Sciences

*Professor Jane L. Winer, Dean*

**Department of Mathematics
and Statistics**

Professor Ronald M. Anderson, Chairperson.

Horn Professor and Ex-Students Association Distinguished Professor Martin; Professors Barnard, Bennett, Chanda, Dayawansa, Duran, Ford, Gilliam, Gustafson, Harris, Lewis, McLaughlin, Newman, Pearce, Ruymgaart, Schovanec, Strauss, and Victory; Associate Professors E. Allen, L. Allen, Byerly, Cordero, Drager, Lee, Mansouri, Mitra, M. Shubov, V. Shubov, Wang, Weinberg, White, Yang, and Zhang; Assistant Professors Avalos, Dunyak, Gornet, Korchagin, Seaquist, and Sun; Visiting Assistant Professor Raquepas.

This department offers study in the following graduate degree programs: MATHEMATICS,
*Master of Arts, Master of Science, *and* Doctor of
Philosophy*; STATISTICS, *Master of
Science*.

Students seeking an advanced degree in mathematics or statistics should consult with the Director of Graduate Studies in Mathematics and Statistics before enrolling in any courses. The department offers a number of graduate courses that are suitable for students who wish to complete a minor in mathematics or statistics.

The Department of Mathematics and Statistics does not have a foreign language requirement for the master's degree. Any foreign language requirement for the Ph.D. degree will be at the discretion of the student's dissertation advisor.

The M.A. degree in mathematics consists of 36 hours of graduate work, including 3 hours of credit for a departmental report. The student must complete three sequences chosen from algebra, analysis, geometry, probability and statistics, number theory, applied mathematics, and computer literacy and programming. This degree is offered primarily for those students who wish to teach mathematics at the preuniversity level.

The M.S. degree in mathematics consists of 36 hours of graduate work, including 3 hours of credit for a departmental report, or 30 hours of graduate work including 6 hours of credit for the master's thesis. The student must complete at least two of the core sequences listed on the Ph.D. program for the 36-hour plan and at least one of the core sequences for the 30-hour plan. In the 36-hour plan a minor of 9 hours is permitted and in the 30-hour plan a minor of 6 hours is permitted. In each case the minor must be approved by the graduate advisor.

A M.S. degree in mathematics with emphasis in computer science is also offered. The degree consists of 36 hours with 3 hours of credit for a departmental report. This plan calls for 18-21 hours of graduate course work in mathematics and 12-15 hours of graduate course work in computer science. Of the 18-21 hours of mathematics course work, at least two sequences from the list in the departmental handbook must be completed. The 12-15 hours of computer science course work constitute adjunct requirements and must be approved by the graduate advisor.

The M.S. degree in statistics consists of 36 hours of graduate work including 3 hours of credit for a departmental report or 6 hours of credit for the master's thesis. Up to 6 hours of computer science may be counted toward this degree, or 6 to 9 hours of graduate work in other areas such as agriculture, biology, business, economics, engineering, psychology, sociology, or fields as approved by the graduate advisor.

Each doctoral student will undergo a preliminary examination as early as possible during graduate training. The examinations will be administered annually in May and the results evaluated by the Graduate Programs and Policies Committee of the Mathematics and Statistics Department. Details concerning the preliminary examinations can be found in the departmental handbook. Each doctoral student must also pass a qualifying examination in a specialty area.

Each degree plan must be approved by the graduate advisor.

**Courses in Mathematics. (MATH)
**

**5101. Seminar in Algebra (1:1:0). ** Discussion of current research and topics of interest in algebra. Must be taken pass-fail. May
be repeated for credit.

**5102. Seminar in Analysis (1:1:0). **Discussion of current research and topics of interest in analysis. Must be taken pass-fail. May
be repeated for credit.

**5103. Seminar in Control Theory (1:1:0).
**Discussion of current research and topics of interest in control theory. Must be taken
pass-fail. May be repeated for credit.

**5104. Seminar in Statistics (1:1:0).
**Discussion of current research and topics of interest in statistics. Must be taken pass-fail. May
be repeated for credit.

**5105. Seminar in Topology (1:1:0). **Discussion of current research and topics of interest in topology. Must be taken pass-fail. May
be

repeated for credit.

**5106. Seminar in Applied Mathematics (1:1:0).
**Discussion of current research and topics of interest in applied mathematics. Must
be taken pass-fail. May be repeated for credit.

**5107. Seminar in Biomathematics (1:1:0).
**Discussion of current research and topics of interest in biomathematics. Must be taken
pass-fail. May be repeated for credit.

**5304. Applied Mathematics for Behavioral and Management Sciences I (3:3:0).
**Topics cover the calculus of a single variable.
Applications emphasized. Computing is introduced and utilized. Not for mathematics, physical science, or engineering majors.

**5305. Applied Mathematics for Behavioral and Management Sciences II (3:3:0).
**Topics cover the calculus of several variables.
Introduction to probability theory and linear algebra. Applications emphasized. Computers and symbolic manipulator software
are utilized. Not for mathematics, physical science, or engineering majors.

**5310. Principles of Classical Applied Analysis I (3:3:0).
**Hilbert space theory, Green's functions, Sturm-Liouville theory, calculus
of variations, and integral equations.

**5311. Principles of Classical Applied Analysis II (3:3:0).
**Functions of a complex variable, Fourier and Laplace transforms, and
partial differential equations.

**5312. Control Theory I (3:3:0). ** Prerequisite: MATH 2360, 3354, 4351, or consent of instructor. Linear dynamical systems,
stability, frequency response and Laplace transform, feedback, state-space description, and geometric theory of linear systems.

**5313. Control Theory II (3:3:0). ** Prerequisite: MATH 5312, 5316, 5318, or consent of instructor. Quadratic regulator for linear
systems, Kalman filtering, nonlinear systems, stability, local controllability, and geometric theory of nonlinear systems.

**5316. Applied Linear Algebra (3:3:0).
**Prerequisite: Consent of instructor. Solution of linear systems, matrix inversion, vector
spaces, projections, determinants, eigenvalues and eigenvectors, Jordan form, computational methods, and applications.

**5318, 5319. Intermediate Analysis I, II (3:3:0 each).
**The real number system, introduction to metric spaces, sequences,
continuity, differentiation, Riemann integration, power series, functions of several variables, and differential forms.

**5320, 5321. Functions of a Complex Variable I, II (3:3:0 each).
**Prerequisite: MATH 4350 or 4356. Analytic functions
as mappings; Cauchy theorems, Laurent series, maximum modulus theorems and ramifications; normal families; Riemann
mapping theorem; Weierstrass factorization theorem; Mittag-Leffler theory; analytic continuation; and harmonic functions.

**5322, 5323. Functions of a Real Variable I, II (3:3:0 each).
**Prerequisite: MATH 5319 or equivalent. General measure
and integration theory, Lp theory, differentiation theory, and basic functional analysis.

**5324, 5325. Topology I, II (3:3:0 each).
**Prerequisite: MATH 4350 or consent of instructor. Point set theory; introduction
to combinatorial topology and homology theory.

**5326, 5327. Modern Algebra I, II (3:3:0 each).
**Prerequisite: MATH 3360 or consent of instructor. Groups; rings; fields;
linear algebra; Galois theory.

**5330, 5331. Theory of Ordinary Differential Equations I, II (3:3:0 each).
**Prerequisite: MATH 4351, 4354, or consent of instructor. Existence and uniqueness results, continuation of solutions, continuous dependence on data, linear equations,
oscillation and comparison theorems, boundary value problems, and stability analysis.

**5332, 5333. Partial Differential Equations I, II (3:3:0 each).
**Prerequisite: MATH 4351, 4354, or consent of instructor.
Topics include first order equations, method of characteristics, parabolic, hyperbolic and elliptic equations, variational and Hilbert
space methods.

**5334, 5335. Numerical Analysis I, II (3:3:0 each).
**Prerequisite: MATH 5316 or equivalent. Stability and error analysis;
numerical solution of ordinary and partial differential equations; integral equations.

**5340, 5341. Functional Analysis I, II (3:3:0 each).
**Prerequisite: MATH 5322. Hilbert and Banach space theory, linear
operator theory, the closed graph theorem, the open mapping theorem, the principle of uniform boundedness, linear functionals, dual
spaces and weak topologies, distribution theory, topological vector spaces, spectral theory of compact and unbounded self-adjoint and
unitary operators, and semigroup theory.

**5342, 5343. Advanced Topics in Analysis I, II (3:3:0 each).
**Prerequisite: Consent of instructor. Current topics in analysis. May
be repeated for credit.

**5344, 5345. Topics in Numerical Analysis I, II (3:3:0 each).
**Prerequisite: MATH 5335. Current advanced topics in
numerical analysis; research work using computers. May be repeated for credit.

**5346. Advanced Topics in Applied Mathematics I (3:3:0).
**Prerequisite: Consent of instructor. Current topics in applied
mathematics. May be repeated for credit.

**5347. Advanced Topics in Control Theory (3:3:0).
** Prerequisite: Consent of instructor. H theory of linear and nonlinear
systems, stochastic control, geometric theory of nonlinear systems, distributed parameter control systems, and computational methods
in control.

**5350. Mathematical Fluid Dynamics (3:3:0).
**Prerequisite: Consent of instructor. Lagrangian and Eulerian descriptions; conservation
of mass, circulation, potential flows, vorticity; conservation of momentum, Newtonian fluids, inviscid flow; conservation of energy,
compressible flow, Navier-Stokes equations.

**5354. Biomathematics I (3:3:0). **Prerequisite: Differential equations and linear algebra or consent of instructor. Qualitative and
quantitative behavior of deterministic biological models are studied.

**5355. Biomathematics II (3:3:0). **Prerequisite: Statistics, differential equations, and linear algebra or consent of instructor.
Qualitative and quantitative behavior of stochastic biological models are studied.

**5356. Topics in Biomathematics (3:3:0).
**Prerequisite: Biomathematics II or consent of instructor. Current topics in biomathematics are
studied such as biomechanics, mathematical epidemiology, mathematical neurology, mathematical opthalmology, and image processing. May be
repeated for credit.

**5360, 5361. Advanced Mathematics for Teachers I, II (3:3:0 each).
**Prerequisite: Consent of instructor. Selected topics
in mathematics. May be repeated for credit.

**5362. Theory of Numbers (3:3:0). **Prerequisite: MATH 4362. Diophantine equations; binary quadratic forms; algebraic
numbers; theory of number-theoretic functions; partitions; the prime number theorem.

**5364, 5365. Computer Literacy and Programming I, II (3:3:0 each).
**Development of computer literacy and programming
ability, algorithms and data structures, and recursion.. A first course for in-service teachers and/or behavioral and management sciences.
Not for engineering students.

**5371. Intermediate Probability Theory (3:3:0).
**Prerequisite: MATH 4342, STAT 5328 or consent of instructor. Random
walks; Markov chains; recurrent events; elementary queuing theory; lattice random variables.

**5382, 5383. Advanced Probability I, II (3:3:0 each).
**Prerequisite: MATH 5319 or consent of instructor. Measure and
integration; axiomatic foundations of probability theory; random variables; distributions and their characteristic functions; stable and
infinitely divisible laws; limit theorems for sums of independent random variables; conditioning; Martingales.

**5399. Advanced Problems (3). **Prerequisite: Graduate standing in mathematics. May be repeated for credit.

**6000. Master's Thesis (V1-6).
**

**6310. Master's Report (3).
**

**7000. Research (V1-12).
**

**8000. Doctor's Dissertation (V1-12).
**

**Courses in Statistics. (STAT)
**

**5302, 5303. Applied Statistics I, II (3:3:0 each).
**Prerequisite: Consent of instructor. Graphical presentation of data,
histograms, confidence intervals for binomial probabilities, one-sample and two-sample t-test, regression and correlation with two variables,
hypothesis testing and confidence intervals; multivariate regression and correlation, partial correlation coefficients, analysis of
variance and covariance, multiple comparison procedures. Emphasis on analysis of research data. Not for mathematics, statistics,
engineering, or physical science majors; these students should take STAT 5384, 5385.

**5328, 5329. Intermediate Mathematical Statistics I, II (3:3:0 each).
**Prerequisite: MATH 2350 or consent of instructor.
Probability space; special families of distribution functions; expectations; conditional distributions; sampling distributions; point and interval
estimation; hypothesis testing; distribution of functions of random variables; regression; nonparametric techniques.

**5370. Decision Theory (3:3:0). **Prerequisite: MATH 4343 or STAT 5329 or consent of instructor. Game theory; statistical
decision; Bayesian statistics.

**5372. Nonparametric Statistical Inference (3:3:0).
**Prerequisite: MATH 4343 or STAT 5329 or consent of instructor.
Statistical inference, rank order statistics; chi-square and slippage tests; Kolmogorov and Smirnov type tests; confidence intervals and bands;
runs tests; applications.

**5373. Design of Experiments (3:3:0).
**Prerequisite: MATH 4343 or STAT 5329 Principles of design and analysis of experiments;
Latin squares; split plots; incomplete block designs; efficiency.

**5374. Theory of Linear Statistical Models (3:3:0).
**Prerequisite: MATH 4343 or STAT 5329. Multivariate normal;
convariance matrix and operations; distribution of quadratic forms; general linear hypothesis of full and non-full rank; specific linear models.

**5375. Statistical Multivariate Analysis (3:3:0).
**Prerequisite: STAT 5329 or consent of instructor. Multivariate normal
distribution; estimation of the mean vector and covariance matrix; distribution of sample correlation coefficients; the generalized
T2 statistic; classification; distribution of the sample covariance matrix.

**5376. Advanced Statistical Methods (3:3:0).
**Prerequisite: MATH 4343 or STAT 5329 or consent of instructor. Applied
regression analysis; cluster analysis; factor analysis; modeling; special topics in designs; sensitivity analysis; nonlinear estimation. May be
repeated for credit.

**5377. Statistical Sampling Theory (3:3:0).
**Prerequisite: MATH 4343 or STAT 5329. Theory of simple random sampling;
stratified random sampling; cluster sampling; ratio estimates; regression estimates; other sampling methods.

**5378. Stochastic Processes (3:3:0).
**Prerequisite: STAT 5329. Markov chains; Markov processes in discrete and continuous
time; diffusion processes; Brownian motion and transformations of Brownian motion; non-Markovian processes.

**5379. Time Series Analysis (3:3:0).
**Prerequisite: STAT 5329 or consent of instructor. Stationary and nonstationary time series;
finite linear models; identification, filtering, and diagnostic checks of such models; spectral analysis of time series data; forecasting
and control.

**5380, 5381. Advanced Mathematical Statistics I, II (3:3:0 each).
**Prerequisite: STAT 5329; STAT 5380 is prerequisite for
STAT 5381. Theory of estimation and tests of statistical hypotheses; sequential analysis.

**5384. Statistics for Engineers and Scientists I (3:3:0).
**Prerequisite: MATH 2350 or consent of instructor. Probability,
descriptive statistics, distributions, estimation, hypothesis testing, nonparametric statistics, data analysis using the computers. Not for
mathematics or statistics majors.

**5385. Statistics for Engineers and Scientists II (3:3:0).
** Prerequisite: STAT 5384 or consent of instructor. Continuation of
STAT 5384; simple and multiple regression analysis; analysis of variance; nonparametric statistics; categorical data analysis; quality
control; reliability; data analysis using the computer. Not for mathematics or statistics majors.

**5386. Statistical Computing and Simulation (3:3:0).
** Prerequisite: Consent of instructor. Methods of approximating functions
and probabilities; computational methods in linear algebra; introduction to theory and applications of random number generation;
testing generators.

**6000. Master's Thesis (V1-6).
**

**6310. Master's Report (3).
**

**7000. Research (V1-12).**

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LAST UPDATE: 12-8-97