Fall 2011 Program

Minnesota State University, Moorhead

October 28-29, 2011

Visitor information can be found (here).

Abstracts are listed at the end of the page. (link)

Friday, October 28

4:30-5:30 Section NExT, How to Protect your Scholarly Research Time, Science Lab 102
6:30-9:15 Registration - Science Lab Atrium
$10; Students, first time attendees and speakers free;
$5 for MAA-NCS Section NExT members.
6:30-8:00 Book Sales, College of Social and Natural Sciences Library, Hagen 113
Evening Session - Science Lab 104, Dr. Wally Sizer, Presiding
7:05-7:25 Prof. Ron Rietz, Gustavus Adolphus College
Integral Solutions of certain 2nd Order Linear Differential Equations
7:30-7:50 Prof. Douglas Anderson, Concordia College
Linear Differential Equations: Variation of Parameters or Reduction of Order?
8:00-9:00 Invited Lecture

Prof. Thomas Halverson, Macalester College
Permutation, Coagulation, and Fragmentation
9:00-10:15 Reception - College of Social and Natural Sciences Library, Hagen 104

Saturday, October 29

8:15-11:00 Registration - Science Lab Atrium
Book Sales - Hagen 113
Morning Session - Science Lab 104, Dr. Betty Midgarden, Presiding
9:00-9:05 Welcome, Dr. Michelle Malott, Dean of the College of Social and Natural Sciences
9:05-9:25 Barry Cipra
The Bricklayer's Challenge
9:30-9:50 Takayuki Yamauchi, Valley City State University
A Proof of the Curl of a Vector Field Based on the Paddle Wheel Model
9:50-10:10 Break-Science Lab Atrium
10:10-10:30 Austin Bren and Kristin Heysse, Concordia College (undergraduate students)
Discrete Approximations of Differential Equations via Trigonometric Interpolation
10:35-10:55 Benjamin Gates and Dylan Heuer, Concordia College (undergraduate students)
Hyers-Ulam stability of second-order linear difference equations
11:00-12:00 Invited Lecture
Prof. Richard Gillman, Valparaiso University
How to find (and keep) neighbors
12:00-1:00 Luncheon Comstock Memorial Union Room 101
1:00-1:30 Business Meeting Science Lab 104, Dr. Michael Hvidsten, Presiding
Afternoon Session - Science Lab 104, Dr. Michael Hvidsten, Presiding
1:35-1:50 Prof. Thomas Q. Sibley, St. John's University/College of St. Benedict
Partial Orders on Zn
1:55-2:15 Prof. Andrew Chen, Minnesota State University Moorhead
Strict Posets Are the Intermediate Stages of Sorting
2:20-2:40 Prof. William Schwalm, University of North Dakota, Dept. of Physics
Electric Circuit Problems
2:45-3:05 Prof. Paul Zorn, St. Olaf College
Taking an Interest in Differential Equations


Invited Addresses

How to find (and keep) neighbors, Richard Gillman, Valparaiso University

This talk explores the implications of our natural instinct to be around other people ‘like ourselves.’ In a major work, Schelling (1971) investigated the equilibrium states possible in bi-cultural housing environments. Young (2001) extended this work by identifying those equilibrium states which are also stochastically stable. Undergraduate students at Valparaiso University (2009, 2011) extended these results to multi-cultural environments. The new results have applications from describing the formation of high school cliques, to the American political landscape, to the stability of post-civil war Libya.

Permutation, Coagulation, and Fragmentation, Thomas Halverson, Macalester College

We will begin by shuffling cards in a silly and slow way. This method of shuffling ---which can be viewed as a random walk on permutations --- has elegant properties and will lead us through a terrific story involving permutations, graphs, and some linear and abstract algebra. We then will add operations that "coagulate" and "fragment" points. These operations will give us a random walk on set partitions and lead us to the "partition algebra." Beautiful combinatorics emerge, ending with a startling connection to alternating sign matrices.

Section NExT

How to Protect your Scholarly Research Time

Now that many of us are more seasoned faculty, much of our time is dedicated to committees, teaching, and office hours. However, we need to continue with some professional development that interests us, and our interests evolve with time. We would like to meet in order to discuss with others what we are currently doing for research and/or what we are interested in pursuing. Discussing our ideas may lead to new discoveries and get us energized.

Contributed Student Talks

Kristin Heysse and Austin Bren, Discrete Approximations of Differential Equations via Trigonometric Interpolation, Concordia College

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.

Dylan Heuer and Ben Gates, Hyers-Ulam stability of second-order linear difference equations, Concordia College

Using a type of variation of constants formula, we establish the stability of second-order linear difference equations in the sense of Hyers and Ulam. That is to say, if an approximate solution of the second-order linear equation exists, then there exists an exact solution to the difference equation that is close to the approximate one.

Contributed Talks

Douglas Anderson, Linear Differential Equations: Variation of Parameters or Reduction of Order?, Concordia College

Historically to solve inhomogeneous second-order linear ordinary differential equations, one can use variation of parameters or reduction of order. We show how these are related and extend these methods to difference equations and general dynamic equations on time scales.

Andrew Chen, Strict Posets Are the Intermediate Stages of Sorting, Minnesota State University Moorhead

Consider the process of establishing an order on a set of distinct elements, commonly known as sorting. Typically this occurs through pairwise comparisons of elements, where the sorting algorithm determines which pairwise comparisons should be made, and in which order. Each comparison can be thought of as creating a new strict partially ordered set by adding the information from the comparison to the previous strict partially ordered set. Thus, sorting can be seen as a sequence of posets which go from totally unordered to totally ordered. We will illustrate this through examples and exposition thereof.

Barry Cipra, The Bricklayer's Challenge

The speaker will describe a combinatorial problem, still largely unsolved, that arises from taking a decidedly non-Gauss approach to summing a series like 1+2+...+100.

Ron Rietz, Integral Solutions of certain 2nd Order Linear Differential Equations, Gustavus Adolphus College

All of the solutions of a 2nd order linear differential equation can be obtained in integral form given one nonzero solution of the corresponding homogeneous equation. Among other results, especially due to the availability of adequate software, this provides a useful alternative to the “conjecture and calculate” method of finding particular solutions when the given equation has constant coefficients.

William Schwalm, Electric Circuit Problems, University of North Dakota (physics)

A circuit is a digraph, which may have multiple edges between a pair of vertices, with a non-negative real function (resistance) defined on the edges. A circuit problem involves two more real valued functions. One (voltage) is defined on the vertices, and the other (current) is defined on the edges. Given sufficient information about the current and/or the voltage, the game is to find the remaining values. Now I may have told the mathematically astute reader more than I know myself. However, several cute problems will be given with solutions. Use is made of permutation symmetry in some of them.

Thomas Q. Sibley, Partial Orders on Zn , St. John’s University/College of St. Benedict

We investigate partial orders on Zn that are compatible with the multiplication. (That is, such a partial order ◄ needs to satisfy if 0 ◄ x and y ◄ z, then xy ◄ xz and the corresponding reversed relation when x ◄ 0.) My students and I found some interesting connections between such partial orders and greatest common divisors.

Takayuki Yamauchi, A Proof of the Curl of a Vector Field Based on the Paddle Wheel Model, Valley City State University

In Calculus texts, the curl of a vector field F is defined and given as Curl(F) = ∇ × F without any physical model or motivation. This definition is then retroactively justified using Stokes' theorem. In this article, ∇ × F is rigorously and directly derived based on the paddle wheel model and the definition of circulation density.

Paul Zorn, Taking an Interest in Differential Equations, St. Olaf College

Among (presumably) the earliest applications of DE-related ideas – growth, accumulation, future value – is the growth of money. In this liberal-arts oriented talk I’ll describe some historical aspects of growth with interest, and relate some of them to differential equations.

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