# Course Descriptions

**MAT 504. (CSC 504) Advanced Discrete Structures (3)** Prerequisite: An undergraduate combinatorics or discrete mathematics course or consent of instructor. Survey of the mathematical foundations of computer science. Mathematical logic, set theory, algebraic structures, lattices and Boolean algebra, graph theory, introduction to computability theory.

**MAT 509. (CSC 509) Design and Analysis of Algorithms (3)** Prerequisite: An undergraduate data structures course. Theory of the design of efficient computer algorithms. Algorithms for sorting, searching, matrix operations, fast Fourier transforms, pattern matching, polynomial arithmetic and operations on graphs. Additional topics selected from NP-completeness, recursion, data structure selection and complexity.

**MAT 511-512. (411-412) Real Analysis (3-3)** Prerequisite: Permission of department. Advanced study of convergence, continuity, differentiation and integration in Euclidean space. The real number system, basic topology of Euclidean spaces; sequences and series; continuity, differentiation of vector-valued functions, uniform continuity; theory of integration; implicit and inverse function theorems, Stokes' Theorem.

**MAT 513. Measure and Integration (3)** Prerequisite: MAT 512. Abstract measure theory. Lebesgue measure, integration, convergence theorems, absolute continuity, differentiation, Radon-Nikodym Theorem, product measures, Fubini's Theorem, Lebesgue spaces, convolution.

**MAT 515. (415) Introduction to Complex Variables (3)** Prerequisite: Advanced calculus or MAT 511. A first study of functions of a complex variable. Algebra of complex numbers, elementary functions with their mapping properties; analytic functions; power series; integration, Cauchy's Theorem, Laurent series and residue calculus; elementary conformal mappings and boundary value problems.

**MAT 516. Complex Analysis (3)** Prerequisite: MAT 511 and 515. Advanced study of complex-valued functions. Holomorphic and harmonic functions, Cauchy's Integral Theorem, Poisson's kernel and the Dirichlet problem, conformality, the Riemann Mapping Theorem, analytic continuation. Additional topics chosen from univalent, entire, meromorphic functions; Riemann surfaces; asymptotic methods; Mittag-Leffler, Runge and Weierstrass factorization theorems.

**MAT 518-519. (418-419) Applied Analytical Methods (3-3)** Prerequisite: Undergraduate differential equations and advanced calculus. A thorough treatment of the solution of initial and boundary value problems of partial differential equations. Topics include classification of partial differential equations, the method of characteristics, separation of variables, Fourier analysis, integral equations and integral transforms, generalized functions, Green's functions, Sturm-Liouville theory, approximations, numerical methods.

**MAT 521. (421) Number Theory (3)** Prerequisite: Permission of department. Use of algebraic techniques to study arithmetic properties of the integers and their generalizations. Primes, divisibility and unique factorization in integral domains; congruences, residues and quadratic reciprocity; diophantine equations and additional topics in algebraic number theory.

**MAT 525. (425) (CSC 525/425) Numerical Analysis (3)** Prerequisite: Undergraduate linear algebra, differential equations, and elementary numerical methods. Introduction to the theoretical foundations of numerical algorithms. Solution of linear systems by direct methods; least squares, minimax, and spline approximations; polynomial interpolation; numerical integration and differentiation; solution of nonlinear equations; initial value problems in ordinary differential equations. Error analysis. Certain algorithms are selected for programming.

**MAT 531. Linear Algebra (3)** Prerequisite: Permission of department. Theory of vector spaces, linear mappings and matrices. Determinants, eigenvalues, canonical forms, the Cayley-Hamilton Theorem, inner product spaces and positive definite matrices.

**MAT 535. (435) Linear Programming (3)** Prerequisite: Undergraduate linear algebra and computing experience. Methods and applications of optimizing a linear function subject to linear constraints. Theory of the simplex method and duality; parametric linear programs; sensitivity analysis; modeling and computer implementation.

**MAT 536. (436) Discrete Optimization (3)** Prerequisite: MAT 535. Theory and applications of discrete optimization algorithms. Transportation problems and network flow problems; integer programming; computer implementation.

**MAT 537. Nonlinear Programming (3)** Prerequisite: Advanced calculus and MAT 535. Theory and applications for constrained and unconstrained nonlinear optimization. Theory of convex sets, convex and concave functions, Kuhn-Tucker conditions, duality, algorithm convergence; computational methods including penalty and barrier functions, gradient projection, and quadratic programming.

**MAT 541. Modern Algebra I (3)** Prerequisite: Permission of department. Introduction to group theory. Binary structures including semigroups and lattices; finite groups, structure theorems, Sylow theorems and applications; group actions; free groups and presentations; structure of abelian groups.

**MAT 542. Modern Algebra II (3)** Prerequisite: MAT 541. Introduction to rings and fields. Modules, integral domains, vector spaces. Structure of polynomial rings and their relation to linear algebra. Field extensions and Galois theory.

**MAT 551. (451) Topology (3)** Prerequisite: Permission of department. A study of the basic concepts of general topology. Metric spaces, continuity, completeness, compactness, connectedness, separation axioms, product and quotient spaces; additional topics in point-set topology.

**MAT 557. (457) Differential Geometry (3)** Prerequisite: Advanced calculus. Theory of curves and surfaces in Euclidean space. Frenet formulas, curvature and torsion, arc length; first and second fundamental forms. Gaussian curvature, equations of Gauss and Codazzi, differential forms, Cartan's equations; global theorems.

**MAT 563. (463) Ordinary Differential Equations (3)** Prerequisite: Undergraduate linear algebra and differential equations. Advanced study of ordinary differential equations. Existence and uniqueness; systems of linear equations, fundamental matrices, matrix exponential; series solutions, regular singular points; plane autonomous systems, stability and perturbation theory; Sturm-Liouville theory and expansion in eigenfunctions.

**MAT 564. Applied Analytical Models (3)** Prerequisite: MAT 519. Topics in applied analysis of current interest. Topics may include tensor analysis and relativity, quantum mechanics, control theory, fluid mechanics, waves, ocean circulation, and mathematical models in biology or economics.

**MAT 565. (465) (STT 565/465) Applied Probability (3)** Prerequisite: A calculus-based statistics course. The formulation, analysis and interpretation of probabilistic models. Selected topics in probability theory. Conditioning, Markov chains, and Poisson processes. Additional topics chosen from renewal theory, queueing theory, Gaussian processes, Brownian motion, and elementary stochastic differential equations.

**MAT 569. (STT 569) Stochastic Processes in Operations Research (3)** Prerequisite: MAT/STT 565. Probabilistic models with applications in operations research. Queueing theory, birth-death processes, embedded Markov chains, finite and infinite waiting room systems, single and multi-server queues, general service distributions; Markov decision processes; reliability.

**MAT 581. (481) Introduction to Mathematical Logic (3)** Prerequisite: Permission of department. The formal study of truth and provability. Propositional calculus; predicate calculus. Gdel's completeness theorem, applications to formal number theory and incompleteness. Additional topics chosen from areas such as undecidability or non-standard analysis.

**MAT 592. Advanced Topics in Mathematics (3)** Prerequisite: Consent of instructor. Advanced topics of current interest in pure and applied mathematics not covered in existing courses.

**MAT 595. Research Seminar (2)** Prerequisite: Consent of instructor. Designed to give the student experience in locating and learning mathematics outside the classroom setting. Use of the major mathematics journals, professional society publications and standard references including Mathematical Reviews. The nature of research in the mathematical sciences and research methodology.

**MAT 596. Research Project (1)** Corequisite: MAT 595. (Not intended for students who write a thesis in mathematics.) Under faculty supervision, each student presents a written exposition of the history, current knowledge, future directions, and bibliography of a mathematical topic.

**MAT 598 . Internship in the Mathematical Sciences** (1) Prerequisite: Permission of the graduate coordinator. Academic training and professional experience through work in a private company or public agency including a written final report. Faculty supervision and evaluation of all study and on-site activity. Grading will be satisfactory (S) or unsatisfactory (U).

**MAT 599. Thesis (1-4)**

STATISTICS

**STT 500. Research Consultation (1-3)** Prerequisite: Consent of instructor. Statistical consultation on graduate thesis research provided through access to the Department of Mathematics and Statistics' Statistical Consulting Center. May be repeated for a total of three credit hours.

**STT 505. Data Analysis (3)** Prerequisite: Any statistics course. Introduction to exploratory data analysis. Use of stem and leaf plots, boxplots. Transformations of data, resistant lines, analysis of two-way tables, residual analysis. Comparison of robust/resistant methods with standard statistical techniques.

**STT 511. (411) Design of Experiments and Analysis of Variance (3)** Prerequisite: Any elementary statistics course. Review of elementary statistics; design of experiments including completely randomized, randomized block, factorial, split-plot, and repeated measures designs; analysis of variance; non-parametric alternative methods of analysis. Statistical software packages will be used as appropriate in problem solving.

**STT 512. (412) Applied Regression and Correlation (3)** Prerequisite: Any elementary statistics course. Review of elementary statistics; linear and multiple regression; correlation. Statistical software packages will be used as appropriate in problem solving.

**STT 520. (420) Biostatistical Analysis (3)** Prerequisite: Statistical programming and consent of instructor. Statistical methods used in epidemiologic studies and clinical trials. Topics include measures of association, logistic regression, covariates, life tables and Cox regression; statistical analysis using SAS.

**STT 525. (425) Categorical Data Analysis (3)** Prerequisite: Statistical programming and consent of instructor. Introduction to the analysis of qualitative data. Basic methods of summary and inference for two and three way contingency tables; introduction to the generalized linear model for binary and Poisson data; focus on multinomial responses (nominal and ordinal) and matched pairs data; statistical analysis using SAS.

**STT 530. (430) Introduction to Non-parametric Statistics (3)** Prerequisite: A calculus-based statistics course. Theory and methods of non-parametric statistics in the one- and two-sample problems and their comparisons with standard parametric procedures. Non-parametric tests for comparing more than two samples; tests of randomness and independence.

**STT 535. (435) Applied Multivariate Analysis (3)** Prerequisite: STT 511, 512. Matrix manipulations; multivariate normal distribution; inference for mean vector and covariance matrix; multivariate analysis of variance; principal components; canonical correlations; discriminant analysis; factor analysis; cluster analysis; statistical analysis using SAS.

**STT 540. (440) Linear Models and Regression Analysis (3)** Prerequisite: A calculus-based statistics course. Theoretical introduction to the general linear model and its application to simple linear regression and multiple regression. Estimation and hypothesis testing of model coefficients; residual analysis; analysis of **covariance.**

**STT 565. (465) (MAT 565/465) Applied Probability (3)** Prerequisite: A calculus-based statistics course. The formulation, analysis and interpretation of probabilistic models. Selected topics in probability theory. Conditioning, Markov chains, and Poisson processes. Additional topics chosen from renewal theory, queueing theory, Gaussian processes, Brownian motion, and elementary stochastic differential equations.

**STT 566-567. (466-467) Mathematical Statistics (3-3)** Prerequisite: A calculus-based statistics course. A rigorous introduction to mathematical statistics. Univariate and multivariate probability distributions; conditional and marginal distributions; theory of estimation and hypothesis testing; limiting distributions and the central limit theorem; sufficient statistics and the exponential class of probability density functions.

**STT 569. (MAT 569) Stochastic Processes in Operations Research (3)** Prerequisite: MAT/STT 565. Probabilistic models with applications in operations research. Queueing theory, birth-death processes, embedded Markov chains, finite and infinite waiting-room systems, single and multi-server queues, general service distributions; Markov decision processes; reliability.

**STT 592. Topics in Statistics (3)** Prerequisite: Consent of instructor. Topics in statistics of current interest not covered in existing courses.