NORTH CENTRAL SECTION of the MAA

**Fall Meeting October 26-27, 2001**

**University of North Dakota, Grand
Forks, ND**

Friday (evening), October 26, 2001

7:00 - 8:00
Gamble Hall Lobby: **Registration** $8.00 for members; students free.

8:00 –
9:00 Gamble Hall Room 1: **INVITED ADDRESS**

Ethnomathematics and the Incas, Thomas E. Gilsdorf, University of North Dakota,

*Abstract*: In this talk we will explain some of the techniques
involved in the study of ethnomathematics. The ideas involved will be
illustrated by discussing the mathematics of the Incas.

9:00 - ? Alumni Center: **Reception**

Saturday, October 27, 2001

8:00 - 12:00
Gamble Hall Lobby: **Registration**

8:00 - 3:00
Gamble Hall Room 150: **MAA Book Sale**

Gamble
Hall Room 1: __Morning Session__

8:50 - 9:00 **Welcome
**from Dr. Martha A Potvin, Dean of Arts and Sciences.

9:05-9:20
Ron Rietz; Gustavus Adolophus College

__Uncountable
Subfields of the Reals__

*Abstract:* A simple proof, using Zorn’s Lemma, that there exist
(proper) uncountable subfields of the reals will be given. We will also point
out a remarkable "interval-avoiding" property possessed by subsets of
such fields.

9:25-9:40 G. Griffith; University of
Saskatchewan

__Solutions to a problem from the 2001
Invitational Challenge for Grade 11 students in Canada__

*Abstract:* The Canadian Mathematics Competition, based in Waterloo,
Ontario, offers annual "Invitational Challenges" to students in
Grades 10 and 11. Some inspirational solutions appeared for one of the
questions on the 2001 paper which may be of general interest.

9:45-10:05 Michael Tangredi; College
of St. Benedict

__Volterra’s Population Models__

*Abstract*: This presentation will examine Volterra’s original
approach to predator-prey problems. In particular, we first look at his
well-known method of analyzing the solutions (which is quite different from the
approach taken in modern textbooks.) Next we present Volterra’s refinement
of this model, which was the origin of the mathematical discipline currently
known as Volterra Integral Equations.

10:10-10:25 Daniel Hill
(Undergraduate student) and Roxana Costinescu; University of Minnesota, Morris

__Learning Sketchpad and Euclidean
Geometry due to a Paper in Hyperbolic Geometry__

*Abstract:* A senior seminar project about a different proof of the
well-known theorem that gives the area of a triangle in hyperbolic geometry
lead to interesting problems of a different nature. This presentation, designed
primarily for undergraduate students exposed to geometry, will describe the
challenges raised by using Sketchpad in illustrating proofs in the Poincaré
half-plane model of hyperbolic geometry.

10:30-10:45 Michael Green;
Metropolitan State University

__Singularities of Hypersurfaces: The
Milnor and Lê Numbers__

*Abstract*: Given an analytic function *f* of n-complex
variables, the set of vectors at which {*f*=0} is called a hypersurface. A
singularity is a point at which the hypersurface is not smooth. One way to
study singularities is to define numbers that can be used to group
singularieties into different "types." We describe two such numbers:
The Milnor number and the Lê numbers. Specific examples and computations will
be given.

10:45-11:00 BREAK

11:00-12:00
Gamble Hall Room 1: **INVITED ADDRESS**

__Touring
a Torus__, Joe Gallian, University of
Minnesota, Duluth.

*Abstract*: This talk concerns
the problem of traversing an m by n directed grid embedded on a torus so that
each vertex is visited exactly once before returning to the starting position.
We will also consider generalizations and variations on this theme. We conclude
with an application to computer graphics.

12:00 – 1:00 Banquet Room
Student Union: **Lunch**

Gamble
Hall Room 1: __Afternoon Session__

1:05 – 1:35 **Business Meeting
**– President Ralph Carr.

1:40-2:00 M.
B. Rao; North Dakota State University

__Number of
Matchings in a Birthday Problem__

Abstract: Not another birthday problem again! My service course classes in Statistics are big. I cannot resist throwing the birthday problem at the students and then set about checking whether or not at least two of them share a common birthday. I have noticed, on many occasions, several sets of students with each set sharing a common birthday. I will talk about this phenomenon of several mathcings.

2:05-2:20 Mizuho Schwalm (Graduate
student); University of North Dakota

__An Exercise in Computing Homology
Groups__

*Abstract*: Homology groups for chain spaces with real coefficients
are computed as an exercise in linear algebra. The essential property of the
boundary is that when boundary operations act successively, the result is zero.
If we form chains with real coefficients, then the chain groups are vector
spaces. Let the matrix representations of successive boundary operators be M
and N so that MN = 0. The vector space representation of the homology group is
then H = *ker* M – *Im* N. Since the matrices MÖ M and NNÖare Hermitian the problem is reduced
essentially to an eigenvalue problem.

2:30-2:50 Dominic Naughton; St.
Cloud State University

__An Unexpected Encounter with some
Famous Numbers__

*Abstract*: The Bernoulli numbers arise when one computes the sums of
powers of consecutive integers. .
The first few 1, , and —are "hidden" in the coefficients of in the formulas and . This talk will briefly describe their discovery and
discoverer and mention some interesting places where they crop up.