Description: P:\MAA\meetings\2007springprogram_files\logo_ncs.jpg

The Mathematical Association of America

North Central Section

Spring 2007 Meeting

April 13-14, 2007

College of Saint Catherine

Saint Paul, MN

 

Preliminary Program

Friday, April 13, 2007 

7:00-8:00  Registration,  Mendel Hall (in the Hallway outside Mendel 106)

                  $10; students, first time attendees, and speakers free; $5 for MAA-NCS section NExT members

7:00-8:00  Book Sale,  Mendel Hall 400

8:00-9:00  Evening Session    Dr. Daniel O'Loughlin presiding,    Mendel Hall 106

      Lecture:  Professor Andrew Odlyzko, University of Minnesota - Twin Cities

                  “Cybersecurity, mathematics, and limits on technology”


Abstract:  Mathematics has contributed immensely to the development of secure cryptosystems and protocols.  Yet our networks are terribly insecure, and we are constantly threatened with the prospect of imminent doom.  Furthermore, even though such warnings have been common for the last two decades, the situation has not gotten any better.  On the other hand, there have not been any great disasters either.  To understand this paradox, we need to consider not just the technology, but also the economics, sociology, and psychology of security.  Any technology that requires care from millions of people, most very unsophisticated in technical issues, will be limited in its effectiveness by what those people are willing and able to do.  This imposes strong limits on what formal mathematical methods can accomplish, and suggests that we will have to put up with the equivalent of baling wire and chewing gum, and to live on the edge of intolerable frustration.

9:00-10:30  Reception,  Mendel Hall 107

 

Saturday, April 14, 2007

8:15-11:00     Registration, Mendel Hall


8:15-11:00     Book Sale,  Mendel Hall 400

9:00               Morning Session    Dr. Daniel O'Loughlin presiding,  Mendel Hall 106

 

9:05

The Social Security Problem, Prof. Jerry Bergum, South Dakota State University

 

9:25

SeVenn, EleVenn, and Beyond, Barry Cipra, Freelance Mathematics Writer

9:45-10:00     Break

10:00-11:00   Lecture:  Professor Allen J. Schwenk, Western Michigan University

                  “Beware of Geeks Bearing Grifts”


Abstract:  We construct nonstandard sets of dice displaying the curious property of “nontransitivity” and the paradox of “perverse reversal”.  We continue to investigate unexpected instances of nontransitivity involving coin tossing and the game of Bingo.  We also observe the paradox of “waiting time reversal”.

 

11:00-12:00  Lunch,  Dining Hall – Coeur de Catherine, first floor (one level below north side entrance)

12:00-12:30    Business Meeting, President Steve Kennedy presiding,  Mendel Hall 106
                 

12:30            Afternoon Session (Faculty Talks)      Dr. Daniel O'Loughlin presiding,  Mendel Hall 106

 

12:35

An Elementary Construction of an RSTS (15), Prof. Joel Iiams, University of North Dakota

 

12:55

Pythagorean Triples, George Bridgman, University of Minnesota Duluth

 

1:20

Comments on Singular Solutions, Prof. William Schwalm, University of North Dakota

 

1:40

Sublimital Analysis, Prof. Thomas Q. Sibley, St. John's University

 

1:55

Break

 

2:10

Area of a Cyclic Polygon in Terms of its Sides, Prof. Ken Yocom, South Dakota State University

 

2:30

Rodrigues Robotic Rotations, Prof. Mike Hvidsten, Gustavus Adolphus College

 

2:55

Medieval Italian Mathematics, Prof. William Branson, St. Cloud State University

 

3:15

The Binomial and Hankel Transformations of a Sequence, Prof. Eric S Egge, Carleton College

 

12:30            Afternoon Session (Student Research)      Dr. Rochelle Pereira presiding,  Mendel Hall 101

 

12:35

Research and Directed Work at Undergraduate Institutions

 

 

Faculty Panel: Eric Egge (Carleton), Tina Garrett (St. Olaf), Peh Ng (UM, Morris), Karen Saxe (Macalester)

 

 

Parallel Student Session A -- Mendel Hall 101

 

Dr. Rochelle Pereira presiding

 

1:25

Using Initial Ideals to Predict Phylogenetic Trees, Matt Moynihan, St. Olaf College

 

1:40

Directed Hypergraphs with Totally Unimodular Matrices, Samuel Potter, University of Minnesota, Morris

 

1:55

Break

 

2:10

Interesting Property of the Symmedian Point, Julia Fisher, Carleton College

 

2:30

The Pfaffian Transformation, Paul Kory, Carleton College

 

2:55

Modeling Influenza for a College Campus, Alanna Hoyer-Leitzel, Kate Tummers, St. Olaf College

 

 

Parallel Student Session B -- Mendel Hall 215

 

Dr. Sue Molnar presiding

 

1:25

Conjugacy Classes of Permutation Products, Josh Campbell, St. Olaf College

 

1:40

Finding Conjugacy Classes of Permutation Products using Commutators, Thomas McConville, St. Olaf College

 

1:55

Break

 

2:10

Streak Hitting: A Gibbs' Sampler Approach, Michael Soma, Tony Hoff, St. Olaf College

 

2:30

Using the Perron-Frobenius Theorem and the Power Method to Rank the Teams in the National Basketball Association, Andrew Sheridan, Macalester College

 

2:55

Chromatic Number of Zero-Divisor Graphs, Erin Manlove, St. Olaf College

 

Abstracts

 

Evening Session Invited Speaker

Andrew Odlyzko, Cybersecurity, Mathematics, and Limits on Technology

Mathematics has contributed immensely to the development of secure cryptosystems and protocols.  Yet our networks are terribly insecure, and we are constantly threatened with the prospect of imminent doom.  Furthermore, even though such warnings have been common for the last two decades, the situation has not gotten any better.  On the other hand, there have not been any great disasters either.  To understand this paradox, we need to consider not just the technology, but also the economics, sociology, and psychology of security.  Any technology that requires care from millions of people, most very unsophisticated in technical issues, will be limited in its effectiveness by what those people are willing and able to do.  This imposes strong limits on what formal mathematical methods can accomplish, and suggests that we will have to put up with the equivalent of baling wire and chewing gum, and to live on the edge of intolerable frustration.

 

Morning Session Invited Speaker

Allen J. Schwenk, Beware of Greeks Bearing Grifts

We construct nonstandard sets of dice displaying the curious property of “nontransitivity” and the paradox of “perverse reversal”.  We continue to investigate unexpected instances of nontransitivity involving coin tossing and the game of Bingo.  We also observe the paradox of “waiting time reversal”.

 

Faculty Speakers

Jerry Bergum, The Social Security Problem

This presentation is dedicated to Professor Sylvan Burgstahler, and gives the solutions to this problem proposed by Professor Knoshaug during her talk at the Fall, 2006 meeting:  Given the integers 1, 2, 3, 4, 5, 6, 7, 8, 9 can you find a social security number ABCDEFGHI such that 2 divides AB, 3 divides ABC, 4 divides ABCD, 5 divides ABCDE, 6 divides ABCDEF, 7 divides ABCDEFG, 8 divides ABCDEFGH, and 9 divides ABCDEFGHI and how many solutions are there?  The presentation also looks at solutions when one allows the integer 0 to be included in the set of digits.  
 

 

William Branson, Medieval Italian Mathematics

Fibonacci wrote the "Liber Abaci" in 1202.  This book is often noted in math history as an attempt to introduce Hindu-Arabic numerals into Europe, but it is perhaps more notable as the foundational text of the scuolo d'abbaco, which were the mathematical schools for merchants. As a social phenomenon, the pupils of these schools came from all classes of society; the teachers of these schools were viewed as middle class.  The language of the schools was Italian, not Latin; the material taught was largely what we call "College Algebra."  All these factors help explain who did mathematical research in 16th century Italy, and why they focused on algebra.

 

George Bridgman, Pythagorean Triples

Pythagorean Triples - positive integers  (a, b, c)  such that --  have been much written about.  This paper deals primarily with primitive Pythagorean triples, where a, b, and c are relatively prime.  One of the results in this paper is a method of generating two different primitive triples with the same hypotenuse c.  For example, (63, 16, 65) and  (33, 56, 65).

 

Barry Cipra, SeVenn, EleVenn, and Beyond

The speaker will report on recent results on the existence of rotationally symmetric Venn diagrams -- a problem first posed, by an  undergraduate, in the 1960s, and finally fully solved, by another undergraduate, almost 40 years later.  Many related open problems remain, perhaps for yet another undergraduate to solve.

 

Eric S. Egge, The Binomial and Hankel Transformations of a Sequence

The binomial transform is a function which maps sequences to sequences, via a simple construction reminiscent of Pascal's triangle. The Hankel transform also maps sequences to sequences, but its definition involves a certain determinant, and is somewhat more involved. After defining both transforms I'll describe their effects on several interesting sequences, including the Catalan and Fibonacci numbers. Then I'll
discuss elementary properties of each transformation, which will lead to a surprising relationship between them.

 

Mike Hvidsten, Rodrigues Robotic Rotations

In this talk we will look at a very elegant solution to a question that has arisen with the development of robotics--how can we mathematically model three dimensional rotations of limbs about joints.  The solution involves a bit of linear algebra, some differential equations, and some clever insights, due to a fellow named Rodrigues.

 

Joel Iiams, An Elementary Construction of an RSTS(15)

A short discussion on the bane of my existence leads to a colorful explanation of more than half of the title. This discussion will be followed by the promised construction. We conclude with an application (or not).

 

Thomas Q. Sibley, Sublimital Analysis

Limits of subsequences have played a supporting role in some important theorems and definitions of analysis.  However, it appears that these "sublimits" haven't ever received star billing. This article redresses this oversight by taking a closer look at subsequences and their sublimits.  While sublimits may well be hidden in the original sequence, they aren't placed there to send subliminal (subconscious) messages, but rather to entice mathematical exploration.

 

William Schwalm, Comments on Singular Solutions

Singular solutions of ODEs can occur where the tangents of different members of the family of general solutions line up.  This can happen at a boundary between two regions across which the number of real solutions changes.  I review some basic things about singular solutions including how to make DEs that have them, how to find them, some properties of them and two, simple physics models that illustrate them.  The two models are (1) shock waves from a supersonic source and (2) caustics formed as lens aberrations in geometrical optics.  In either case the model is very simple and one need not try to follow a long story or complicated analysis.

 

Ken Yocom, Area of a Cyclic Polygon in Terms of its Sides

This is an expository talk on the areas of a cyclic polygon in terms of its sides from Heron’s formula for the triangle, Brahmagupta’s formula for the cyclic quadrilateral and ending with the recent work of David Robbins and others on Cyclic n-gons for  n = 5, 6, 7 and 8.

 

Panel Discussion

Eric Egge, Tina Garrett, Peh Ng, Karen Saxe, Research and Directed Work at Undergraduate Institutions

Panel participants will each talk for 10 minutes about the various forms that independent undergraduate research takes at their institutions. This includes undergraduate research, capstone projects, as well as directed work in upper-division courses. There will be time for questions and discussion.

 

Student Speakers

Joshua Campbell, Conjugacy Classes of Permutation Products

Take t elements from a group G and consider all the different orders the product of these elements can be taken in, e.g. , and so on. The set of all these permutation products is denoted . When the elements of  are in n conjugacy classes, we say the t-tuple () binds  to n conjugacy classes. For a general group G, it was established that the maximum number of conjugacy classes that a t-tuple can bind  to is (t − 1)! and the binding conditions were completely determined for a t-tuple () where is in the center of G. The conditions on elements of dihedral groups that cause  to bind to different numbers of conjugacy classes were developed. Also, it was found that for a class of extra-special groups, a t-tuple () always binds  to one conjugacy class when is not in the center.

 

Julia Fisher (with Adam Carr, Andrew Roberts, and David Xu),  An Interesting Property of the Symmedian Point

One of the natural questions to consider when studying triangles is how we might group them into families.  Clearly, many such groupings are possible.  We will focus our attention on the families of triangles which share the same circumcenter, centroid, and circumradius.  Many interesting properties arise from these groupings.  One such property concerns a triangle center known as the symmedian point, defined to be the isogonal conjugate of the centroid of a given triangle.  Specifically, we find that the locus of symmedian points of one of our families of triangles always lies on a circle.

 

Alanna Hoyer-Leitzel and Kate Tummers (with Matt Moynihan and Jennifer March), Modeling Influenza for a College Campus

In this talk, we will discuss three models used to examine a typical epidemic outbreak of the influenza virus on a college campus.  Based on a complexified SIR model, the models take into account differences between faculty and students, vaccination rates, situations like musical or athletic groups where a number of students all return to campus sick after a tour or event, and the academic calendar of college institutions. When examined together the models, a deterministic model, a stochastic parameters model and a discrete population stochastic model, give a more complete picture of a typical flu season on a college campus than one model could alone.

 

Paul Kory (with Tracale Austin, Hans Bantilan, and Isao Jonas), The Pfaffian Transformation

We define a new transformation on sequences using the concept of the Pfaffian of a skew-symmetric matrix, which is an analogue to the ordinary determinant. We then develop an algorithm that reduces sequences which satisfy linear recurrences to sequences with only a finite number of non-zero terms while preserving (most of) the effects of the Pfaffian transformation. In particular, we use tools from linear algebra, graph theory, and combinatorics to show how the effects of the Pfaffian transformation on sequences that satisfy linear recurrences can be understood by examining its effects on these reduced sequences. In doing so, we prove that the Pfaffian transformation takes any sequence that satisfies a linear recurrence to another sequence that satisfies a linear recurrence.

 

Erin Manlove (with Dan Endean and Kristin Henry), Chromatic Number of Zero-Divisor Graphs

We examine the vertex coloring of zero-divisor graphs.  We begin by exploring the chromatic number of the zero-divisor graphs of Zn, and then we move to the zero-divisor graphs of direct products of Zn's. 

 

Thomas McConville, Finding conjugacy classes of permutation products using Commutators

We show that if the conjugacy class of a permutation product in  is sufficiently large then every element in  will be in the same conjugacy class.  Using this, we determine some cases in which sets of permutation products are bound to one when formed by three elements in a general semi-direct product of cyclic groups.  These findings are then applied to a particular Metacyclic group.

 

Matt Moynihan, Using Initial Ideals to Predict Phylogenetic Trees

We explore the utility of initial ideals in testing phylogenetic trees.  Using the two-state general Markov model we prove that given an unrooted bifurcating phylogenetic tree, if a flattening produces a grouping of identical taxa then the initial ideal of polynomial invariants corresponding to that flattening vanishes when evaluated at appropriate frequency data. We also provide a counterexample for our conjecture that the initial ideal of polynomial invariants generated by the product of the diagonal entries of the 3x3 submatrices for each flattening will vanish when evaluated at the frequency data for an evolutionarily correct tree. Hence the vanishing of the initial ideal is not a useful predictor of an evolutionarily correct tree as it is with the full ideal of polynomial invariants.

 

Samuel Potter, Directed Hypergraphs with Totally Unimodular Matrices

A directed hypergraph is a generalization of a directed graph G, where arcs have multiple tails or multiple heads, i.e. G=(V,H) where V is a set of vertices and H is a set of hyperarcs, which have multiple heads and tails. A network optimization problem on a directed hypergraph is the linear programming problem: , where c is the cost vector, b is the net supply vector, A is the vertex-hyperarc incidence matrix of G, and x is the vector of variables. A condition under which integer optimal solutions exist for the optimization problem  for any possible c and any integer vector b is that the coefficient matrix, A, is totally unimodular. We found forbidden structures in certain classes of directed hypergraphs that will yield totally unimodular vertex-hyperarc incidence matrices.

 

Andrew J. Sheridan, Using the Perron-Frobenius Theorem and the Power Method to Rank the Teams in the National Basketball Association

The games played in a round-robin tournament can be expressed in graph theory as directed edges of a complete simple graph. Similarly, the games played in a league can be expressed as directed edges of a complete multi-graph. Creating an adjacency matrix from these graphs produces a non-negative matrix with a unique, dominant eigenvalue. The eigenvector corresponding to this dominant eigenvalue will contain all real entries of the same sign and will serve as a vector of relative strengths for teams 1, 2… n. These relative strengths can be used to create a ranked order of all teams in the league.

 

Mike Soma, Streak Hitting: A Gibbs' Sampler Approach

The existence of “Streak hitting”, the idea that baseball players tend to have periods of increased hitting ability, has been a controversial topic in baseball, psychology, and statistics.  In an attempt to detect streak hitting among major league baseball players, a Gibbs’ sampling algorithm was used to estimate the probability of staying in the same hot or cold state in subsequent at-bats for both artificially generated hitting data and the results of real-life batters.  The algorithm was usually capable of detecting various levels of streak hitting in the artificial data, and only detected marginal degrees of streakiness in real-life players.