**North Central
Section**

**Mathematical**

**Association of
America**

Fall Meeting Ÿ October 17-18, 2008

Concordia College

Moorhead, Minnesota

**Friday, October
17**

7:00 – 8:00 Registration
– Knutson Campus Center, Lounge outside Jones
Conference Center A&B (2^{nd}

floor)

$10 registration fee;

$5 for MAA-NCS section NExT members; Students, first time
attendees and speakers free.

7:00 – 8:00 Book
Sales – Jones Conference Center C&D

8:00 – 9:00 **Invited Lecture **Jones
Conference Center A&B, Dr. Dan Biebighauser, presiding

**Dr. Karen Saxe, Macalester College**

** Mathematics and
Politics**

9:00 –10:30 Reception: Lounge outside Jones Conference Center
A&B

9:00 –10:30 Student
Reception Area: Jones Conference Center B

**Saturday, October 18**

8:15 – 11:00 Registration
- Lounge outside Jones Conference Center A&B

8:15 – 11:00, 12:00-3:05 Book Sales – Jones
Conference Center C&D.

**Morning
Session **Jones Conference
Center A&B,
Dr. Jessie Lenarz, presiding

9:00 Greetings,
Dr. Heidi Manning, Chair of the Division of Science and Mathematics

**Morning
Session A **Jones Conference Center A, Dr. Jessie Lenarz,
presiding

9:10
– 9:30 **Prof. Joel Iiams, University
of North Dakota**

** **Paradoxes from Poker with Low and/or Hole
Card Wild

9:35
– 9:55 **Prof. Michael C. Mangini, Concordia College**

When Will I Ever Use Math? A
Few Answers from this Cognitive Psychologist

10:00 – 10:20 **Prof. Din Chen, South Dakota State
University**

A
generalized Nonlinear Mixed Model for Binomial Dose-response Modeling

10:20-10:40** Prof.
Daniel P. Biebighauser, Concordia
College, Moorhead**

Burnside’s Lemma
and Color Cycling

**Morning Session B **Jones Conference
Center B, Dr. Jerry
Heuer, presiding

9:10
– 9:30 **Prof. In-Jae Kim, Minnesota
State University, Mankato**

On Fiedler - and
Parter-vertices of Trees

9:35
– 9:55 **Prof. Doug Anderson, Concordia
College Moorhead**

Global
Attractivity for Nonlinear Delay Dynamic Equations

10:00
– 10:20 **William Hall, Concordia College (undergraduate student)**

Oscillation
Criteria for Systems of First-Order Equations

10:20 – 10:40 **David Mathisen, Minnesota State University
Moorhead, (undergraduate student)**

Permutation
Statistics and *q*-Fibonacci Numbers

10:45 – 11:00 **Break**

11:00 – 12:00 **Invited Lecture **Jones Conference Center A&B,
Dr. Su Doree, presiding

**Dr. Betty Mayfield, Hood College**

**Women and Mathematics at the Time of Euler**

12:00 – 1:00 **Luncheon**, Anderson
Commons (1^{st} floor)

1:00 – 1:30 **Business
Meeting** Jones
Conference Center A&B, President Su Doree, presiding

**Afternoon Session A **Jones Conference Center A, Tim Peil, presiding

1:30-1:50 **Prof. Sarah
Jahn, **Concordia
University- St. Paul

Why Not Transform the Axes
Instead of the Graph?

1:55-2:10 **Prof. Dan Kemp, South Dakota
State University**

Triangle Area Ratios: An
Undergraduate Research Project

2:15-2:35 **Prof. William Schwalm, University of North
Dakota**

** **Eigenvalues and Eigenvectors of Certain
Pretty Matrices

2:40-3:00 **Prof. Co Livingston, Prof. Randy Westhoff, Bemidji State University**

STEM Camp

3:05-3:25 **Prof. Wojciech Komornicki, Hamline University **

Another Look at the
Exponential Function

**Afternoon Session B **Jones Conference Center B, Xueqi Zeng, presiding

1:30-1:50
**Prof. ****Mike Weimerskirch, St. Olaf College**

An Algorithmic Approach to Nim-like
Games

1:55-2:10 **Prof. Sri Pudipeddi, Augsburg College**

Radial Solutions
to a Superlinear Dirichlet Problem Using Bessel Functions

2:15-2:35 **Prof. Lisa Rezac, University of St. Thomas**

An Introduction
to Measure Theory with a Special Application: What was Feynman Conjecturing?

2:40-3:00
**Prof. Eric Errthum, Winona State College**

Finding Minimal
Polynomials with a Norm Calculator

3:05-3:25 **Prof. Byungik Kahng, University of Minnesota Morris**

The Invariant
Set Theory of Discrete Time Control Dynamical Systems with Deterministic
Disturbance

**Student
Session** Anderson Commons private room (1^{st} floor),
Dr. Peh Ng, presiding

1:35-2:20 **Flatland: The Movie**

2:15-3:15** Prof.
Aaron Wangberg,** **Winona State
University **

The Mathematics Behind
Constructing - and Viewing - Four-dimensional Shapes.

**Abstracts**

__Invited Speakers__

**● ****Dr. Betty
Mayfield, Women and Mathematics at the Time of Euler**

In 2007 mathematicians around the world focused on All Things
Euler: his life, his work, his legacy. We were treated to special conferences,
books, papers, posters, a study tour, and sessions at national meetings. We
will examine a slightly different topic: female contemporaries of Leonhard
Euler (1707 - 1783), some famous, some not so famous. We will look at their
lives and their work, at mathematics that was written by and – surprisingly –
for women in the time of Euler. This
talk grew out of an experimental summer research project with a group of
undergraduate students in the history of mathematics.

**●**** Dr. ****Karen Saxe,
Mathematics and Politics**

"...democracy is the worst form of government except all
those other forms that have been tried from time to time." -- Winston
Churchill

The cornerstone for a democracy is, arguably, the electoral system
chosen. How do we elect our president? Are there better alternatives? We will
consider mathematical approaches to these questions, and discuss the advantages
and disadvantages of the many different electoral systems that are used by
democratic countries around the world.

__Morning Session A__

**●**** Joel Iiams, Paradoxes from Poker with Low and/or Hole Card Wild**

It’s
been known for some time that introducing wild cards into the game of poker
skews the frequencies of hands which may lead to paradoxes. Yet there is a
well-educated acquaintance of mine who refuses to stop playing poker with wild
cards. His refuge is a game called seven-card stud low hole card wild. We
consider this and several related games. In each case we produce a paradox.
There is also an interesting surprise!

**●**** Michael C. Mangini, When Will I Ever Use Math? A Few Answers from this
Cognitive Psychologist**

Cognitive
Psychology, the study of how humans and animals process information, has set
for itself the difficult task of learning the representations and operations
that occur in the mind. I will discuss
two ways in which mathematics has influenced my work. First, a group of researchers suggest that
the brain is essentially a machine for capturing the natural statistics of its
sensory world. In this context, I will
discuss my research showing vector space representations are predictive for
human face recognition. Second, I will
discuss how a simple method of statistical inference makes explicit the
ineffable qualities of visual decision-making.

**● Din Chen, ****A generalized Nonlinear Mixed Model for Binomial Dose-response Modeling**

The Limit of detection (LOD) has attracted wide
attention in the literature of environmental protection and from various
regulatory agencies. Bioassays are often
used to estimate LOD as a measure of the sensitivity using dose-response
modeling. In this talk, a nonlinear mixed model is proposed to model this
dose-response relationship for binomial mortality data to incorporate various
random effects due to the characteristics of the living organism etc. An extended
quasi-likelihood method is used to estimate the model parameters along with the
investigation of over-dispersion with simulation studies and real data to
demonstrate the applicability of this approach.

**●**** Daniel P. Biebighauser, **** Burnside’s Lemma and
Color Cycling**

A
standard application of Burnside’s Counting Lemma is to count the number of
indistinguishable colorings of the vertices of a geometric object, where two
colorings of the object are indistinguishable if there is a permutation from
the group of symmetries of the object that sends one coloring to the other
coloring. In this talk, we explore the indistinguishable colorings that
arise from symmetries and from permuting the colors themselves. In
addition to discussing the general case, we use technology to display the
indistinguishable colorings for specific objects and specific color
permutations.

__Morning Session B__

**●****In-Jae Kim, On Fiedler - and
Parter-vertices of Trees**

Fiedler-
and Parter-vertices are defined in terms of multiplicities of an eigenvalue of
an n by n symmetric matrix and its principal submatrix of order n-1. In this talk we provide geometric
characterizations of Fiedler- and Parter-vertices of acyclic matrices. Furthermore, we describe a structure of an
acyclic matrix by those vertices, which enables us to construct an acyclic
matrix of a desired form according to the locations of Fiedler- and
Parter-vertices. This is a joint work
with Bryan Shader at University of Wyoming.

**●**** Doug Anderson, Global Attractivity for
Nonlinear Delay Dynamic Equations**

Conditions under which
solutions of a first-order nonlinear variable-delay dynamic equation go to zero
at infinity are given, for arbitrary time scales that are unbounded above. In
two examples, we apply our techniques to dynamic equations on isolated,
unbounded time scales, including a logistic model and a food-limited model.

**● ****William Hall (student), Oscillation Criteria for Systems of First-Order
Equations**

Oscillation criteria for two-dimensional difference
systems of first-order linear difference equations are generalized and extended
to arbitrary dynamic equations on time scales. This unifies under one theory
corresponding results from differential systems, and includes second-order
self-adjoint differential, difference, and q-difference equations within its
scope. Examples are given illustrating a key theorem.

**● **** David Mathisen (student),
Permutation Statistics and q-Fibonacci
Numbers**

We
consider the distributions of permutation statistics of restricted sets of
permutations. We shall focus on the
distribution of the *inv *statistic
over reverse layered permutations. This
distribution will give us a q-analogue of the Fibonacci numbers, Fn(q). We will use these q-Fibonacci numbers to
bijectively prove q-analogues of Fibonacci identities.

__Afternoon
Session A__

**● Sarah Jahn, ****Why Not Transform the Axes Instead of the Graph?**

Once
students know the shape of basic graphs we typically teach them how to
transform these basic graphs. Most
students memorize rules about how to transform the graphs without any clear
understanding of which order they should perform the transformations or why the
rules for horizontal transformations are the opposite of the rules for vertical
transformations. Would students
understand better if we looked at the equation in terms of linear
transformations on x and y and transformed the axes instead of the graph?

**● ****Dan Kemp, Triangle Area Ratios: An Undergraduate Research Project**

In an attempt to do some undergraduate
research at SDSU a group of four students was assembled. They discovered the following: In triangle ABC if points D, E, and F are
chosen on the sides such that

DA/BA
= EB/CB = CF/CA = k and the cevians AE, BF, CD intersect to form triangle GHI,
then area(GHI)/area(ABC) is constant. A
proof was found using some geometrical interpretations of complex numbers. This will be a report of their
accomplishments.

**● ****William Schwalm, Eigenvalues and Eigenvectors of Certain Pretty
Matrices**

Usually
when I want a problem in which the students should find eigenvectors or
eigenvalues, either the matrix is pretty to start with, in which case the
solution is ugly, or else I make a problem with a pretty solution, in which
case the problem is ugly. There is a way
around this dilemma. I present a
strategy for making interesting looking problems with nice solutions. Several examples are given.

**● ****Co Livingston, Randy Westhoff, STEM Çamp**

In
June 2008, we conducted a one-week STEM Camp for high school students, with
support from a TENSOR-SUMMA grant and a MnSCU IPESL Grant. We will discuss recruitment, curriculum, and
activities.

**● ****Wojciech Komornicki, Another Look at the Exponential Function**

We
investigate the definition of the exponential function as a function which
agrees with exponentiation when exponents are integers. The development is via the inverse of the
natural log function and uses only elementary differential and integral
calculus. As a generalization we look at functions f satisfying the functional
equation f(xy) = f(x) + f(y) and show the uniqueness of the solutions under
very mild conditions. In particular, if
f is continuous at a point, the only such functions are multiples of the
natural log function.

__Afternoon
Session B__

**●**** ****Mike Weimerskirch, An Algorithmic Approach to Nim-like
Games**

The winning strategy for the ancient Indian game of
Nim has been known for a century in both the normal play (unable to move loses)
and misere play (unable to move wins) versions.
Nim is one of a larger class of games called impartial games, for which
the generalized normal play strategy was discovered in the 1930s. This talk describes an algorithmic approach
to finding strategies for certain impartial misere games. This algorithm was implemented by three
students in a computer algorithms course at St. Olaf, which lead to an award
winning paper at the 2008 MICS symposium.

**● ****Sri Pudipeddi, Radial Solutions
to a Superlinear Dirichlet Problem Using Bessel Functions**

We look for radial solutions of a superlinear
dirichlet problem in a ball. We show that for if n is a suﬃciently large nonnegative
integer, then there is a solution u which has exactly n interior

zeroes.

**●**** Lisa Rezac, An Introduction to Measure Theory with a Special
Application: What was Feynman Conjecturing?**

We
introduce the idea of measure on the real line and abstraction to measures on
other spaces. We will consider some
“ideal” properties of a measure, and compare the development of Lebesgue
measure on the real line with Wiener measure on the space of continuous
functions on a closed interval: C[a,b].
We close by highlighting a (perhaps) surprising result about C[a,b] and
some implications for making Feynman’s proposed path integral mathematically
rigorous.

**●**** Eric Errthum, Finding Minimal Polynomials with a Norm Calculator**

Given
an algorithm that outputs norms of algebraic numbers, is it possible to
reconstruct these numbers’ minimal polynomials? This talk will demonstrate a
technique that does so using some basic Galois theory, special information
about when the numbers are rational, and a little bit of brute force. This
method is applied to the situation of singular moduli on Shimura curves, though
no prior knowledge of such structures is required.

**●**** Byungik Kahng, The Invariant Set Theory of Discrete Time Control
Dynamical Systems with Deterministic Disturbance**

Invariant
set theory is an important tool in control and automation theory. In this talk, we focus upon the invariant set
theory of discrete time control dynamical systems with deterministic
disturbance and explain how it gives rise to the invariant set theory of
multiple valued iterative dynamical systems.
Time permitting, the controllability problems of the maximal invariant
sets will also be discussed.

__Afternoon Student Session__

**●** **Flatland: the
Movie**

This
is an animated film inspired by Edwin A. Abbott's classic novel, Flatland. Set
in a world of only two dimensions inhabited by sentient geometrical shapes, the
story follows Arthur Square and his ever-curious granddaughter Hex. When a
mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms
with the truth of the third dimension, risking dire consequences from the evil
Circles that have ruled Flatland for a thousand years.

**●** **Aaron Wangberg,** **The
Mathematics Behind Constructing - and Viewing - Four-dimensional Shapes.**

When
Arthur Square is visited by a being from the third dimension in Edwin A.
Abbot's "Flatland", he realizes there are higher-dimensional shapes
which he can partially view in two dimensions.
What would a

four-dimensional
shape look like? In this session, we'll
explore this question and use mathematics to construct 4-dimensional shapes.
We'll also explore the mathematical techniques which will allow us to "see"
these objects as they pass through our 3-dimensional world. This interactive
session will be accessible to all undergraduate students.