George Mason University 1997-98 Catalog Catalog Index
Course Descriptions

Search the 1997-1998 Catalog:


Mathematical Sciences Courses (MATH)


Mathematical Sciences

Knowledge of high school algebra is a prerequisite for all mathematics courses. In exceptional cases the prerequisite for a course above the calculus sequence may be waived at the discretion of the instructor.

101 Algebra and Elementary Functions (0:3:0). Meets three hours per week. No degree credit is given; however, tuition and fees are charged for three hours. Basic algebraic manipulations. Linear equations and graphing. A study of the properties of polynomial, logarithmic, exponential, and trigonometric functions.

105 Precalculus Mathematics (3:3:0). Prerequisites: High school Algebra I, Algebra II, and Geometry, and successful completion of Math Placement Test given by the Testing Center, or successful completion of self-paced Algebra Tutorial Program offered by the Mathematics Literacy Center. (Call the Mathematical Sciences Department at 993-1460 for details.) A review of mathematics skills essential to the study of calculus. Topics covered are equations, inequalities, absolute values, graphs, functions, exponential and logarithmic functions, and trigonometry. May not be used as credit toward the B.A. or B.S. in Mathematical Sciences or toward satisfying Area B of the university core requirements or the Analytical Reasoning requirement for the B.A. degree in College of Arts and Sciences. May not be taken for credit after receiving a grade of C or better in any MATH course numbered 113 or higher.

106 Concepts of Mathematics (3:3:0). Important ideas in mathematics and their development.

108 Introductory Calculus with Business Applications (3:3:0). Prerequisite: Successful completion of Math Placement Test given by the Testing Center, or successful completion of self-paced Algebra Tutorial Program offered by the Mathematics Literacy Center. (Call the Mathematical Sciences Department at 993-1460 for details.) Functions, limits, the derivative, the integral. Applications of differentiation and integration. Students who have received credit for MATH 113 or 114 may not receive credit for this course.

110 Finite Mathematics (3:3:0). Elementary set theory, probability, and statistics.

111 Finite Mathematics (3:3:0). Matrix algebra, linear programming, Markov chains, and game theory. Applications.

113 Analytic Geometry and Calculus I (4:4:1). Prerequisites: Thorough understanding of high school algebra and trigonometry and successful completion of Math Placement Test offered through the Testing Center, or a grade of C or better in MATH 105. Functions, limits, the derivative, maximum and minimum problems, the integral, transcendental functions.

114 Analytic Geometry and Calculus II (4:4:1). Prerequisite: Grade of C or better in MATH 113. Methods of integration, conic sections, parametric equations, infinite series, and power series.

125 Discrete Mathematics I (3:3:0). Introduction to the ideas of discrete mathematics and combinatorial proof techniques including mathematical induction, sets, graphs, trees, recursion, and enumeration.

203 Matrix Algebra (3:3:0). Prerequisite: MATH 114 or permission of instructor. Systems of linear equations, linear independence, linear transformations, inverse of a matrix, LU factorization, determinants, vector spaces, eigenvalues, and eigenvectors.

209 Introduction to Mathematical Modeling (3:3:0). Prerequisite: Grade of C or better in MATH 114. Calculus-based introduction to discrete and continuous models in the natural and social sciences. Topics include approximations, data analysis, model evaluation, optimization, difference and differential equations, deterministic and stochastic models.

213 Analytic Geometry and Calculus III (3:3:0). Prerequisite: Grade of C or better in MATH 114. Partial differentiation, multiple integrals, line and surface integrals, and three-dimensional analytic geometry.

214 Elementary Differential Equations (3:3:0). Prerequisites: MATH 213 and a knowledge of a scientific programming language. First-order ODEs, higher-order ODEs, Laplace transforms, linear systems, nonlinear systems, numerical approximations, and modelling.

215 Vector Calculus (3:3:0). Prerequisite: MATH 114. Vectors and vector-valued functions, partial differentiation, multiple integrals, line integrals, surface integrals, transformation of coordinates.

216 Theory of Differential Equations (3:3:0). Prerequisite: MATH 213 or 215. Matrix algebra, first- and second-order equations, existence-uniqueness of solutions, systems of linear differential equations, power series solutions.

271 Mathematics for the Elementary School I (3:3:0). Concepts and theories underlying elementary school mathematics, including sets, logic, systems of numeration, whole numbers and integers, operations with integers, equations, and inequalities. Intended for school educators; does not count toward a major in mathematics.

272 Mathematics for the Elementary School II (3:3:0). Continuation of MATH 271; MATH 271 is recommended before enrolling in MATH 272. Topics include elementary number theory, rational and real numbers, intuitive geometry, and measurement, including the metric system. Intended for school educators; does not count toward a major in mathematics.

290 Foundations of Mathematics (3:3:0). Prerequisite: MATH 114. Axiomatic set theory; graphs; functions; equivalence relations and partitions; partially ordered sets; induction; construction of the natural, rational, real and complex number systems; well-ordering principle; and cardinality. Primarily intended for mathematics majors.

301 Number Theory (3:3:0). Prerequisite: Six hours of math. Prime numbers, factorization, congruences, Diophantine equations.

302 Geometry (3:3:0). Prerequisite: Six hours of math. Fundamental concepts of incidence. Axioms of Euclidean geometry and the resulting theory, axioms and development of non-Euclidean and projective geometry.

313, 314 Introduction to Applied Mathematics (3:3:0), (3:3:0). Prerequisite: MATH 214 or 216. Vector fields and integration, complex analysis, Fourier and orthogonal series, Laplace and Fourier transforms, Green's functions, partial differential equations, boundary value problems, special functions, and other topics arising in mathematical physics.

315 Advanced Calculus I (3:3:0). Prerequisites: MATH 213 and 290. The number system, functions, sequences, limits, continuity, differentiation, integration, trancendental functions, infinite series.

316 Advanced Calculus II (3:3:0). Prerequisite: MATH 315. Sequences of functions, Taylor series, vectors, functions of several variables, implicit functions, multiple integrals, surface integrals.

321 Abstract Algebra (3:3:0). Prerequisites: MATH 290 and 215. Theory of groups, rings, fields.

322 Linear Algebra (3:3:0). Prerequisites: MATH 290 and 203 or 216. Abstract vector spaces, linear independence, bases, linear transformations, matrix algebra, inner product, special topics.

325 Discrete Mathematics II (3:3:0). Prerequisite: MATH 125. Advanced counting, binomial identities, generating functions, advanced recurrence, inclusion-exclusion, network flows.

351 Probability (3:3:0). Prerequisite: MATH 213 or 215. Random variables, probability functions, special distributions, limit theorems.

352 Statistics (3:3:0). Prerequisite: MATH 351. Estimation, decision theory, testing hypothesis, correlation, linear models, and design.

382 Introduction to Stochastic Processes (3:3:0). Prerequisite: MATH 351. General notion of stochastic processes, finite and infinite Markov chains, discrete and continuous Markov processes, stationary processes, random walk problems, birth and death processes, waiting line and serving problems, Brownian motion.

411 Functions of a Complex Variable (3:3:0). Prerequisite: MATH 214 or 216. Analytic functions, contour integration, residues, and applications to such topics as integral transforms, generalized functions, and boundary value problems.

413 Modern Applied Mathematics I (3:3:0). Prerequisites: MATH 216 (or MATH 203 and MATH 214) and MATH 315. Synthesis of "pure mathematics" and "computational mathematics." The interplay between discrete and continuous mathematics is emphasized throughout. Mathematical structure is revealed from equilibrium models in discrete and continuous systems.

414 Modern Applied Mathematics II (3:3:0). Prerequisite: MATH 413. Continuation of MATH 413, which involves a synthesis of "pure mathematics" and "computational mathematics." Fourier analysis and its role in applied mathematics is developed (e.g., differential equations and approximations). Discrete aspects are emphasized in computational models.

431 Topology (3:3:0). Prerequisite: MATH 315. Metric spaces, topological spaces, compactness, connectedness.

441 Operations Research I (3:3:0). Prerequisite: MATH 203 or 216, or permission of instructor. Survey of deterministic methods for solving "real-world" decision problems. The programming model and simplex method of solution, duality and sensitivity analysis, transportation and assignment problems, shortest path and maximal flow problems, project networks including PERT and CPM, introduction to integer and nonlinear programming, dynamic programming and game theory. Emphasis on modeling and problem solving.

442 Operations Research II (3:3:0). Prerequisite: MATH 351 or permission of instructor. Survey of probabilistic methods for solving "real-world" decision problems. Probability review, queuing theory, inventory theory, Markov decision processes, reliability, decision theory, simulation. Emphasis on modeling and problem solving.

446 Numerical Analysis I (3:3:0). Prerequisites: MATH 203 or 216 and knowledge of a scientific programming language. Significant figures, round-off errors, iterative methods of solution of nonlinear equations of a single variable, solutions of linear systems, iterative techniques in matrix algebra, interpolation and polynomial approximation.

447 Numerical Analysis II (3:3:0). Prerequisites: MATH 216 (or 214) and 446. Numerical differentiation and integration, initial-value and boundary-value problems for ordinary differential equations, methods of solution of partial differential equations, iterative methods of solution of nonlinear systems, approximation theory.

491, 492 Reading and Problems (1-3:0:0), (1-3:0:0). For mathematical sciences majors only. Independent study in math. Must be arranged with instructor before registering.

493 Topics in Applicable Mathematics (3:3:0). Prerequisite: Six credits of math at or above the 310 level. Topics that have been successfully used in applications of mathematics. Subject determined by instructor.

494 Topics in Pure Mathematics (3:3:0). Prerequisite: Six hours of math at or above the 310 level. Topics of pure math not covered in other courses. Topics might include Galois theory, cardinal and ordinal arithmetic, measure theory, mathematical logic, and differential geometry. Subject determined by instructor. A double number separated by a comma (MATH 555, 556) indicates that both graduate courses normally constitute a sequence and that the first semester is prerequisite to the second. The prerequisite may be waived by permission of the department chair. See also STAT and OR courses.

551 Regression and Time Series (3:3:0). (Replaces MATH 451.) Prerequisite: MATH 352, STAT 652, SOA Exam 110, or permission of instructor. Mathematics of regression, exponential smoothing, time series, and forecasting. Material covered corresponds to the Society of Actuaries Exam: Applied Statistical Methods.

554 Mathematics of Compound Interest (3:3:0). (Formerly MATH 360.) Prerequisite: MATH 113; corequisite: MATH 114. Simple and compound interest, annuities, present and future value, yield rates, capital budgeting, amortization schedules, mortgages, and bonds. Material covered corresponds to the Society of Actuaries Exam: Mathematics of Compound Interest. Not appropriate for graduate science and engineering majors not considering an actuarial or financial career.

555, 556 Actuarial Mathematics I, II (3:3:0), (3:3:0). (Formerly MATH 460, 461.) Prerequisites: MATH 554 and either MATH 351 or STAT 344. A two-semester sequence covering the material for Society of Actuaries Exam: Actuarial Mathematics. Topics include survival distribution and life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life models, pensions, insurance models including expense, and nonforfeiture benefits and cash values.

557 Credibility and Loss Distribution (3:3:0). Prerequisite: MATH 352, SOA Exam 110, or permission of instructor. Statistical concepts of location and dispersion, inferences from insurance data. Bayesian statistics and its relation to credibility theory and their use in modeling insurance claims. This is the material covered in the Casualty Actuarial Society Exam: Credibility Theory and Loss Distributions.

558 Demography (3:3:0). Prerequisite: MATH 555; corequisite: MATH 654, or permission of instructor. Measures of mortality and fertility, demographic characteristics and trends in Canada and the United States, mathematics of population projections, mathematics of stable and stationary populations. The material covered in this course corresponds with the Society of Actuaries Exam: Mathematics of Demography.

559 Mathematics of Graduation (3:3:0). Prerequisites: Courses in probability, statistics, matrix algebra, and numerical analysis. Graduation is the process of revising estimates. Methods include moving weighted averages, Whittaker, and Bayesian, parametric, and smooth junction interpolation. Material covered corresponds to the Society of Actuaries Exam: Mathematics of Graduation.

600 Special Topics in Mathematics (1-6:1-6:0). Prerequisite: Permission of instructor. Mathematical workshops, special courses, or other projects.

611 Intermediate Analysis (3:3:0). Development of the number system; review of the highlights of calculus, sequences, and series of functions. Credit not applicable toward the 30 credits required for the M.S. in Mathematical Sciences, but can be counted toward the M.Ed.

612 Intermediate Algebra (3:3:0). Linear algebra, vector spaces, linear independence, linear transformations, and matrix operations. Credit not applicable toward the 30 credits required for the M.S. in Mathematical Sciences, but can be counted toward the M.Ed.

619 Topics in Mathematical Logic (3:3:0). Prerequisite: Permission of instructor. Special topics in the foundations of mathematics not included in the regular mathematics curriculum. May be repeated for credit.

621 Algebra I (3:3:0). Groups, linear algebra, matrix groups.

622 Algebra II (3:3:0). Prerequisite: MATH 621 or permission of instructor. Rings, fields, Galois theory.

623 Algebraic Coding Theory (3:3:0). Prerequisite: MATH 203, 303, or permission of instructor. Introduction to the mathematical theory of error-correcting codes including linear block codes such as Hamming, Golay, BCH, and Reed-Muller. Also included are the MacWilliams equations and t-designs.

625 Numerical Linear Algebra (3:3:0). Prerequisites: A course in linear algebra and some programming ability. Computational procedures for linear systems, least-square problems, and eigenvalue problems, with an emphasis on error analysis.

629 Topics in Algebra (3:3:0). Prerequisite: Permission of instructor. Special topics in pure or applied algebra not covered in the regular algebra sequence. May be repeated for credit.

631, 632 Point Set Topology I, II (3:3:0), (3:3:0). Topological spaces, connectedness, compactness, compactifications, uniform spaces, function spaces.

637, 638 Non-Euclidean Geometry I, II (3:3:0), (3:3:0). Prerequisite: Permission of instructor. Affine, protective, hyperbolic, elliptical, differential geometry; transformations and elementary combinatorics.

639 Topics in Topology and Geometry (3:3:0). Prerequisite: Permission of instructor. Special topics in topology and geometry not covered in the regular topology and geometry sequence. May be repeated for credit.

641 Combinatorics and Graph Theory (3:3:0). Prerequisite: Permission of instructor. Study of fundamental concepts in combinatorics and graph theory. Various methods of enumerative combinatorics, including the principle of inclusion-exclusion, the multinomial theorem, generating functions, recurrence relations, graphs and subgraphs, trees, connectivity, planar graphs, coloring, and matching.

644 Combinatorics and Convexity (3:3:0). Prerequisite: Permission of instructor. Separation theory of convex sets, polarity, duality theorems of convex optimization, valuation theory, combinatorial aspects of convexity, and applications to linear and integer programming.

651 Probability Theory (3:3:0). Axioms for a probability space, conditional probability, random variables, distribution functions, moments, characteristic functions, modes of convergence, limit theorems.

652 Mathematical Statistics (3:3:0). Prerequisite: MATH 651. Sampling distributions, point and interval estimation (Cramer-Rao theorem), testing of hypotheses (Neyman-Pearson tests, uniformly most powerful tests, sequential tests), linear models, distribution free methods.

653 Risk Theory (3:3:0). Prerequisite: MATH 351 or STAT 644 required. MATH 555 recommended but not required. Economics of insurance, individual risk models for short term, collective risk models for single period, collective risk models over an extended period, and applications of risk theory. Material included in this course corresponds to the Society of Actuaries Exam: Risk Theory.

654 Survival Models and Construction of Tables (3:3:0). Prerequisite: MATH 556 or permission of instructor. The nature and properties of survival models, methods of estimates from complete and incomplete data, tabular and parametric models, and practical issues in survival model estimation. Material covered corresponds to the Society of Actuaries Exam: Survival Models and Construction of Tables.

655 Pension Valuation (3:3:0). Prerequisite: MATH 556, SOA exam P-360U or EA-1A, or permission of instructor. Basic mathematics used in pension actuarial work without regard to pension law. This is the material covered in the Society of Actuaries Exam P-36OU (EA-1B).

661, 662 (771, 772; 511, 512) Complex Analysis I, II (3:3:0, 3:3:0). Prerequisite: MATH 316. Topology of complex numbers, holomorphic functions, series, complex integration. Meromorphic, multivalued, and elliptic functions.

671 Fourier Analysis (3:3:0). The study of fundamental ideas in Fourier analysis. Topics include orthonormal systems, Fourier series, continuous and discrete Fourier transform theory, generalized functions, and an introduction to spectral analysis. Applications to the physical sciences, linear systems theory, and signal processing are used to motivate and integrate these topics.

675 Linear Analysis I (3:3:0). Metric spaces, normed linear spaces, completeness, compactness, continuous (bounded) linear transformations, Banach spaces, Hilbert spaces, orthogonal series.

676 Linear Analysis II (3:3:0). Prerequisite: MATH 675 or permission of instructor. Analysis of bounded and unbounded operators, spectral theorems, differential operators, applications. A brief account of Lebesque integration theory may be included.

677 Ordinary Differential Equations (3:3:0), (3:3:0). Prerequisites: MATH 203 and 214. General properties of differential equations. Linear systems. Properties of solutions. Related topics.

678 Partial Differential Equations (3:3:0). Prerequisites: MATH 203 or 303 and 214 or 304. Physical examples, characteristics, boundary-value problems, integral transforms, and other topics, such as variational, perturbation, and asymptotic methods.

679 Topics in Analysis (3:3:0). Prerequisite: Permission of instructor. Special topics in analysis not covered in the regular analysis sequence. May be repeated for credit.

680 Industrial Mathematics (3:3:0). Prerequisite: Permission of instructor. Students take examples from industry, and go through the complete solution process: formulation of a mathematical model of the problem; solution of the mathematical model (possibly by numerical approximation), interpretation and presentation of the results. The course emphasizes working in groups, relating mathematics to concrete situations, and communication and presentation skills.

683 Modern Optimization Theory (3:3:0). Prerequisite: Permission of instructor. Introduction to the basic mathematical ideas and methods for solving linear and nonlinear programming problems, with emphasis on the mathematical aspects of optimization theory. Along with reviewing the classical topics of linear programming, the course covers the recent developments in linear programming, including the interior point method, and considers basic results in nonlinear programming, including very recent developments in this field.

685 Numerical Analysis (3:3:0). Prerequisites: Linear algebra, advanced calculus or its equivalent, and some programming ability. Study of computational methods with an emphasis on error analysis in linear algebra, approximation theory, nonlinear equations, and numerical differentiation and integration.

686 Numerical Solutions of Differential Equations (3:3:0). Prerequisites: MATH 446 or MATH 685 and some knowledge of ordinary differential equations. Finite difference methods for initial value problems, two-point boundary value problems, Poisson equation, heat equation, and first-order partial differential equations.

687 Variational Methods (3:3:0). Prerequisites: MATH 446 or MATH 685 and some knowledge of partial differential equations. Weak formulation of partial differential equations, energy principles, Galerkin approximations, and finite element methods. Review and development of the necessary analysis is included.

688 Topics in Actuarial Mathematics (3:3:0). Prerequisite: Permission of instructor. Special topics in actuarial science not covered in the regular actuarial mathematics sequence. May be repeated for credit.

689 Topics in Applied Mathematics (3:3:0). Prerequisite: Permission of instructor. Special topics in applied math not covered in the regular applied math sequence. May be repeated for credit.

697 Independent Reading and Research (1-3:0:0). Prerequisites: Graduate standing and permission of instructor. In areas of importance, but with insufficient demand to justify a regular course, an individual student may undertake a course of study under the supervision of a consenting faculty member. A written statement of the content of the course and a tentative reading list is normally submitted by the student as part of the request for approval to take the course. A literature review, project report, or other written product is normally required. May be repeated for a maximum of nine credits.

795 Seminar (3:3:0). May be repeated for credit.

799 Thesis (1-6:0:0). Original or compilatory work to be evaluated by a committee of three faculty members.

800 Studies for the Doctor of Philosophy in Education (varied credit). Prerequisite: Admission to the Ph.D. in Education program to study in mathematical sciences. Program of studies designed by student's discipline director and approved by student's doctoral committee, which brings the student to participate in the current research of the discipline director and results in a paper reporting the original contributions of the student. Enrollment may be repeated.


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