Illinois Section of the MAA

Abstracts of Talks at the 1999 ISMAA Annual Meeting

Planimeters and Isoperimetric Inequalities

Robert Foote, Wabash College
1:30p.m. Friday April 9, 1999, Olin Auditorium

A planimeter is a mechanical instrument used to determine the area of a region in the plane. As the boundary of the region is traced, a wheel attached to the instrument partially rolls and partially slides, recording a component of its motion on the plane. The area of the region is a simple function of the net roll of the wheel. I will show how a planimeter works, and then use the ideas involved to give a simple proof of the isoperimetric inequality in the plane. I will also mention how a planimeter would work on the sphere and hyperbolic plane, and how this is related to the isoperimetric inequalities in these cases. A planimeter will be available for those who want to try it. For pictures and additional information on planimeters, see This talk will be accessible to undergraduates.

Innovations in Undergraduate Teaching: Internet Distance Learning Programs

Debra Woods, Illinois Net Math Director; Jerry Uhl, University of Illinois at Champaign; Lisa Novak, Undergrad Lead Net Math Mentor; David Clydesdale, Sauk Valley Community College; Tom Anderson, Rock Falls; Several Current Students
3:00p.m. Friday April 9, 1999, Olin, Room 306

Panelists, providers and users, will describe the computer-based distance education courses which have been developed and offered by the University of Illinois during the past six years.

Issues in Undergraduate Education: Assessment

James Olsen, Western Illinois University; Iraj Kalantari, Western Illinois University; Kevan Hastings, Knox College; Herb Lyon, Black Hawk College
3:00p.m. Friday April 9, 1999, Olin, Room 201

Panelists will consider the current state of assessment for accreditation purposes and include experience from the college side and the visiting accrediting team side.

Trends in Mathematics: Tangent Sequences, World Records, Pi, and The Meaning of Life

Ira Rosenholtz, Eastern Illinois University
3:00p.m. Friday April 9, 1999, Olin, Room 202

A calculus student's question about whether the sequences {tan(n)/n}, {tan(n)/n^2}, {tan(n)/n^3}, ... go to 0 as n goes to infinity leads to some number theory answers and HUGE integers. Come play, "I can think of a larger important integer than you" and be prepared to lose!

Permanents, SDR's and Finitely Generated Abelian Groups

Victor K. Akatsa, Chicago State University
3:00p.m. Friday April 9, 1999, Olin, Room 307

The property of a non-negative matrix having a non-zero permanent is important in combinatorics. For example, it is used to determine whether, or not, a system of m subsets of an n-set (m <= n) has a system of distinct representatives ( SDR' s). In this paper, using permanents, we give a finiteness condition on the structure of abelian groups, defined by their generators and matrix of relations.

Math Modeling: A Project-Oriented Course

Svetlana Butler, Northern Illinois University
3:20p.m. Friday April 9, 1999, Olin, Room 307

Northern Illinois University first offered a mathematical modeling course (prerequisite calculus II) twenty five years ago. This paper describes the present organization of the course, which places strong emphasis on projects. It contains some statistics on students who have taken the course over the past few years and the results of a survey given to students.

Chinese Mathematics: Original Theories and Cultural Transmissions

Suzanne Hamon, Augustana College
3:40p.m. Friday April 9, 1999, Olin, Room 307

China's mathematics was influenced by India, the Arabs and the West. However, the Chinese have also contributed mathematics of significance. There is a general lack of awareness of these contributions.

Native American Mathematics

Robert Megginson, University of Michigan
ISMAA Annual Banquet, Friday April 9, 1999, College Center

Though mathematics is often portrayed as a subject invented and developed (until comparatively recent times) in the Eastern Hemisphere, in fact some very good mathematics was done in the Americas prior to the arrival of the Europeans. This talk will focus on the numbering systems of the Maya and Aztecs, as well as some of the number theory implicit in the computations they did. For example, it will be shown why the Native peoples of Central America considered the natural life span of a human being to be 52 years, based on some elementary number theory.

Travels with Selfridge

John Selfridge, Northern Illinois University
8:30a.m., Saturday April 10, 1999, Olin Auditorium

Since retirement, the speaker for this session continues to be involved in mathematics throughout the world. Within the past year or so he has been spotted at conferences from San Antonio to Australia. He will focus on the works of another famous mathematical traveler, Paul Erdös.

Innovations in Undergraduate Teaching: Using Projects and Labs with Technology to Teach Introductory Statistics

Lannette Poteete-Young, Judson College; Dave Klanderman, Trinity Christian College
9:45a.m., Saturday, April 10, 1999, Olin, Room 201

Two different innovations in teaching introductory statistics will be presented - using a laboratory component and using team research projects. Included in the presentation will be a discussion of the types of technology used for each innovation.

Issues in Undergraduate Education: An Articulation Dialog in Progress

Sandra Dawson, Glenbrook High School; Jim Trefzger, Parkland College; Larry Morley, Western Illinois University
9:45a.m., Saturday, April 10, 1999, Olin, Room 306

Can we make the mathematics experience from high school to college "seamless" for ALL our students? Do the new Illinois Learning Standards provide a framework for building a bridge? Is college algebra" still a connecting link in the bridge for ALL our students?

Trends in Mathematics: Trends in the Theory of Impulsive Differential Equations

Zahia Drici, Illinois Wesleyan University
9:45a.m. Saturday April 10, 1999, Olin, Room 202

Many real world phenomena experience an abrupt change of state and behave in a discontinuous fashion. These evolution processes may encounter short term perturbations whose durations are negligible in comparison to the duration of the process. Hence, it is natural to assume that the perturbations act instantaneously, that is, in the form of impulses. An adequate apparatus for the mathematical modeling of such phenomena is the impulsive differential equation. The talk will survey the trends that have emerged in the theory of impulsive differential equations and will reflect results in differential inequalities, monotone iterative techniques, stability or solutions, quenching phenomena, and oscillation. Some open problems will be formulated.

Innovations in Undergraduate Teaching: Totally Asynchronous Courses, Not For Every One but Perfect for Some

Jerry Uhl and Ben Halperin, University of Illinois at Urbana-Champaign; Carrie Eldridge, Lead Undergraduate Class Assistant; Several Current Students
10:50a.m., Saturday, April 10, 1999, Olin, Room 306

Differential equations and linear equations are currently being offered with a totally asynchronous option. Advantages and disadvantages will be discussed along with a consideration of the computer based materials needed to support them.

Issues in Undergraduate Education: Placement

Mary Kilbride, Augustana College; Lowell Doerder, Black Hawk College; John Beachy, Northern Illinois University; Richard Wilders, North Central College
10:50a.m., Saturday, April 10, 1999, Olin, Room 201

A panel, moderated by Mary Kilbride, will discuss what's being used and how it's working.

Trends in Mathematics: Geometry Revisited

Peter Andrews, Eastern Illinois University
10:50a.m., Saturday, April 10, 1999, Olin, Room 202

Often neglected in our entire education system, elementary geometry is a rich source of beautiful, and even useful, results. Many of these have been made very accessible to students and non-specialists through advances in technology such as Geometers Sketchpad. This talk will look at three examples spanning a period of over 2000 years, all of which can be easily understood with only high school geometry: Apollonius' Three Tangent Theorem, Feuerbach's Theorem, and Bankoff's Asymmetric Propeller Theorem.

An Examination of Sixth Order Diophantine Equations

Timothy Huber, Northern Illinois University
10:50a.m., Saturday, April 10, 1999, Olin, Room 307

Consider equations of the form (X_1)^k + (X_2)^k + ... + (X_r)^k = (Y_1)^k + (Y_2) + ... + (Y_r)^k where X_i is not equal to Y_j for all X_i , Y_j. Specifically, consider three-sums of six-powers (k = 6, r = 3). For many of these sums, the solutions obtained are also solutions to the quadratic three-sum (k = 2, r = 3). For example, 3^6 + 19^6 + 22^6 = 10^6 +15^6 + 23^6 and 3^2 + 19^2 + 22^2 = 10^2 +15^2 +23^2. There do exist counterexamples such that (X_1)^6 + (X_2)^6 + (X_3)^6 = (Y_1)^6 + (Y_2)^6 +(Y_3)^6, but (X_1)^2 + (X_2)^2 + (X_3)^2 is not equal to (Y_1)^2 + (Y_2)^2 + (Y_3)^2. A computer search is used to find these rare counterexamples. Computational methods will also be discussed.

Measure Chains and LaSalle's Invariance Principles

Anders Floor, Illinois Wesleyan University
11:10a.m., Saturday, April 10, 1999, Olin, Room 307

Measure chains are special subsets of the real line. The real line itself and all its discrete subsets are examples of measure chains but many subsets containing combinations of continuous intervals and discrete points are also measure chains. The calculus on measure chains is thus an extension of the differential and difference calculuses. The axioms defining measure chains will be given, and basic concepts and theorems in the measure chain calculus will be presented. Some results on stability (which culminate in Lasalle's Invariance Principle) will be presented in both a differential calculus and a difference calculus context; the hope is to extend these results to the measure chain calculus.

A Seemingly Novel Computation of the Extra Sum of Squares Due to a Regressor Variable in Linear Regression Partial F-Test

Sarah Dalpiaz, Western Illinois University
11:30a.m., Saturday, April 10, 1999, Olin, Room 307

The Extra Sum of Squares Principle is well known in linear regression as a method for calculating the variation of the dependent variable that can be explained by an independent variable. A seemingly novel approach to the calculation involves the following:

  1. partial residuals of the dependent variable on the set of all regressor variables with the exception that the regressor variable of interest is excluded,
  2. partial residuals of the regressor variable of interest when regressed on all other regressor variables, and
  3. the regression on these two partial residuals.

The sum of squares attached to such a partial residual regression model is identical to that attached to the models involving the Extra Sum of Squares Principle. Interpretations of an alternative approach are explored.

Ideals Attaining a Given Hilbert Function

Matthew J. Rodriguez, University of Illinois at Champaign
10:50a.m., Saturday, April 10, 1999, Olin, Room TBA

Ideals are investigated which attain given Hilbert functions. For each Hilbert function, the minimals among the graded resolutions of the ideals attaining it are of interest. The current literature cites only one example of such incomparable resolutions and in this case the sums of the graded betti numbers are the same. The aim is to produce more examples of incomparable minimal resolutions, and further to discover cases in which the minimal betti numbers are also incomparable. For this purpose, first a program is written in Macaulay 2 to produce all monomial ideals that attain a given Hilbert funtion with h(d) eventually zero. Each resulting ideal's resolution is calculated and their betti diagrams are compared. The existence of incomparable resolutions of ideals for a Hilbert function in four variables is shown.

Eigenvalues and Eigenvectors of the Coefficent Matrix of 2-Dimensional Linear Differential Equation Systems

Nathan Linger, University of Illinois at Champaign
11:10a.m., Saturday, April 10, 1999, Olin, Room TBA

A result is presented which was discovered in the study of differential equations. The result is that the coefficient matrix of 2-dimensional linear equation systems has complex eigenvalues and eigenvectors if and only if there are regions of attraction. A graphical field plot representation of such systems will be utilized. The used of Mathematica for both algebraic calculation and generation of these field plots enable the proof to arise quite naturally.

Multicultural Education and College Professors

Mariah Birgen, Wartburg College
12:00 noon, Olin Auditorium

Wartburg College, in an attempt to fulfill a strong commitment to Multicultural Education, sends faculty and staff to many areas in the world to participate in Cultural Immersion programs. The college believes that we will be unable to convince our students of the value of diversity if our faculty attitudes do not reflect that value. This past summer I spent three weeks in South Dakota participating in an immersion in the Lakota culture in order to learn cultural attitudes that will be beneficial to the culture of the college.

April 3, 1999