﻿ Undergraduate Catalog: Mathematical Sciences - WPI

# Mathematical Sciences

### MA 143X. CALCULUS III: A THEORETICAL APPROACH

This course will cover the same material as MA1023 but from a different perspective. A more rigorous study of sequences and series will be undertaken: starting from the least upper bound property in R, the fundamental theorems for convergent series will be proved. From there, convergence criteria for series will also be rigorously justified. L'Hopital's rule will also be introduced and proved. Homework problems will include a blend of computational exercises as usually assigned in MA1023, and problems with a stronger theoretical flavor. Recommended background: MA1021 and 1022 or equivalent. Note: Students can receive credit for both MA1023 and for MA143X.

### MA 144X. CALCULS IV: A THEORETICAL APPROACH

This course will cover the same material as in MA1024 from a more mathematically rigorous perspective. The course will start with the rigorous introduction of differentiation and integration for functions of one variable. After introducing vector functions, differentiation and integration will be extended to functions of several variables. Recommended background: MA143X. Note: Students can receive credit for both MA1024 and MA144X.

### MA 489X. APPLIED DIFFERENTIAL GEOMETRY

Topics covered: Geometry of curves and surfaces in R^3, tensor analysis, Riemannian geometry in n dimensions, (Geodesics, Covariant Differentiation, Riemann-Christoffel and Ricci tensors), applications in general relativity (Bianchi identity and Einstein tensor, Schwarzschild solution), differential forms, differential manifolds, Grassmann Manifolds and projective geometry, differentiation on manifolds, vector fields on manifolds (tangent and cotangent bundles),integration on manifolds. Recommended background: MA3831

### MA 1020. CALCULUS I WITH PRELIMINARY TOPICS

Cat. I (14-week course) This course includes the topics of MA 1021 and also presents selected topics from algebra, trigonometry, and analytic geometry. This course, which extends for 14 weeks and offers 1/3 unit of credit, is designed for students whose precalculus mathematics is not adequate for MA 1021. Although the course will make use of computers, no programming experience is assumed. Students may not receive credit for both MA 1020 and MA 1021.

### MA 1021. CALCULUS I

Cat. I This course provides an introduction to differentiation and its applications. Topics covered include: functions and their graphs, limits, continuity, differentiation, linear approximation, chain rule, min/max problems, and applications of derivatives. Recommended background: Algebra, trigonometry and analytic geometry. Although the course will make use of computers, no programming experience is assumed. Students may not receive credit for both MA 1021 and MA 1020.

### MA 1022. CALCULUS II

Cat. I This course provides an introduction to integration and its applications. Topics covered include: inverse trigonometric functions, Riemann sums, fundamental theorem of calculus, basic techniques of integration, volumes of revolution, arc length, exponential and logarithmic functions, and applications. Recommended background: MA 1021. Although the course will make use of computers, no programming experience is assumed.

### MA 1023. CALCULUS III

Cat. I This course provides an introduction to series, parametric curves and vector algebra. Topics covered include: numerical methods, indeterminate forms, improper integrals, sequences, Taylor?s theorem with remainder, convergence of series and power series, polar coordinates, parametric curves and vector algebra. Recommended background: MA 1022. Although the course will make use of computers, no programming experience is assumed.

### MA 1024. CALCULUS IV

Cat. I This course provides an introduction to multivariable calculus. Topics covered include: vector functions, partial derivatives and gradient, multivariable optimization, double and triple integrals, polar coordinates, other coordinate systems and applications. Recommended background: MA 1023. Although the course will make use of computers, no programming experience is assumed.

### MA 1033. INTRODUCTION TO ANALYSIS III

Cat. I This course develops the theory of integration and provides an introduction to series of numbers and series of functions. Topics covered include the Fundamental Theorem of Calculus, integration by parts, change of variable, series, convergence tests, rearrangements of series, sequences and series of functions, power series, Taylor series. Recommended background: MA 1032

### MA 1034. INTRODUCTION TO ANALYSIS IV

Cat. I The course provides a rigorous introduction to multivariable analysis. Topics covered include vector algebra, functions of several variables, partial derivatives, gradient, multiple integrals, Green?s theorem, Stokes? theorem, divergence theorem. Recommended background: MA 1033

### MA 1120. CALCULUS II (SEMESTER VERSION)

Cat.I The topics for integral calculus (MA 1022) are covered in this course: the concept of the definite integral, the Fundamental Theorem of Calculus, integration techniques, and applications of integration. Applications include: area, volume, arc length, center of mass, work, force, and exponential growth and decay. Logarithmic and exponential functions are studied in depth. Arithmetic and geometric sequences and series will also be covered. Key historical events in the development of integral calculus are examined. Technology will be used as appropriate to support the material being studied. This course extends for 14 weeks and offers 1/3 unit of credit. It is designed for students who would benefit from additional contact hours and who need to strengthen their mathematical background. Although the course will make use of computers, no programming experience is assumed. Students may not receive credit for both MA 1120 and MA 1022 or MA 1102.

### MA 1971. BRIDGE TO HIGHER MATHEMATICS

Cat.I The principal aim of this course is to introduce and enhance mathematical thinking. The course is intended not only for beginning mathematics, statistics or actuarial students, but also for students seeking to further their mathematical interests and those simply curious about logic and reason. Students in the course will be expected to explain, justify, defend, disprove, conjecture and verify mathematical ideas, both verbally and in writing. One expected by-product of this training is that students will develop concrete proof-writing skills which will improve their prospects for success in more advanced mathematics courses. When appropriate, course discussion will touch on current events in the mathematical sciences, including recently solved problems and open challenges facing today?s scientists. Recommended background: at least two courses in Mathematical Sciences at WPI, or equivalent.

### MA 2051. ORDINARY DIFFERENTIAL EQUATIONS

Cat. I This course develops techniques for solving ordinary differential equations. Topics covered include: introduction to modeling using first-order differential equations, solution methods for linear higher-order equations, qualitative behavior of nonlinear first-order equations, oscillatory phenomena including spring-mass system and RLC-circuits and Laplace transform. Additional topics may be chosen from power series method, methods for solving systems of equations and numerical methods for solving ordinary differential equations. Recommended background: MA 1024.

### MA 2071. MATRICES AND LINEAR ALGEBRA I

Cat. I This course provides a study of computational techniques of matrix algebra and an introduction to vector spaces. Topics covered include: matrix algebra, systems of linear equations, eigenvalues and eigenvectors, least squares, vector spaces, inner products, and introduction to numerical techniques, and applications of linear algebra. Recommended background: None.

### MA 2073. MATRICES AND LINEAR ALGEBRA II

Cat. I This course provides a deeper understanding of topics introduced in MA 2071 and also continues the development of those topics. Topics covered include: abstract vector spaces, linear transformations, matrix representations of a linear transformation, characteristics and minimal polynomials, diagonalization, eigenvalues and eigenvectors, inner product spaces. This course is designed primarily for Mathematical Science majors and those interested in the deeper mathematical issues underlying linear algebra. Undergraduate credit may not be earned both for this course and for MA 3071. Recommended background: MA 2071.

### MA 2201. DISCRETE MATHEMATICS

Cat I. This course serves as an introduction to some of the more important concepts, techniques, and structures of discrete mathematics providing a bridge between computer science and mathematics. Topics include functions and relations, sets, countability, groups, graphs, propositional and predicate calculus, and permutations and combinations. Students will be expected to develop simple proofs for problems drawn primarily from computer science and applied mathematics. Intended audience: computer science and mathematical sciences majors. Recommended background: None.

### MA 2210. MATHEMATICAL METHODS IN DECISION MAKING

This course introduces students to the principles of decision theory as applied to the planning, design and management of complex projects. It will be useful to students in all areas of engineering, actuarial mathematics as well as those in such interdisciplinary areas as environmental studies. It emphasizes quantitative, analytic approaches to decision making using the tools of applied mathematics, operations research, probability and computations. Topics covered include: the systems approach, mathematical modeling, optimization and decision analyses. Case studies from various areas of engineering or actuarial mathematics are used to illustrate applications of the materials covered in this course. Recommended background: MA 1024. Suggested background: Familiarity with vectors and matrices. Although the course makes use of computers, no programming experience is assumed. Students who have received credit for CE 2010 may not receive credit for MA 2210. Industrial Engineering majors cannot receive credit for both MA 2210 and BUS 2080.

### MA 2251. VECTOR AND TENSOR CALCULUS

Cat. I This course provides an introduction to tensor and vector calculus, an essential tool for applied mathematicians, scientists, and engineers. Topics covered include: scalar and vector functions and fields, tensors, basic differential operations for vectors and tensors, line and surface integrals, change of variable theorem in integration, integral theorems of vector and tensor calculus. The theory will be illustrated by applications to areas such as electrostatics, theory of heat, electromagnetics, elasticity and fluid mechanics. Recommended background: MA 1024.

### MA 2271. GRAPH THEORY

This course introduces the concepts and techniques of graph theory? a part of mathematics finding increasing application to diverse areas such as management, computer science and electrical engineering. Topics covered include: graphs and digraphs, paths and circuits, graph and digraph algorithms, trees, cliques, planarity, duality and colorability. This course is designed primarily for Mathematical Science majors and those interested in the deeper mathematical issues underlying graph theory. Undergraduate credit may not be earned both for this course and for MA 3271. Recommended background: MA 2071.

### MA 2273. COMBINATORICS

This course introduces the concepts and techniques of combinatorics? a part of mathematics with applications in computer science and in the social, biological, and physical sciences. Emphasis will be given to problem solving. Topics will be selected from: basic counting methods, inclusion-exclusion principle, generating functions, recurrence relations, systems of distinct representatives, combinatorial designs, combinatorial algorithms and applications of combinatorics. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying combinatorics. Undergraduate credit may not be earned both for this course and for MA 3273. Recommended background: MA 2071.

### MA 2431. MATHEMATICAL MODELING WITH ORDINARY DIFFERENTIAL EQUATIONS

Cat. I This course focuses on the theoretical foundations of ordinary equations while building models for physical and biological systems. Mathematical topics may include methods for solving systems of ordinary differential equations, existence and uniqueness theory, stability theory, phase-plane analysis and limit cycles. Examples will be chosen from electrical and mechanical oscillations, control theory, ecological models and reaction kinetics. Students will learn how to turn a real-life physical or biological problem into a mathematical one and to interpret the mathematical results. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying mathematical modeling. Undergraduate credit may not be earned both for this course and for MA 3431. Recommended background: MA 1024, MA 2051 and MA 2071.

### MA 2610. APPLIED STATISTICS FOR THE LIFE SCIENCES

Cat. I This course is designed to introduce the student to statistical methods and concepts commonly used in the life sciences. Emphasis will be on the practical aspects of statistical design and analysis with examples drawn exclusively from the life sciences, and students will collect and analyze data. Topics covered include analytic and graphical and numerical summary measures, probability models for sampling distributions, the central limit theorem, and one and two sample point and interval estimation, parametric and non-parametric hypothesis testing, principles of experimental design, comparisons of paired samples and categorical data analysis. Undergraduate credit may not be earned for both this course and for MA 2611. Recommended background: MA 1022.

### MA 2611. APPLIED STATISTICS I

Cat. I This course is designed to introduce the student to data analytic and applied statistical methods commonly used in industrial and scientific applications as well as in course and project work at WPI. Emphasis will be on the practical aspects of statistics with students analyzing real data sets on an interactive computer package. Topics covered include analytic and graphical representation of data, exploratory data analysis, basic issues in the design and conduct of experimental and observational studies, the central limit theorem, one and two sample point and interval estimation and tests of hypotheses. Recommended background: MA 1022.

### MA 2612. APPLIED STATISTICS II

Cat. I This course is a continuation of MA 2611. Topics covered include simple and multiple regression, one and two-way tables for categorical data, design and analysis of one factor experiments and distribution-free methods. Recommended background: MA 2611.

### MA 2621. PROBABILITY FOR APPLICATIONS

Cat. I This course is designed to introduce the student to probability. Topics to be covered are: basic probability theory including Bayes theorem; discrete and continuous random variables; special distributions including the Bernoulli, Binomial, Geometric, Poisson, Uniform, Normal, Exponential, Chisquare, Gamma, Weibull, and Beta distributions; multivariate distributions; conditional and marginal distributions; independence; expectation; transformations of univariate random variables. Recommended background: MA 1024.

### MA 2631. PROBABILITY

Cat. I The purpose of this course is twofold: ? To introduce the student to probability. Topics to be covered will be chosen from: axiomatic development of probability; independence; Bayes theorem; discrete and continuous random variables; expectation; special distributions including the binomial and normal; moment generating functions; multivariate distributions; conditional and marginal distributions; independence of random variables; transformations of random variables; limit theorems. ? To introduce fundamental ideas and methods of mathematics using the study of probability as the vehicle. These ideas and methods may include systematic theorem-proof development starting with basic axioms; mathematical induction; set theory; applications of univariate and multivariate calculus. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying probability theory. Recommended background: MA 1024. Undergraduate credit may not be earned both for this course and for MA 2621.

### MA 3211. THEORY OF INTEREST

Cat. I An introduction to actuarial mathematics is provided for those who may be interested in the actuarial profession. Topics usually included are: measurement of interest, including accumulated and present value factors; annuities certain; amortization schedules and sinking funds; and bonds. Recommended background: MA 1024 and the ability to write computer programs.

### MA 3212. LIFE CONTINGENCIES

Cat. I A continuation of a study of actuarial mathematics with emphasis on the theory and application of contigency mathematics in the areas of life insurance and annuities. Topics usually included are: survival functions and life tables; life insurance; life annuities; net premiums; and premium reserves. Recommended background: MA 3211 and either MA 2621 or MA 2631.

### MA 3231. LINEAR PROGRAMMING

Cat. I This course considers the formulation of real-world optimization problems as linear programs, the most important algorithms for their solution, and techniques for their analysis. Topics covered include: the primal and dual simplex algorithms, duality theory, parametric analysis, network flow models and, as time permits, bounded variable linear programs or interior methods. Recommended background: MA 2071.

### MA 3233. DISCRETE OPTIMIZATION

Discrete optimization is a lively field of applied mathematics in which techniques from combinatorics, linear programming, and the theory of algorithms are used to solve optimization problems over discrete structures, such as networks or graphs. The course will emphasize algorithmic solutions to general problems, their complexity, and their application to real-world problems drawn from such areas as VLSI design, telecommunications, airline crew scheduling, and product distribution. Topics will be selected from: Network flow, optimal matching, integrality of polyhedra, matroids, and NP-completeness. Undergraduate credit may not be earned both for this course and for MA 4233. Recommended background: At least one of MA 2271, MA 2273 or MA 3231.

### MA 3257. NUMERICAL METHODS FOR LINEAR AND NONLINEAR SYSTEMS

Cat. I This course provides an introduction to modern computational methods for linear and nonlinear equations and systems and their applications. Topics covered include: solution of nonlinear scalar equations, direct and iterative algorithms for the solution of systems of linear equations, solution of nonlinear systems, the eigenvalue problem for matrices. Error analysis will be emphasized throughout. Recommended background: MA 2071. An ability to write computer programs in a scientific language is assumed.

### MA 3457. NUMERICAL METHODS FOR CALCULUS AND DIFFERENTIAL EQUATIONS

Cat. I This course provides an introduction to modern computational methods for differential and integral calculus and differential equations. Topics covered include: interpolation and polynomial approximation, approximation theory, numerical differentiation and integration, numerical solutions of ordinary differential equations. Error analysis will be emphasized throughout. Recommended background: MA 2051. An ability to write computer programs in a scientific language is assumed. Undergraduate credit may not be earned for both this course and for MA 3255/CS 4031.

### MA 3471. ADVANCED ORDINARY DIFFERENTIAL EQUATIONS

The first part of the course will cover existence and uniqueness of solutions, continuous dependence of solutions on parameters and initial conditions, maximal interval of existence of solutions, Gronwall?s inequality, linear systems and the variation of constants formula, Floquet theory, stability of linear and perturbed linear systems. The second part of the course will cover material selected by the instructor. Possible topics include: Introduction to dynamical systems, stability by Lyapunov?s direct method, study of periodic solutions, singular perturbation theory and nonlinear oscillation theory. Recommended background: MA 2431 and MA 3832.

### MA 3475. CALCULUS OF VARIATIONS

This course covers the calculus of variations and select topics from optimal control theory. The purpose of the course is to expose students to mathematical concepts and techniques needed to handle various problems of design encountered in many fields, e. g. electrical engineering, structural mechanics and manufacturing. Topics covered will include: derivation of the necessary conditions of a minimum for simple variational problems and problems with constraints, variational principles of mechanics and physics, direct methods of minimization of functions, Pontryagin?s maximum principle in the theory of optimal control and elements of dynamic programming. Recommended background: MA 2051.

### MA 3627. APPLIED STATISTICS III

Cat. II This course continues the exploration of statistics for scientific and industrial applications, begun in MA 2611 and MA 2612. Topics covered include the design and analysis of general factorial experiments, two-level factorial and fractional factorial experiments, Taguchi methods, response surface analysis, and statistical quality control. Recommended background: MA 2612. This course will be offered in 2013-14, and in alternating years thereafter.

### MA 3631. MATHEMATICAL STATISTICS

Cat. I This course introduces students to the mathematical principles of statistics. Topics will be chosen from: Sampling distributions, limit theorems, point and interval estimation, sufficiency, completeness, efficiency, consistency; the Rao- Blackwell theorem and the Cramer-Rao bound; minimum variance unbiased estimators and maximum likelihood estimators; tests of hypotheses including the Neyman-Pearson lemma, uniformly most powerful and likelihood radio tests. Recommended background: MA 2631.

### MA 3823. GROUP THEORY

Cat. II This course provides an introduction to one of the major areas of modern algebra. Topics covered include: groups, subgroups, permutation groups, normal subgroups, factor groups, homomorphisms, isomorphisms and the fundamental homomorphism theorem. Recommended background: MA 2073. This course will be offered in 2014-15 and in alternating years thereafter.

### MA 3825. RINGS AND FIELDS

This course provides an introduction to one of the major areas of modern algebra. Topics covered include: rings, integral domains, ideals, quotient rings, ring homomorphisms, polynomial rings, polynomial factorization, extension fields and properties of finite fields. Recommended background: MA 2073. Undergraduate credit may not be earned both for this course and for MA 3821.

### MA 3831. PRINCIPLES OF REAL ANALYSIS I

Cat. I Advanced Calculus is a two-part course giving a rigorous presentation of the important concepts of classical real analysis. Topics covered in the two-course sequence include: basic set theory, elementary topology of Euclidean spaces, limits and continuity, differentiation Reimann-Stieltjes integration, infinite series, sequences of functions, and topics in multivariate calculus. Recommended background: MA 2051 and MA 2071.

### MA 3832. PRINCIPLES OF REAL ANALYSIS II

Cat. I MA 3832 is a continuation of MA 3831. For the contents of this course, see the description given for MA 3831. Recommended background: MA 3831.

### MA 4213. RISK THEORY

This course covers topics in risk theory as it is applied, under specified assumptions, to insurance. Topics covered include: economics of insurance, short term individual risk models, single period and extended period collective risk models, and applications. Recommended background: MA 2631.

### MA 4214. SURVIVAL MODELS

Survival models are statistical models of times to occurrence of some event. They are widely used in areas such as the life sciences and actuarial science (where they model such events as time to death, or to the development or recurrence of a disease), and engineering (where they model the reliability or useful life of products or processes). This course introduces the nature and properties of survival models, and considers techniques for estimation and testing of such models using realistic data. Topics covered will be chosen from: parametric and nonparametric survival models, censoring and truncation, nonparametric estimation (including confidence intervals and hypothesis testing) using right-, left-, and otherwise censored or truncated data. Recommended background: MA 3631.

### MA 4235. MATHEMATICAL OPTIMIZATION

This course explores theoretical conditions for the existence of solutions and effective computational procedures to find these solutions for optimization problems involving nonlinear functions. Topics covered include: classical optimization techniques, Lagrange multipliers and Kuhn-Tucker theory, duality in nonlinear programming, and algorithms for constrained and unconstrained problems. Recommended background: Vector calculus at the level of MA 2251.

### MA 4237. PROBABILISTIC METHODS IN OPERATIONS RESEARCH

This course develops probabilistic methods useful to planners and decision makers in such areas as strategic planning, service facilities design, and failure of complex systems. Topics covered include: decisions theory, inventory theory, queuing theory, reliability theory, and simulation. Recommended background: Probability theory at the level of MA 2621 or MA 2631.

### MA 4291. APPLICABLE COMPLEX VARIABLES

Cat. I This course provides an introduction to the ideas and techniques of complex analysis that are frequently used by scientists and engineers. The presentation will follow a middle ground between rigor and intuition. Topics covered include: complex numbers, analytic functions, Taylor and Laurent expansions, Cauchy integral theorem, residue theory, and conformal mappings. Recommended background: MA 1024 and MA 2051.

### MA 4411. NUMERICAL ANALYSIS OF DIFFERENTIAL EQUATIONS

This course is concerned with the development and analysis of numerical methods for differential equations. Topics covered include: well-posedness of initial value problems, analysis of Euler?s method, local and global truncation error, Runge-Kutta methods, higher order equations and systems of equations, convergence and stability analysis of one-step methods, multistep methods, methods for stiff differential equations and absolute stability, introduction to methods for partial differential equations. Recommended background: MA 2071 and MA 3457/CS 4033. An ability to write computer programs in a scientific language is assumed.

### MA 4451. BOUNDARY VALUE PROBLEMS

Cat. I Science and engineering majors often encounter partial differential equations in the study of heat flow, vibrations, electric circuits and similar areas. Solution techniques for these types of problems will be emphasized in this course. Topics covered include: derivation of partial differential equations as models of prototype problems in the areas mentioned above, Fourier Series, solution of linear partial differential equations by separation of variables, Fourier integrals and a study of Bessel functions. Recommended background: MA 1024 or and MA 2051.

### MA 4473. PARTIAL DIFFERENTIAL EQUATIONS

The first part of the course will cover the following topics: classification of partial differential equations, solving single first order equations by the method of characteristics, solutions of Laplace?s and Poisson?s equations including the construction of Green?s function, solutions of the heat equation including the construction of the fundamental solution, maximum principles for elliptic and parabolic equations. For the second part of the course, the instructor may choose to expand on any one of the above topics. Recommended background: MA 2251 and MA 3832.

### MA 4603. STATISTICAL METHODS IN GENETICS AND BIOINFORMATICS

This course provides students with knowledge and understanding of the applications of statistics in modern genetics and bioinformatics. The course generally covers population genetics, genetic epidemiology, and statistical models in bioinformatics. Specific topics include meiosis modeling, stochastic models for recombination, linkage and association studies (parametric vs. nonparametric models, family-based vs. population-based models) for mapping genes of qualitative and quantitative traits, gene expression data analysis, DNA and protein sequence analysis, and molecular evolution. Statistical approaches include log-likelihood ratio tests, score tests, generalized linear models, EM algorithm, Markov chain Monte Carlo, hidden Markov model, and classification and regression trees. Recommended background: MA 2612, MA 2631 (or MA 2621), and one or more biology courses.

### MA 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 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.)

Cat. I