Applied Mathematics in the Undergraduate Curriculum

Abstracts

• Using Spreadsheets in a Computer Classroom

C. Edward Sandifer, Western Connecticut State University

In this workshop, we will demonstrate some laboratory exercises for a computer classroom that have been used successfully in various mathematics classes.  Exercises include a heat flow problem from Calculus, a cryptography problem from a Liberal Arts Math course, a population problem from a Modeling course, and a financial problem from a Business Math course.   The point of the workshop is not to teach you how to use Excel, but rather to show the kinds of problems you can ask students to do in a laboratory and to demonstrate the kind of support and encouragement you have to give students so that they can be successful with those problems.  Participants should already know at least a little about Excel.  The workshop will be in a computer laboratory/classroom.  Bring a disk.

Ed Sandifer teaches mathematics at Western Connecticut State University, host of this MAA meeting.  He particularly enjoys the history of mathematics, and has taught himself Latin so that he can read and translate the works of Leonhard Euler.  He is editing the next edition of the MAA's Basic Library List, the list of 3000 books that the MAA recommends as essential to an undergraduate library collection.  He is also an avid runner, and has completed the last 26 Boston Marathons.
 

• College Mathematics Departments' Need to Respond to State Frameworks and Mastery Testing

Kathy Bavelas, Manchester Community Technical College (CT)

This talk will focus primarily on the impact that the recently adopted Connecticut Frameworks in Mathematics, the Connecticut Mastery Tests (CMTs) and the Connecticut Academic Performance Test (CAPT) are having on the K - 12 mathematics curricula. Other states in New England have similar testing programs. We will address as many of the following questions as time permits. Should mastery certificates affect admissions decisions, placement testing and/or placement in mathematics courses?  Do we need to reconsider the questions asked on collegiate placement exams? Do we need to reexamine the course content of developmental courses and college level mathematics courses ? Are our pre-service teacher mathematics courses meeting the needs of our prospective teachers? Are these future teachers prepared for the BEST Program? Are these teachers, once hired, able to meet the expectations of the Boards of Education that hired them?
 
Kathy Bavelas has taught mathematics at Manchester Community-Technical  College since 1985.  She chaired the Mathematics Department there from 1989 to 1996 and will co-chair it this year.  She has a B.A. from the University of Connecticut, an M.S. from Central Connecticut State University, and an M.A.L.S. from Wesleyan University.  During her career, she has also taught at the middle-school level, at the high-school level, and part-time at Central Connecticut State University.  She is a member of the MAA, NCTM, AMATYC, MATYCONN,  ATOMIC, and AWM.  In 1996-97 she was the Distinguished Professor in Residence at the Connecticut Academy for Education in Mathematics, Science, and Technology.  For the past five years, she has been involved in writing and providing professional development for the field test teachers for Mathematical Connections, a secondary core curriculum project.
 

• Nonlinear Oscillations in Suspension Bridges

 P. Joseph McKenna, University of Connecticut
 
We review the early history of suspension bridges and  especially the famous collapse of the Tacoma Narrows bridge. We show  how new insights from computation and nonlinear analysis can explain much of the previously mysterious behavior of the bridge.
 
P. Joseph McKenna did his undergraduate work in Dublin (at U.C.D.) and his graduate work in Ann Arbor. He is currently a professor of mathematics at the University of
Connecticut, having previously worked at the University of Wyoming and the University of Florida.  For most of his professional life, he has been involved in research in partial differential equations, applying them to diverse problems in soil physics, fluid dynamics, optics, biology, and most recently on flexing in bridges and ships.
 
• Who Gets What? -- Apportionment and Fashion

Theresa Sandifer, Southern Connecticut State University

Many of the texts that are used to teach Liberal Arts math courses have a chapter on apportionment algorithms.  Students could not care less how Congress is apportioned now, let alone in the 1800s!  This talk will describe an application of apportionment algorithms that was used in the retail industry. The company that used the algorithm dominated the market during the 1980's.  The algorithms themselves are simple, but looking at them in a different context is something that students do not often see.  We will look at the problem that needed to be solved and talk about how the algorithm was modified to solve the problem.  We will also look at extensions of the problem and talk about how you can encourage students to "think outside the box".
 
Theresa Sandifer has a Ph.D. in Operations Research from the University of Massachusetts.  She has consulted on a variety of mathematical modeling and statistics problems.  Terry has written two workbooks for learning statistics using Excel and Minitab and is working on a statistics textbook that will be published in 1999.  She has also been a  judge for the Contest in Mathematical Modeling for the past three years.
 

• Succeeding in Industry: What Do We and Our Students Need?

Paul Davis, Worcester Polytechnic Institute

Most of our students will be working in business, industry, or government, not teaching or doing research.  What skills and attitudes do they need to succeed?  Which of those skills
can we teach in math classes, especially when the teacher is someone like me whose entire career has been spent behind ivy covered walls? There are no rigid prescriptions, but there are plenty of ideas and suggestions.  Some of them could seem a bit subversive.  Others may open up rewarding new classroom experiences.  Ultimately, meeting the challenge of preparing students for nonacademic work can mean better education and better mathematics.

Paul Davis has been a member of the Mathematical Sciences Faculty at Worcester Polytechnic Institute since 1970, when he received his Ph.D. in applied mathematics from Rensselaer Polytechnic Institute.  He has worked with various applications of differential equations and, more recently, with optimization problems in electric power networks.  Driven by his research interests, by his experience as an industrial consultant, and by his desire to prepare students for a range of careers, his teaching emphasizes modeling, applications, and open-ended problem solving.  He is in the last stages of completing a new differential equations text, Differential Equations: Modeling with MATLAB, to be published by Prentice Hall.  He is currently Director of WPI's London Project Center.

Saturday, November 21

• The Crisis in Inner-city Public School Mathematics: What College Math Departments Can Do.

Robert Case, Northeastern University

The majority of inner-city secondary students are denied access to strong secondary school mathematics programs. The problem has technical (pedagogy and content), normative (expectations of students, cultural issues), and resource (strength and strategy of schools themselves) dimensions. Colleges and universities have a key role to play in responding. The concrete model of Boston's new calculus courses will be used to provide examples.

Bob Case was chairman of the Department of Mathematics at St. Anselm College from 1967-70. In the seventies, he developed and taught graduate courses in logic at Northeastern University, where he is Associate Professor of Mathematics. In the eighties, he was involved in calculus reform, and in the nineties, he is working with high school teachers in Boston, where state-of-the-art calculus courses have begun in five schools which previously offered no calculus whatsoever.  Dr. Case is this year's teaching award honoree for the  Northeastern Section.
 

• CHRISTIE LECTURE:     Cosine Transforms and Wavelet Transforms and Applications

Gilbert Strang (Massachusetts Institute of Technology)

 Each Discrete Cosine Transform uses N real basis vectors whose components are cosines.  These basis vectors are orthogonal and the transform is extremely useful in image processing.  The cosine series is quickly computed by the FFT.  But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are in applications.  We prove orthogonality in a different way.  Each DCT comes from the eigenvectors of a symmetric "second-difference matrix".  By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 (and also four more orthogonal bases of cosines).  The boundary condition determines the centering (at a meshpoint or a midpoint) and decides on the entries cos [j or j+0.5] [k or k+0.5] pi/[N or N+...].   Then we discuss bases from filter banks and wavelets.  The key is to create a banded block Toeplitz matrix whose inverse is also banded.  We will describe how wavelets are created and how they are used in signal and image processing.  They are surprisingly interesting.

Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford.  His Ph. D. was from UCLA and since then he has taught at MIT.  He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences.   He is President-Elect of SIAM during 1998 (to take office in 1999-2000).  Professor Strang has published a monograph with George Fix, An Analysis of the Finite Element Method, and six textbooks: Introduction to Linear Algebra  (1993,1998), Linear Algebra and Its Applications (1976,1980,1988), Introduction to Applied Mathematics (1986), Calculus (1991), Wavelets and Filter Banks, with Truong Nguyen (1996), and Linear Algebra, Geodesy, and GPS, with Kai Borre (1997).  His web home page is http://www-math.mit.edu/~gs
 

• STUDENT WORKSHOP:     Wavelets and Applications

Vasily Strela, Dartmouth College

The purpose of this workshop is to get some insight into why wavelets are useful for many practical applications. We demonstrate that wavelet bases have an ability to analyze signals on different scales. Then we connect mathematical properties of wavelet functions such as approximation order, symmetry, orthogonality, and smoothness with the performance of filter banks in signal processing.  The MATLAB Wavelet Tool-box will be used to illustrate the talk, and active  participation of the audience is expected.  Participants may get an introduction to wavelets by attending Gilbert Strang's talk at 10:35 a.m.

Vasily Strela received his M.Sc. degree in applied mathematics from Moscow Institute of Physics and Technology in 1992 and his  Ph.D. degree in mathematics from Massachusetts Institute of Technology in 1996. Currently  he is with the  Department of Mathematics at  Dartmouth College.  His interests include theory and applications of wavelets, signal processing, Toeplitz matrices, and circulant matrices.
 

• Panel discussion on undergraduate programs that link students to industrial/applied problems :  how internships and undergraduate research involving industrial/real-world applied problems can be implemented in the undergraduate curriculum.

Paul Davis (Worcester Polytechnic Institute), moderator.

David Chris Arney (United States Military Academy at West Point) will discuss the Interdisciplinary Lively Application Projects (ILAPs) written and used by schools in the INTERMATH Consortium,  James Hefferon (St. Michael's College) will discuss the internship program at St. Michaels's, and  Bogdan Vernescu (Worcester Polytechnic Institute) will discuss WPI's Center for Industrial Mathematics and Statistics (CIMS) and related student projects.

David Chris Arney has an undergraduate degree from the United States
Military Academy (USMA) and a Ph.D. from Rensselaer Polytechnic Institute.  He has taught mathematics at USMA for 17 years and is the author of several mathematics textbooks and laboratory manuals.  Recently, he edited two books containing applications of mathematics:  Interdisciplinary Lively Application Projects (ILAPs) published by the MAA and Military Mathematical Modeling.  He is the book review editor for the Mathematics and Computer Education Journal.  His areas of research interest include applied mathematics, numerical analysis, number theory, and history of mathematics.  His teaching interests include using computers, writing, and interdisciplinary applications in the mathematics curriculum.  He has given numerous workshops on curriculum reform and the use of technology and applications in teaching and learning mathematics.  He is the Head of the Department of Mathematical Sciences at USMA.

Jim Hefferon earned his undergraduate and graduate degrees at the University of
Connecticut, finishing with a thesis in Logic.  He spent three years visiting Union College, and a year in industry and at the University of Connecticut.  He's been at Saint Michael's College for eight years and is an Associate Professor.  He has been the coordinator of  mathematics internships there for two years.

Bogdan Vernescu has been a faculty member in the Department of Mathematical Sciences at Worcester Polytechnic Institute since 1991. He received a Ph.D. in applied mathematics from the Institute of Mathematics of the Romanian Academy.  His research work is in the field of applied partial differential equations in continuum mechanics.  He has developed and advised several undergraduate and graduate projects with corporate sponsorship. He established and is currently the Director of the Center for Industrial Mathematics and Statistics at WPI. This past summer he organized the first NSF sponsored Research Experience for Undergraduates Program in Industrial Mathematics and Statistics at WPI.
 

• The SLOB in Baseball:  A Mathematical Model for Judging Offensive Value

Helen Salzberg, Rhode Island College, and  David Abrahamson, Rhode Island College

Wouldn't it be nice if a baseball fan (or scout or owner) could, by looking at an easily interpretable single number, determine a player's contribution to his team's offense?  In this paper, we report on such a statistic, present the relationship between it and team runs scored, and compare the correlation  coefficient with that of the more usually used offensive statistics.

Helen Salzberg is a professor of Mathematics and Chair of the Department of Mathematics and Computer Science at Rhode Island College.  She received her BA from Brooklyn College, MA from the University of Wisconsin, and did doctoral work at Cornell University.  She teaches a wide range of courses, with abstract algebra as her favorite.  Baseball is a passion since her  childhood with the Brooklyn Dodgers.

Dave Abrahamson received his Ph.D. in Applied Mathematics from Brown University in 1981 and has taught at Rhode Island College since 1986.  He has been a dedicated fan of baseball, football, and basketball his entire life.  His academic interests include differential and difference equations, mathematical modelling, and mathematics education.
 

• Constraint Modeling Issues for a Restructured Electric Power Industry

Lucy Kimball, Bentley College
 
In a deregulated environment an electric power generation company will submit offer prices for which it is willing to supply a given capacity on an hourly basis. These prices will be determined in part by an economic dispatch. The economic dispatch determines the optimal generating level for each generator which minimizes the generation costs, subject to a set of generator and network constraints. Existing economic dispatch algorithms either ignore important constraints such as ramping and energy constraints or treat them in a heuristic manner.  An economic dispatch algorithm which includes time coupling constraints as well as transmission constraints will be described. The algorithm can be used to illustrate some caveats to relying on traditional economic dispatch algorithms. Some of its predictions raise issues about operation and modeling that managers of generation companies must consider as they enter the deregulated electricity market. For example, electric power utility planners and system operators typically model water rates, or the efficiency of hydro turbine systems as linear. A close examination of actual hydro-turbine efficiency curves shows that the water rates are not linear. Though the discharge rates are fairly flat, even a small curvature can dramatically change the dispatch and the total generation costs.

Lucy Kimball earned her B.S. from the University of Massachusetts Lowell, and her M.S. and Ph.D. in Applied Mathematics from Worcester Polytechnic Institute. Her research interests include optimization problems for large scale electric power systems. She is currently an Assistant Professor at Bentley College in Waltham, MA, and a Project NeXT fellow.