propertiesof the special functions.
- Emphasis on Bessel, Legendre, and hypergeometric functions.
Linear and Nonlinear Waves I (MAP 5514)
- AS 3(3,0)
- PR: MAP 2302; MAP 4363; or C. I.
- Equations of motion in inviscis and viscous fluids, energy equation and energy flux, linear theory of gravityand capilliary-gravity waves, variational principles for water waves.
Advanced Mathematics for Engineers (MAP 5435)
- AS 3(3,0)
- PR: MAP 2302.
- Linear algebra and matrix methods, ordinary differential equations, Fourier series, partial differential equations,numerical methods for differential equations, applications to engineering.
Mathematical Foundations for Industrial Engineering (MAP 5404)
- AS 3(3,0)
- PR: MAP 2302, STA 5156 or equivalent; ESI 4312, or C. I.
- Methods of proof, set theory; basic elements of topology, real analysis, graph theory and matrix analysis.
Introduction to Nonlinear Dynamics (MAP 6104)
- AS 3(3,0)
- PR: MAP 5336, PHY 3048 or equivalent, or C.I.
- Nonlinear differential equations;
bifurcation theory; Hamiltonian dynamics;
integrable systems and breakdown of
integrability; chaos in conservative and
dissipative systems.
Measure and Probability (MAP 6110)
- AS 3(3,0)
- PR: MAA 5210 or C.I.
- Measure and integration, probability
measures, random variables, distribution
and characteristic functions. Convergence
in LP, probability, distrubition and with
probability one.
Mathematical Statistics (MAP 6111)
- AS 3(3,0)
- PR: MAS 6110 (Measure & 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.
Asymptotic Methods in Mathematical Statistics (MAP 6112)
- AS 3(3,0)
- PR: MAP 6111 (Mathematical Statistics) or C.I.
- Large sample theory, martingale
sequences, probability measures on metric
spaces, absolute continuity and
singularity, Hellinger distance,
functions fo statistics, asymptotic
theory of estimation.
Optimization Theory (MAP 6207)
- AS 3(3,0)
- PR: MAA 4226 - Advanced Calculus I; or C.I.
- Lagrangian function and duality,
Kuhn-Tucker theorem, quadratic
programming and Wolfe's theorem, Griffith
and Stewart's method, search methods for
unconstrained optimization.
Partial Differential Equations (MAP 6356)
- AS 3(3,0)
- PR: MAP 4364 or MAP5435 or C.I.
- First and second order linear equations;
classification and analytical methods of
solution; Green's functions and integral
representations; applications in
engineering and physical sciences.
Numerical Solutions of PDE (MAP 6385)
- AS 3(3,0)
- PR: MAP 6356; MAP 5385 or C.I.
- Numerical solution of linear and
nonlinear partial differential equations
of parabolic, elliptic and hyperbolic
type using finite difference and spectral
methods.
Applied Mathematics II (MAP 6408)
- AS 3(3,0)
- PR: MAP 2302, MAA 5405 or C.I.
- Asymptotic series, asymptotic expansion
of integrals, regular and singular
perturbation expansions, boundary layer,
multiple scales, WKB theory.
Advanced Transform Methods (MAP 6419)
- AS 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.
Generalized Functions (MAP 6420)
- AS 3(3,0)
- PR: MAP 6506; C.I.
- Spaces of test functions and their duals,
calculus of distributions, convolution
and tempered distributions, Fourier
transforms of distributions, and
applications to PDE's.
Integral Equations (MAP 6421)
- AS 3(3,0)
- PR: MAA 5405; C.I.
- Successive approximations, Volterra
equations, Fredholm theory,
Hilbert-Schmidt theory, Newmann series,
singular integral equations, the
Riemann-Hilbert problem.
Transform Methods (MAP 6424)
- AS 3(3,0)
- PR: MAA 5405 or C.I.
- Laplace, Fourier, Hankel and other
integral transforms, inversion theorems;
the Z transform; applications to physical
problems.
Advanced Complex Analysis and Applications (MAP 6425)
- AS 3(3,0)
- PR: MAA 5404 or C.I.
- Schwarz-Christoffel and Jaukowsky
transformations, entire functions,
Weierstrass factorization theorem and
Blaschke's product, meromorphic functions
and Mittag-Leffler theorem.
Approximation Techniques (MAP 6445)
- AS 3(3,0)
- PR: MAA 5210 or C.I.
- Normed linear spaces; Weierstrass
approximation theorem; Tchebycheff
approximation by polynomials;
trigonometric approximation; orthogonal
expansions and least squares
approximations.
Orthogonal
Polynomials and Digital Signal Processing (MAP 6455)
- AS 3(3,0)
- PR: MAA 6306; MAA 6404; or C.I.
- Orthogonal polynomials, Szego's
orthogonal polynomials, Toeplitz matrix,
Caratheodory functions and Schur
functions, Levinson algorithim,
associated Szego polynomials.
Doubly Stochastic Measures (MAP 6463)
- AS 3(3,0)
- PR: MAP 6506; MAA 5416; MAP 6111; or C.I.
- Doubly stochastic matrices, Birkhoff's
theorem, doubly stochastic measures,
Douglas-Linden-strauss theorem, copulas,
Frechet bounds, dependence of random
variables, Markov operators.
Wavelets and Their Applications (MAP 6465)
- AS 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.
Functional Analysis (MAP 6506)
- AS 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.
Linear and Nonlinear Waves II (MAP 6515)
- AS 3(3,0)
- PR: MAP 5514, Linear and Nonlinear Waves
I; or C.I.
- Nonlinear shallow water waves and
solitons, inverse scattering transform,
Lie group methods, nonlinear dispersive
waves, solitary waves and the nonlinear
Schrodinger equations.
Fractal Image Compression (MAP 6520)
- AS 3(3,0)
- PR: MTG 4302; MAA 5416 or MAA6306; or
C.I.
- Hausdorff metric H, Hutchenson maps,
contraction maps on H, the collage
theorem, measures and IFS with
probabilities, fractal image compression,
Huffman codes, addresses on fractals.
Multivariate Splines and Surface Fitting (MAP 6938)
- AS 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.
Thesis (MAP 6971)
Advanced Nonlinear Dynamics (MAP 7119)
- AS 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.
Advanced Topics in Partial Differential Equations (MAP 7357)
- AS 3(3,0)
- PR: MAP 6356; C.I.
- Variational techniques, perturbation and
asymptotic methods, hyperbolic systems,
Lie group methods, parabolic, elliptic,
or free boundary value problems, spectral
analysis.
Advanced Linear Algebra and Matrix Theory (MAS 5145)
- AS 3(3,0)
- PR: MAS 3105 or C.I.
- LU and LDU decompositions, linear spaces,
inner product spaces, systems of linear
equations, eigenvalues and canonical
forms, variational principles and
applications.
Doubly Stochastic Measures (MAS 6463)
- AS 3(3,0)
- PR: MAA 6506; MAP 5416; MAP 6111; MAP
6110; 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.
|
