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Hudson Colloquium Series - 2013
Dr. Matt Noble, Mercer University
A Few Odds and Ends on Euclidean Distance Graphs in Q^n
Abstract: Let’s start with a little notation. For any X \subset R^n and any d>0, we denote by G(X, D) the graph with vertex set X where any two vertices are adjacent if and only if they are Euclidean distance d apart. We let χ(X, d) be the chromatic number of such a graph – that is to say, the minimum number of colors needed to color the vertices of G(X, d) such that no two adjacent vertices receive the same color. Determining the exact value χ(R^2, 1) – the so-called “chromatic number of the plane” – is a fairly well-known problem and has frustratingly been open for over sixty years. In this talk I will give a short history of this problem and then give descriptions of related questions that have popped up in my own research. The better part of the talk will be spent considering chromatic numbers (along with the underlying structure) of Euclidean distance graphs with vertex set Q^n ,the n-dimensional rational space. It should be fun
Commutative Rings and the Invariant Basis Property.
Abstract: Commutative algebra is an immense and vibrant field (no pun intended!) that grew primarily out of number theory and algebraic geometry. Essentially, it is the study of sets where elements can be added, subtracted, and multiplied, but not necessarily divided. We wish to convey some of the subject's appeal by illustrating a few of its key ideas, including the notions of prime ideals and R-modules, certain universal mapping problems, and the importance of Zorn's Lemma. As an example of the power of these selected topics, we will use them to prove that the invariant basis property applies to every commutative ring, a result which generalizes an important theorem from linear algebra.
Dr. Sungkon Chang
The p-adic numbers, Part 1: Introduction to Modular Arithmetic and the p-adic numbers.
Abstract: The modular arithmetic is an algebraic method of investigating integer solutions of equations such as 2x2 - y2 = 1. In general the process of this investigation is not guaranteed to terminate, and in this potentially endless investigation the systems of numbers called the p-adic numbers rise naturally. In this talk we shall introduce the modular arithmetic and the basic algebra of p-adic numbers.
Like the system of the real numbers, the systems of the p-adic numbers are “complete” as well, which means that we cannot invent any more numbers that can fit “in between” the numbers in the current system. The theory of calculus is built on top of the completeness of the real numbers, and hence, we may attempt to develop the theory of calculus on the p-adic numbers as well. In Part 2 we shall introduce the basic topology and the elementary calculus on the p-adic numbers.
Dr. Sean Eastman and Dr. Jared Schlieper
The volume of the spatial region corresponding to n x n correlation matrices
Abstract: Correlations between random variables can be displayed in a matrix, where the i,j entry in the matrix given the correlation between the i-th and j-th variables. An n x n correlation matrix can be identified with a point in n(n-1)/2 dimensional space, and so the set of all valid correlation matrices of a given size determines a region in this space. In this talk, we investigate a method for determining the volume of this shape for any n.
Dr. Paul Hadavas
Dawn of the MOOC
Abstract: In the last two years, the latest craze to hit the education sector has been the rise of MOOCs. These are Massive Open Online Courses. Free to anyone with an internet connection and desire to learn. Now the talk has changed to how to offer these classes and get college credit. Over the summer, I explored three of the more popular MOOC platforms by taking 5 classes. In this talk, I will discuss my experience in general, some neat Calculus examples that I came across in particular, and what's next for MOOCs and Armstrong.
Dr. Sabrina Hessinger, Dr. Andi Beth Mincer, Dr. Jared Schlieper, Dr. Michael Tiemeyer, Moses*, Piepergerdes*
Meta-cognitive Enhancement of Cooperative Learning for Concepts of Calculus
Abstract: Metacognition, simply defined as the act of thinking about one’s thinking, can be more concretely understood as thinking of 1) what one knows, 2) what one is currently doing, and 3) one’s current cognitive state. Researchers have established questioning strategies in order to facilitate the development of metacognition in students in order to bring student to deeper levels of learning.
Our research group is currently implementing a pilot test of five cooperative learning modules, each of which is designed around an essential concept in calculus and enhanced with meta-cognitive thinking strategies. Module design is based on four proven research principles: 1) cooperative learning improves student performance on higher level mathematical tasks; 2) there is a solid connection between active learning and student attainment of conceptual knowledge; 3) meta-cognition improves problem solving skills; and 4) authentic tasks increase student interest and motivation in mathematics. Hence, each instructional module is designed around an essential calculus concept and uses an authentic task. As an example, the concept of limits uses the task of long term analysis of growth rates in the workforce. The five concepts are: 1) limits at infinity; 2) definition of derivative; 3) chain rule for derivatives; 4) Riemann sums; and 5) the Fundamental Theorem of Calculus.
Next year, our team will implement a more rigorous research study using the piloted cooperative learning modules. Our expected outcomes are improved performance on authentic tasks, improved understanding of calculus concepts, maintenance of computational skill, and increased passing rates in calculus. In this presentation we will share the content modules and discuss faculty and student observations collected thus far this semester as well as present in detail the project research design.
The Putnam Team
Reflections on the 73rd Annual Putnam Mathematical Competition
Abstract: In December 3, 2012, an intrepid team of Armstrong students spent most of their Saturday working on a dozen frighteningly challenging mathematical problems. Why? Just another installment of the notoriously difficult William Lowell Putnam Mathematical Competition. Members of the team and faculty members will discuss solutions of their favorite problems from the most recent competition.
March 6, 2013
Dr. Tim McMillan
Abstract: This colloquium will feature the geometry software Geometer's Sketch Pad. Dr. McMillan will do a handful of geometry constructions. Some of the problems he will consider are not usually thought of as geometry problems, but they have nice geometric interpretations and solutions.
GSP will be used regularly in the Modern Geometry class that will run in the fall. So this presentation is something of a shameless commercial for that course.
We also hope to make the point that the learning hurdle for GSP is not so substantial. So it can be a handy tool to illustrate many kinds of problems.
Dr. Travis Trentham
An Introduction to Half-Factorial Domains
Abstract: Shortly after we learn to multiply we learn that we can break down some integers into products of other ones. That is, we learn to factor. We quickly learn that there are certain integers that cannot be broken down and these are called the prime integers. And what's more, we learn that not only is 2(2)(3) a prime factorization of twelve, but that this is pretty much the only way twelve can be factored. That is, twelve factors uniquely into a product of primes. And wonder of wonders, twelve is not so special in this regard. The Fundamental Theorem of Arithmetic guarantees us that every positive integer greater than one enjoys this property. Hence, the integers constitute what is nowadays called a unique factorization domain, or UFD for short. As it happens, many rings enjoy this property. For example, given any polynomial with real coefficients and degree greater than zero, then this polynomial admits a unique factorization into polynomials that mimic primality in the integers. The set of all polynomials with real coefficients form a UFD. The Gaussian integers would be yet another example of a UFD. The UFD property is so nice and is encountered so frequently, that it is tempting to think that unique factorization is the norm. Unfortunately, this is not the case. Indeed, many of the rings encountered in number theory and elsewhere exhibit nightmarish factorization properties. Our discussion will not be a tour through this exotic world, but rather we will focus on what is probably the next best thing to the UFD property, at least from a factorization point of view. In this talk we will be examining UFD's and the class of rings called half-factorial domains, or HFD's. Emphasis will be placed on accessible examples and how these two classes of rings differ. We will close with some recent developments and an open question.
Dr. Lynn Williams
A Variation on the Three-Coin Toss Experiment
Abstract: When a balanced coin is tossed three times, there are eight equally likely outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT). Two friends, Bert and Ernie, select a 3-coin toss outcome and toss the coin 3 times to determine the winner. Approximately 3/4 of the time, neither Bert nor Ernie wins. Each one wins approximately 1/2 of the other times. They decide to alter the game slightly by not starting over if one of the two chosen outcomes does not appear in the first three tosses. They continue to toss the coin until one of the chosen outcomes appears as the last three tosses in the sequence. If, for example, Bert chose the sequence HTH and Ernie chose the sequence HTT, then the sequence THHTH of five tosses would represent a win for Bert. They discover rather quickly that in this new version of the game, not all pairs of choices are equally likely. For example, if Bert selects HHH and Ernie selects THH, then Bert only wins when the initial three tosses are heads, which occurs about 1/8 of the time. Bert always selects his sequence first. Ernie knows Bert’s choice before making his selection. Can Bert make a selection that will always give him an even chance of winning no matter what Ernie chooses? If the answer is no and they decide to bet on the outcome, what odds should Bert request and what choice should he make so that it is potentially a fair game?