Description: P:\MAA\meetings\2005springprogram_files\logo_ncs.jpg

The Mathematical Association of America

North Central Section

Spring 2005 Meeting

April 22-23, 2005

Saint Olaf College, Northfield, MN

Preliminary Program

Friday, April 22, 2005 

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Prior to the Spring MAA-NCS Section Meeting, there will be a meeting of the North Central Section NExT, beginning at 1:00 pm on April 22.  Please visit the North Central Section NExT website for additional details concerning this event.

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7:00-8:00  Registration, Science Center Lobby

7:00-8:00  Book Sales, Science Center 188

Friday Evening Program, Science Center 282

Dr. Matt Richey, presiding

8:00-8:05  Welcome by Dr. Dave Van Wylan, Associate Dean, Faculty of Natural Sciences and Mathematics

8:05-9:00  Invited Lecture “Coloring the Generalized Tower of Hanoi Graphs”

                 Dr. Suzanne Doree, Augsburg College (2004 NCS MAA Distinguished Teaching Award winner)

Abstract:  The state graphs of the multi-peg version of the Towers of Hanoi puzzle are complex, recursively constructed (self-similar) graphs that are highly non-planar.  This talk will describe the generalized problem and properties of the state graphs, including exhibiting a natural minimal vertex coloring.  I will also offer open questions suitable for undergraduate research problems.

9:00-?      Reception, Science Center Lobby

 

Saturday, April 23, 2005

8:15-10:45     Registration, Science Center (outside Science Center 182)


8:15-10:55     Book Sales, Science Center 188

Morning Program

  Concurrent Session I, Science Center 182

Dr. Katherin Crowley, presiding

 

9:00  When a Group Has Itself as an Automorphism Group, Jason Saccomano, St. Olaf College
9:20  Variae Observationes Circa Series Infinitas, Dr. Milan Lukic, Viterbo University
9:40  Generating Prime Numbers Efficiently, Dr. George Bridgman, retired (Hamline University, UM Duluth)
10:00  How Honeybees Prove Fibonacci Identities, Dr. Danrun Huang (presenter) and Kyung H. Sun, St. Cloud State University

10:20  When Multiplication Mixes Up Digits, Dr. David Wolfe, Gustavus Adolphus College

 

  Concurrent Session II, Science Center 184

Dr. Paul Zorn, presiding

 

9:00  Counting Crooked Paths—A Tangential Approach, Dr. Ralph Carr, St. Cloud State University
9:20  Fractal Measures of Heart Rate Dynamics Applied to Epileptic Seizures, Rick Powers (presenter), Dr. Ernest Boyd, Minnesota State University, Mankato; Dr. Andrew Reeves, Mayo (Mankato)

9:40  Making Change and Crossing Bridges, Dr. Bill Schwalm, University of North Dakota
10:00  A Pedestrian Study of the Roots in y of the Polynomial p(x) = (5x-1)y+17xy+ 4xy - x+ 1 as Functions of x, Cody Nitschke, University of North Dakota

10:20  Computer Search for Antipodal Binary Gray Codes, Dr. Todd Will, Mike Dodge, University of Wisconsin LaCrosse

 

10:35-11:00  Break

 

11:00-12:00  Invited Lecture, Buntrock Center Viking Theater

                     “The MAA American Mathematics Competitions:  Easy Problems, Hard Problems, History and Outcomes”

                         Dr. Steven Dunbar, University of Nebraska—Lincoln


Abstract:  The MAA has continuously sponsored a sequence of nationwide high-school level math contests since 1952.  The sequence of contests now spans 5 different contests at increasing levels of mathematical sophistication.  Students who succeed at the top level on these contests become the team representing the U.S. at the annual International Mathematical Olympiad.  I’ll survey the history and organization of the contests, along with the outcomes and some notable mathematicians whose early indications of their talent came on these contests.  Along the way, I’ll showcase some of the interesting, easy, and hard mathematical problems that occur on these contests.

 

12:00-1:00  Lunch, Buntrock Commons Stav Hall ($6.25 per person)

12:00-3:00     Book Sales, Science Center 188

 

Afternoon Program, Buntrock Commons Viking Theater

President Jennifer Galovich, presiding

1:00-1:30  Business Meeting
                 

1:30          MAA Highlights Presentation, Dr. Steven Dunbar, University of Nebraska—Lincoln

1:50          Comparing Interior Point Methods and the Out-of-Kilter Algorithm, Roxanne Toupal, Minnesota State University, Mankato
2:10          Stochastic Apportionment, Kuniko Suzuki, Cecilia Chandra, St. Cloud State University

2:30          A Brief Introduction to Maplets, Dr. Keith Agre, St. Cloud State University

 

 

Abstracts

Friday Evening Program

Suzanne Doree, “Coloring the Generalized Tower of Hanoi Graphs”

The state graphs of the multi-peg version of the Towers of Hanoi puzzle are complex, recursively constructed (self-similar) graphs that are highly non-planar.  This talk will describe the generalized problem and properties of the state graphs, including exhibiting a natural minimal vertex coloring.  I will also offer open questions suitable for undergraduate research problems.

 

Morning Program, Concurrent Session I

Jason Saccomano, “When a Group Has Itself as Automorphism Group”

Consider a group G and the set of isomorphisms from G to itself, Aut(G) or the automorphism group of G.  We will develop some interesting examples of groups that satisfy Aut(G) = G.

 

Milan Lukic, “Variae Observationes Circa Series Infinitas”

In a 1737/1744 paper whose title I am borrowing, Leonhard Euler presented nineteen theorems, plus numerous corollaries, providing closed sums for some quite amazing infinite series.  Euler’s proofs rely on a bit liberal use of divergent series.  I will present a rigorization of one of Euler’s proofs.

 

George Bridgman, “Generating Prime Numbers Efficiently”

I present two methods of generating prime numbers “efficiently”.  An efficient method is one in which each nonprime number is crossed out only once, rather than multiple times.

 

Danrun Huang (presenter) and Kyung H. Sun, “How Honeybees Prove Fibonacci Identities”

There are many beautiful combinatorial proofs of Fibonacci identities, especially those collected in the MAA bestseller Proofs That Really Count by Benjamin and Quinn.  In this talk, we will prove many Fibonacci identities by using honeybee wisdom, or “honeycombology”.  Some identities proved here may not have been combinatorially proved before.

 

David Wolfe, “When Multiplication Mixes Up Digits”

It’s long been observed that if you multiply the number 123456789 by 2, 4, 5, 7, or 8 that the result permutes all 9 digits:

            123456789 x 2 = 246913578

            123456789 x 4 = 493827156

            123456789 x 5 = 617283945

            123456789 x 7 = 864197523

            123456789 x 8 = 987654312

In this talk we address why this is the case by generalizing the problem to b-1 base b numbers.  Only grade school mathematics is assumed in the talk.

 

Morning Program, Concurrent Session II

Ralph Carr, “Counting Crooked Paths—A Tangential Approach”

This is a nice problem suitable for a beginning discrete mathematics course that counts a certain class of permutations on Z, called Zigzag permutations.  Using simple divide-and-conquer techniques a nonlinear recurrence relation emerges from which a nifty generating function is produced, and students are led from the discrete world back into calculus, a nice way to reinforce the connections among branches of mathematics that may seem unconnected to beginning students.  Sometimes going off on a tangent is the right thing to do.

 

Rick Powers (presenter), Dr. Ernest Boyd, Minnesota State University, Mankato; Dr. Andrew Reeves, Mayo (Mankato), “Fractal Measures of Heart Rate Dynamics Applied to Epileptic Seizures”

This paper is an exploration of heart rate dynamics for persons before, during, and after epileptic seizures.  We have received data on fourteen patients.  We will compute different mathematical measures of heart rate dynamics in this data and look for markers of abnormality that have been reported by other researchers.  We hope that this paper will supplement the current literature about applying nonlinear time-series analysis to the study of epilepsy.

 

Bill Schwalm, “Making Change and Crossing bridges”

            I apply generating functions to two simple counting problems where generating functions are often used:

1.      If I have two dimes, five quarters, five nickels and eight pennies, how many different ways can I make change for a dollar?

2.      If the Koenigsberg bridges and the bodies of land they connect are all labeled, and if I cross bridge A six times, Bridge B three times, etc., how many different routes can I take starting form island 1 and arriving at island 2?  What if I relax some conditions, so for instance suppose I don’t say how many times I cross bridge N?

 

Cody Nitschke, “A Pedestrian Study of the Roots in y of the Polynomial p(x) = (5x-1)y+17xy+ 4xy - x+ 1 as functions of x

I study behavior of the roots y(x) as functions of x.  For example, these can be expanded in power series.  Then the coefficients for different k, or in other words for different root functions, are related by permutations.  It is interesting to see what happens when two or more roots coalesce.

 

Todd Will, “Computer Search for Antipodal Binary Gray Codes”

A binary Gray code is a cyclical listing of the 2 binary strings of length n in which successive entries differ in only one bit position.  In such a listing a binary string and its complement must appear at least n steps apart.  If all complementary pairs appear exactly n steps apart the code is antipodal.  Savage and Killian proved that antipodal codes exist when n is a power of 2 and that antipodal codes do not exist when n is odd.  We discuss computer based searches for the case n = 10.

 

 

Saturday Invited Address

Steven Dunbar, “The MAA American Mathematics Competitions:  Easy Problems, Hard Problems, History and Outcomes”

The MAA has continuously sponsored a sequence of nationwide high-school level math contests since 1952.  The sequence of contests now spans 5 different contests at increasing levels of mathematical sophistication.  Students who succeed at the top level on these contests become the team representing the U.S. at the annual International Mathematical Olympiad.  I’ll survey the history and organization of the contests, along with the outcomes and some notable mathematicians whose early indications of their talent came on these contests.  Along the way, I’ll showcase some of the interesting, easy, and hard mathematical problems that occur on these contests.

 

 

Afternoon Program

Roxanne Toupal, “Comparing Interior Point Methods and the Out-of-Kilter Algorithm”

The Primal Affine Scaling (PAS) Method and The Primal Dual Interior Point (PDIP) Method, interior point algorithms suggested by Bhatti in Practical Optimization Methods, are applicable to a large class of linear programming problems.  We develop these methods by applying the Karush-Khum-Tucker conditions to a standard linear programming problem.  We discuss duality, null and range spaces, and derivation of the algorithms plus the purification algorithm.  We apply the PAS and PDIP methods to a transshipment problem presented by Lee-Lee Liu (1987) in The Transportation Problem Using the Out-of-Kilter Algorithm.  Finally we discuss algorithmic and computational efficiency as well as practical cost considerations in order to compare the PAS and PDIP methods to the out-of-kilter algorithm.

 

Kuniko Suzuki, Cecilia Chandra, “Stochastic Apportionment”

The problem of apportionment has been a subject of extensive study and debate by both mathematicians and politicians.  To state the problem simply, given a house size S and state populations P, we need to find a nonnegative, integer valued allocation {A} of house seats to states where = S.  In this talk we describe a stochastic method of apportionment introduced in 2002 by Geoffrey Grimmett.  Prior to Grimmett, many static methods of apportionment were proposed but one that we are particularly interested in is Hamilton’s method because it is the only static method that satisfies a fairness condition called quota rule.  We will compare the apportionment methods of Hamilton and Grimmett and discuss other stochastic methods.

 

Keith Agre, “A Brief Introduction to Maplets”

Many instructors want to include technology within their math courses.  However, some instructors are unwilling to spend the time needed for students to become proficient with its use.  As a result, instructors sometimes include technology only as a lecture tool rather than as a tool for student investigation.  Recent versions of Maple include a Maplets package which allows students to use the power of Maple to investigate difficult concepts with minimal knowledge of Maple syntax.  Sample Maplets will be introduced to illustrate how an instructor might utilize Maplets within a Calculus II course.