Spring 2011 Program

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

Friday, April 15

3:30-5:30 Introduction to WebWork - CMC 201
Jonathan Rogness (University of Minnesota-Twin Cities)
and Aaron Wangberg (Winona State University)
Open to **ALL**
6:30-8:00 Registration - Boliou Lobby
$10; Students, first time attendees and speakers free;
$5 for MAA-NCS Section NExT members.
6:30-9:30 Book Sales - CMC 210
Evening Session - Boliou 104, Professor Jon Armel, Presiding
7:10-7:15 Welcome, Dr. Stephen Kennedy, Chair of Mathematics, Carleton College
7:15-7:30 Prof. In-Jae Kim, Minnesota State University-Mankato
Path cover number and maximum corank of a tree
7:35-7:55 Prof. Wally Sizer, Minnesota State University-Moorhead
Period Two Solution to some Systems of Rational Difference Equations
8:00-8:05 Welcome
Dean Beverly Nagel, Carleton College
8:05-9:05 Invited Lecture

Professor Jonathan Rogness, University of Minnesota-Twin Cities
When Mathematics meets YouTube
9:00-10:30 Reception - Boliou Lobby

Saturday, April 16

7:00-8:30 Executive Board Meeting, CMC 328
8:00-11:00 Registration - Boliou Lobby
Book Sales - CMC 210
Morning Session A - Boliou 104, Professor Gail Nelson, Presiding
9:05-9:25 Dr. Barry Cipra
Factor Subtractor
9:30-9:45 Prof. John Holte, Gustavus Adolphus College
Small-sample confidence intervals for proportions
9:50-10:10 Break - Boliou Lobby
10:10-10:25 Prof. George Bridgman, University of Minnesota-Duluth (retired)
Differentiability of a Function of 2 or more Real Variables
10:30-10:50 Prof. Roger Kirchner, Carleton College (retired)
Rigorous Treatment of max/min Problems in Calculus
Morning Session B - CMC 206, Professor Helen Wong, Presiding
9:05-9:25 Prof. Jody Sorenson and John Zobitz, Augsburg College
The Real Story of the Lorenz System
9:30-9:45 Mr. John Hanenburg
Observations of a Generalized Fibonacci Series
9:50-10:10 Break - Boliou Lobby
10:10-10:30 Prof. Thomas Halverson, Macalester College
Motzkin Numbers
10:35-10:50 Prof. Ron Rietz, Gustavus Adolphus College
An Elementary Integral Formula for Products of Functions with Poporational Second Derivatives
11:00-12:00 Invited Lecture - Boliou 104, Professor Gail Nelson, Presiding

Prof. Doug Ensley, Shippensburg University
Invariants under Group Actions to Amaze Your Friends!
12:00-1:00 Luncheon Severance Hall Tea Room (Fireside end)
1:00-1:30 Business Meeting Boliou 104, Dr. Jason Douma, Presiding
Afternoon Session A - Boliou 104, Dr. Jason Douma, Presiding
1:30-1:50 Prof. Adam McDougall, St. Olaf College
Regarding Rational Density, Part 1
1:55-2:15 Nathan Bishop (undergraduate)
Regarding Rational Density, Part 2
2:20-2:35 Prof. Aaron Wangberg and Ben Johnson (undergraduate), Winona State University
Exploring Multivariable Calculus Concepts on Real, Tangible Surfaces
Afternoon Session B - CMC 206, Professor Gail Nelson, Presiding
1:30-1:50 Matthew Friedrichson (undergraduate), St. Olaf College
Structure and Statistics of the Self-Power Map
1:55-2:10 Jorge Banuelos (undergraduate), Macalester College
A Simultaneous Random Walk Game
2:15-2:30 Wanyi Li (undergraduate), Macalester College
Elastohydrodynamic Instabilities in Gravity-Driven Flow



Jonathan Rogness and Aaron Wangberg, Introduction to Webwork

WebWork is an online homework system originally developed at the University of Rochester and now supported by the MAA through a multi-year NSF grant. Unlike WebAssign or Aleks, WebWork is not tied to publishers, and can be used for free in your department, either by running a server at your own school or making use of the MAA's hosting services. Depending on your level of computer savviness, you can either write your own problems, or make use of the National Problem Library, which contains over 20,000 problems in calculus, precalculus and other courses. This workshop will give you an overview of WebWork's features and show you how to get started creating homework assignments for your courses.

Invited Addresses

Jonathan Rogness, When Mathematics meets YouTube

What happens when 1.8 million people encounter high-level mathematics on YouTube? “Möbius Transformations Revealed” is a short film that illustrates the beauty of Möbius Transformations and shows how moving to a higher dimension makes them easier to understand. After winning an award from the National Science Foundation and Science magazine the video went viral, with unexpected and entertaining results. This talk will describe the behind-the-scenes making of the movie, explore the mathematics it illustrates, and show the reactions of YouTube users who discover the visual allure of mathematics.

Doug Ensley, Invariants under Group Actions to Amaze Your Friends!

By understanding invariant properties of a group action (shuffling!) on a deck of cards, a magician can find order where the spectator believes he or she has created disorder. This presentation will introduce some general mathematical principles of group actions by looking at some specific card tricks that anyone can do.

Contributed Student Talks

Jorge Banuelos, A Simultaneous Random Walk Game

Two tokens are placed on vertices of a graph. At each time step, one token is chosen and moved to a random neighboring point. Prior research analyzed the Angel strategy for bringing the tokens together as quickly as possible, and the Demon strategy for delaying their collision as long as possible. We study a game version of this process.

In our game, Angel and Demon take turns choosing the token to move. We present optimal strategies for both players on stars, different types of directed cycles, and paths. Our proofs employ couplings of random walks as well as strategy stealing arguments.

Nathan Bishop, Regarding Rational Density Part 2

It's generally accepted that Lebesgue measure is the best way to measure subsets of R because, when applied to any real interval, it agrees with an interval's length. But what if we wanted to measure subsets of Q? Lebesgue measure proves to be unhelpful, so instead, we use density theory to study subsets of Q and find that an interval's density is represented by its length.

Matthew Friedrichsen, Structure and Statistics of the Self-Power Map

We investigate the structure of a cryptographic object called the self-power map, given by x → xn mod p, for p a prime. As a variation of the Discrete Log Problem, the self-power map is thought to be difficult to solve in the inverse and therefore considered safe for use in some versions of the ElGamal digital signature algorithm. Nonetheless, utilizing functional graphs to represent the map has revealed non-random structural properties, which we describe primarily through number theory and statistics.

Ben Johnson, Exploring Multivariable Calculus Concepts on Real, Tangible, Surfaces

The concepts and procedures studied in multivariable calculus are very geometric in nature and are easily measured, constructed, and explored using surfaces cut on a CNC machining center. In this talk, we’ll share how the six different surfaces, designed and constructed with an undergraduate student, have helped calculus students discover the geometry behind many concepts including gradient vectors, directional derivatives, and the method of Lagrange multipliers.

Wanyi Li, Elastohydrodynamic Instabilities in Gravity-Driven Flow

The dynamics and stability of gravity-driven flow of a thin Newtonian fluid down an inclined plane covered with either a topographical trench feature (Case i) or a deformable elastic gel (Case ii) is studied. Mathematical models are derived starting from the Navier-Stokes equations coupled with appropriate boundary conditions. Lubrication theory and non-dimensionalization are applied to obtain the coupled nonlinear PDEs describing the time evolution of the interfaces. Steady interfacial shapes are found and used as a base state. The dynamics is simulated in Matlab. Linear stability analysis is performed to identify potential unstable parameter sets and verify the numerical code.

Contributed Talks

George Bridgman, Differentiability of a Function of 2 or more Real Variables: Examples and Counterexamples

I present a theorem connecting (1) differentiability of a function f of 2 or more real variables at a point P = (a,b,...), and (2) continuity of the partial derivatives of f at P. I give some examples and counterexamples.

Barry Cipra, Factor Subtractor

The speaker will describe a new combinatorial game, Factor Subtractor, which is intended to give children a motivated excuse to practice elementary arithmetic, while providing mathematicians some problems to analyze.

Thomas Halverson, Motzkin Numbers

Although not as ubiquitous as their relatives the Catalan numbers, the Motzkin numbers (1, 1, 2, 4, 9, 21, 51, 127, 323,...) show up in a variety of combinatorial settings. They count nonintersecting chords on a circle, certain paths in the integer lattice, up-down-level walks on the natural numbers, and a family of pattern avoiding permutations. We will examine some of the bijections between these sets, and we will uncover an algebraic structure (the "Motzkin algebra") in which they appear.

John Hanenburg, Observations of a Generalized Fibonacci Series

The definition of the Fibonacci sequence can be expanded: In general, starting with a series of 0’s and a 1 and each number in the sequence will be the sum of a set of preceding numbers. The number in the sequence F(Na, Nb, Nc,...,Ni) will be the sum of the numbers Na, Nb, Nc,...,Ni where Ni is the preceding i places number. The traditional Fibonacci sequence, F(N1, N2) is the sum of N1, the preceding 1 place number, and N2, the preceding 2 places number. I will discuss:

John Holte, Small-sample confidence intervals for proportions

The standard confidence interval for a population proportion involves a couple of approximations. The first is the approximation of the exact binomial distribution by a normal distribution. The second is the approximation of the standard deviation of the sample proportion, even though this is not necessary. I have yet to see a statistics textbook that includes the elementary derivation of the confidence interval without making the second approximation. In this talk I’ll present this confidence interval as well as others and compare them with the exact confidence intervals for small sample sizes.

In-Jae Kim, Path cover number and maximum corank of a tree

For a tree T on n vertices, it has been shown that the path cover number, p(T), of T is equal to the maximum corank, MaxMult(T), of a real symmetric matrix whose graph is T. In this talk we provide an alternative proof of the result, p(T)=MaxMult(T), by showing that the two quantities of interest can be computed systematically in the same manner.

Roger B. Kirchner, Rigorous Treatment of Max/Min Problems in Calculus - Three Wolfram Demonstrations in Tribute of J. L. Walsh

J. L. Walsh, Professor of Mathematics at Harvard (1921 to 1966), published "A Rigorous Treatment of the First Maximum Problem in the Calculus", a January 1947 Classroom Note, and a booklet, Rigorous Treatment of Maximum-Minimum Problems in the Calculus, in 1962. "Tin Box With Maximum Volume", "Swim, Swim and Walk, or Walk?", and "Maximum Area Field with a Corner Wall" are Wolfram Demonstrations which generalize three problems of Walsh. They are problems where extrema occur at an endpoint or where the derivative does not exist, as well as where it is zero. The Corner Wall Problem generalizes the Wall Problem of Pierre Malraison, for which a computer generated film and slides were produced in a 1973 NSF workshop at Carleton College.

Adam McDougall, Regarding rational density part 1

What is the probability of choosing a rational number at random that, when reduced, is of the form odd/even? even/odd? odd/odd? Instinctively, we might guess 1/3 for all three questions; however, the fact that rationals can be reduced gives reason for us to pause. In fact, it so happens that it isn't possible to define a "reasonable" probability on Q. Instead, we show that we can rely upon (generalized) natural density to answer such questions about Q.

Ron Rietz, An Elementary Integral Formula for Products of Functions With Proportional Second Derivatives

Let f and g be real-valued functions defined on the same interval such that D2f = Af and D2g = Bg, where A and B are constants with A ≠ B. We will consider a formula for the indefinite integral of fg which compares favorably with the usual integration by parts result.

Walter Sizer, Period Two Solutions to some Systems of Rational Difference Equation

We consider period two solutions to various cases of the difference equations
x(n+1) = (a + b x(n) + c y(n))(d + e x(n) + f y(n)), y(n+1) = (j + k x(n) + l y(n))(p + q x(n) + r y(n))
We give several examples of period two solutions and in sme cases categorize such solutions.

Jody Sorenson, and John Zobitz, The Real Story of the Lorenz System

Mathematical lore describes how in 1963 Edward Lorenz used a computer to model the weather and discovered sensitive dependence on initial conditions. The “strange” attracting set of the Lorenz system of differential equations is one of the hallmarks of the mathematical field of chaos theory. In popular culture sensitive dependence on initial conditions is often described in terms of the “butterfly effect”. The truth, as always, is a little more complicated. In this talk we will clarify the history behind the legend of the origins of chaos, and delve in to the meaning of the three variables in Lorenz's equations.

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