Joint Meetings of the Iowa MAA, ASA, and IMATYC

Simpson College, Indianola, Iowa
April 14 and 15, 2000

Abstracts

Computer Workshop:

Maple6 and Linear Algebra
Gene Herman, Grinnell College     herman@math.grinnell.edu

Get an early hands-on look at the new Maple6 with its completely rewritten Linear Algebra package.  This package is the major change from MapleV, Release 5.1, to Maple6, and it offers exciting opportunities for teaching and learning Linear Algebra.  Not only is the new package much easier to use than the old one, but it has much stronger numerical capabilities that come from the alliance between Maple and NAG, the Numerical Algorithms Group.

The presenter is a Maple6 beta tester and coauthor of "Linear Algebra: Modules for Interactive Learning Using Maple."  This is a book and CD to be published this fall by Addison Wesley Longman that will exploit the new Maple6 Linear Algebra package.  The book and CD will contain 28 Maple worksheets (i.e., "modules") that cover the topics in a standard first course in linear algebra.  Instructors can use these modules as frequent or occasional labs in a linear algebra course, or they can use the entire collection as the sole textbook in a lab-based version of the course.  Workshop participants will be able to use the new Linear Algebra package and preview a sampling of the 28 modules.

Although the workshop will be designed for faculty who have had some previous exposure to Maple, all are welcome.  If you have questions, contact the presenter directly:  herman@math.grinnell.edu or 515-269-4202.
 

Mathematics Contributed Papers:

The Interpretation of the Lambdas of Lagrange Multipliers
Eric Canning, Morningside College     epc002@morningside.edu

When using the method of Langrange multipliers in optimization problems, the values of the lambdas are often found but then discarded.  To quote from a popular calculus text, "The associated values of lambda may be revealed as well but often are not of much interest."  It turns out that the lambdas do have meaning.  In particular, each lambda is equal to the change in the optimal value of the function with respect to the condition imposed on the constraint associated with that particular lambda.
 

The History and Current Status of the Collatz Conjecture
Marc Chamberland, Grinnell College     chamberl@math.grin.edu

The Collatz Conjecture, also widely known as the 3x+1 problem, is a very easily-stated problem which has withstood a solution for fifty years.  In this talk I will give an overview of the work done on this problem, including approaches from the areas of number theory, functional equations and dynamical systems.
 

There Should be a Cubic Solver in Your Life
A. M. Fink, Iowa State University     fink@math.iastate.edu

Part of the undergraduate curriculum should include education in the best sense, that of imparting the culture of our discipline and the connections between various areas.  Cubic solvers can be talked about in at least three different courses, each with a different emphasis, and each teaching something about the subject.
 

Round Table Discussion for MAA Student Chapter Advisors
Cathy Gorini, Maharishi University of Management       cgorini@mum.edu

Come share your successes, ideas, and questions about MAA Student Chapters and other forms of math clubs.  Information about starting MAA Student Chapters as well as Pi Mu Epsilon chapters will be available.
 

Illustrating Functional Programming with Mathematica
Al Hibbard, Central College   hibbarda@central.edu

In this talk, we will see how to use some of the interesting Mathematica constructs that accomplish functional programming.  These include Map, Apply, Nest, Fold, NestWhile, and others.  Each of these will be used to illustrate various mathematical concepts (eg, subgroup generated by an element).

Note: No previous Mathematica experience is required.
 

Update on College Algebra Reform
Scott Herriott, Maharishi University of Management      herriott@mum.edu

This presentation will review the discussions on college algebra reform that were held at the national MAA meetings in 1999 and 2000.  Topics include the motivations for reform, the directions that new textbooks are taking, issues in implementing reform, and efforts to get federal support for a national conference on college algebra reform.
 

Quadrilaterals with Integer Sides
Charles Jepsen, Grinnell College      jepsen@math.grin.edu

There are known formulas for the number of triangles with integer sides and prescribed perimeter.  If one wishes to count the number of such quadrilaterals, restrictions are needed since a quadrilateral is not uniquely determined by its four sides.  We find counting formulas for two special cases:  a) cyclic quadrilaterals, and b) trapezoids.
 

The Marriage of College Algebra and Physical Science
Steve Nimmo, Morningside College     sdn001@alpha.morningside.edu

We are in the fifth semester of teaching a course called Math in the Physical World.  This is a course in the interdisciplinary portion of our new core curriculum.  It was developed by the mathematics and physics professors at Morningside College.  In this talk, I will discuss how the course came to be, some of its strengths and some of the pitfalls.
 

Some Nonconstructive Existence Proofs
Dave L. Renfro, Drake University     dlrenfro@pop.gateway.net

A survey of some interesting nonconstructive existence proofs.
 

A Look at Subposets of Noether Lattices
Rick Spellerberg, Simpson College     speller@simpson.edu

In 1962, R. P. Dilworth introduced Noether lattices as an abstraction of the lattice of ideals of a commutative Noetherian ring.  Noether lattices form one of the most important classes of multiplicative lattices.  In this presentation I will focus on the subposets of regularly generated, regular, and faithful elements of a Noether lattice L, (Lrg, Lr, Lf).  Similarities with the analogous terms in L(R), the lattice of ideals of a commutative Noetherian ring are discussed.
 

A Parable of Computational Geometry
Leon Tabak, Cornell College     l.tabak@ieee.org

A Voronoi diagram is a solution to the Post Office Problem:  given the location of all post offices in a country, divide the country into regions such that every house within a region is closer to the single post office in its own region than it is to any other post office.  This problem asks for an assignment of responsibilities to post offices.  The solution sensibly has each house receiving mail from the closest post office.

A post office becomes a model in the more general description of the problem.  Scientists working in many disciplines, including crystallography, meteorology, botany, and zoology, have repeatedly rediscovered this problem in its many guises over the course of the last hundred years.  However, only in the past quarter century have mathematicians devised fast methods for drawing Voronoi diagrams.

Three approaches to the solution of the Post Office Problem each improve upon its predecessor and illustrate general principles that apply in many domains.  In turn, these algorithms compare everything to everything, divide and conquer, and add a dimension to the apparently two dimensional problem.  The speaker will construct a solution to an instance of the Post Office Problem, discuss the problem's relevance to other questions, and point to relevant literature and software now available on the web.
 

Maple Laboratory in Calculus III and Differential Equations
Murphy Waggoner, Simpson College     waggoner@simpson.edu

In 1997 the Simpson Mathematics Department received an NSF ILI grant to develop a computer laboratory component in the Calculus III and Differential Equations courses.  This talk outlines the work to develop the laboratory component including the structure of the new course, the choice of laboratory assignments, and the instructional philosphy.

Mathematics Student Papers:

Number Theory, Balls in Boxes, and the Asymptotic Uniqueness of Maximal Discrete Order Statistics
Jayadev S. Athreya, Iowa State and Lukasz Fidkowski, Michigan Tech REU     jayadev@iastate.edu

Abstract Not Available At This Time.
 

A Qualitative Investigation of the Discretized Lorenz System
Ian Besse, Grinnell College     besse@grinnell.edu

The Lorenz equations are a system of three nonlinear coupled differential equations which model convection that have been extensively studied since 1963.  Though much is known theoretically about the periodic structure within the Lorenz attractor, and of its fractal nature, little has been accomplished in uncovering that structure and calculating the exact fractal dimension of the Lorenz attractor.  These open questions were the compulsion behind the investigation I conducted on the periodic behavior present in the Euler discretization of the Lorenz system.  Periodic behavior for a variety of time-steps was cataloged and analyzed in hopes that as the time-step tended to zero, periodic structures in the discrete case might approach periodic structures in the continuous case.  Ultimately the intention was to determine if identifying a sufficient number of periodic orbits on the Lorenz attractor, would yield insight into possible methods for obtaining the fractal dimension of the attractor.  Although computational constraints hindered analysis of cycles of length greater than two, much was discovered about the behavior of the system's two-cycles as the time-step was varied and also about appropriate methods and equipment necessary for a study of this nature.
 

The Stone Balance Problem
Micah James, Wartburg College     jamesm@wartburg.edu

The Stone Balance problem is one wherein a 40 pound stone has been broken into four pieces.  The weights of the smaller stones have the special property that, when used on a simple balance, one can balance any integer weight from 1 to 40 pounds.  Included is an algorithm for determining the necessary positions of stones for a given weight along with a generalization to accommodate non-simple balances.  Explored as well is a way to easily explore variations of the problem with generating functions and Maple.
 

Maximum Degree and Color Extension
Grace Lewis, Grinnell College     lewisg@grinnell.edu

I discuss the question of how maximum degree and color extension are related.  The coloring extension problem concerns taking certain vertices or cliques in an r-colorable graph G and pre-coloring each of them with one of r+1 colors.  If these vertices or k-cliques are a distance of 4k or more apart, then Kostochka showed that any pre-coloring extends to an (r+1)-coloring of G.  This paper investigates whether or not putting a bound on the maximum degree of G can reduce the distance pre-colored vertices must be apart to guarantee a color extension.  Results include new critical distances for graphs where the maximum degree is less than or equal to the chromatic number of G, and graphs where the maximum degree is one more than the chromatic number of G.
 

A Look Into Apportionment and Voting Power at Drake University
Kari Meyering, Drake University

The Student Athletic Advisory Board at Drake is currently a "one team, one vote" committee.  A more representative committee might be beneficial to the student-athletes and Drake University.  This possible change in apportionment and its influence on voting power will be explored.
 

Coloring Extensions of Planar Graphs
Rob Park, Grinnell College     parkp@grinnell.edu

Suppose that P is a subset of the vertices of a graph G with chromatic number r, where P induces a set of k-cliques and all of the vertices of P are precolored.  There exists a finite critical distance so that for any graph, if the k-cliques are at least this distance apart, then any (r+1)-coloring of P will extend to an (r+1)-coloring of G.  In this talk, I will describe several results found while looking for such a critical distances for planar graphs.  I give an example of a planar graph in which a certain precoloring of P does not extend to a coloring of G.  This example gives a lower bound for a critical distance.  I will also look at upper bounds for this critical distance.