2002-2003 Undergraduate Catalog

Mathematical Sciences

H. F. WALKER, HEAD
PROFESSORS: P. R. Christopher, P. W. Davis, M. Humi, R. P. Lipton, K. A. Lurie, R. Y. Lui, J. D. Petruccelli, D. Tang, H. F. Walker
ASSOCIATE PROFESSORS: M. Chen, W. Farr, J. D. Fehribach, A. C. Heinricher, W. J. Martin, B. Nandram, B. Servatius, D. Vermes, B. Vernescu
ASSISTANT PROFESSORS: B. Doytchinov, R. Jordan, C. Larsen, M. Sarkis, S. Weekes
VISITING FACULTY: D. Damian, W. Hwang, D. Pasca, A. Swift, S. Mathur, Y. Boissy
ACADEMIC STAFF: J. Goulet, A. Wiedie

INTRODUCTION

Study in the mathematical sciences requires hard work and discipline to attain the clarity, precision, logic, and economy of thought that recruiters from business, industry and academia value so highly in our graduates. And the rewards are substantial: mathematical science careers such as actuary, statistician and mathematician consistently rank at the top of lists of the most desirable professions. If what we've said so far interests you, read on to learn more about what mathematical scientists do, and particularly about how our programs-Mathematical Sciences and Actuarial Mathematics-can prepare you for a challenging and rewarding career.

PROGRAM IN MATHEMATICAL SCIENCES

Study in the mathematical sciences offers a broad spectrum of opportunities to the prospective major. While filled with open problems and subject to intense research in well-established areas, the mathematical sciences are constantly undergoing renewal as new questions and challenges arise in applications. Indeed, whole new areas of mathematical sciences are continually being born and existing areas are reinvigorated as surprising connections are discovered to the physical, biological, and social sciences, to computer science, and to engineering, business, industry, and finance. Not surprisingly, many of our majors are double majors who seek to apply mathematics to problems in their other fields. Such students are trained in the modeling, analysis, and computation necessary for solving problems in their other fields of interest.

Career opportunities for majors are many and varied. Recent graduates have embarked on careers in business and industry (e.g., Microsoft Corporation, Raytheon Company, Polaroid Corporation, MITRE Corporation, Fidelity Investments, Aetna Insurance, and Sun Life Financial) or have entered graduate school (e.g., Purdue University, University of California at Berkeley, Harvard University, Stanford University, Northwestern University, WPI) in such diverse disciplines as mathematics, statistics, law, management, physics, nuclear engineering, civil engineering, and education. More on career and employment opportunities can be found at the web sites of the Mathematical Association of America http://www.maa.org/students/career.html, or the American Statistical Association http://www.amstat.org/careers/index.html.

PROJECTS/INDEPENDENT STUDIES

Some of the most active career directions in the mathematical sciences are reflected in the MQP areas around which the department's offerings are organized: Algebraic and Discrete Mathematics, Computational and Applied Analysis, Operations Research, and Probability and Statistics. As early as practical, and certainly no later than the sophomore year, the mathematical sciences major should begin exploring these different areas. The transition courses, MA 2073, 2271, 2273, 2431, and 2631, are specifically designed to introduce the four MQP areas while preparing the student for advanced courses and the MQP.

While most students choose MQPs in one of the four areas mentioned above, it is possible to design an MQP that does not fit into any one area. In these cases, students will want to take special care to plan their programs carefully with their advisors so that sufficient backgrounds are obtained by the time the students begin their MQPs.

Many MQPs involve the solution of real-world problems proposed by industrial sponsors. Details can be found at http://www.wpi.edu/~cims.

A current listing of specific available projects with their descriptions is available at the department office, and at http://www.wpi.edu/Academics/Projects/available.html.

Independent studies are a good way for students to learn topics that are not taught in regularly-scheduled courses. Interested students should approach faculty with requests for independent studies. Some independent study areas and faculty advisors are listed below. In what follows, you will find for each MQP area:

• A brief description of the area including the kinds of challenges likely to be encountered by MQP students and mathematical scientists working there.
• Courses of interest.
• MQP and independent study topics and advisors.
• Examples of recent MQPs.

ALGEBRAIC AND DISCRETE MATHEMATICS
Algebraic and discrete mathematics is recognized as an increasingly important and vital area of mathematics. Many of the fundamental ideas of discrete mathematics play an important role in formulating and solving problems in a variety of fields ranging from ecology to computer science. For instance, graph theory has been used to study competition of species in ecosystems, to schedule traffic lights at an intersection, and to synchronize parallel processors in a computer. Coding theory has been applied to problems from the private and public sectors where encoding and decoding information securely is the goal. In turn, the problems to which discrete mathematics is applied often yield new and interesting mathematical questions. The goal of a project in discrete mathematics would be to experience this interaction between theory and application. To begin, a typical project team would assess the current state of a problem and the theory that is relevant. Once this is done, the project team's objective would be to make a contribution to solving the problem by developing new mathematical results.

In working in discrete mathematics, one may be writing algorithms, using the computer as a modeling tool, and using the computer to test conjectures. It is important that a student interested in this area have some computer proficiency. Depending on the project, an understanding of algorithm analysis and computational complexity may be helpful.

Courses of Interest
MA 2271 Graph Theory
MA 2273 Combinatorics
MA 3231 Linear Programming
MA 3233 Discrete Optimization
MA 3823 Group Theory
MA 3825 Rings and Fields
MA 4891 Topics in Mathematics (when appropriate)
CS 2005 Data Structures and Programming Techniques
CS 4120 Analysis of Algorithms
CS 4123 Theory of Computation

COMPUTATIONAL AND APPLIED ANALYSIS
This area of mathematics concerns the modeling and analysis of continuous physical or biological processes that occur frequently in science and engineering. Students interested in this area should have a solid background in analysis which includes the ability to analyze ordinary and partial differential equations through both analytical and computational means.

In most circumstances, an applied mathematician does not work alone but is part of a team consisting of scientists and engineers. The mathematician's responsibility is to formulate a mathematical model from the problem, analyze the model, and then interpret the results in light of the experimental evidence. It is, therefore, important for students to have some experience in mathematical modeling and secure a background in one branch of science or engineering through a carefully planned sequence of courses outside of the department.

With the increase in computational power, many models previously too complicated to be solvable, can now be solved numerically. It is, therefore, recommended that students acquire enough computer proficiency to take advantage of this. Computational skill is growing in importance and should be a part of every applied mathematician's training. Students may learn these skills through various numerical analysis courses offered by the department. An MQP in this area will generally involve the modeling of a real-life problem, analyzing it, and solving it numerically.

Courses of Interest
MA 2251 Vector and Tensor Calculus for Engineers
MA 2431 Mathematical Modeling with Ordinary Differential Equations
MA 3231 Linear Programming
MA 3257 Numerical Methods for Linear and Nonlinear Systems
MA 3457 Numerical Methods for Calculus and Differential Equations
MA 3471 Advanced Ordinary Differential Equations
MA 3475 Calculus of Variations
MA 4235 Mathematical Optimization
MA 4255 Numerical Analysis II
MA 4291 Applicable Complex Variables
MA 4411 Numerical Analysis of Differential Equations
MA 4451 Boundary Value Problems
MA 4473 Partial Differential Equations

OPERATIONS RESEARCH
Operations research is an area of mathematics which seeks to solve complex problems that arise in conducting and coordinating the operations of modern industry and government. Typically, operations research looks for the best or optimal solutions to a given problem. Problems within the scope of operations research methods are as diverse as finding the lowest cost school bus routing that still satisfies racial guidelines, deciding whether to build a small plant or a large plant when demand is uncertain, or determining how best to allocate timesharing access in a computer network.

Typically, these problems are solved by creating and then analyzing a mathematical model to determine an optimal strategy for the organization to follow. Often the problem requires a statistical model, and nearly always the analysis - whether optimizing through a set of equations or simulating the behavior of a process - involves the use of a computer. Finally, operations researchers must be able to interpret and apply the results of their analyses in an appropriate manner.

In addition to a solid background in calculus, probability and statistics, and the various operations research areas, prospective operations researchers should be familiar with computer programming and managerial techniques.

Courses of Interest
MA 2271 Graph Theory
MA 2273 Combinatorics
MA 3231 Linear Programming
MA 3233 Discrete Optimization
MA 3627 Applied Statistics III
MA 3631 Mathematical Statistics
MA 4235 Mathematical Optimization
MA 4237 Probabilistic Methods in Operations Research
MA 4631 Probability and Mathematical Statistics I
MA 4632 Probability and Mathematical Statistics II
MG 2500 Management Science I: Deterministic Decision Models
MG 3501 Management Science II: Risk Analysis
MG 3760 Simulation Modeling and Analysis

PROBABILITY AND STATISTICS
In many areas of endeavor, decisions must be made using information which is known only partially or has a degree of uncertainty attached to it. One of the major tasks of the statistician is to provide effective strategies for obtaining the relevant information and for making decisions based on it. Probabilists and statisticians are also deeply involved in stochastic modeling - the development and application of mathematical models of random phenomena. Applications to such areas as medicine, engineering, and finance abound.

Students interested in becoming probabilists or mathematical statisticians should consider additional study in graduate school. While graduate study is an option for students whose goals are to be applied statisticians, there are also career opportunities in business, industry, and government for holders of a B.S. degree. More information about careers in statistics can be found at the American Statistical Association's web site http://www.amstat.org/profession/index.html .

Students planning on graduate studies in this area would be well advised to consider, in addition to the courses of interest listed below, additional independent study or PQP work in probability and statistics, or some of the department's statistics graduate offerings.

Courses of Interest
MA 2611 Applied Statistics I
MA 2612 Applied Statistics II
MA 2631 Probability
MA 3627 Applied Statistics III
MA 3631 Mathematical Statistics
MA 4237 Probabilistic Methods in Operations Research
MA 4631 Probability and Mathematical Statistics I
MA 4632 Probability and Mathematical Statistics II

PROGRAM DISTRIBUTION REQUIREMENTS FOR THE MATHEMATICAL SCIENCES MAJOR

The normal period of residency at WPI is 16 terms. In addition to the WPI requirements applicable to all students, completion of a minimum of 10 units of study is required as follows:

NOTES:

1. Must include MA 3831-3832, or their equivalents, at least one of MA 3257, MA 3457, or equivalent, and at least one of MA 3823, MA 3825, or equivalent.
2. Must include at least three of the following: MA 2073, MA 2271, MA 2273, MA 2431, MA 2631, or their equivalents.
3. At least 7/3 units must consist of MA courses at the 3000 level or above.
4. May not include both MA 2631 and MA 3613.
5. At least 1/3 of the 2/3 CS units must require programming in a scientific language such as C, C++, Java, or Fortran.

MATHEMATICAL SCIENCES MAJOR DISTRIBUTION REQUIREMENTS CHART

PROGRAM IN ACTUARIAL MATHEMATICS

An actuary is a business professional who uses mathematical skills to define, analyze, and solve financial and social problems. Preparation for a career as an actuary requires mathematical aptitude, but actuarial work involves a practical type of mathematical ability mixed with business skills. An actuary deals with real-life problems rather than theoretical ones, must be curious, have sound judgment, and be able to think logically and creatively. The goal of the program in actuarial mathematics is to prepare students for positions in life and health insurance companies, property and casualty insurance companies, consulting firms, or state or federal government agencies.

The most widely accepted standard of professional qualification to practice as an actuary in the United States is a Fellowship in either the Society of Actuaries (SoA) or the Casualty Actuarial Society (CAS). Each organization administers a series of examinations leading to Fellowship. The first few in this series are mathematical in nature covering topics in calculus and linear algebra, probability, mathematical and applied statistics. Students interested in the actuarial mathematics program should read the latest SoA Associateship Catalog for more information. This catalog may be obtained from the Department of Mathematical Sciences, or at http://www.soa.org.

The actuarial mathematics program at WPI provides the first steps in preparing for these examinations and an introduction to fundamentals in business and economics. Students with mathematical aptitude should be able to pass the first two SoA examinations before graduation.

After graduation, most actuarial training is through self-study combined with on-the-job experience. Many employers rotate their actuarial trainees through various assignments exposing them to different aspects of business operations. In addition, companies frequently maintain actuarial libraries, sponsor group study sessions, and give trainees study time during work hours.

Brief descriptions of the project opportunities, distribution requirements, and the actuarial examinations are given below.

PROJECTS/INDEPENDENT STUDIES

Off-campus qualifying projects are regularly done in collaboration with insurance companies, and have in the past been sponsored by Aetna, Allmerica Financial, Blue Cross Blue Shield of Massachusetts, John Hancock Mutual Insurance, Premier Insurance, and Travelers Property Casualty. These projects give real-world experience of the actuarial field by having students involved in solving problems faced by professional actuaries. Instead of choosing a project already posed by a company/advisor team, students may instead seek out industry-sponsored projects on their own (often through internship connections) and propose them to a potential faculty advisor. Alternatively, students may choose to complete any other project in mathematics.

Students should select MQP and independent study topics which are related to their areas of preparation and interest. Some project and independent study areas are given below. In addition, a current listing of specific available projects with their descriptions is available at the department office, and at http://www.wpi.edu/Academics/Projects/available.html.

PROGRAM DISTRIBUTION REQUIREMENTS FOR THE ACTUARIAL MATHEMATICS MAJOR

The normal period of residency at WPI is 16 terms. In addition to the WPI requirements applicable to all students, completion of a minimum of 10 units of study is required as follows:

Requirements	Minimum Units
1. Mathematics (including MQP) (See notes 1-6).	7
2. Management (See note 7).	4/3
3. Additional courses or independent studies (except MS, PE courses, and other degree requirements) from any area (See note 8).	5/3

NOTES:

1. Must include MA 3831-3832, or their equivalents, at least one of MA 3257, MA 3457, or equivalent, and at least one of MA 3631, MA 4632, or equivalent.
2. Must include two of the following: MA 2073, MA 2271, MA 2273, MA 2431, MA 2631, or their equivalents.
3. Must include three of the following: MA 3211, MA 3212, MA 4213, MA 4214, or their equivalents.
4. May not include independent studies directed toward Society of Actuaries exams.
5. May not include either MA 2201 or MA 2210.
6. May not include both MA 2631 and MA 3613.
7. Must include MG 2101 and MG 2200 or their equivalents.
8. Must include 2/3 units of computer science.

ACTUARIAL MATHEMATICS MAJOR DISTRIBUTION REQUIREMENTS CHART

WPI COURSES AND THE SOCIETY OF ACTUARIES (SoA) EXAMINATIONS

The formulation of the distribution requirements for the program in actuarial mathematics was in large part motivated by the nature of the sequence of examinations that lead to Fellowship in the SoA or CAS. In particular, there are a number of WPI courses that cover fundamental topics that are included on the first few exams in this sequence.

Society of Actuaries Examination	WPI Courses
1. Mathematical Foundations of Actuarial Science	MA 1021 to MA 1024, MA 2631
2. Interest Theory, Economics and Finance	MA 3211, SS 1110, SS 1120, MG 1100, MG 2200
3. Actuarial Models	MA 3212, MA 4213,MA 4214, MA 2431,MA 4237
4. Actuarial Modeling	In addition to topics in 3.: MA 2071, MA 2611, MA 2612, MA 3627,

It must be emphasized that course work alone is not sufficient preparation for the examinations listed above; passage requires additional self-study. Several publications of the Society of Actuaries are available in the mathematics department office, and comprehensive information may be found at http://www.soa.org and http://www.casact.org. In addition, requests for information about the actuarial profession can be sent to the Society of Actuaries, 475 North Martingale Road, Suite 800, Schaumburg, IL 60173-2226.

STATISTICS MINOR

Statistical methods are widely used in science, engineering, business, and industry. The Statistics Minor is appropriate for all WPI students with interests in experimental design, data analysis, or statistical modeling. The minor is designed to enable a student to properly design studies and analyze the resulting data, and to evaluate statistical methods used in their field of study.

The statistics minor consists of completion of at least 2 units of work, which must consist of

1. At least 5/3 units of coursework, which must be drawn from the following lists of Foundation and Upper-Level Courses, and which must include successful completion of at least 2/3 units from each list:
   
   

   Courses for Statistics Minor (5/3 Unit Required)

   Foundation Courses (2/3 Unit Required)	Upper-Level Courses (2/3 Unit Required)
   o MA 2073 Matrices and Linear Algebra II o MA 2611 Applied Statistics I o MA 2612 Applied Statistics II o MA 2631 Probability, or    MA 3613 Probability for Applications	o MA 3627 Applied Statistics III o MA 3631 Mathematical Statistics o MA 4213 Risk Theory o MA 4214 Survival Models o MA 4237 Probabilistic Methods in Operations Research o MA 4631 Probability and Mathematical Statistics I o MA 4632 Probability and Mathematical Statistics II o Any statistics graduate course:    MA 509 or any course numbered MA 540 through MA 559

   
   
2. Capstone Experience
   The capstone experience usually consists of completion of MA 4658, Statistical Consulting. In this course, undergraduate students work with statistics faculty and graduate students to learn statistical practice and provide statistical advice to clients from the WPI community. Alternatively, students may arrange an independent study with one of the statistics faculty.

For information about the Statistics Minor, see any of the statistics faculty: Professors Joseph D. Petruccelli, Balgobin Nandram, Ming-Hui Chen, Doris Damian, Sunil Kumar Mathur, or Ann H. Wiedie.