The Mathematical Association of America
North Central Section
Spring 2003 Meeting
April 25-26, 2003
Macalester College, St. Paul, MN
7:00-8:00 Registration, basement of Ruth Strikler
Dayton Campus Center
Evening Session, John B. Davis Auditorium, Karen Saxe presiding
8:00 Welcome by Professor Tom Halverson, chair of the Mathematics Department
Teaching and Using Linear Algebra, Gil Strang, MIT
Abstract: Linear algebra is a crucial subject in the teaching and the applications of mathematics. We hope to suggest new ideas in its presentation, including a range of problems whose exploration leads to the major themes of linear algebra. The pure and applied parts of this subject will be intertwined as they are in reality.
8:00-11:00 Registration, Smail Gallery, Olin-Rice Bldg.
8:00-3:30 Book Sales, 241 Olin-Rice Bldg.
9:00 The Yeast Genome, the Hat Problem, and
Some Extensions, M. B. Rao, NDSU
9:25 Voronoi Diagrams and Scheduling Forest Rehabilitation, Colleen Livingston (presenter), Bemidji State University, and Steven Morics, University of Redlands
9:50 A Recursion Formula for the Partition Function, Jason Saccomano, St. Olaf College
10:15 The Mathematics of the Math Awareness Month Poster, Doug Dunham, University of Minnesota, Duluth
Morning Session II, David Bressoud presiding, 150 Olin-Rice
9:00 What Are the Odds?, Marc
Brodie, College of St. Benedict
9:25 Polynomial Graphs, George Bridgman
9:50 Bessel Functions and Vibrations of an Infinite Line of Springs and Weights, W. Schwalm, UND
10:15 A Catalan Related Sequence, Matt Handley, Janette Herbers, T. Noah Loome, Adam McDougall (presenter), St. Olaf College
John B. Davis Auditorium, Ruth Stricker Dayton Campus Center,
Wayne Roberts presiding
11:00-12:00 Forbidden Symmetry—Relaxing the
Crystallographic Restriction, Frank Farris, Santa Clara University
Abstract: If you look at enough swatches of wallpaper, you will see centers of 2-fold, 3-fold, 4-fold, and 6-fold rotation. Why not 5-fold centers? They cannot occur, according to the Crystallographic Restriction, a fundamental result about wallpaper patterns, which are defined to be invariant under two linearly independent translations. Even so, we offer convincing pictures that appear to show wallpapers with 5-fold symmetry.
12:00-1:00 Lunch, Smail Gallery, Olin-Rice Bldg.
1:00-1:30 Business Meeting, M. B. Rao presiding,
John B. Davis Auditorium
John B. Davis Auditorium, M. B. Rao presiding
1:30 –2:30 A Favorite Lesson: Introduction to Statistical Estimation, Robert Lacher, SDSU (2002 NCS Distinguished Teaching Award winner)
Abstract: In our introductory course in calculus-based statistics, I have evolved a short set of lessons based on sampling the uniform distribution on the interval [0,b). It is interesting to see how this seemingly most simple of populations can lead to the use of such a variety of techniques and concepts. I will describe how I use this example to introduce the topics of maximum likelihood, matching moments, the concept of bias, and ultimately how to construct interval estimates and a test of hypothesis.
Afternoon Session I, M. B. Rao presiding, 250 Olin-Rice
2:40 Odometer-like Actions of the Symmetric
Group on Restricted Growth Functions, Jennifer Galovich, St.
John’s University and the College of St. Benedict
3:05 Granting Mathematical Absolution: Dropping the Lowest Quiz, Tom Sibley, St. John’s University and the College of St Benedict
Afternoon Session II, David Bressoud presiding, 150 Olin-Rice
2:40 How to
Add Perfect Shuffles, Danrun Huang,
St. Cloud State
3:05 Teaching Confidence Intervals, John Holte, Gustavus Adolphus College
M. B. Rao, The Yeast Genome, the Hat Problem, and Some Extensions
the Hat Problem arose in issues surrounding genomic complexity. The Hat Problem straddles probability,
optimization, algebra, and Hamming distances.
The talk will begin with an outline of the original Hat Problem and optimal
strategies. Extensions will be presented
along with some open problems. A
connection to the Deletion Problem of the Yeast Genome will be pointed out.
Colleen Livingston and Steven Morics, Voronoi Diagrams and Scheduling Forest Rehabilitation
diagrams partition the plane into zones of nearest proximity to sites in a preselected
set. We will describe a classroom module
that introduces Voronoi diagrams and puts them to use in a scheduling problem
concerning reforestation. This work was
begun at the DIMACS Reconnect 2002 Conference.
Jason Saccomano, A Recursion Formula for the Partition Function
Doug Dunham, The Mathematics of the Math Awareness Month Poster
1958, H. S. M. Coxeter sent the Dutch graphic artist M. C. Escher a reprint of
a paper that contained a figure showing a triangle tessellation of the
hyperbolic plane. Escher expressed
“shock” upon seeing that figure, since it showed him the long-desired solution
to his problem of designing repeating patterns in which the motifs become ever
smaller toward a circular limit.
Escher’s shock set in motion a sequence of events that led to the design
of the pattern on the 2003 Math Awareness Month poster. First, Escher created his four “circle Limit”
patterns in the Poincare circle model of hyperbolic geometry. About 20 years later, inspired by Escher, I
set out to create a computer program that could reproduce those patterns. Then, after succeeding in this task, the
program evolved to be able to draw more general patterns, including the one on
the MAM poster, which is based on the fish motif of Escher’s “Circle Limit III”
Brodie, What Are the Odds?
will investigate how the odds printed on the back of lottery playslips are
George Bridgman, Polynomial Graphs
look at the possible shapes which the graph of, say, a 4th degree
polynomial can assume, and we’ll note certain restrictions on the shapes of
W. Schwalm, Bessel Functions and Vibrations of an Infinite Line of Springs and Weights
functions arise in connection with modes of a circular drum head, theory of
frequency modulation, vibrations of a hanging chain and so on. The modes of vibration of a linear chain of
springs and weights, on the other hand, are easily found by symmetry. However, the initial condition in which a
single mass is displaced and all masses are at rest at t = 0 admits a solution
in terms of Bessel functions. The
complete solution, for instance by a generating function, is both elementary
Matt Handley, Janette Herbers, T. Noah Loome, and Adam McDougall, A Catalan Related Sequence
studied a sequence that is related to the Catalan numbers. The Catalan numbers count the number of
0-dominated strings of 0’s and 1’s—where a 0-dominated string of length l
has the property that the first k digits of the string contain at least as many
0’s and 1’s for k = 1, . . . , l. These
sequences correspond to certain paths in an integer lattice, and we will
explore this and other interpretations.
Using these interpretations, we were able to generalize some identities
which are known for the Catalan numbers.
For example, this new sequence, which we are calling the “tri-Catalan”
numbers, corresponds to the number of strings with 2n 0’s and n 1’s that are
doubly 0-dominated (at least two 0’s for each l in the first k entries). A known formula for the Catalan numbers is
C(n) = C(2n,n) – C(2n, n+1), while the explicit formula of this form for the
tri-Catalan numbers is D(n) = C(2n,n) -- 2C(2n, n+1). (Here C(2n,n) is the number of combinations
of 2n things taken n at a time).
Jennifer Galovich, Odometer-like Actions of the Symmetric Group on Restricted Growth Functions
John Holte, Teaching Confidence Intervals
An easy way to teach the concept of a “confidence interval”
is presented. It involves a class in the
simulation of confidence intervals, it provides striking visual imagery, and it
lends itself to discussion of related issues.
For the instructor or the advanced student, the theory behind
order-statistic-based interval estimates of quantiles is also presented.
Danrun Huang, How to Add Perfect Shuffles
A perfect shuffle divides a deck of 2n cards exactly in
half, and then interleaves the top n cards with the bottom n perfectly. Mathematicians, statisticians, computer
scientists, and magicians have studied perfect shuffles extensively. The purpose of this talk is to introduce
a new, simple, algebraic and dynamic
model for perfect shuffles. It is an
interesting and very vivid application of undergraduate abstract algebra. Using this new model, students were able to
use the knowledge of abstract algebra to prove many of their conjectures
arising from the patterns they discovered in shuffling cards. This model also provides simple proofs of
some well-known orbit rules of perfect shuffles, as well as some new results,
such as how to add two perfect shuffles.
Tom Sibley, Granting Mathematical Absolution: Dropping the Lowest Quiz
How can we model the effect of dropping a student’s lowest
quiz grade before averaging the others?
Are we mostly providing a psychological boost to students, or are we
protecting our students from their more inglorious moments? Instead of the pedestrian solution of
consulting my grade book, I attempt a more mathematically interesting
approach. After an overly naïve example,
I turn to integrals involving correlated random variables, which quickly
surpass closed form techniques. Thus my
explorations soon depend on computer approximations and simulations.