Professors: Larry P. Ammann, M. Ali Hooshyar, Louis R. Hunt, George Kimeldorf, Patrick L. Odell (Emeritus), Istvan Ozsvath, Ivor Robinson, Robert Serfling, John W. Van Ness, John Wiorkowski
Associate Professors: Raimund Ober, Janos Turi
Assistant Professors: Michael I. Baron, Tiberiu Constantinescu, Viswanath Ramakrishna
Associated Faculty: Thomas R. Butts (Science/Mathematics Education)
Senior Lecturers: Frank R. Allum, Joanna R. Robinson, H. Edward Stone
The Mathematical Sciences program at The University of Texas at Dallas offers graduate study in four majors: applied mathematics, engineering mathematics, mathematics, and statistics. The degree programs offer students the opportunity to prepare for careers in these disciplines themselves or in any of the many other fields for which these disciplines are such indispensable tools. As other sciences develop, problems which require the use of these tools are numerous and pressing.
In addition to a wide range of courses in mathematics and statistics, the Mathematical Sciences Program offers a unique selection of courses that consider theoretical and computational aspects of engineering and scientific problems. This orientation is enhanced by the activities of the Center for Engineering Mathematics in which faculty members of both the Program in Mathematical Sciences and the Erik Jonsson School of Engineering and Computer Science work in an interdisciplinary effort on engineering problems.
The Master of Science degree program is designed for persons seeking specializations in applied mathematics, engineering mathematics, mathematics, or statistics.
The Doctor of Philosophy degree program covers two basic areas of concentration: statistics, and applied mathematics. The curriculum also includes optional emphasis on applications to engineering problems.
(Administered by Science/Mathematics Education Program)
The Master of Arts in Teaching degree program, which stresses both the art of teaching and advanced knowledge of mathematics, is designed for persons who are teaching in grades 6-12. The Master of Science degree (above) is available for those teachers in grades 6-12 who wish to significantly increase their knowledge in the mathematical sciences. Persons who are teaching or plan to teach mathematics and/or mathematical sciences above the remedial level at a community college or at a college or university are strongly encouraged to pursue the Master of Science degree (as a minimum since an earned doctorate is sometimes required). For information concerning the Master of Arts in Teaching in Mathematics see Science/Mathematics Education.
Faculty and students in Mathematical Sciences have access to state-of-the-art scientific workstations and supercomputers. Faculty and staff offices in Mathematical Sciences are equipped with Sun Sparc-stations or X-terminals, and all Teaching Assistant offices are equipped with X-terminals, connected via Ethernet to the Mathematical Sciences' three Sun SPARCserver 10's. Two of the SPARCserver 10's are servers for the X-terminals, while the third SPARCserver 10 is configured for large computational projects. Mathematical Sciences students also have direct access to the Center for Engineering Mathematics, which has additional Sun Sparcstations and X-terminals. A large collection of mathematical and statistical software is maintained on the SPARCserver 10's for educational and research use. Mathematical Sciences also has access via the Academic Computer Center to Silicon Graphics graphical workstations and to The University of Texas System Cray Y-MP supercomputer.
The Center for Engineering Mathematics, a joint organization of the Programs in Mathematical Sciences, School of Natural Sciences and Mathematics, and the Erik Jonsson School of Engineering and Computer Science, provides additional facilities. Its purpose is to encourage research interaction between mathematical sciences and engineering.
(For general degree and admission requirements, see the sections headed "General Academic Regulations" beginning on page 22.) Specific degree requirements for students in Mathematical Sciences follow. Students lacking undergraduate prerequisites for graduate courses in their area must complete these prerequisites or receive approval from the graduate adviser and the course instructor before registering.
Students seeking a Master of Science in Mathematical Sciences must complete a total of 12 three-credit hour courses. In some cases a three-credit hour waiver is approved for good mathematics background. The student may choose a thesis plan or a non-thesis plan. In the thesis plan, the thesis replaces two elective courses with completion of an approved thesis (six thesis hours). The thesis is directed by a Supervising Professor and must be approved by the Head of the Mathematical Sciences Program.
Each student must meet a 3.3 minimum GPA requirement in the set of core courses listed below corresponding to the student's area of concentration, OR must earn a 3.0 minimum GPA in the courses listed for the student's program as a whole plus a 6000/7000 level approved elective course taken beyond the degree program requirements.
Students seeking a Master of Science in Mathematical Sciences with concentration in Applied Mathematics, Engineering Mathematics, or Mathematics must complete the following core courses:
MATH 6301 Real Analysis I
MATH 6302 Real Analysis II, or
MATH 6331 Linear Systems and Signals
MATH 6315 Ordinary Differential Equations I
MATH 6303 Theory of Complex Functions I
MATH 6311 Abstract Algebra I
MATH 6313 Numerical Analysis I
Additional requirements for each of the above concentrations are as follows.
A minimum of three courses from the following:
MATH 6314 Numerical Analysis II
MATH 6316 Ordinary Differential Equations II
MATH 7313 Partial Differential and Integral Equations I
MATH 7316 Wave Propagation with Applications
MATH 7317 Inverse Problems and Applications
MATH 7318 Numerical Analysis of Differential Equations
STAT 6341 Numerical Linear Algebra and Statistical Computing
A minimum of three courses from the following:
MATH 6332 Advanced Control
MATH 6336 Nonlinear Control Systems
MATH 6339 Control of Distributed Parameter Systems
STAT 7338 Time Series Modeling and Filtering
STAT 6347 Applied Time Series Analysis
MATH 7316 Wave Propagation with Applications
MATH 7317 Inverse Problems and Applications
A minimum of two courses from the following:
MATH 6304 Theory of Complex Functions II
MATH 6306 General Topology
MATH 7301 Differential Geometry
MATH 7313 Partial Differential and Integral Equations I
MATH 7319 Functional Analysis I
MATH 7320 Functional Analysis II
Students seeking a Master of Science in Mathematical Sciences with a major in Statistics must complete the following core courses:
STAT 6331 Statistical Inference I
STAT 6337-38 Statistical Methods I, II
STAT 6339 Linear Statistical Models
STAT 6341 Numerical Linear Algebra and Statistical Computing
and one course from each of any two of the following sets of courses:
{STAT 6348, STAT 7331} Multivariate Analysis
{STAT 6347, STAT 7338} Time Series Analysis
{STAT 6329, STAT 6343} Stochastic Processes or Experimental Design
Students must choose remaining courses from among the following electives:
MATH 6301 (strongly recommended), MATH 6302, MATH 6313, MATH 6331 or any 6300- 7300-level statistics courses. Also, a maximum of two of the following prerequisite 5000-level courses may be counted as electives: MATH 5301, 5302, Elementary Analysis I, II and STAT 5351, 5352 Probability and Statistics I, II.
The remaining required credit hours are electives approved by the graduate adviser. Typically these electives are 6000 and 7000 level mathematical sciences courses. Courses from other disciplines may also be used upon approval.
Substitutions for required courses may be made if approved by the graduate adviser. Instructors may waive stated prerequisites for students with equivalent experience.
Each Doctor of Philosophy degree program is tailored to the student. The student must arrange a course program with the guidance and approval of the graduate adviser. Adjustments can be made as the student's interests develop and a specific dissertation topic is chosen.
MATH 6301 Real Analysis I
MATH 6302 Real Analysis II, or MATH 6331 Linear Systems and Signals
STAT 6331 Statistical Inference I
STAT 6344 Probability Theory
MATH 6303 Theory of Complex Functions I
MATH 6306 General Topology
MATH 6311 Abstract Algebra I
MATH 6313 Numerical Analysis I
MATH 6315, 6316 Ordinary Differential Equations I, II
MATH 7319, 7320 Functional Analysis I, II
MATH 7313 Partial Differential and Integral Equations I
STAT 6332 Statistical Inference II
STAT 6337, 6338 Statistical Methods I, II
STAT 6339 Linear Statistical Models
STAT 7330 Decision Theory
STAT 7331 Multivariate Analysis
STAT 7334 Nonparametric Statistics
STAT 7338 Time Series Modeling and Filtering
STAT 7345 Stochastic Processes
MATH 6303 Theory of Complex Functions I, or MATH 6313 Numerical Analysis I, or MATH 6315 Ordinary Differention Equations I, or MATH 7319 Functional Analysis I
An additional 18-24 credit hours designed for the student's area of specialization are taken as electives in a degree plan designed by the student and the graduate adviser. This plan is subject to approval by the Program Head. After completion of the first 3 or 4 academic semesters of the course program, the student must pass a Ph.D. Qualifying Examination in order to continue on to the research and dissertation phase of the Ph.D. program. Finally, a dissertation is required and must be approved by the graduate program. Areas of specialization include:
Other specializations are possible, including interdisciplinary topics. There must be available a dissertation research adviser or group of dissertation advisers willing to supervise and guide the student. A dissertation Supervising Committee should be formed with at least four members from the Mathematical Sciences faculty. The dissertation may be in Mathematical Sciences exclusively or it may involve considerable work in an area of application.
(3 semester hours) Algebraic and analytical mathematics for mathematics in the social, behavioral and management sciences. The course also prepares for MATH 5404. No credit allowed to mathematical sciences majors. (3-0)
(3 semester hours) Real numbers, differentiation, integration, metric spaces, basic point set topology, power series, analytic functions, Cauchy's theorem. Prerequisite: calculus through multivariable calculus. (3-0)
(3 semester hours) Continuation of MATH 5301. Prerequisite: MATH 5301. (3-0)
(4 semester hours) Techniques of mathematical analysis applicable to the social, behavioral and management sciences. Graphical representations, differential and integral calculus of one and many variables. No credit allowed to mathematical sciences majors. Three lecture hours and two discussion hours a week. Prerequisite: MATH 5300 or equivalent. (3-1)
(3 semester hours) Topics in modern Euclidean geometry including distinguished points of a triangle, circles including the nine point circle, cross ratio, transformations; introduction to projective geometry. No credit allowed to mathematical sciences majors except those in M.A.T. program. Prerequisite: Junior level mathematics course. (3-0)
(3 semester hours) The relations among elliptic, Euclidean and hyperbolic geometries, Euclidean models of elliptic and hyperbolic geometries. No credit allowed to mathematical sciences majors except those in M.A.T. program. Prerequisite: Junior level mathematics course. (3-0)
(1-6 semester hours) ([1-6]-0)
(3 semester hours) Vector spaces, modules, linear transformations, dual spaces, groups, rings, fields. Prerequisite: Undergraduate linear algebra. (3-0)
(3 semester hours) Study of modern algebra involving groups, rings, fields and Galois theory. No credit allowed to mathematical sciences majors except those in M.A.T. program. Prerequisite: Junior level mathematics course. (3-0)
(3 semester hours) Measure theory and integration. Hilbert and Banach spaces. Fourier transforms. Prerequisites: Undergraduate analysis course or MATH 5301/5302; undergraduate course in linear algebra or MATH 5311. (3-0)
(3 semester hours) Continuation of MATH 6301. Prerequisite: MATH 6301. (3-0)
(3 semester hours) Complex integration, Cauchy's theorem, calculus of residues, power series, entire functions, Riemann mapping theorems. Riemann surfaces, Hardy spaces, interpolation theory, conformal mapping with applications. Prerequisite: Advanced calculus. (3-0)
(3 semester hours) Continuation of MATH 6303. Prerequisite: MATH 6303. (3-0)
(3 semester hours) Topological spaces, product and quotient spaces, compactness, connectedness, continuity, metric spaces, function spaces and fixed-point theorems. Prerequisite: Advanced calculus or MATH 5302. (3-0)
(3 semester hours) Basic properties of groups, rings, fields, and modules. Topics selected from group representations, rings with minimal condition, Galois theory, local rings, algebraic number theory, classical ideal theory, basic homological algebra, and elementary algebraic geometry. Prerequisite: Undergraduate algebra course or equivalent such as MATH 5311. (3-0)
(3 semester hours) A study of numerical methods including the numerical solution of non-linear equations, linear systems of equations, interpolation, iterative methods and approximation by polynomials. Prerequisites: Linear algebra and advanced calculus. (3-0)
(3 semester hours) Continuation of MATH 6313 including numerical differentiation, numerical integration, and numerical solutions of differential equations. Prerequisites: MATH 6313 and ordinary differential equations. (3-0)
(3 semester hours) The theory of ordinary differential equations with emphasis on existence, uniqueness, and stability. Prerequisites: undergraduate course in linear algebra or MATH 5311; undergraduate analysis course or MATH 5301/5302, and undergraduate course in ordinary differential equations. (3-0)
(3 semester hours) Continuation of MATH 6315. Prerequisite: MATH 6315. (3-0)
(3 semester hours) Introduction to theoretical and practical concepts of optimization using a functional analytic framework: Hilbert and Banach spaces, least-squares estimation, optimization of functionals, local and global theory of constrained optimization, iterative methods. Prerequisites: Ordinary differential equations and linear algebra. (3-0)
(3 semester hours) Basic principles of systems and control theory: state space representations, stability, observability, controllability, realization theory, transfer functions, feedback. Prerequisites: Undergraduate course in linear algebra or MATH 5311 and undergraduate analysis course or MATH 5301, 5302. (3-0)
(3 semester hours) Theoretical and practical aspects of modern control methodologies in state space and frequency domain, in particular LQG and H-infinity control: coprime factorizations, internal stability, Kalman filter, optimal regulator, robust control, sensitivity minimization, loop shaping, model reduction. Prerequisite: MATH 6331. (3-0)
(3 semester hours) Differential geometric tools, controllability, observability, stability, feedback linearization, output injection, input-output linearization. The theory will be used to discuss engineering applications. Prerequisites: MATH 6331, MATH 6315. (3-0)
(3 semester hours) Theoretical and technical issues for control of distributed parameter systems in the context of linear infinite dimensional dynamical systems: Evolution equations and control on Euclidean space, elements of functional analysis, semigroups of linear operators, abstract evolution equations, control of linear infinite dimensional dynamical systems, approximation techniques. Prerequisites: MATH 6316 and MATH 6331. (3-0)
(3 semester hours) Manifolds, Lie groups, fiber bundles and multilinear algebra. Prerequisite: Undergraduate analysis course. (3-0)
(3 semester hours) General theory of partial differential and integral equations, with emphasis on existence, uniqueness and qualitative properties of solutions. Prerequisites: MATH 6301 and MATH 6302. MATH 6315 recommended. (3-0)
(3 semester hours) Continuation of MATH 7313. Prerequisite: MATH 7313. (3-0)
(3 semester hours) Study of the wave equation in one, two and three dimensions, the Helmholtz equation, associated Green's functions, asymptotic techniques for solving the propagation problems with applications in physical and biomedical sciences and engineering. Prerequisites: MATH 6303, MATH 7313, and MATH 6314 or MATH 7318. (3-0)
(3 semester hours) Exact and approximate methods of nondestructive inference, such as tomography and inverse scattering theory in one and several dimensions, with applications in physical and biomedical sciences and engineering. Prerequisite: MATH 7316. (3-0)
(3 semester hours) Practical and theoretical aspects of numerical methods for both ordinary and partial differential equations are discussed. Topics to be covered include: initial value problems for Ordinary Differential Equations, two-point boundary value problems, projection methods, finite difference, finite element and boundary element approximations for Partial Differential Equations. Prerequisites: MATH 6314, MATH 6315, MATH 7313. (3-0)
(3 semester hours) Elements of operator theory, duality theory, spectral theory, Banach algebras, further topics. Prerequisites: MATH 6301/6302. MATH 6303 recommended. (3-0)
(3 semester hours) Continuation of MATH 7319. Prerequisite: MATH 7319. (3-0)
(1-6 semester hours) ([1-6]-0)
(1-6 semester hours) ([1-6]-0)
(1-9 semester hours) Open to students with advanced standing subject to approval of the Graduate Adviser. ([1-9]-0)
(3-9 semester hours) May be repeated for credit. ([3-9]-0)
(3-9 semester hours) May be repeated for credit. ([3-9]-0)
(1-6 semester hours) ([1-6]-0)
(1 semester hour) Introduction to use of major statistical packages such as SAS, BMD, and Minitab. Based primarily on self-study materials. No credit allowed to mathematical sciences majors. Prerequisite: One semester of statistics. (1-0)
(1 or 3 semester hours) Theory and methods of statistics used in management and business. Topics include: frequency distributions, measures of location, measures of variation, probability, Bayes theorem, sampling distributions, point and interval estimation, statistical decisions (hypotheses testing), correlation and regression. This course may only be taken by students seeking a management degree. Prerequisite: MATH 5304 or equivalent. ([1 or 3]-0)
(3 semester hours) Intermediate statistical theory and methods used in management and business. Emphasis on the concepts and use of linear statistical models. Topics include: multiple regression, analysis of variance and multiple comparisons, and analysis of covariance. Real-life problems will be presented to illustrate these methods and statistical computer packages will be used extensively to handle these problems. This course may only be taken by students seeking a management degree. Prerequisite: STAT 5311 or equivalent. (3-0)
(3 semester hours) A mathematical treatment of probability theory. Random variables, distributions, conditioning, expectations, special distributions and the central limit theorem. The theory is illustrated by numerous examples. This is a basic course in probability and uses calculus extensively. Prerequisite: Calculus through multivariable calculus. (3-0)
(3 semester hours) Theory and methods of statistical inference. Sampling, estimation, hypothesis testing, analysis of variance, and regression with applications. Prerequisite: STAT 5351. (3-0)
(3 semester hours) Statistical methods and theory with emphasis on applications in the social and natural sciences. Concepts of variability, sampling, point and interval estimation, testing hypotheses, correlation, regression, and analysis of variance. Use of a computer package. The concentration will be on applicability, appropriateness, and utility of statistics. This course may not be taken for credit by mathematical sciences majors. Prerequisite: College algebra. (3-0)
(3 semester hours) Designed for users of statistics. Emphasis on the appropriate use, utility and limitations of the methods discussed. Use of a computer package is stressed. Topics from applied multiple regression and correlation; residual analysis; multi-way analysis of variance; multi-way contingency table analysis, and analysis of covariance. Prerequisite: STAT 5353 or equivalent. This course may not be taken for credit by mathematical sciences majors. (3-0)
(3 semester hours) Application of statistical methods to the problem of controlling quality of production. Topics include control charts, sampling methods, and process control techniques. Prerequisite: STAT 5311, or STAT 5351, or equivalent. (3-0)
(3 semester hours) Introduction to survey sampling theory and methods. Simple random, stratified, systematic and cluster sampling. Estimation of means, proportions, variances, ratios, and other parameters for a finite population. Prerequisite: STAT 5351. (3-0)
(3 semester hours) Basic random processes used in stochastic modeling including Poisson, Gaussian, and Markov processes with an introduction to queuing theory. Measure theory not required. Prerequisite: Probability theory. (3-0)
(3 semester hours) Introduction to fundamental concepts and methods of statistical modeling and decision-making. Exponential families of models, sufficiency, estimation, hypothesis testing, likelihood methods, optimality, analysis of variance, linear models, nonparametric methods, decision theory. Prerequisites: Advanced calculus, STAT 5351 or equivalent and MATH 5302 or equivalent. MATH 6301 strongly recommended, before or concurrently. (3-0)
(3 semester hours) Continuation of STAT 6331. Prerequisites: STAT 6331 and STAT 6344 should be taken either before or concurrently. (3-0)
(3 semester hours) Statistical methods most often used in the analysis of data. Study of statistical models, including multiple regression, nonlinear regression, stepwise regression, balanced and unbalanced analysis of variance, analysis of covariance and log-linear analysis of multiway contingency tables. Prerequisites: Calculus and STAT 5352 or STAT 6331. (3-0)
(3 semester hours) Continuation of STAT 6337. Prerequisite: STAT 6337. (3-0)
(3 semester hours) Vectors of random variables, multivariate normal distribution, quadratic forms. Theoretical treatment of general linear models including the Gauss-Markov theorem, estimation, hypotheses testing, and polynomial regression. Introduction to the analysis of variance and analysis of covariance. Prerequisites: STAT 6331, and MATH 5311 or equivalent. (3-0)
(3 semester hours) A study of computational methods used in statistics. Topics to be covered include the simulation of stochastic processes, numerical linear algebra, and graphical methods. Prerequisite: STAT 5352 or STAT 6337. (3-0)
(3 semester hours) This course focuses on the planning, development, implementation and analysis of data collected under controlled experimental conditions. Repeated measures designs, Graeco-Latin square designs, randomized block designs, balanced incomplete block designs, partially balanced incomplete block designs, fractional replication and confounding. The course requires substantive use of computer facilities. Prerequisite: STAT 6338 or equivalent knowledge of fixed and random effects crossed ANOVA designs. (3-0)
(3 semester hours) A measure theoretic coverage of mathematical probability theory. Students are assumed to have had at least one semester of measure theory. Prerequisite: MATH 6301. (3-0)
(3 semester hours) Methods and theory for the analysis of data collected over time. The course covers techniques commonly used in both the frequency domain (harmonic analysis) and the time domain (autoregressive, moving average models). Prerequisite: STAT 6337 or STAT 6339 or equivalent. (3-0)
(3 semester hours) The most frequently used techniques of multivariate analysis. Topics include T/T2, MANOVA, principal components, discriminant analysis and factor analysis. Prerequisite: STAT 5352 or STAT 6331. (3-0)
(1-3 semester hours) Practical experience in collaboration with individuals who are working on problems which are amenable to statistical analysis. Problem formulation, statistical abstraction of the problem, and analysis of the data. Course may be repeated but a maximum of three hours may be counted toward the requirements for the master's degree. Prerequisite: Consent of instructor. ([1-3]-0)
(3 semester hours) Statistical decision theory and Bayesian inference are developed at an intermediate mathematical level. Prerequisites: Undergraduate analysis course and either STAT 6331 or STAT 6338. (3-0)
(3 semester hours) The multivariate normal distribution. Estimation and sampling distributions of estimators of parameters of a multivariate population. Derivation and distributions of likelihood ratio statistics for hypothesis tests including linear models MANOVA, sphericity and independence. Various methods of derivation of null hypothesis distributions are examined. Prerequisite: STAT 6331 or equivalent. (3-0)
(3 semester hours) Nonparametric and robust methods of statistics: order statistics, rank tests, M-estimates, L-statistics, and goodness of fit. Prerequisite: STAT 6331 or equivalent. (3-0)
(3 semester hours) Theory of correlated observations observed sequentially in time. Stationary processes, power spectra, stationary models fitting, correlation analysis and regression. Prerequisite: STAT 6331 or equivalent. (3-0)
(3 semester hours) Main topics include Kolmogorov's existence theorem, Markov Processes, Martingales, and Brownian motion. Prerequisite: STAT 6344. (3-0)
(1-6 semester hours) ([1-6]-0)
(1-6 semester hours) ([1-6]-0)
(1-9 semester hours) Open to students with advanced standing, subject to approval of the graduate adviser. ([1-9]-0)
(3-9 semester hours) (May be repeated for credit.) ([3-9]-0)
(3-9 semester hours) (May be repeated for credit.) ([3-9]-0)