Department of Mathematics

College of Natural Sciences and Mathematics

## MATH 518/* Sampling Survey Theory and Its Applications 3 cr.

Directed to the student who is or will be doing quantitative research, commissioning large-scale surveys, and evaluating the results. Sampling techniques and statistical principles underlying their use will be introduced. Consideration will be given to the practical problems associated with implementation. Prerequisites: (for non-Math majors) MATH 214, MATH 216, MATH 217, or GSR 516.

## MATH 521-522 Advanced Calculus I, II 3 cr.

A rigorous investigation of continuity, differentiation, and integration on real p-dimensional space. The Riemann-Stieltjes integral, infinite series, and infinite series of functions are also studied. Prerequisite: Permission of the advisor.

## MATH 523/* Complex Variables I 3 cr.

Introduces fundamental concepts of complex analysis and includes the following topics: complex numbers, functions, sequences, analytic functions, elementary functions, complex integration, power series, Laurent series, singular points, calculus of residues, infinite product and partial fraction expansion, conformal mapping, and analytic continuation. Prerequisite: Permission of the advisor.

## MATH 525/* Applied Mathematical Analysis I 3 cr.

Provides the necessary background for an understanding of mathematical programming, proofs of convergence of algorithms, convexity, and factorable functions. Develops necessary concepts in matrix theory which are required to develop efficient algorithms to solve linear and nonlinear programming models. Prerequisite: Calculus sequence, introductory linear algebra, or permission of the instructor.

## MATH 527/* Topology 3 cr.

Basic topological concepts, including some topological invariant relationships between topology and other disciplines of mathematics, are discussed. Prerequisites: Differential and Integral Calculus.

## MATH 545/* Deterministic Models in Operations Research 3 cr.

An introductory course on using the basic tools of solving deterministic models in operations research. Topics include optimization techniques and applications such as linear programming, nonlinear and dynamic programming, transportation models, and network models. In addition, sensitivity analysis, duality, simplex methods, and integer programming are discussed. Students will use technology to solve problems and interpret the results. Prerequisites: Two semesters of calculus and one semester of linear algebra.

## MATH 546/* Probabilistic Models in Operations Research 3 cr.

A survey of probabilistic methods for solving decision problems under uncertainty. Probability review, decision theory, queuing theory, inventory models, and Markov chains are covered. Students will use technology to solve problems and interpret the results. Prerequisites: Two semesters of calculus, one semester of introductory linear algebra, and introductory probability and statistics.

## MATH 547/* Modeling and Simulation 3 cr.

Construction and solution of mathematical models. Emphasis is on applications in areas such as logistics, natural and social sciences, and manufacturing. Discrete and continuous system models are analyzed using mathematical and computer-based methods. Introduction to computer simulation. Introductory course in differential equations is recommended but not required. Prerequisites: Two semesters of calculus, one semester of introductory linear algebra, and introductory probability and statistics.

## MATH 551/* Numerical Methods for Supercomputers 3 cr.

Supercomputers make use of special computer architectures—vector and parallel processors—in order to achieve the fastest processing speed currently available. Students will be introduced to these features and will learn how numerical algorithms can be constructed to exploit supercomputers’ capabilities. Students will gain practical experience in programming for the Cray YMP, in incorporating existing scientific software packages into user-written programs, in submitting remote jobs to the Pittsburgh Supercomputer Center, and in producing animated graphical output to summarize the typically large volume of output data generated by large scientific programs. Prerequisite: Permission of the instructor.

## MATH 553/* Theory of Numbers 3 cr.

Elementary properties of divisibility, congruences, Chinese remainder theories, primitive roots and indices, quadratic reciprocity, diophantine equations, and number theoretic functions. Prerequisites: Differential and Integral Calculus.

## MATH 563/* Mathematical Statistics I 3 cr.

Probability theory necessary for an understanding of mathematical statistics is developed; applications of the theory are given, with emphasis on binomial, Poisson, and normal distributions. Sampling distributions and the central limit theorem are developed. Prerequisites: Differential and Integral Calculus.

## MATH 564/* Mathematical Statistics II 3 cr.

Multivariate distributions, properties of the moment generating function, change of variable technique. Chi-square distribution, estimation, confidence intervals, testing hypotheses, contingency tables, goodness of fit. Many practical applications. Use of calculating machines and computers where appropriate. Prerequisite: MATH 563.

## MATH 571/* Linear Algebra 3 cr.

Theory of vector spaces and linear transformations, applications to linear equations, determinants, and characteristic roots are studied.

## MATH 576-577 Abstract Algebra I, II 3 cr.

Basic algebraic structures such as groups, rings, integral domains, and fields. Designed to develop ability to construct formal proofs and work within an abstract axiomatic system. Polynomial rings, factorization, and field extension leading up to Galois theory.

## MATH 581-583 Topics Seminars in Mathematics 3 cr.

Special Topics going beyond the scope of regularly offered courses. Offered per student interest/available staff. Students may take more than one topic seminar with approval of advisor. Prerequisite: Consent of instructor.

## MATH 610 Real-Data Approach to Teaching Mathematics 3 cr.

Presents the content knowledge as well as effective teaching strategies to incorporate real data in the teaching of grades K-12 mathematics curriculum. Students will learn to integrate real data into the teaching of numerical concepts, pre-algebra, algebra, probability, statistics, geometry, and advanced mathematics. The intended audience is K-12 teachers who wish to learn content and teaching methods to integrate real data into the teaching of mathematics. Prerequisite: Permission of the instructor.

## MATH 625 Analysis for Applied Mathematics 3 cr.

A graduate-level introduction to classical applied mathematics. Topics include vector spaces and orthogonality, eigenvalue problems, quadratic forms, vector calculus in n-space, infinite series and applications, Fourier series, least squares approximation, and systems of differential equations. Prerequisites: Calculus sequence and introductory linear algebra or permission of the instructor.

## MATH 631 Foundations of Mathematics 3 cr.

Designed to acquaint students with logical techniques used in proof and set theory. Topics include symbolic logic, rules of inference, validity of arguments, algebra of sets, cardinal numbers, the well-ordering property, and the Axiom of Choice.

## MATH 640 Numerical Mathematics 3 cr.

Intended for graduate students in mathematics and the sciences, this course will cover solving mathematical problems using computer algorithms, in particular root-finding methods, direct and iterative methods for linear systems, nonlinear systems, eigenvalue problems, and differential equations. Prerequisites: Calculus sequence, introductory linear algebra, and programming literacy, or permission of the instructor.

## MATH 641 Ordinary and Partial Differential Equations 3 cr.

Solution techniques for linear and solvable nonlinear ordinary and partial differential equations are covered. A variety of methods including series solutions, operator methods, Laplace transforms, characteristics, and separation of variables are demonstrated for numerous applications to physical problems. Systems of differential equations, associated phase plane, and stability theory are addressed. Solutions and applications for the equations of mathematical physics are discussed, including the heat equations, Laplace’s equations, and the wave equation. Prerequisite: MATH 625 or permission of the instructor.

## MATH 643 Graphs, Networks, and Combinatorics 3 cr.

Introduces elementary concepts of graph theory and its applications and the fundamentals of combinatorics. Systematic methods for counting are given via the study of arrangements and generating functions through the use of classical and analytical techniques. Prerequisite: Calculus sequence.

## MATH 645 Nonlinear Programming Models 3 cr.

Examines algorithms for solving nonlinear programming (optimization) models. Also concerned with the theory of nonlinear optimization and with characteristics of optimal points. Optimization models of real-world problems which can be solved by nonlinear programming methodology are also presented. Prerequisites: MATH 525 and MATH 545 or equivalent courses.

## MATH 647 Advanced Simulation 3 cr.

An in-depth study of computer simulation techniques using simulation software. Emphasis is on discrete-event systems, although continuous-event systems will also be modeled. Model validation and verification including statistical analysis. Prerequisites: MATH 545 and MATH 563.

## MATH 650 History of Mathematics 3 cr.

People and ideas that have shaped the course of events in mathematics. Major attention given to developing activities for the secondary school mathematics classroom which incorporate the historical viewpoint.

## MATH 651 Seminar in Teaching Junior High School Mathematics 3 cr.

Explores problems of teaching mathematics at junior high level. Emphasis on a discovery, lab-oriented approach to teaching. Prerequisite: Permission of instructor.

## MATH 652 Seminar in Teaching Senior High School Mathematics 3 cr.

National and international forces shaping today’s mathematics programs, curriculum development and research, art of generating interest, formation of concepts, proof, problem solving, generalization, and evaluation. Special attention to teaching topics from algebra and calculus and modern approaches to teaching geometry and trigonometry. Prerequisite: Permission of instructor.

## MATH 654 Curriculum and Supervision in Mathematics 3 cr.

Basic principles underlying effective mathematics curriculum from both a theoretical and an experimental viewpoint. Investigates supervisor’s role as source of stimulation, leadership, and expertise in teaching mathematics.

## MATH 655 Projective Geometry 3 cr.

Introduces Klein’s formulation of geometry of the invariant theory of a given set under a given group of transformations and develops projective spaces of one and two dimensions and conics and quadratic forms. Prerequisite: Undergraduate courses in linear algebra and geometry.

## MATH 665 Applied Regression Analysis and Design of Experiments 3 cr.

Designed as an applied course in regression analysis, analysis of variance, and experimental design. The student is introduced to least squares, the matrix approach to linear regression, the examination of residuals, dummy variables, the polynomial model, best regression equations, multiple regression, and mathematical model building. Statistical software is used for the data analysis. Analysis of variance (ANOVA) and design of experiments including one- and two-factor analysis, randomized block designs, and Latin squares are covered. Both the ANOVA and regression approaches to these concepts are introduced, as well as the appropriate nonparametric alternatives. Prerequisite: MATH 564 or permission of the instructor.

## MATH 667 Applied Statistical Methods 3 cr.

Focus will be on the understanding and the application of statistical techniques in sampling, categorical data analysis, and time series. Statistical software is used for data analysis. Prerequisite: MATH 564 or permission of the instructor.

## MATH 681-683 Special Topics in Mathematics 3 cr.

Special Topics in graduate mathematics beyond the scope of regularly offered graduate classes. Offered as student interest and available staff permit. With approval of the advisor, more than one Special Topics class may be taken. Prerequisite: Consent of the instructor.

## MATH 684 Topics in Operations Research 3 cr.

Special Topics in operations research beyond the scope of regularly offered graduate courses. Offered as student interest and available staff permit. Prerequisite: Permission of the instructor.

## MATH 685 Topics in Probability and Statistics 3 cr.

Special Topics in probability and statistics beyond the scope of regularly offered graduate classes. Offered as student interest and available staff permit. Prerequisite: Permission of the instructor.

## MATH 688 Topics in Applied Mathematics 3 cr.

Special Topics in applied mathematics beyond the scope of regularly offered gradate classes. Offered as student interest and available staff permit. Prerequisite: Permission of the instructor.

## MATH 698 Internship 1-6 cr.

Positions with participating companies or agencies provide students with experience in mathematics-related work under the supervision of agencies and faculty.

## MATH 699 Independent Study in Mathematics 3 cr.

Under the guidance of faculty member, a student may study some area of mathematics not included in the regular courses.

## MATH 795 Thesis 3 cr.

*Indicates dual-listed class