Mathematical Sciences
All courses are 3 credits unless otherwise noted.
MA 501. Engineering Mathematics
This course develops mathematical techniques used in the engineering disciplines. Preliminary concepts will be reviewed as necessary, including vector spaces, matrices and eigenvalues. The principal topics covered will include vector calculus, Fourier transforms, fast Fourier transforms and Laplace transformations. Applications of these techniques for the solution of boundary value and initial value problems will be given. The problems treated and solved in this course are typical of those seen in applications and include problems of heat conduction, mechanical vibrations and wave propagation. (Prerequisite: A knowledge of ordinary differential equations, linear algebra and multivariable calculus is assumed.)
MA 503-504. Analysis I and II
Topics covered include open and closed sets, compactness, continuity, upper and lower semicontinuity, Lebesgue measure, integration, functions of bounded variation, absolute continuity, the fundamental theorem of calculus for Lebesgue integrals, Banach spaces, classical L p spaces, the Hölder and Minkowski inequalities, the Riesz-Fischer theorem, and the Riesz representation theorem. (Prerequisite: basic knowledge of undergraduate analysis is assumed.)
This course will provide a rigorous and thoroughtreatment of the theory of functions of one complex variable. The topics to be covered include complex numbers, complex differentiation, the Cauchy-Riemann equations, analytic functions, Cauchy’s theorem, complex integration, the Cauchy integral formula, Liouville’s theorem, the Gauss mean value theorem, the maximum modulus theorem, Rouche’s theorem, the Poisson integral formula, Taylor-Laurent expansions, singularity theory, conformal mapping with applications, analytic continuation, Schwarz’s reflection principle and elliptic functions. (Prerequisite: knowledge of undergraduate analysis.)
This course introduces mathematical model building using dimensional analysis, perturbation theory and variational principles. Models are selected from the natural and social sciences according to the interests of the instructor and students. Examples are: planetary orbits, springmass systems, fluid flow, isomers in organic chemistry, biological competition, biochemical kinetics and physiological flow. Computer simulation of these models will also be considered. (Prerequisite: knowledge of ordinary differential equations and of analysis at the level of MA 501 is assumed.)
MA 509. Stochastic Modeling
This course gives students a background in the theory and methods of probability, stochastic processes and statistics for applications. The course begins with a brief review of basic probability, discrete and continuous random variables, expectations, conditional probability and basic statistical inference. Topics covered in greater depth include generating functions, limit theorems, basic stochastic processes, discrete and continuous time Markov chains, and basic queuing theory including M/M/1 and M/G/1 queues. (Prerequisite: knowledge of basic probability at the level of MA 2631 and statistics at the level of MA 2612 is assumed.)
MA 510/CS 522. Numerical Methods
This course is an introduction to modern numerical techniques. It is suitable for both mathematics majors and students from other departments. It covers material not treated in either MA 512 or MA 514, and it introduces the main ideas of those two courses. Topics covered may include interpolation by polynomials, roots of nonlinear equations, approximation by various types of polynomials, orthogonal polynomials, least-squares approximation, trigonometric polynomials and fast Fourier transforms, piecewise polynomials and splines, numerical differentiation and integration, unconstrained optimization including Newton’s method and the conjugate direction method, and an introduction to the solution of systems of linear equations and initial value problems for ordinary differential equations. Both theory and practice are examined. Error estimates, rates of convergence and the consequences of finite precision arithmetic are also discussed. Other topics may include integral equations or an introduction to boundary value problems. In the course of analyzing some of the methods, topics from elementary functional analysis will be introduced. These include the concept of a function space, norms and inner products, operators and projections. (Prerequisite: knowledge of undergraduate linear algebra and differential equations, and a higher-level programming language is assumed.)
MA 511. Applied Statistics for Engineers and Scientists
This course is an introduction to statistics for graduate students in engineering and the sciences. Topics covered include basic data analysis, issues in the design of studies, an introduction to probability, point and interval estimation and hypothesis testing for means and proportions from one and two samples, simple and multiple regression, analysis of one and two-way tables, one-way analysis of variance. As time permits, additional topics, such as distribution-free methods and the design and analysis of factorial studies will be considered. (Prerequisites: Integral and differential calculus.)
MA 512. Numerical Differential Equations
This course begins where MA 510 ends in the study of the theory and practice of the numerical solution of differential equations. Central topics include a review of initial value problems, including Euler's method, Runge-Kutta methods, multi-step methods, implicit methods and predictor- corrector methods; the solution of two-point boundary value problems by shooting methods and by the discretization of the original problem to form systems of nonlinear equations; numerical stability; existence and uniqueness of solutions; and an introduction to the solution of partial differential equations by finite differences. Other topics might include finite element or boundary element methods, Galerkin methods, collocation, or variational methods. (Prerequisites: graduate or undergraduate numerical analysis. Knowledge of a higher-level programming language is assumed.)
MA 514. Numerical Linear Algebra
This course provides students with the skills necessary to develop, analyze and implement computational methods in linear algebra. The central topics include vector and matrix algebra, vector and matrix norms, the singular value decomposition, the LU and QR decompositions, Householder transformations and Givens rotations, and iterative methods for solving linear systems including Jacobi, Gauss-Seidel, SOR and conjugate gradient methods; and eigenvalue problems. Applications to such problem areas as least squares and optimization will be discussed. Other topics might include: special linear systems, such as symmetric, positive definite, banded or sparse systems; preconditioning; the Cholesky decomposition; sparse tableau and other least-square methods; or algorithms for parallel architectures. (Prerequisite: basic knowledge of linear algebra or equivalent background. Knowledge of a higherlevel programming language is assumed.)
MA 521. Partial Differential Equations
This course considers a variety of material in partial differential equations (PDE). Topics covered will be chosen from the following: classical linear elliptic, parabolic and hyperbolic equations and systems, characteristics, fundamental/Green's solutions, potential theory, the Fredholm alternative, maximum principles, Cauchy problems, Dirichlet/ Neumann/Robin problems, weak solutions and variational methods, viscosity solutions, nonlinear equations and systems, wave propagation, free and moving boundary problems, homogenization. Other topics may also be covered. (Prerequisites: MA 503 and MA 504.)
MA 525. Optimal Control and Design with Composite Materials I
Modern technology involves a wide application of materials with internal structure adapted to environmental demands. This, the first course in a two-semester sequence, will establish a theoretical basis for identifying structures that provide optimal response to prescribed external factors. Material covered will include basics of the calculus of variations: Euler equations; transversality conditions; Weierstrass-Erdmann conditions for corner points; Legendre, Jacobi and Weierstrass conditions; Hamiltonian form of the necessary conditions; and Noether's theorem. Pontryagin's maximum principle in its original lumped parameter form will be put forth as well as its distributed parameter extension. Chattering regimes of control and relaxation through composites will be introduced at this point. May be offered by special arrangement.
MA 526. Optimal Control and Design with Composite Materials II
Topics presented will include basics of homogenization theory. Bounds on the effective properties of composites will be established using the translation method and Hashin-Shtrikman variational principles. The course concludes with a number of examples demonstrating the use of the theory in producing optimal structural designs. The methodology given in this course turns the problem of optimal design into a problem of rigorous mathematics. This course can be taken independently or as the sequel to MA 525.
MA 530. Discrete Mathematics/ CS 501. Discrete Structures
This course provides the student of mathematics or computer science with an overview of discrete structures and their applications, as well as the basic methods and proof techniques in combinatorics. Topics covered include sets, relations, posets, enumeration, graphs, digraphs, monoids, groups, discrete probability theory and propositional calculus. (Prerequisites: college math at least through calculus. Experience with recursive programming is helpful, but not required.)
MA 533. Discrete Mathematics II
This course is designed to provide an in-depth study of some topics in combinatorial mathematics and discrete optimization. Topics may vary from year to year. Topics covered include, as time permits, partially ordered sets, lattices, matroids, matching theory, Ramsey theory, discrete programming problems, computational complexity of algorithms, branch and bound methods.
MA 535. Algebra
Fundamentals of group theory: homomorphisms and the isomorphism theorems, finite groups, structure of finitely generated Abelian groups. Structure of rings: homomorphisms, ideals, factor rings and the isomorphism theorems, integral domains, factorization. Field theory: extension fields, finite fields, theory of equations. Selected topics from: Galois theory, Sylow theory, Jordan- Hölder theory, Polya theory, group presentations, basic representation theory and group characters, modules. Applications chosen from mathematical physics, Gröbner bases, symmetry, cryptography, error-correcting codes, number theory.
MA 540/4631. Probability and Mathematical Statistics I
Intended for advanced undergraduates and beginning graduate students in the mathematical sciences, and for others intending to pursue the mathematical study of probability and statistics. Topics covered include axiomatic foundations, the calculus of probability, conditional probability and independence, Bayes' Theorem, random variables, discrete and continuous distributions, joint, marginal and conditional distributions, covariance and correlation, expectation, generating functions, exponential families, transformations of random variables, types of convergence, laws of large numbers the Central Limit Theorem, Taylor series expansion, the delta method. (Prerequisite: knowledge of basic probability at the level of MA 2631 and of advanced calculus at the level of MA 3831/3832 is assumed.)
MA 541/4632. Probability and Mathematical Statistics II
This course is designed to provide background in principles of statistics. Topics covered include estimation criteria: method of moments, maximum likelihood, least squares, Bayes, point and interval estimation, Fisher's information, Cramer-Rao lower bound, sufficiency, unbiasedness, and completeness, Rao-Blackwell Theorem, efficiency, consistency, interval estimation pivotal quantities, Neyman-Person Lemma, uniformly most powerful tests, unbiased, invariant and similar tests, likelihood ratio tests, convex loss functions, risk functions, admissibility and minimaxity, Bayes decision rules. (Prerequisite: knowledge of the material in MA 540 is assumed.)
MA 542. Regression Analysis
Regression analysis is a statistical tool that utilizes the relation between a response variable and one or more predictor variables for the purposes of description, prediction and/or control. Successful use of regression analysis requires an appreciation of both the theory and the practical problems that often arise when the technique is employed with real-world data. Topics covered include the theory and application of the general linear regression model, model fitting, estimation and prediction, hypothesis testing, the analysis of variance and related distribution theory, model diagnostics and remedial measures, model building and validation, and generalizations such as logistic response models and Poisson regression. Additional topics may be covered as time permits. Application of theory to real-world problems will be emphasized using statistical computer packages. (Prerequisite: knowledge of probability and statistics at the level of MA 511 and of matrix algebra is assumed.)
MA 546. Design and Analysis of Experiments
Controlled experiments—studies in which treatments are assigned to observational units—are the gold standard of scientific investigation. The goal of the statistical design and analysis of experiments is to (1) identify the factors which most affect a given process or phenomenon; (2) identify the ways in which these factors affect the process or phenomenon, both individually and in combination; (3) accomplish goals 1 and 2 with minimum cost and maximum efficiency while maintaining the validity of the results. Topics covered in this course include the design, implementation and analysis of completely randomized complete block, nested, split plot, Latin square and repeated measures designs. Emphasis will be on the application of the theory to real data using statistical computer packages. (Prerequisite: knowledge of basic probability and statistics at the level of MA 511 is assumed.)
MA 547. Design and Analysis of Observational and Sampling Studies
Like controlled experiments, observational studies seek to establish cause-effect relationships, but unlike controlled experiments, they lack the ability to assign treatments to observational units. Sampling studies, such as sample surveys, seek to characterize aspects of populations by obtaining and analyzing samples from those populations. Topics from observational studies include: prospective and retrospective studies; overt and hidden bias; adjustments by stratification and matching. Topics from sampling studies include: simple random sampling and associated estimates for means, totals, and proportions; estimates for subpopulations; unequal probability sampling; ratio and regression estimation; stratified, cluster, systematic, multistage, double sampling designs, and time permitting, topics such as modelbased sampling, spatial and adaptive sampling. (Prerequisite: knowledge of basic probability and statistics, at the level of MA 511 is assumed.)
MA 548. Quality Control
This course provides the student with the basic statistical tools needed to evaluate the quality of products and processes. Topics covered include the philosophy and implementation of continuous quality improvement methods, Shewhart control charts for variables and attributes, EWMA and Cusum control charts, process capability analysis, factorial and fractional factorial experiments for process design and improvement, and response surface methods for process optimization. Additional topics will be covered as time permits. Special emphasis will be placed on realistic applications of the theory using statistical computer packages. (Prerequisite: knowledge of basic probability and statistic, at the level of MA 511 is assumed.)
MA 549. Analysis of Lifetime Data
Lifetime data occurs frequently in engineering, where it is known as reliability or failure time data, and in the biomedical sciences, where it is known as survival data. This course covers the basic methods for analyzing such data. Topics include: probability models for lifetime data, censoring, graphical methods of model selection and analysis, parametric and distribution-free inference, parametric and distribution-free regression methods. As time permits, additional topics such as frailty models and accelerated life models will be considered. Special emphasis will be placed on realistic applications of the theory using statistical computer packages. (Prerequisite: knowledge of basic probability and statistics, at the level of MA 511 is assumed.)
MA 550. Time Series Analysis
Time series are collections of observations made sequentially in time. Examples of this type of data abound in many fields ranging from finance to engineering. Special techniques are called for in order to analyze and model these data. This course introduces the student to time and frequency domain techniques, including topics such as autocorrelation, spectral analysis, and ARMA and ARIMA models, Box-Jenkins methodology, fitting, forecasting, and seasonal adjustments. Time permitting, additional topics will be chosen from: Kalman filter, smoothing techniques, Holt-Winters procedures, FARIMA and GARCH models, and joint time-frequency methods such as wavelets. The emphasis will be in application to real data situations using statistical computer packages. (Prerequisite: knowledge of MA 511 is assumed. Knowledge of MA 541 is also assumed, but may be taken concurrently.)
MA 552. Distribution-Free and Robust Statistical Methods
Distribution-free statistical methods relax the usual distributional modeling assumptions of classical statistical methods. Robust methods are statistical procedures that are relatively insensitive to departures from typical assumptions, while retaining the expected behavior when assumptions are satisfied. Topics covered include, time permitting, order statistics and ranks; classical distribution-free tests such as the sign, Wilcoxon signed rank, and Wilcoxon rank sum tests, and associated point estimators and confidence intervals; tests pertaining to one and two-way layouts; the Kolmogorov-Smirnov test; permutation methods; bootstrap and Monte Carlo methods; M, L, and R estimators, regression, kernel density estimation and other smoothing methods. Comparisons will be made to standard parametric methods. (Prerequisite: knowledge of MA 541 is assumed, but may be taken concurrently.)
MA 554. Applied Multivariate Analysis
This course is an introduction to statistical methods for analyzing multivariate data. Topics covered are multivariate sampling distributions, tests and estimation of multivariate normal parameters, multivariate ANOVA, regression, discriminant analysis, cluster analysis, factor analysis and principal components. Additional topics will be covered as time permits. Students will be required to analyze real data using one of the standard packages available. (Prerequisite: knowledge of MA 541 is assumed, but may be taken concurrently. Knowledge of matrix algebra is assumed.)
MA 556. Applied Bayesian Statistics
Bayesian statistics makes use of an inferential process that models data summarizing the results in terms of probability distributions for the model parameters. A key feature is that in the Bayesian approach, past information can be updated with new data in an elegant way in order to aid in decision making. Topics included in the courses: statistical decision theory, the Bayesian inferential framework (model specification, model fitting and model checking); computational methods for posterior simulation integration; regression models, hierarchical models, and ANOVA; time permitting, additional topics will include generalized linear models, multivariate models, missing data problems, and time series analysis. (Prerequisites: knowledge of MA 541 is assumed.)
MA 559. Statistics Graduate Seminar
1 credit
This seminar introduces students to issues and trends in modern statistics. In the seminar, students and faculty will read and discuss survey and research papers, make and attend presentations, and participate in brainstorming sessions toward the solution of advanced statistical problems.
MA 560. Graduate Seminar
0 credits
Designed to introduce graduate students to study of original papers and afford them opportunity to give account of their work by talks in the seminar.
MA 562 A and B. Professional Master's Seminar
0 credits
This seminar will introduce professional master's students to topics related to general writing, presentation, group communication and interviewing skills, and will provide the foundations to successful cooperation within interdisciplinary team environments. All full-time students will be required to take both components A and B of the seminar during their professional master's studies.
MA 571. Financial Mathematics I
Introduction to arbitrage-based pricing of derivative securities, and their uses for hedging and risk management. Topics include securities markets, futures, options, swaps and other derivatives; arbitrage and risk-neutral pricing; binomial trees, martingales, stochastic difference equations; Black-Scholes formula and partial differential equation via limit transition; pricing of American options, convertible bonds, options on dividendpaying stock and on futures; sensitivity measures ("greeks"), implied and estimated volatilities; use of derivatives for hedging and risk management.
MA 572. Financial Mathematics II
This course introduces the advanced mathematical concepts and terminology used at the professional quantitative financial workplace and in the literature, and provides students with the background necessary to work in the rapidly expanding fixed income securities sector. The first part of the course is devoted to the concepts, terminology and methods of continuous-time mathematical finance. Topics include Brownian motion, continuous- time martingales. Stochastic differential equations, Ito calculus; risk-neutral valuation in terms of equivalent martingale measures. The power of the new tools is demonstrated on the derivation of the Black-Scholes and foreign exchange option pricing formulas. The second part of the course is devoted to fixed income securities and the termstructure of interest rates. Topics covered in this part include fixed income markets, instruments, risks and the term structure of interest rates; yield curve models, calibration and fitting; pricing of interest rate derivatives using models based on short rates (Vasicek, Cox-Ingersoll-Ross), and on the static and dynamic term-structure of interest rates (Ho-Lee, Black-Derman- Toy, Hull-White and Heath-Jarrow-Morton); pricing of corporate bonds, mortgage-backed securities and insurancelinked bonds; implementation of pricing models; derivative strategies for hedging and risk management in the fixed income sector. (Prerequisites: MA 503, MA 540 and MA 571.)
MA 573. Computational Methods of Financial Mathematics
Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two 7- week blocks covering the following topics.
Part I: Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives.
Part II: Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniques—antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models. (Prerequisites: MA 571, undergraduate level familiarity with numerical methods and basic programming skills.)
MA 574. Portfolio Valuation and Risk Management
Balancing returns vs. risks is one of the fundamental tasks of quantitative financial management. This course presents the most important mathematical concepts, methods and models used to value assets; select, maintain and optimize portfolios; and to manage risks. Topics covered include the following: returns, risks and utilities; quantification of risk—variance, shortfall risk, value at risk; portfolio analysis, diversification, correlations, principal components, sensitivity measures ("greeks"); asset valuation and pricing methods as capital markets theory, capital asset pricing model, efficient frontiers, arbitrage pricing theory, consumption/ accumulation and equilibrium models; risk management techniques—diversification, immunization, insurance/reinsurance, hedging; optimal asset allocation, portfolio optimization and dynamic delta hedging. The quantitative techniques covered in this course are used to support decisions by trading desk managers, corporate investment strategists, mutual companies, utilities, and of companies with commodities or foreign exchange risk exposures. (Prerequisite: MA 571.)
Courses on special topics are offered under this number. Contact the Mathematical Sciences Department for current offerings. See the supplement section of the on-line catalog for descriptions of courses to be offered in this academic year.
MA 595. Independent Study
1 to 3 credits
Supervised independent study of a topic of mutual interest to the instructor and the student.
MA 596. Master's Capstone
1 or more credits
The Master's Capstone is designed to integrate classroom learning with real-world practice. It can consist of a project, a practicum, a research review report or a research proposal. A written report and a presentation are required.
MA 598. Professional Master's Project
1 or more credits
This project will provide the opportunity to apply and extend the material studied in the coursework to the study of a real-world problem originating in the industry. The project will be a capstone integrating industrial experience with the previously acquired academic knowledge and skills. The topic of the project will come from a problem generated in industry, and could originate from prior internship or industry experience of the student. The student will prepare a written project report and make a presentation before a committee including the faculty advisor, at least one additional WPI faculty member and representatives of a possible industrial sponsor. The advisor of record must be a faculty member of the WPI Mathematical Sciences Department. The student must submit a written project proposal for approval by the Graduate Committee prior to registering for the project.
MA 599. Thesis
1 or more credits
Research study at the master's level.
MA 698. Ph.D. Project
1 or more credits
Ph.D. project work.
MA 699. Dissertation
1 or more credits
Research study at the Ph.D. level.
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