56:645:503-504
Theory of Functions of a Complex Variable (3,3)
Analytic functions and the Cauchy Integral Theorem. Conformal mappings. Laplace transforms.
|
56:645:505-506
Analysis (3,3)
Infinite series, especially Fourier series. Epsilon-delta proofs of continuity and differentiability. Convergence tests. Measure theory and integration.
|
56:645:507
Contemporary Issues-Teaching Beyond Regurgitation (3)
Discusses traditional as well as contemporary approaches to teaching mathematics. Comparisons within these contexts are investigated. The intricate connections between geometry and algebra serve as a segue to a deeper analysis of calculus and linear and abstract algebra. Selected readings from NCTM publications are a course requirement.
|
56:645:508
Mathematical Reasoning (3)
This
course develops two fundamental components of writing mathematics:
reasoning (thinking about the proof) and writing (formulating and
writing the ideas precisely using logical statements). The course begins with
illustrative examples and general guidelines.
|
56:645:510
Mathematical Communication and Technology (3)
New technologies for doing and teaching mathematics and for creating mathematical documents for print and World Wide Web distribution.
|
56:645:511
Computer Science (3)
A survey of computer science, both theoretical and practical, for the pure mathematician. Topics could include time-complexity of algorithms, NP-completeness, Turing machines, factoring and primality testing, Strassen's matrix reduction algorithm, and the fast Fourier transform.
|
56:645:527-528
Methods of Applied Mathematics (3,3)
Derivation of the heat and wave equations. Existence theorems for ordinary differential equations, series solutions. Bessel and Legendre equations. Sturm-Liouville Theorem.
Pre- or corequisite: 56:645:549.
|
56:645:530
Manifolds (3)
Topological and differential manifolds. Surfaces. Fundamental groups and coverings. Differential forms and de Rham cohomology.
|
56:645:531
Geometry (3)
Review and reevaluation of Euclid's geometry. Axiomatic development of Euclidean and hyperbolic geometries. The parallel postulate. The impossibility of trisecting an angle or duplicating a cube.
|
56:645:532
Differential Geometry (3)
Curves and surfaces in Euclidean space. Riemannian manifolds, connections, and curvature.
|
56:645:533-534
Introduction to the Theory of Computation I,II (3,3)
645:533: Introduction to formal languages, automata, and computability: regular languages and finite state automata; context-free grammars and languages; pushdown automata; the Church-Turing thesis; Turing machines; decidability and undecidability; Rice's theorem. 645:534: Second course in the sequence; addresses key topics in computability and complexity theory, such as recursive and recursively enumerable sets; the Recursion Theorem; Turing reductions and completeness; Kolmogorov complexity; Space and Time complexity; NP-completeness; hierarchy theorems; probabilistic complexity classes, and interactive proof systems.
|
56:645:535-536
Algebra for Computer Scientists I,II (3,3)
Linear and abstract algebra, including group theory, with applications to image processing, data compression, error correcting codes, and encryption.
|
56:645:537
Computer Algorithms (3)
Algorithm design techniques: divide-and-conquer, greedy method, dynamic programming, backtracking, and branch-and-bound. Advanced data structures, graph algorithms, and algebraic algorithms. Complexity analysis, complexity classes, and NP-completeness. Introduction to approximation algorithms and parallel algorithms.
|
56:645:538
Combinatorial Optimization (3)
Algorithmic techniques for solving optimization problems over discrete structures, including integer and linear programming, branch-and-bound, greedy algorithms, divide-and-conquer, dynamic programming, local optimization, simulated annealing, genetic algorithms, and approximation algorithms.
|
56:645:540
Computational Number Theory and Cryptography (3)
Primes and prime number theorems and numerical applications; the Chinese remainder theorem and its applications to computers and Hashing functions; factoring numbers; cryptography; computation aspects of the topics emphasized. Students required to do some simple programming.
|
56:645:541
Introduction to Computational Geometry (3)
Algorithms and data structures for geometric problems that arise in various applications, such as computer graphics, CAD/CAM, robotics, and geographical information systems (GIS). Topics include point location, range searching, intersection, decomposition of polygons, convex hulls, and Voronoi diagrams.
|
56:645:542
Parallel Supercomputing (3)
Fundamental issues in the design and development of programs for parallel supercomputers; programming models and performance optimization techniques; application examples and programming exercises on a contemporary parallel machine; cost models and performance analysis and evaluation.
|
56:645:545
Topology (3)
Point set topology, fundamental group and coverings. Singular homology and cohomology, the Brouwer degree and fixed-point theorems, the sphere retraction theorem, invariance of domains.
|
56:645:549-550
Linear Algebra and Applications (3,3)
Finite dimensional vector spaces, matrices, and linear operators. Eigenvalues, eigenvectors, diagonalizability, and Jordan canonical form. Applications.
|
56:645:551-552
Abstract Algebra (3,3)
Introductory topics in rings, modules, groups, fields, and Galois theory.
Pre- or corequisite: 56:645:549.
|
56:645:554
Applied Functional Analysis (3)
Infinite dimensional vector spaces, especially Banach and Hilbert vector spaces. Orthogonal projections and the spectral decomposition theorem. Applications to differential equations and approximation methods.
|
56:645:555
Glimpses of Mathematics (3)
The intuitive beginnings and modern applications of key ideas of mathematics, such as polyhedra and the fundamental theorem of algebra. Extensive use of computer-generated films to help visualize the methods and results.
|
56:645:556
Visualizing Mathematics by Computer (3)
Introduction to symbolic computational packages and scientific visualization through examples from calculus and geometry. Covers 2-D, 3-D, and animated computer graphics using Maple, Mathematica, and Geomview. No programming knowledge required.
|
56:645:557
Signal Processing (3)
Signal modeling: periodic, stationary, and Gaussian signals. System representation: Volterra representation, state space representation, simulation. Themes in system design: least square estimation, system identification, adaptive signal processing. Representation of discrete causal signals: role of Fourier analysis, convolutions, fast Fourier transforms. Realization of linear recurrent structures: controllability, observability and minimal realization, frequency domain analysis of signals, and the role Laplace transforms. Stability analysis: Lyapunov and linearization methods. Prediction, filtering, and identification: linear prediction, the LQR problem, Kalman filter.
|
56:645:558
Theory and Computation in Probability and Queuing Theory (3)
Basic probability structures, probability distributions, random number generations and simulations, queuing models, analysis of single queue, queuing networks, applications of queuing theory.
|
56:645:560
Industrial Mathematics (3)
Monte Carlo methods, Wavelets, data acquisition and manipulation, filters, frequency domain methods, fast Fourier transform, discrete Fourier transform.
|
56:645:561
Optimization Theory (3)
Linear programming: optimization, simplex algorithm, nonlinear programming, game theory.
|
56:645:562
Mathematical Modeling (3)
Perturbation methods, asymptotic analysis, conservation laws, dynamical system and chaos, oscillations, stability theory. Applications may include traffic flow, population dynamics, combustion.
|
56:645:563
Statistical Reasoning (3)
Random variables, uniform, Gaussian, binomial, Poisson distributions, probability theory, stationary processes, central limit theorem, Markov chains, Taguchi quality control.
|
56:645:570
Special Topics in Pure Mathematics (3)
Topics vary from semester to semester.
Prerequisite: Permission of instructor. Course may be taken more than once.
|
56:645:571-572
Computational Mathematics I,II (3,3)
Newton's method, curve and surface fitting. Numerical solutions of differential equations and linear systems, eigenvalues and eigenvectors. Fast Fourier transform.
|
56:645:574
Control Theory and Optimization (3)
Controllability, observability, and stabilization for linear and nonlinear systems. Kalman and Nyquist criteria. Frequency domain methods, Liapunov functions.
|
56:645:575
Qualitative Theory of Ordinary Differential Equations (3)
Cauchy-Picard existence and uniqueness theorem. Stability of linear and nonlinear systems. Applications to equations arising in biology and engineering.
|
56:645:577
Quality Engineering (3)
Introduction to statistical tools, such as data analysis, and their use in the testing of product design and minimization of uncontrollable variation.
|
56:645:580
Special Topics in Applied Mathematics (3)
Topics vary from semester to semester.
Prerequisite: Permission of instructor. Course may be taken more than once.
|
56:645:698
Independent Study in Pure Mathematics (3)
Study of a particular subject independently but with frequent consultations with a faculty member.
|
56:645:699
Independent Study in Applied Mathematics (3)
Study of a particular subject independently but with frequent consultations with a faculty member.
|
56:645:700
Thesis in Pure Mathematics (3)
Expository paper written under the close guidance of a faculty member.
|
56:645:701
Thesis in Applied Mathematics (3)
Expository paper written under the close guidance of a faculty member.
|
56:645:800
Matriculation Continued (0)
Continuous registration may be accomplished by enrolling for at least 3 credits in standard course offerings, including research courses, or by enrolling in this course for 0 credits. Students actively engaged in study toward their degree who are using university facilities and faculty time are expected to enroll for the appropriate credits.
|
56:645:877
Teaching Assistantship (E6)
|
