Fall 2012 Program

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

Friday, October 19

4:30-5:30 Section NExT
Bioinformatics and Potential Curricular Possibilities - Solon Campus Center (SCC) #21
Prof. Marshall Hampton University of Minnesota, Duluth
Open to **ALL**
6:30-9:15 Registration - Wedge area Solon Campus Center (SCC)
$\$$10; Students, first time attendees and speakers free;
$\$$5 for MAA-NCS Section NExT members.
6:30-8:00 Book Sales - Solon Campus Center (SCC) #130
Internet Access: SCC #118; wireless access throughout campus
Evening Session - Solon Campus Center (SCC) #120, Dr. Zhuangyi Liu, Presiding
7:00-7:25 Prof. Dalibor Froncek, University of Minnesota, Duluth
Benjamin Franklin and Tournament Scheduling
7:30-7:50 Prof. Mike Hvidsten, Gustavus Adolphus College
Interactive Math on the Web – It’s as Easy as $\pi$!
8:00-9:00 Invited Lecture

Beth Skubak Wolf, University of Wisconsin, Madison
Polynomials, Ellipses, & Matrices: Three Questions, One Answer
9:00-10:15 Reception - Wedge area of Solon Campus Center (SCC)

Saturday, October 20

8:15-11:00 Registration - Wedge area Solon Campus Center (SCC)
Book Sales - Solon Campus Center (SCC) #130
Morning Session - Solon Campus Center (SCC) #120, Dr. Bruce Peckham, Presiding
9:00-9:05 Welcome
Dr. James Riehl, Dean of College of Science and Engineering, UMD
9:05-9:25 Prof. Ruijun Zhao, Minnesota State University, Mankato
What makes the eradication of malaria difficult?
9:30-9:50 Prof. John Greene, University of Minnesota, Duluth
Traces of matrix products
9:50-10:10 Break - Buenger Education Center Lobby
10:10-10:30 Prof. Christopher Phan, Winona State University
Uniquely reducing polynomials: Gröbner bases and the diamond lemma
10:35-11:05 Prof. Doug Dunham, University of Minnesota, Duluth
Patterns on Triply Periodic Polyhedra
11:10-12:00 Invited Lecture - Solon Campus Center #120, Dr. Dan Kemp, Presiding

Prof. Joseph A. Gallian, University of Minnesota, Duluth
Getting Undergraduates Involved in Research
12:00-1:00 Luncheon Kirby Student Center, Garden Room
1:00-1:30 Business Meeting Solon Campus Center #120, Dr. Dan Kemp, Presiding
Afternoon Session I - SCC #120, Dr. Marshall Hampton, Presiding
1:35-1:50 Prof. Dan Kemp, South Dakota State University
A Project to Evaluate the Gaussian Integral $\int_0^{\infty}e^{-x^2}dx$ using Calculus II techniques
1:55-2:15 Prof. Lindsay Erickson, Concordia College, Moorhead
Nim on multipartite graphs
2:20-2:40 Prof. Ron Rietz, Gustavus Adolphus College
Edge Subgroups of the Reals
2:45-3:05 Prof. Wenhao Gui, University of Minnesota, Duluth
A Symmetric Component Alpha Normal Slash Distribution: Properties and Inferences
Afternoon Session II - SCC #21, Dr. Dalibor Froncek, Presiding
1:35-1:50 Prof. Aaron Wangberg, Winona State University
Relatively simple identifications of strange subalgebras of the Exceptional Lie Algebra $e6$
1:55-2:10 Robert Vaselaar1, Hyun Lim1*, Jung-Han Kimn1, Dongming Mei2
*presenter (undergraduate student) 1South Dakota State University, 2University of South Dakota
Trigonometric Space-Time Discretization of the Gauge-Free Dirac Equation and its Parallel Implementation
2:20-3:05 Math Jeopardy
(organized by Professors Stephen Kennedy, Kris Nairn Jody Sorensen and Aaron Wangberg)
Student Activity - Faculty members are welcome to watch and cheer students!


Invited Addresses

Joseph A. Gallian, Getting Undergraduates Involved in Research, University of Minnesota, Duluth

Although involving undergraduates in research has been a long standing practice in the experimental sciences, it has only been fairly recently that undergraduates have been involved in research in mathematics in significant numbers. In this talk I discuss in general terms such things as how faculty can get started in involving undergraduates in research, the benefits of undergraduate research to faculty and students, how to find suitable research problems, and what is considered to be undergraduate research.

Beth Skubak Wolf, Polynomials, Ellipses, & Matrices: Three Questions, One Answer, University of Wisconsin, Madison

Given two points $a$, $b$ in the unit disk, when is there a cubic polynomial with roots on the circle with $a$, $b$ as critical points? I'll describe the connection between this question and two others, one geometric and another concerning matrices, and give the one concise answer for all three questions. The result and its proof extend very naturally to any finite number of points by using a specific type of rational function, called a finite Blaschke product, both as a proof tool and as a link between the three scenarios. This work was the culmination of an undergraduate research project and honors thesis at Bucknell University under Professor Pamela Gorkin.

Section NExT

Marshall Hampton, Bioinformatics and Potential Curricular Possibilities, University of Minnesota, Duluth

This talk will survey some broad areas of bioinformatics and how they can be incorporated in various types of courses. Bioinformatics is a mix of biology with mathematics, statistics, and computer science, any combination of which can be emphasized. Some of these topics also tie in to mathematical biology and modeling.

Contributed Talks

Doug Dunham, Patterns on Triply Periodic Polyhedra, University of Minnesota, Duluth

Artists have created patterns on symmetric closed polyhedra. However to our knowledge no one has created patterns on triply periodic polyhedra in Euclidean 3-space. We show a few such patterns and explain how some of them are related to triply periodic minimal surfaces (TPMS). These patterns are also related to repeating patterns of the hyperbolic plane that are based on regular tessellations $\{p,q\}$ composed of regular $p$-gons meeting $q$ at each vertex. We will explain these relationships, and examine interesting geometric facts that link the patterns on triply periodic polyhedra, the corresponding TPMS's, and hyperbolic patterns.

Lindsay Erickson, Nim on multipartite graphs, Concordia College, Moorhead

The game of Nim on graphs as originally described by Masahiko Fukuyama is a 2-player combinatorial game of increasing popularity. This paper expands previous bipartite graph results to include the uniform weight case of the $K_{m,n}$ for all $m$, $n$. We solve the $K_{m,n}$ under what we call the “uniformly weighted $V$” condition and discuss the arbitrary weight case in the most general setting. We also include new results on multipartite graphs solving entirely the uniform weight case, the uniformly weighted $V$ case, and briefly discuss the arbitrary weight case. We pose some suitable research questions for any beginning researchers.

Dalibor Froncek, Benjamin Franklin and tournament scheduling, University of Minnesota, Duluth

Scheduling a complete round robin tournament is a routine task if you know a little bit of graph theory. Scheduling an incomplete tournament which is fair or equal chance (which is not the same in our terminology) is not so well known. We will present a relatively easy method based on graph labelings. We may also present handicap tournaments, where the underdogs have easier schedules than the top teams. More precisely, the strengths of the schedules are ordered according to the team strengths.

John Greene, Traces of matrix products, University of Minnesota, Duluth

Given two matrices, $A$ and $B$, it is well known that AB and BA have the same trace, as do cyclic permutations of products of $A$'s and $B$'s. We show here that for $2 \times 2$ matrices $A$ and $B$, whose elements are independent random variables with standard normal distributions, the probability that $\mbox{Tr}(ABAB) > \mbox{Tr}(AABB)$ is $1/\sqrt{2}$. For $n \times n$ matrices, we give tables from computations suggesting that the probability that $\mbox{Tr}(ABAB) > \mbox{Tr}(AABB)$ is still roughly 0.7, though we do not know the exact value when $n > 2$.

Wenhao Gui, A Symmetric Component Alpha Normal Slash Distribution: Properties and Inferences, University of Minnesota, Duluth

In this talk, we introduce a new class of symmetric bimodal distribution. We define the distribution by means of a stochastic representation as the mixture of a symmetric component alpha normal random variable with respect to the power of a uniform random variable. The proposed distribution is more flexible in terms of its kurtosis. It can successfully capture the bimodality. We illustrate it with a real application by maximum likelihood procedure.

Mike Hvidsten, Interactive Math on the Web – It’s as Easy as $\pi$!, Gustavus Adolphus College

There has been a convergence of two different technologies that now provide a simple, yet powerful, platform for the development of interactive math on the web. These freely available technologies include MathJax -- an open source JavaScript display engine for mathematics, and JSXGraph -- an open source JavaScript library for interactive geometry. In this talk, the presenter will discuss these new technologies and give several examples of how simple they are to use.

Dan Kemp, A Project to Evaluate the Gaussian Integral $\int_0^{\infty}e^{-x^2}dx$ using Calculus II, South Dakota State

It’s well known, in the sense that numerous articles have been published on the topic, that the Gaussian Integral $I_0 = \int_0^{\infty}e^{-x^2}dx$, can be evaluated exactly as $I_0 = \frac{\sqrt{\pi}}{2}$. However the techniques are not so well known that they commonly appear in calculus text books. Indeed, every calculus text book I have looked at evaluates $I_0$ by using a multiple integral technique, similar to what Laplace used in his Théorie Analytique des Probabilités. However the integral is commonly used in a chapter on Probability coming long before multiple integrals. This talk will show a Project to evaluate $I_0$ that I have successfully used in Calculus II for the past several years. I discovered the technique in the MAA book ‘A Garden of Integrals’ which also referred me to Young’s wonderful ‘Excursions in Calculus’, another MAA book. The tools used are integration by parts, mathematical induction, Wallis’ formula, and an inspired idea from Thomas Jan Stieltjes.

Ron Rietz, Edge Subgroups of the Reals, Gustavus Adolphus College

Let $R$ be the group of real numbers under addition. A subgroup $H$ of $R$ is an Edge Subgroup in case there exists a number $x$ in $R$ such that $H$ is maximal with respect to the property of not containing $x$. We will investigate some properties of these subgroups.

Robert Vaselaar1, Hyun Lim1*, Jung-Han Kimn1, Dongming Mei2, Trigonometric Space-Time Discretization of the Gauge-Free Dirac Equation and its Parallel Implementation, *presenter (undergraduate student) 1South Dakota State University, 2University of South Dakota

We study a space-time finite element discretization of the gauge-free Dirac equation based on C1 trigonometric interpolation functions. Numerical experiments are used to demonstrate convergence to analytic results without modification to the Dirac operator. Our work is implemented using PETSc, the “Portable, Extensible, Toolkit for Scientific Computation”, a general purpose suite of tools for the scalable solution of linear problems developed by Argonne National Laboratory. The PETSc platform is designed for high performance computing and parallelization, and also has built-in iterative solvers like the Biconjugate gradient stabilized (BiCGStab) method, which is used to solve the discrete problem above.

Christopher Phan, Uniquely reducing polynomials: Gröbner bases and the diamond lemma, Winona State University

How do you know when two polynomials are equal modulo an ideal? Or, to put it in geometric terms, how do you know when two polynomial functions are equal on a (special) subset? One way is to create a unique representation for each polynomial. Gröbner bases and the diamond lemma are a way to do that. We can even extend these ideas to a noncommutative context.

Aaron Wangberg, Relatively simple identifications of strange subalgebras of the Exceptional Lie Algebra $e6$, Winona State University

Rotation groups acting on $n$-dimensions naturally contain chains of smaller subgroups acting on lower dimensional spaces. The Octonions, the largest of the division algebras, can be used to construct a 78-dimensional Lie Group $E6$ and its associated Lie Algebra $e6$. In this talk, I’ll share how we used chains of division algebras and subspaces to not only identify large subalgebras of $e6$ but also find 8 of the possible 5 “different” real forms of $e6$.

Ruijun Zhao, What makes the eradication of malaria difficult?, Minnesota State University, Mankato

In this talk, I will present a mathematical model of malaria control, particularly investigating the efficacy of imperfect vaccination. The model consists of deterministic ordinary differential equations. Based on our model, a backward bifurcation very likely occurs, which suggests that using the basic reproduction number as the threshold to eradicate the disease is questionable. This finding might provide some valuable suggestions to health policy makers.

Math Jeopardy, (organized by Stephen Kennedy, Kris Nairn, Jody Sorensen and Aaron Wangberg)

Teams of students participate in this mathematics trivia game and must provide their answers to the clues in the form of a question. The content ranges from throughout the undergraduate curriculum. All students are invited to participate.

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