**North Central
Section**

**Mathematical**

**Association of
America**

Spring Meeting Ÿ April 24-25, 2009

Hamline University

St. Paul, Minnesota

**Friday, April
24**

6:15 – 7:00 Registration
– Robbins Science Basement Lobby

$10 registration fee;

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

6:15 – 7:00 Book
Sales – Robbins Science 05

**Friday Evening
Session **Drew Science 118, Presider: Dr. Arthur Guetter

7:00 – 7:05 **Greetings: Dean Fernando Delgado, College
of Liberal Arts**

7:05 – 7:25 **Prof. In-Jae Kim, Minnesota State University, Mankato**

Symmetric Sign Patterns With Maximal Inertias

7:30 – 7:50** Prof. Tom Sibley, St. John’s University
and College of St. Benedict**

Puzzling Groups

8:00 – 9:00 **Invited Lecture**

**Dr. Jason Douma, University of Sioux Falls**

**Gotta Have My Dots: Strategies for Dot Voting**

9:00 –10:30 **Reception****:**
Robbins Science Basement Lobby

**Saturday, April 25**

8:15 – 11:00 Registration
- Robbins Science Basement Lobby

8:15 – 11:00, 12:00-3:05 Book Sales – Robbins
Science 05

**Morning
Session A **Drew Science 118, Presider: Dr. Frank Shaw

9:05
– 9:25 **Prof. Dan Kemp, South Dakota State University**

** ** Pi
Really Is Transcendental

9:30
– 9:50 **Prof. Roger B. Kirchner, Carleton College**

The Crease Length Problem,
Revisited

9:55 – 10:15 **Prof. William Schwalm, University of North
Dakota**

Nine Point Circle

10:20 – 10:35 **Maajida Murdock, University of North Dakota
**(graduate student)

Geometric Optics of Concave
Mirrors

**Morning
Undergraduate Student Session
Robbins** Science 11** , **Dr.
Peh Ng, presiding

10:20 – 10:40 **Jeremy
Davis, University of Minnesota at Morris** (undergraduate)** **

Regular Sierpinski Fractals and Their Maximal
Hausdorff Dimensions

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10:40 – 11:00 **Break**

11:00 – 12:00 **Invited Lecture**,
**Drew Science 118 ,**** Presider: Dr. Matthew Haines**

** Dr. George
Andrews, The Pennsylvania State University**

** The Lost
Notebook of Ramanujan**

12:00 – 1:00 **Luncheon**,
Sorin Dining Hall A and B.

1:00 – 1:30 **Business
Meeting** Drew
Science 118, Presider: Su Doree, NCS President

**Afternoon Session **Drew Science 118, Presider: Dr.
Wojciech Komornicki

1:35-1:55 **Adam
McDougall, University of Iowa **(graduate student)

A
Rational Approach to Picking a Random Fraction

2:00-2:20 **Prof. ****Ken Takata, Hamline University **

Individualized Additional
Instruction for Calculus

2:25-2:45 **Prof.** **Andrew Beveridge,
Macalester College**

Extremal Random Walks on
Trees

2:50-3:05** Craig Erickson, Minnesota State
University, Mankato** (graduate
student)

On Nilpotence Indices of Sign Patterns

3:10-3:30 **John Greene, University of Minnesota, Duluth **

Traces of
Matrix Products

** **

**Special Session in Number
Theory and Combinatorics, Robbins** Science 12,** ** Presider: Dr. Tina Garrett

1:35-1:55 **Prof. ****Igor Pak, University of
Minnesota**

Long cycles in
abc-permutations

2:00-2:20 **Prof.** **Dennis Stanton, University
of Minnesota**

The negative q-binomial

2:25-2:45 **Prof. Jennifer Galovich, **

**St. John's
University and the College of St. Benedict**

MacMahon’s Other Maj

2:50-3:05** ****Prof.**** Drew Armstrong,** University of Minnesota

** ** Parking Functions and Noncrossing Partitions

**Afternoon
Undergraduate Student Session** Robbins
Science 11, Presider:
Dr. Peh Ng

1:35-1:55 **Charles P. Rudeen, University of Minnesota Morris **(undergraduate)** **

The Maximum Weight Connected
Subgraph Problem

2:00
– 2:45** Flatland:
The Movie**

2:50
– 3:30** Prof.
Aaron Wangberg,** **Winona State University **

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

**Abstracts**

__Invited Speakers__

**● ****Dr. George Andrews, The Lost Notebook of
Ramanujan**

In 1976 quite by accident, I
stumbled across a collection of about 100 sheets of mathematics in Ramanujan's
handwriting; they were stored in a box in the Trinity College Library in
Cambridge. I titled this collection
"Ramanujan's Lost Notebook" to distinguish it from the famous
notebooks that he had prepared earlier in his life. On and off for the past 32 years, I have
studied these wild and confusing pages.
Some of the weirder results have yielded entirely new lines of research
in number theory are related topics. I
will try to provide a gentle account of where these efforts have led. I will conclude the talk with an account of
Ramanujan at the Tribeca Film Festival.

**● Dr. ****Jason Douma, Gotta Have My Dots: Strategies for Dot Voting**

Dot voting is a popular form of
approval voting, in which each voter is awarded a fixed number of “dots” that
may be allocated strategically by the voter among the various selections being
considered. Our own section employed dot
voting last spring as a way of identifying priorities which would be
recommended to MAA leadership at the national level. In a dot voting environment, a voter might
choose to concentrate her dots in one or two favorite selections, as a way to
help ensure that her top choice is elected; alternatively, a voter may choose
to spread out her dots among many desirable selections, hoping to push multiple
choices across the threshold of victory. This talk will employ game-theoretic
concepts to examine risk-averse and risk-loving behavior in the dynamics of dot
voting.

__Friday Evening
Session__

**● In-Jae Kim, Symmetric Sign
Patterns With Maximal Inertias**

The inertia of an n by n symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order $n$. In this talk we classify all the maximal inertias for symmetric sign patterns of order n and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.

**●** **Tom Sibley, Puzzling Groups**

Bob Earles from St. Cloud State University used some puzzles in his abstract algebra class. The puzzles consist of rotating slotted wheels and pieces fitting into the slots that are shifted around as the wheels rotate. Like many puzzles, these puzzles are at heart permutation groups in disguise. We'll determine the groups of these puzzles and consider some generalizations.

__Morning Session
A__

**● ****Dan
Kemp, ****Pi Really Is Transcendental**

Everyone ‘knows’
that pi is a transcendental number, but how many of us have actually seen a
proof. In 1939 Ivan Niven published a
proof of the transcendence of pi in the AMM that depends upon some properties
of the elementary symmetric function, some unintuitive procedures, and
calculus. This proof should be
understandable to undergraduates.

**●** **Roger B. Kirchner, The
Crease Length Problem, Revisited**

Fold a corner of an 8.5 x 10.812167… (ratio of length to width is the square root of the Golden Ratio) sheet of paper to the opposite longer edge and crease. There are two distinct longest creases and two distinct shortest creases. The “classic” shortest crease is a local minimum, not an absolute minimum. The things you can discover with Mathematica!

**● William Schwalm,
Nine Point Circle**

In high school I learned of two chemicals that when mixed would ignite and burn spontaneously. More interesting, in retrospect, was the nine point circle theorem:

*For
any triangle there is a circle passing through (a) the midpoints of the sides
(b) the feet of the altitudes, and (c) the midpoints of the lines connecting
the vertices to the orthocenter where the altitudes intersect. Its center is the midpoint of the line
between the orthocenter and the circumcenter*.

Pedoe (*Geometry, a
comprehensive course*) gives a proof based on central dilatation. Can this be extended to give a theorem about
tetrahedra?

**●** **Maajida Murdock, ****Geometric Optics of Concave Mirrors **(graduate student)

I present a simple
form of ray optics and to apply this to a concave spherical mirror. Using similar triangles to locate an image
point defined by two rays gives the standard formula *q f + p f = p q, where p, q, f are*
object, image and focal distances from
the mirror. To insure an image is
actually formed, one should check what happens to near-by rays. This involves the external angles theorem and
small angle approximation. Discussion of
real and virtual images is given. These
ideas could motivate student problems or projects in geometry or pre-calculus.

__Afternoon
Session__

**● ****Adam McDougall, A Rational Approach to Picking a Random
Fraction **(graduate
student)

Ways to
‘randomly’ select a whole number have been studied in depth. In contrast, not
much thought has been given to how one might randomly select a fraction. As a
warm-up, ways of randomly selecting whole numbers will be discussed. Then we
analyze some common (but arguably incorrect) methods of randomly selecting
fractions and give other more ‘mathematically correct’ methods of randomly
selecting a fraction.

**● ****Ken Takata, Individualized Additional Instruction for Calculus**

College
students enrolling in the calculus sequence have a wide variance in their
preparation and abilities, yet they are usually taught from the same
lecture. We describe another pedagogical
model of *Individualized Additional
Instruction *(IAI) that assesses each student frequently and prescribes
further sessions based on the student's performance. Our study compares two calculus classes, one
taught with mandatory remedial IAI and the other without. The class with mandatory remedial IAI did
significantly better on comprehensive multiple-choice exams, participated more
frequently in classroom discussion, and showed greater interest in
theorem-proving and other advanced topics.

**●**** Andrew Beveridge, Extremal
Random Walks on Trees**

In
a random walk on a graph, we iteratively move from one vertex to a randomly
chosen adjacent vertex. Considering intelligent stopping rules which “look
where they are going,” we can obtain an example sample from the stationary
distribution. We define the mixing time to be the expected length of an optimal
stopping rule. For a fixed tree size, we determine the tree structure that
minimizes/maximizes the mixing time. We show that the star is the unique
minimizing structure and the path is the unique maximizing structure. Joint work with Meng Wang, Macalester ’09.

**●** **Craig
Erickson, On Nilpotence Indices of Sign Patterns** (graduate student)

A sign pattern is called potentially nilpotent
if it allows nilpotence. Eschenbach and Li (1999) listed four 4-by-4
potentially nilpotent sign patterns which they conjectured have nilpotence
index at least 3. We confirm that these sign patterns have nilpotence index 3.
We also generalize these sign patterns of order 4 so that we provide classes of
n-by-n sign patterns of nilpotence indices at least 3, if they are potentially
nilpotent. Furthermore, it is shown that if a full sign pattern A of order n
has nilpotence index k with , then A has nilpotent realizations of nilpotence indices k,
k + 1,… n.

**●** **John Greene,
Traces of Matrix Products**

Given two non-commuting 2x2 matrices A and B,
what can be said about traces of products of these two matrices? It is well known that AB and BA have the same
trace. This easily generalizes to cyclic
permutations. For example, AABB, BAAB,
BBAA and ABBA all have the same trace.
However, AABB and ABAB usually have different traces. We show that there is another
symmetry: reversal. That is, AABBAB and BABBAA have the same
trace even though they are not cyclic permutations of each other. We also address problems of the following
type: Which is usually larger, tr(ABAB) or tr(AABB)?

__Special
Session on Number Theory and Combinatorics__

● **Igor
Pak, Long cycles in abc-permutations**

An abc-permutation is a permutation generated by exchanging an initial block of length a and a final block of length c of {1...n}, where n=a+b+c. In this note we compute the limit of the probability that a random abc-permutation is a long cycle. This resolves an open problem V. Arnold posed in 2002. Joint work with A. Redlich.

● **Dennis Stanton, The negative q-binomial**

Interpretations for
the negative q-binomial coefficient are given. Two positivity conjectures in a
(q,t)-universe are proposed. Authors: V. Reiner and D. Stanton, University
of Minnesota

● **Jennifer
Galovich, MacMahon’s Other Maj**

MacMahon’s classic paper on the indices of permutations introduces the (now) well-known major index. In a not-so-well-known appendix to that paper, MacMahon proposes a weighted version of the major index and finds the distribution in some special cases. Regarding a more general result, he remarks: “The actual determination of this expression is reserved for a future occasion.”

Although this talk is not quite that “future occasion”, I will discuss this problem, its relationship to partitions, and the Andrews connection(s), together with some conjectures and suggestions for further work.

● **Drew Armstrong, Parking Functions and
Noncrossing Partitions**

We consider a
nonhomogeneous version of Haiman's parking function module and study its
expansion in terms of the complete homogeneous basis. It turns out that the
coefficients in this basis count noncrossing partitions in terms of a new
statistic, which we call the ``reduced type'' of the partition. Using this
connection we obtain an explicit formula for these numbers. This is joint work
with Sen-Peng Eu.

__Undergraduate Student Session__

**●** **Jeremy Davis, Regular Sierpinski Fractals and Their Maximal Hausdorff
Dimensions **(undergraduate)

We study the invariant fractals in regular N-gons given by N identical pure contractions at its vertices with non-overlapping images, which we will call regular Sierpinski fractals. We provide the explicit formulae for the maximal contraction ratio and the maximal Hausdorff dimension for each regular Sierpinski fractal. We use N-gram decomposition as the main tool. Also, we verify our results through various numerical computations and discuss some ways to expand our research beyond Sierpinski fractals.

**●** **Charles P. Rudeen, The Maximum Weight Connected Subgraph Problem **(undergraduate)

The *Maximum-Weight
Connected-Subgraph Problem* (MWCSP) is defined as follows. Given a connected graph G = (V, E) and any
rational valued weight function on the edges, find a connected subgraph of G
with the maximum total sum of all included edge weights. It has been shown that the MWCSP is NP-Hard,
meaning there is no known efficient algorithm to solve it and there probably
never will be. Our goal was to find
subclasses of graphs where we can provably solve the MWSCP to optimality. We will present an algorithm to solve
instances of the MWCSP on select subclasses to optimality.

**●** **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 that 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 that 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.