Mathematical Sciences (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.

105 Precalculus Mathematics (3:3:0). Prerequisites: High school Algebra I, Algebra II, and Geometry, and specified score on the Math Placement Test, or successful completion of self-paced Algebra Tutorial Program offered by the Math Literacy Center. Call the Mathematical Sciences Department at (703) 993-1460 for details. 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 the 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 Quantitative Reasoning (3:3:0). Prerequisite: Specified score on the Math Placement Test or successful completion of self-paced Basic Math Program offered by the Math Literacy Center. Quantitative skills for the real world. Topics include critical thinking, modeling by functions, graphs, growth, scaling, probability, and statistics.

108 Introductory Calculus with Business Applications (3:3:0). Prerequisite: Specified score on the Math Placement Test or successful completion of self-paced Algebra Program offered by the Math Literacy Center. Call the Mathematical Sciences Department at (703) 993-1460 for details. Functions, limits, the derivative, and 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 Introductory Probability and Statistics (3:3:0). Prerequisite: Specified score on the Math Placement Test or successful completion of self-paced Basic Math Program offered by the Math Literacy Center. Elementary set theory, probability, and statistics.

111 Linear Mathematical Modeling (3:3:0). Prerequisite: Specified score on the Math Placement Test or successful completion of self-paced Basic Math Program offered by the Math Literacy Center. Matrix algebra, systems of linear equations, Markov chains, difference equations, and data fitting.

113 Analytic Geometry and Calculus I (4:4:1). Prerequisites: Thorough understanding of high school algebra and trigonometry and specified score on the Math Placement Test or a grade of C or better in MATH 105. Functions, limits, the derivative, maximum and minimum problems, the integral, and 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.

115 Analytic Geometry and Calculus I (Honors) (4:4:1). Prerequisite: Placement or permission of department. More challenging version of MATH 113. Functions, limits, the derivative, maximum and minimum problems, the integral, and transcendental functions.

116 Analytic Geometry and Calculus II (Honors) (4:4:1). Prerequisite: Successful completion of MATH 115 or A in MATH 113 and recommendation of MATH 113 instructor. More challenging version of MATH 114. Methods of integration, conic sections, parametric equations, infinite series, and power series.

125 Discrete Mathematics I (3:3:0). Prerequisite: Specified score on the Math Placement Test or successful completion of the self-paced Algebra Program offered by the Math Literacy Center. 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, determinants, vector spaces, eigenvalues, eigenvectors, and orthogonalization.

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). Prerequisite: MATH 213 or 215. First-order ODEs, higher-order ODEs, Laplace transforms, linear systems, nonlinear systems, numerical approximations, and modeling.

215 Vector Calculus (3:3:0). Prerequisites: Permission of instructor and MATH 113, 114. Vectors and vector-valued functions, partial differentiation, multiple integrals, line integrals, surface integrals, and transformation of coordinates.

216 Theory of Differential Equations (3:3:0). Prerequisites: MATH 203 and either MATH 213 or 215. First- and second-order equations, existence-uniqueness of solutions, systems of differential equations, and phase plane analysis.

271 Mathematics for the Elementary School I (3:3:0). Concepts and theories underlying elementary school mathematics, including problem solving, whole numbers and numeration, whole numbers operations and properties, number theory, fractions, decimals, ratio and proportion, and integers.

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 rational and real numbers, introduction to algebra, geometry, statistics, and probability. Intended for school educators; does not count toward a major in mathematics.

290 Foundations of Mathematics (3:3:0). Prerequisite: MATH 114. 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 credits of math. Prime numbers, factorization, congruences, and Diophantine equations.

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

313 Introduction to Applied Mathematics (3:3:0). Prerequisite: MATH 213. Vector differential calculus, vector integral calculus, Fourier analysis, and complex analysis.

314 Introduction to Applied Mathematics (3:3:0). Prerequisite: MATH 214 or 216. Series solutions of differential equations, Bessel and Legendre equations, Sturm-Liouville problems, and partial differential equations.

315 Advanced Calculus I (3:3:0). Prerequisites: MATH 213 and 290. Number system, functions, sequences, limits, continuity, differentiation, integration, trancendental functions, and 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, and 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. Abstract vector spaces, linear independence, bases, linear transformations, matrix algebra, inner product, and special topics.

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

351 Probability (3:3:0). Prerequisite: MATH 213 or 215. Random variables, probability functions, special distributions, and 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, and 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 203 and 216 (or 214). Synthesis of pure mathematics and computational mathematics. 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, and 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. 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: MATH203 and CS 112. 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 credits 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.

Prior knowledge of linear algebra and calculus (single and multivariable) is assumed in all math graduate courses. Adouble 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). (Formerly 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). (Formerly MATH 460, 461.) Prerequisites: MATH 554 and either MATH351 or STAT 344. 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.

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

601, 602 Analysis I, II for Teachers (3:2:1). Prerequisites: Open to in-service teachers of mathematics at the middle or secondary level. Others may enroll with permission of instructor. A background in mathematics is desirable but not necessary. While some needed topics from college algebra will be reviewed in class, a thorough understanding of high school algebra and trigonometry is expected. Develops the continuous ideas of calculus with particular emphasis on concepts as opposed to computational aspects of calculus. Specific topics include decimal representation of real numbers, sequences, series, and limits; differentiation to find speed, slopes of curves, and tangents; integration to find volumes and distances and area under curves. Optimization problems including maximization of area and volume and the modeling of these concepts. Graphing techniques will be supported by both the theory of calculus and graphing utilities such as the TI-83 calculator or if possible computer software such as Maple.

604 Geometry for Teachers (3:2:1). Prerequisites: Open to in-service teachers of mathematics at the middle or secondary level. Others may enroll with permission of instructor. Background in mathematics is desirable but not necessary. Covers standard topics from Euclidean geometry and includes a discussion of non-Euclidean geometries. Emphasizes an informal and explorative approach to geometry and makes use of the geometry sketchpad. Other topics include geometric constructions and the role of proof in geometry.

605 Discrete/Finite Mathematics for Teachers (3:2:1). Prerequisites: Open to in-service teachers of mathematics at the middle or secondary level. Others may enroll with permission of instructor. Background in mathematics is desirable but not necessary. A thorough understanding of high school algebra is assumed. Discusses finite mathematics in juxtaposition to the continuous ideas of calculus. Specific topics may be chosen from a variety but might consist of elementary counting and combinatorics including recursion and difference equations and their analogy to calculus. A different emphasis beginning with counting might lead to a more thorough discussion of probability and central measures of statistics. Other topics may include graph theory and its connection to geometry. Audience for whom the course is intended will to some extent guide the choice of topics.

607 Algebraic Structure for Teachers (3:2:1). Prerequisites: Open to in-service teachers of mathematics at the middle school level. Others may enroll with permission of instructor. Background in mathematics is desirable but not necessary. A thorough understanding of high school algebra is assumed. Expands on the customary operations on the integers and rationals to discuss systems that mimic these operations, thereby enhancing one's understanding of the former. Emphasis is placed on the concepts of multiplicative and additive inverses and their corresponding identities as they occur in other systems. Topics might include the integers modulo n and their connections to elementary number theory, permutation groups, rigid transformations, groups of symmetry of the plane and their connection to geometry, and matrices treated as linear transformations and their connections to solutions of systems of equations.

608 Problem Solving in Mathematics (3:2:1). Prerequisites: Open to in-service teachers of mathematics at the middle school level. Others may enroll with permission of instructor. Background in mathematics or science is desirable but not necessary. It is assumed that a student will have been exposed to most of the topics covered in MATH 601, 604, 605, and 607. Introduces a variety of challenging mathematical problems appropriate for the middle school student for the purpose of analyzing and solving problemsperhaps in a variety of waysusing the mathematics learned in the previous courses. In addition, students are asked to search for such problems and orally present their solutions. The specific topics to be covered in such problems might be any of the courses listed as prerequisites.

619 Topics in Mathematical Logic (3:3:0). 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, and matrix groups.

625/CSI 740 Numerical Linear Algebra (3:3:0). Prerequisite: Computer literacy, including some programming experience. Theory and development of numerical algorithms for the solution of a variety of matrix problems: linear systems, least squares problems, eigenvalue problems, and the singular value decomposition. Direct and iterative method, analysis of sensitivity to rounding errors, and applications.

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

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

639 Topics in Topology and Geometry (3:3:0). 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). 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). Separation theory of convex sets, polarity, duality theorems of convex optimization, valuation theory, combinatorial aspects of convexity, and applications to linear and integer programming.

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, and 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. 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 ofTables.

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 Complex Analysis I (3:3:0). Topology of complex numbers, holomorphic functions, series, complex integration. Meromorphic, multivalued, and elliptic functions.

671 Fourier Analysis (3:3:0). 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.

672 Wavelet Theory (3:3:0). Prerequisites: Some knowledge of advanced calculus and computer literacy. Study of the theory and computational aspects of wavelets and the wavelet transform. Emphasizes computational aspects of wavelets, defining the Fast Wavelet Transform in one and two dimensions. Developing the appropriate numerical algorithms. Includes developing the theory of wavelet bases on the real line, discussing multi-resolution analysis, splines, time-frequency localization, and wavelet packets.

673 Dynamical Systems (3:3:0). Prerequisites: Elementary courses in linear algebra and differential equations. Contemporary topics in the field of nonlinear dynamical systems are illustrated in mathematical models from physics, ecology, and population dynamics. Traditional qualitative analysis of difference and differential equations provides the background for understanding chaotic behavior when it occurs in these models. Topics include stability theory, fractals, lyapunov exponents, and chaotic attractors.

674 Stochastic Differential Equations (3:3:0). Prerequisites: MATH 214 and MATH 351. Introduction to stochastic calculus and differential equations. Wiener process, Ito and Stratonovich integrals, Ito formula, martingales, diffusions, and applications. Simulations and numerical approximations of solutions.

675 Linear Analysis I (3:3:0). Prerequisite: MATH 315 (Advanced Calculus) or its equivalent. Metric spaces, normed linear spaces, completeness, compactness, continuous (bounded) linear transformations, Banach spaces, Hilbert spaces, and 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, and applications. Brief account of Lebesque integration theory may be included.

677 Ordinary Differential Equations (3:3:0). Elementary differential equations course. Qualitative and quantitative theory of ordinary differential equations. Phase portrait analysis of linear and nonlinear systems, including classification of stable and unstable equilibrium states and periodic orbits. Poincare-Bendixson theorem, Lyapunov stability and Lyapunov functions, and bifurcation theory. Optional topics include averaging and perturbation methods, numerical solution techniques, and chaos.

678 Partial Differential Equations (3:3:0). Prerequisite: An elementary differential equations course. 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). Special topics in analysis not covered in the regular analysis sequence. May be repeated for credit.

680 Industrial Mathematics (3:3:0). 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.

682/OR 641 Linear Programming (3:3:0). Prerequisite: OR 541 or permission of instructor. An in-depth look at the simplex method. Computational enhancementsthe revised simplex method; sparse-matrix techniques; bounded variables and generalized upper bounds; and large-scale decomposition methodsare also covered. Other topics include computational complexity of the simplex algorithm, and the Khachian and Karmarkar algorithms.

683 Modern Optimization Theory (3:3:0). 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). Prerequisite: Computer literacy, including some programming experience. Computational techniques for the solution of problems arising in science and engineering. Includes theoretical development as well as implementation, efficiency, and accuracy issues in using algorithms and interpreting the results. Specific topics include linear and nonlinear systems of equations, polynomial interpolation, numerical integration, and an introduction to numerical solution of differential equations.

686 Numerical Solutions of Differential Equations (3:3:0). Prerequisites: MATH 446 or 685 and an elementary differential equations course. 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 685 and an elementary differential equations course. 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). 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). Inareas 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. 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. Literature review, project report, or other written product is normally required. May be repeated for a maximum of nine credits.

795 Seminar (1-3:1-3:0). 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. Graded S/NC.

800 Studies for the Doctor of Philosophy in Education (variable 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.