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The Mathematical Association of America

North Central Section

Spring 2003 Meeting

April 25-26, 2003

Macalester College, St. Paul, MN

Preliminary Program

Friday, April 25, 2003 

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

8:05-9:05 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.

9:05-?  Reception

Saturday, April 26, 2003

8:00-11:00  Registration, Smail Gallery, Olin-Rice Bldg.
8:00-3:30  Book Sales, 241 Olin-Rice Bldg.

  Morning Session I, Tom Halverson presiding, 250 Olin-Rice

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

10:30-11:00 Break

 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

 

Abstracts

M. B. Rao, The Yeast Genome, the Hat Problem, and Some Extensions

Initially, 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

Voronoi 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

  Euler first studied the partition function P(n) which counts the number of ways a number can be expressed as a sum of positive integers.  Since, the number theoretic function has drawn the likes of Hardy and Ramanujan into the immense task of discovering its properties.  In most attempts to study this function, the worlds of analysis and discrete mathematics collide as the generating function is probably the most useful tool in attacking the problem.  Combining the infinite series and techniques from elementary calculus, I have discovered a relationship between the divisor function σ (n) and P(n), which has traditionally been very difficult (it took one of Hardy’s brightest students over two months to calculate P(100)).  The proof for the recursion is a taste of the wonderful uses of generating functions in attacking problems from the theory of numbers.

Doug Dunham, The Mathematics of the Math Awareness Month Poster

In 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” print.

Marc Brodie, What Are the Odds?

We will investigate how the odds printed on the back of lottery playslips are determined.

George Bridgman, Polynomial Graphs

We’ll 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 polynomial graphs.

W. Schwalm, Bessel Functions and Vibrations of an Infinite Line of Springs and Weights

Bessel 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 and interesting.

Matt Handley, Janette Herbers, T. Noah Loome, and Adam McDougall, A Catalan Related Sequence

We 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

  A restricted growth function gives a way of representing a set partition as a word.  It is known that certain odometer-like actions of Sn preserve the distribution of a large family of statistics on  words.  What happens if we try the same procedure (using analogous statistics) on just those words which happen to correspond to 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.