Mathematical Sciences

Graduate Courses

The Panther Web Access System offers up to date information on the course descriptions and course schedule each semester as well. Course offerings:

Applied Mathematics

Operations Research

Applied Mathematics

MTH 5007 INTRODUCTION TO OPTIMIZATION (3 credits). An applied treatment of modeling, analysis and solution of deterministic (e.g., nonprobabilistic) problems. Topics include model formulation, linear programming, network flow, discrete optimization and dynamic programming. (Requirement: At least one upper-level undergraduate math course.)

MTH 5009 INTRODUCTION TO PROBABILISTIC MODELS (3 credits). An applied treatment of modeling, analysis and solution of problems involving probabilistic information. Topics chosen from decision analysis, inventory models, Markov chains, queuing theory, simulation, forecasting models and game theory. (Requirement: Instructor approval or prerequisite course.) Prerequisites: MTH 2401.

MTH 5050 SPECIAL TOPICS (3 credits). Contents may vary depending on the needs and interests of the students and the fields of expertise of the faculty. (Requirement: Instructor approval.)

MTH 5051 APPLIED DISCRETE MATHEMATICS (3 credits). Logic fundamentals, induction, recursion, combinatorial mathematics, discrete probability, graph theory fundamentals, trees, connectivity and traversability. Applications from several fields of science and engineering, including computer science, operations research, and computer and electrical engineering. Prerequisites: MTH 2051.

MTH 5070 EDUCATIONAL STATISTICS (3 credits). Includes sampling procedures, frequency distributions, measures of central tendency, estimation of variability, the normal distribution, differences between two groups, analysis of variance and correlation. Also includes nonparametric techniques, multivariate techniques and computer analysis of educational data.

MTH 5101 INTRODUCTORY ANALYSIS (3 credits). Rigorous treatment of calculus. Includes sequences and series of real numbers, limits of functions, topology of the real line, continuous functions, uniform continuity, differentiation, Riemann integration, sequences and series of functions, Taylor’s theorem, uniform convergence and Fourier series. Prerequisites: MTH 2001, MTH 2201.

MTH 5102 LINEAR ALGEBRA (3 credits). Linear algebra, systems of linear equations and Gauss elimination method; inverses, rank and determinants; vector spaces; linear transformations, linear functional and dual spaces; eigenvalues, eigenvectors; symmetric, Hermitian and normal transformations; and quadratic forms. (Requirement: Undergraduate course in multivariable calculus or linear algebra.)

MTH 5107 OPTIMIZATION MODELS AND METHODS (3 credits). Surveys popular optimization models and algorithms. Topics chosen from linear, integer, nonlinear, dynamic and combinatorial optimization. (Requirement: At least one upper-level undergraduate math course.)

MTH 5111 REAL VARIABLES 1 (3 credits). Studies basic topology, continuous and semicontinuous functions, metric spaces, differentiation, measures, product measure, Lebesgue integration, Radon-Nikodym Theorem, Lp-spaces and measures on topological spaces. Prerequisites: MTH 5101.

MTH 5112 REAL VARIABLES 2 (3 credits). Studies basic topology, continuous and semicontinuous functions, metric spaces, differentiation, measures, product measure, Lebesgue integration, Radon-Nikodym Theorem, Lp-spaces and measures on topological spaces. Prerequisites: MTH 5111.

MTH 5115 FUNCTIONAL ANALYSIS (3 credits). Banach spaces, Hilbert spaces, topological vector spaces, bounded and unbounded linear operators, spectral theory. Prerequisites: MTH 5101.

MTH 5125 APPLIED COMPLEX VARIABLES (3 credits). Analytic functions, Cauchy-Reimann equations, contour integration, Cauchy theorem, Cauchy integral formula, Taylor and Laurent series, residue theorem and applications, linear fractional transformations, conformal mapping, Schwarz-Christoffel transformation. Inversion integral for Laplace transform with complex argument; inverse Laplace transforms. Prerequisites: MTH 2001, MTH 2201.

MTH 5130 THEORY OF COMPLEX VARIABLES (3 credits). Topology of the complex plane, analytic functions, Cauchy’s integral formula, Liouville’s theorem, maximum modulus theorem, Taylor and Laurent series, singularities, residue theorem, analytic continuation, entire functions, infinite product representation and conformal mapping. Prerequisites: MTH 2201, MTH 4101.

MTH 5201 MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING 1 (3 credits). Fourier series and their convergence properties; Sturm-Liouville eigenfunction expansion theory; Bessel and Legendre functions; solution of heat, wave and Laplace equations by separation of variables in Cartesian coordinates. Prerequisites: MTH 2001, MTH 2201.

MTH 5202 MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING 2 (3 credits). Solution of heat, wave and Laplace equations by separation of variables in cylindrical and spherical coordinates. Associated Legendre functions, hypergeometric functions and spherical harmonics. Fourier transforms and separation of variables for heat and wave equations on infinite intervals. Vector integral calculus. Prerequisites: MTH 5201.

MTH 5203 MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING 3 (3 credits). General perturbation techniques for linear and nonlinear ordinary differential equations, boundary layer theory, WKB methods, multiple scale analysis, approximate methods of solution, asymptotic expansion of integrals, asymptotic power series solutions of linear ODEs near irregular singular points. Prerequisites: MTH 5125, MTH 5201.

MTH 5220 THEORY OF ORDINARY DIFFERENTIAL EQUATIONS (3 credits). Includes basic existence theory, differential and integral inequalities,qualitative and quantitative theory, and Lyapunov’s second method. Prerequisites:MTH 2201, MTH 4101.

MTH 5230 PARTIAL DIFFERENTIAL EQUATIONS (3 credits). Includes the Hamilton-Jacobi equation; and elliptic, parabolic and hyperbolic problems, Green function methods, transform methods, maximum principle. Prerequisites: MTH 2001, MTH 2201, MTH 4101.

MTH 5301 NUMERICAL ANALYSIS (3 credits). Includes Gaussian elimina¬tion and solution of linear systems of equations, root finding methods, systems of nonlinear equations, interpolation, numerical integration, initial value problems for ODEs and fast Fourier transform. Prerequisites: CSE 1502 or CSE 1503 or CSE 2050, MTH 2201.

MTH 5305 NUMERICAL LINEAR ALGEBRA (3 credits). Covers iterative methods of solution of systems of linear equations, numerical methods for computing eigenvalues and eigenvectors, and singular value methods for least squares problems. Prerequisites: MTH 5301.

MTH 5310 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS (3 credits). Numerical methods for initial value problems, boundary value problems and eigenvalue problems for ordinary differential equations. Runge-Kutta methods, multistep and adaptive methods, stiff equations and A-stable methods, collocation. Prerequisites: MTH 5301.

MTH 5315 NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS (3 credits). Covers finite difference and finite element methods for partial differential equations. Prerequisites: MTH 3201, MTH 5301.

MTH 5320 NEURAL NETWORKS (3 credits). Introduces architectures, algorithms and applications. Includes single and multilayer perceptrons, counterpropagation, Kohonen self-organization, adaptive resonance theory, neocognition, probabilistic neural networks and Boltzmann machines with and without learning, recurrent neural networks. Prerequisites: CSE 1502 or CSE 1503 or CSE 2050, MTH 2201.

MTH 5401 APPLIED STATISTICAL ANALYSIS (3 credits). Covers statistical distributions, statistical tests for data, least squares and regression, estimations, tests of hypotheses, analysis of variance, planning and designing research experiments, randomized blocks, Latin and Graeco-Latin squares and data reduction, analysis using ANOVA (analysis of variance) and other methods. Prerequisites: MTH 2001.

MTH 5411 MATHEMATICAL STATISTICS 1 (3 credits). Covers discrete and continuous random variables, generating and moment generating functions, multivar¬iate distributions, covariance and correlation, sums of independent random variables, conditional expectation, Central Limit Theorem, Markov and Chebyshev inequalities and the Law of Large Numbers. (Requirement: Undergraduate courses in multivariable calculus and linear algebra.)

MTH 5412 MATHEMATICAL STATISTICS 2 (3 credits). Includes maximum likelihood and Bayes estimators, confidence intervals, testing hypotheses, uniformly most powerful tests, nonparametric methods (chi-square and Kolmogorov-Smirnov goodness-of-fit tests) and regression analysis. Prerequisites: MTH 5411.

MTH 5420 THEORY OF STOCHASTIC PROCESSES (3 credits). Includes discrete- and continuous-time stochastic processes, point and counting processes and Poisson counting process; as well as compound Poisson process, nonstationary Poisson process, renewal theory, regenerative processes and Markov chains. Prerequisites: MTH 5411.

MTH 5425 THEORY OF STOCHASTIC SIGNALS (3 credits). Covers univariate and multivariate distributions, generating and moment generating functions; autocorrelation, wide-sense, strict-sense stationary, voltage, Poisson, Wiener, random telegraph signal and white noise processes; Direc delta function, Fourier transform, system response, transfer function and spectral analysis. (Requirement: Instructor approval.)

MTH 5430 QUEUING THEORY (3 credits). Includes queuing processes; imbedded and continuous time parameter processes; Markov, semi-Markov and semi-regenerative processes; single-server and multiserver queues; and processes of servicing unreliable machines. Controlled stochastic models. Prerequisites:MTH 5411.

MTH 5434 STOCHASTIC ANALYSIS OF FINANCIAL MARKETS 1 (3 credits). Lays the foundation for mathematical concepts widely applied in financial markets. Uses economical theory with stochastics (martingales, Wiener, Markov, Ito processes, stochastic differential equations) to derive fair option prices and to hedge call options. Also uses fluctuation theory to predict stocks’ crossing of critical levels. Prerequisites: MTH 5411 or MTH 5425.

MTH 5436 STOCHASTIC ANALYSIS OF FINANCIAL MARKETS 2 (3 credits). Offers multidimensional stochastics applied to financial markets. Continues with multivariate Ito processes and multidimensional Feynman-Kac theorems, hedging of American and exotic call options and forward exchange rates. Introduces time-sensitive analysis of stocks, and risk theory. Prerequisites: MTH 5434 or ORP 5025.

MTH 5899 FINAL SEMESTER THESIS (0-2 credits). Variable registration for thesis completion after satisfaction of minimum registration requirements. (Requirements: Accepted petition to graduate and approval by Office of Graduate Programs.)

MTH 5999 THESIS (3-6 credits). Individual work under the direction of a member of the graduate faculty on a selected topic in the field of mathematics. (Requirement: Instructor approval.)

MTH 6050 RESEARCH IN APPLIED MATHEMATICS (1-6 credits). Research conducted under the guidance of a member of the faculty in a selected area of mathematics. (Requirement: Instructor approval.)

MTH 6100 SELECTED TOPICS IN NONLINEAR ANALYSIS (3 credits). Advanced topics in nonlinear analysis emphasizing recent developments. May vary depending on the needs and interests of the student and the fields of expertise of the faculty. (Requirement: Instructor approval.)

MTH 6200 SELECTED TOPICS IN APPLIED ANALYSIS (3 credits). Advanced topics in applied analysis emphasizing recent developments. May vary depending on the needs and interests of the student and the fields of expertise of the faculty. (Requirement: Instructor approval.)

MTH 6300 SELECTED TOPICS IN NUMERICAL AND COMPUTATIONAL MATHEMATICS (3 credits). Advanced topics in numerical and computational mathematics with emphasis on recent developments. May vary depending on the needs and interests of the student and the fields of expertise of the faculty. (Requirement: Instructor approval.)

MTH 6350 SPECIAL TOPICS IN PARALLEL PROCESSING (3 credits). Specific contents vary, but focuses on selected aspects of parallel processing algo¬rithms and architectures. (Requirement: Instructor approval.)

MTH 6899 FINAL SEMESTER DISSERTATION (0-2 credits). Variable registration for dissertation completion after satisfaction of minimum registration requirements. (Requirements: Accepted candidacy and approval by Office of Graduate Programs.)

MTH 6999 DISSERTATION RESEARCH (3-12 credits). Research and prepara¬tion of the doctoral dissertation. (Requirement: Instructor approval.)

 

Operations Research Program

ORP 5001 DETERMINISTIC OPERATIONS RESEARCH MODELS (3 credits). An applied treatment of modeling, analysis and solution of deterministic operations research problems. Includes model formulation, linear programming, network flow and transportation problems and algorithms, integer programming and dynamic programming. (Requirement: At least one upper-level undergraduate math course.)

ORP 5002 STOCHASTIC OPERATIONS RESEARCH MODELS (3 credits). An applied treatment of modeling, analysis and solution of probabilistic operations research problems. Topics chosen from decision analysis, game theory, inventory models, Markov chains, queuing theory, simulation, forecasting models. (Requirement: At least one upper-level undergraduate math course, preferably probability and statistics.)

ORP 5003 OPERATIONS RESEARCH PRACTICE (3 credits). Includes OR methodology, how an OR analyst interacts with clients, and preparation and presenta¬tion of oral reports. Students form teams to analyze real cases where each student gets an opportunity to be a team leader and present oral reports. Prerequisites: ORP 5001, ORP 5002.

ORP 5010 MATHEMATICAL PROGRAMMING (3 credits). Surveys popular optimization techniques. Topics chosen from linear, integer, nonlinear, dynamic and network flow programming; combinatorial graph algorithms. (Requirement: Prerequisite course or instructor approval.) Prerequisites: MTH 5102 or ORP 5001.

ORP 5011 DISCRETE OPTIMIZATION (3 credits). Studies combinatorial optimization and integer programming. Prerequisites: MTH 5051, ORP 5001.

ORP 5020 THEORY OF STOCHASTIC PROCESSES (3 credits). Introduces stochastic models, discrete- and continuous-time stochastic processes, point and counting processes, Poisson counting process, compound Poisson processes, nonsta¬tionary Poisson processes, renewal theory, regenerative processes and Markov chains. (Requirement: Instructor approval or prerequisite course.) Prerequisites: MTH 5411.

ORP 5021 QUEUING THEORY (3 credits). Includes queuing processes; imbedded and continuous time parameter processes; Markov, semi-Markov and semi-regenerative processes; single-server and multiserver queues; processes of servicing unreliable machines and computer applications; and controlled stochastic models. (Requirement: Instructor approval or prerequisite course.) Prerequisites: MTH 5411.

ORP 5025 STOCHASTIC ANALYSIS OF FINANCIAL MARKETS 1 (3 credits). Lays the foundation for mathematical concepts widely applied in financial markets. Uses economic theory with stochastics (martingales, Wiener, Markov, Ito processes, stochastic differential equations) to derive fair option prices and hedge call options. Also uses fluctuation theory to predict stocks’ crossing of critical levels. Prerequisites: MTH 5411 or MTH 5425.

ORP 5026 STOCHASTIC ANALYSIS OF FINANCIAL MARKETS 2 (3 credits). Offers multidimensional stochastics applied to financial markets. Continues with multivariate Ito processes and multidimensional Feynman-Kac theorems, hedging of American and exotic call options and forward exchange rates. Introduces time-sensitive analysis of stocks, and risk theory. Prerequisites: MTH 5435 or ORP 5025.

ORP 5030 DECISION ANALYSIS (3 credits). Covers normative models of decisions under certainty, risk and uncertainty; assessment of subjective probability and utility functions; Bayesian decision analysis and the value of information; influence diagrams; and descriptive aspects of decision making. (Requirement: Undergraduate statistics course.)

ORP 5031 MULTIOBJECTIVE DECISION ANALYSIS (3 credits). Covers normative models of decisions considering multiobjective and multiattribute models. Includes multiattribute utility theory, the analytical hierarchy process, linear multiobjective programming and goal programming. Prerequisites: ORP 5001, ORP 5030.

ORP 5040 QUALITY ASSURANCE (3 credits). Covers the principles and application of statistical quality control and statistical process control. (Requirement: Undergraduate statistics course.)

ORP 5041 RELIABILITY ANALYSIS (3 credits). Covers the principles of reliability analysis and assessment; reliability probability models; combinatorial and system reliability; and reliability estimation. (Requirement: Instructor approval or prerequisite course.) Prerequisites: MTH 5411.

ORP 5042 RELIABILITY, AVAILABILITY AND MAINTAINABILITY (3 credits). Discusses maintainability concepts relating to system effectiveness and support-system design. Includes basic mathematical concepts, design concepts and data analysis used in quantifying availability, maintainability and reliability as measures of operational readiness and system effectiveness. Prerequisites: ORP 5041.

ORP 5050 DISCRETE SYSTEM SIMULATION (3 credits). Covers the principles of building and using a discrete event simulation; construction and statistical testing of random variate generators; statistical analysis and validation of results; design of simulation projects; and variance reduction methods. (Requirement: Instructor approval or prerequisite course.) Prerequisites: MTH 5411.

ORP 5051 APPLIED EXPERT SYSTEMS (3 credits). Covers the concepts and methods of rule-based expert systems; methods of knowledge representation; and use of an expert system shell to build a small expert system. Noncredit for CS majors.

ORP 5070 SEQUENCING AND SCHEDULING (3 credits). Bridges the gap between scheduling theory and its application in manufacturing and service environments. Emphasizes basic scheduling principles and uses selected readings and case studies to illustrate the use of these concepts in industrial environments.

ORP 5090 SPECIAL TOPICS IN OPERATIONS RESEARCH 1 (3 credits). Content variable depending on the fields of expertise of the faculty and the desire and needs of the students.

ORP 5091 SPECIAL TOPICS IN OPERATIONS RESEARCH 2 (3 credits). Content variable depending on the fields of expertise of the faculty and the desire and needs of the students. Prerequisites: ORP 5090.

ORP 5899 FINAL SEMESTER THESIS (0-2 credits). Variable registration for thesis completion after satisfaction of minimum registration requirements. (Requirements: Accepted petition to graduate and approval by Office of Graduate Programs.)

ORP 5999 THESIS RESEARCH (3-6 credits). Individual research under the direction of a major adviser approved by the chair of the program. A maximum of six credits may be credited toward the master’s degree.

ORP 6010 ADVANCED TOPICS IN MATHEMATICAL PROGRAMMING (3 credits). Overviews selected topics in the theory of optimization. Unifies much of the field by use of a few principles of linear vector space theory. The concepts of distance, orthogonality and convexity play fundamental roles in this development. Prerequisites: MTH 5101, MTH 5102, ORP 5010.

ORP 6030 ADVANCED TOPICS IN DECISION MODELS (3 credits). Discusses current methods and research in decision analysis. May include large-scale multicriteria decision analysis, behavioral analysis of decision making, methods of uncertainty representation and decision making in the public domain. (Requirement: Instructor approval or prerequisite course.) Prerequisites: ORP 5031.

ORP 6095 PREPARATION FOR CANDIDACY/OPERATIONS RESEARCH (1-6 credits). Research under the guidance of a member of the operations research faculty in a selected area of operations research. Repeatable as required. (Requirement: Program chair approval.)

ORP 6899 FINAL SEMESTER DISSERTATION (0-2 credits). Variable registration for dissertation completion after satisfaction of minimum registration requirements. (Requirements: Accepted candidacy and approval by Office of Graduate Programs.)

ORP 6999 DISSERTATION RESEARCH (3-12 credits). Research and preparation for the doctoral dissertation. (Requirement: Admission to doctoral candidacy.)