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

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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.