The Hudson Colloquium Series
In 1988, at the initiative of Dr. Anne Hudson, the then Department of Mathematics and Computer Science at Armstrong State College began a near-weekly luncheon colloquium. Students and faculty would gather in the luxurious confines of Hawes 203 for hot dogs, spaghetti, taco salad, etc., and an enjoyable talk on some topic in mathematics or computer science. In 2003 this luncheon-colloquium series was named in honor of Anne and Sigmund Hudson.
Today, the colloquium is sponsored by the Department of Mathematics and takes place on Wednesdays at 12:00 (noon) in University Hall, room 157 (unless otherwise noted). For a donation of a dollar—$2 for faculty and other non-students—you can enjoy a delicious light lunch, invigorating conversation with students and faculty members, and a lecture, demonstration, or other event arranged by faculty, students and/or visitors. Please come.
Please contact Dr. Selwyn Hollis if you are interested in giving a presentation. Also, please send your email address to Dr. Hollis if you would like to be added to the mailing list. If you're interested in helping with lunch preparation, please contact Dr. Hollis. His email is Selwyn.Hollis@armstrong.edu
Fall 2011October 4
Tricia Muldoon Brown
The Robinson-Schensted Algorithm
Abstract: We introduce two familiar combinatorial objects, partitions and permutations, and give a bijection between sets of these objects. An algorithm due to Robinson and Schensted that makes use of another combinatorial object, Young diagrams, defines this bijection. The talk concludes with some questions.
The Axiom of Choice
Abstract: In this discussion we will be looking at the Axiom of Choice and some of the statements which are equivalent to it. In particular, we will be looking at Zorn's Lemma and the Well-Ordering Principle as well as some applications of all three. These applications will highlight not only the utility of the Axiom but also how it tends to keep popping up in a surprisingly broad spectrum of mathematical disciplines. Additionally, we will look at how the Axiom arises in a way that makes its usage seem both inevitable and necessary. However, there are a number of remarkable (some would say "disturbing") consequences of its usage, which we will be looking at that might make you want to completely reject its veracity. This discussion will be example-heavy and should quite accessible to an undergraduate audience.
Folding Graphs into Polyhedra
Abstract: In the branch of mathematics called graph theory, a graph is a collection of dots, called vertices, some of which are connected by line segments or curves, called edges. In this talk we will explore the question of which graphs can be folded, by gluing together various vertices, and perhaps stretching edges as needed, into three-dimensional frames of particular polyhedra. To whet your appetite, can you see how you might fold the bug-shaped graph on the left into a cube? How about the “hexastar” graph on the right? Can you also fold the hexastar graph into an octahedron? The talk will focus on finding graphs that can be folded into two different polyhedra.
Spring 2011April 20
A Non-Least-Squares Method For Estimation of Broken-line Regression to Determine Plateau Time of Outputs In Some Animal Science Studies
Abstract: Broken-line regression has been used to estimate parameters in increasing or decreasing linear trends that later plateau. Most approaches employ least squares methods that under usual assumptions lead to maximum likelihood estimates (Gill, 2004). In Hoffman, Knofczynski and Clark (2010) an estimation procedure called MMNPR is derived and simulation studies show that parameter estimates of the broken-line regression perform better than the MLEs in small sample situations when data is from distributions where error terms are ill-behaved. A review of applied science literature revealed many studies using broken-line regression. We note three animal studies that report parameters based on least squares approaches: 1) determining times of steady milk production in cows (Sahinler, 2009), 2) transport of urea in mice (Marini, Lee and Garlick, 2006) and 3) the maximum sustained effect of dietary copper on chicks (Robbins, Norton and Baker, 1979). We investigate the appropriateness of underlying assumptions concerning correlation and homoscedacity. We then revisit the analysis of the animal data using the MMPNR technique, discussing differences between parameter estimates based on least squares and those based on the MMNPR technique.
Generating Lattice Polygons With Specified Number of Interior Points
Abstract: How many unique lattice polygons are there with three interior points? The answer is... God knows! Finding the number of nonisomorphic two-dimensional convex lattice polygons with a specified number of interior points i is surprisingly hard: the number of inequivalent polygons with i ≥ 2 is still an open problem. We propose a new way to construct a complete set of lattice inequivalent polygons for any i, based on the relative minimality or maximality of polygons instead of the number of sides, and demonstrate it for i = 1 and i = 2. The algorithm presented allows insight into which polygons can be generated from which other polygons, inducing a directed graph structure into any set of inequivalent polygons for a value of i. Some combinatorial properties of these graphs will also be discussed. Finally, we discuss applications to cryptography, the theory of modular forms, and several problems in computational geometry.
Sungkon Chang and others
Reflections on the 71th Annual Putnam Mathematical Competition
Abstract: In December 4, 2010, 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.
Four-Cycle Frames of the Complete Multipartite Graph
Abstract: PDF version
February 9 (postponed from January 26)
Exploring Rational Residue Graphs
Abstract: During the early 1960's, Horst Sachs introduced Paley graphs to the world of mathematics. Paley graphs tie together graph theory and number theory. We shall explore the symmetry of these rational residue graphs alongside the calculation of the number of triangles and the number of edge-disjoint hamiltonian cycles.
Fall 2010November 10
Abstract: Krull dimension is a notion of central importance in commutative algebra. One can think of Krull dimension as a measure of "how high" one can continue to stack up prime ideals in a ring. In this sense, we get an idea as to the size of the ring. In Dr. Trentham's doctoral thesis, he presented a generalization of the idea of Krull dimension. This generalization allows us to assign a unique cardinal number to every commutative ring with identity. Thus, in the infinite dimensional case, we are able to say what the Krull dimension of a ring actually is instead of being cornered into referring to what it is not, i.e., finite. In this discussion we will consider Krull dimension in both its traditional sense as well as this generalization. We will also perform some computations using this generalization.
Tricia Muldoon Brown
The Partial n-Queens Problem
Abstract: The n-queens problem asks: Given a positive integer n, can n non-attacking queens be placed on an n x n chessboard? This problem and and its extensions have been studied by mathematicians for over 200 years. In this talk we discuss the history and several of the variations of the n-queens problem. We focus on the problem of finding a maximum solution given a partial arrangement of k non-attacking queens and demonstrate a computer algorithm which solves the partial problem for small n.
The Algebra of Grand Unified Theory
Abstract: The Standard Model representation in particle physics describes the three forces: the strong force, the weak force, and the electromagnetic force via the actions of the Lie algebras of Lie groups. A grand unified theory (GUT) is an attempt to describe the three forces via the action of the Lie algebra of a simple Lie group, thereby offering an elegant unified mathematical description of the action of the three forces. In this talk, the mathematics behind SU(5)-GUT will be introduced. Although SU(5)-GUT is no longer accepted in physics, it is an excellent prototype of GUTs, from which one can see how modern algebra plays a role in some fundamental part of physics.
The Power of Proofs
Abstract: In this talk, we will see why we need mathematical proofs, and then discuss various proofs for one of the most popular theorems of all time. A bonus "proof without words" for another well-known theorem will also be included.
Building Dynamic Demonstrations with Mathematica
Abstract: With Mathematica, a remarkably small amount of code can result in a small interactive application, or "demonstration," with a sophisticated user interface. This talk will focus on the basic ideas involved in building demonstrations, using a few elementary classroom topics as examples.
Spring 2010April 21
Sungkon Chang, William Nathan Hack, Selwyn Hollis, Scott King, Tim McMillan
Reflections on the 70th Annual Putnam Mathematical Competition
Abstract: On December 5, 2009, an intrepid team of four 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.
Dr. Joshua K. Lambert
Abstract: The squares of an m by n checkerboard are alternately colored black and red. Two squares are called neighbors if they belong to the same row or the same column and there is no square between them. It is conjectured that we can place coins on some of the red squares such that every black square neighbors an odd number of coins. We shall discuss how this checkerboard problem relates to the newly conceived concept in graph theory of modular chromatic numbers.
Moshe Cohen, Louisiana State University
Domino Tilings, Perfect Matchings on Graphs, and the Alexander Polynomial of a Knot
Abstract: The goal of this talk is to investigate how well-understood problems in combinatorics interact with a polynomial from knot theory. Combinatorics (the art of counting) asks questions like "How many ways can we cover a checkerboard with dominoes?". A knot is a circle embedded in three dimensional space. Knot theory asks "How can we tell two knots apart?". The Alexander polynomial is one example of a knot invariant (that is, if the Alexander polynomials of two knots are not the same, the knots must be different). This polynomial is the determinant of a matrix, and we'll construct this matrix using techniques from combinatorics.
Dr. Sabrina Hessinger
Differential Galois Theory: An Area for Mathematicians Loving the Pure and the Applied!
Abstract: In this talk we will introduce the basic constructs and ideas behind the field of Differential Algebra with a specific focus on Differential Galois Theory. This relatively young field of mathematics was formally developed by E.R. Kolchin et.al. in the mid 1970s. It involves differential equations, abstract algebra, computer algebra, linear algebra and topology. We’ll explore the various avenues of inquiry in differential Galios theory and discuss example results.
Dr. Jared Schlieper
Free, As In Beer
Abstract: With all the talk of budget cuts, "free" has such a nice ring to it. In fact, there are many free open source programs that we can use to enhance our teaching. We will discuss using various open source mathematics software available and spend more time focused on two programs in particular: SAGE and WeBWorK. We will give a brief overview of each one along with classroom experiences from the past couple semesters.
Dr. Patricia Brown
How Deep Is Your Playbook?
Abstract: Math Awareness Month is coming up in April and this year's theme is Math & Sports. In order to support MAM, this talk will include algebra, combinatorics, and football. We will discuss 4 basic defensive formations utilized in the National Football League and consider how many options a coach has when calling his play. No advanced knowledge of math (or football!) is necessary for this talk.
Dr. Mark Budden with Scott King and Alex Moisant
Permutations of Rational Residues II
Abstract: Reciprocity laws in number theory relate the residue symbols of distinct primes with one another. Mathematicians' attempts to extend such laws have guided the direction of algebraic number theory for hundreds of years and their results have implications throughout mathematics. In this talk, we will provide an overview of the natural setting for proving rational reciprocity laws and will explain how an extension of Zolotarev's 1872 proof of the Law of Quadratic Reciprocity may be generalized to proving more recent generalized rational laws.
Dr. Jared Schlieper
Abstract: Dr. Schlieper will take a brief excursion into financial math from the simple (interest) to the random. Financial mathematics has many topics that everyone will encounter at some point in their life (e.g., auto loans or mortgages). The area also includes stochastic models of interest rates and financial derivatives. Dr. Schlieper will go over a few examples to give an idea of the mathematics and statistics involved. (Disclaimer: The presenter takes no responsibility for your financial losses after the talk, but expects to be compensated for your gains.)
Dr. Sungkon Chang
The Equal Circle Packing Problem
Abstract: For a closed convex region S in the Euclidean plane and a positive integer k, the equal circle packing problem is to find the largest radius a for which k open disks of radius a can be inscribed in S such that the disks do not intersect each other. This problem, which was introduced in the 1960's, is an interesting NP-hard optimization problem. With the aid of computer technology, the literature on computational results and algorithmic developments has been recently rich and active. This talk will introduce the theoretical aspects of the problem and will also introduce results for the six circle case. All students are invited, and especially those who are interested in research experience are encouraged to come and learn about the opportunity.
Dr. Sean Eastman
Using Numerical Analysis to Re-Visit Calculus
Abstract: A standard undergraduate course in numerical methods assumes that students are very familiar with a number of theoretical ideas from calculus, such as the Intermediate Value Theorem, the Mean Value Theorem, and Taylor's Theorem. All too often, these ideas get short shrift in beginning calculus, as students generally tend to focus most of their effort on learning techniques of symbol manipulation. This talk will give an overview of a new approach to teaching numerical analysis that utilizes a constructive approach to mathematics, which allows the student to take a new look at these calculus ideas in the context of algorithm construction. The talk will also include a constructive proof of the Mean Value Theorem.
Dr. Sungkon Chang and William Nathan Hack
Maximizing the Minimum Mutual Distance (Preliminary Report)
Abstract: In this talk we shall introduce the problem of maximizing the minimum mutual distance. This problem is in fact in the heart of coding theory, but its obvious analogue to Euclidean space is also very interesting. We began to investigate some cases, and the main part of the talk will be a preliminary report on our investigation.
The Armstrong Putnam Team
Reflections on the 69th Annual Putnam Mathematical Competition
Abstract: On December 6, 2008, 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 will discuss solutions of their favorite problems from the most recent competition.
Dr. Jared Schlieper, Mathematics
An Introduction to Convex Bodies
Abstract: If we slice the cube through its center, in which direction will the slice have greatest area? We will begin by answering the classic problem of slicing a cube in R3 and then move on to Rn. We then see how the problem led to the recent use of the Fourier transform in solving some classic problems with volumes of convex bodies. Finally, we will examine some recent research efforts related to the cube slicing problem.
Jeanette Olli, University of North Carolina at Chapel Hill
An Introduction to Dynamical Systems
Abstract: There are many things that are unpredictable that people try to predict, such as the weather. Dynamical systems involve studying a system's long term behavior. After providing a definition of what a dynamical system is, we will look at several examples of dynamical systems and properties of them that we can study. One type of dynamical system is a substitution system, which is defined by a particular substitution in either one or two dimensions. A 2-dimensional example of this is a tiling system, which is a covering of the plane by particular tiles that fit together with no overlap. We'll also look at several examples of those and some of their properties.
Hunting the Hidden Dimension: A NOVA film about fractals.
Abstract: You may not know it, but fractals, like the air you breathe, are all around you. Their irregular, repeating shapes are found in cloud formations and tree limbs, in stalks of broccoli and craggy mountain ranges, even in the rhythm of the human heart. In this film, NOVA takes viewers on a fascinating quest with a group of maverick mathematicians determined to decipher the rules that govern fractal geometry.
Dr. Sungkon Chang
The Birch and Swinnerton Dyer Conjecture and Quadratic Twists of an Elliptic Curve.
Abstract: Solving an equation is a fundamental problem in mathematics, and in number theory, solving a polynomial equation of two variables for rational solutions is a well-known difficulty problem. In this presentation Dr. Chang will introduce the basic theory of cubic equations of two variables known as elliptic curves, and present his research results in this area.
The immensity of the arithmetic of elliptic curves was revealed to the mathematics community when the proof of Fermat's Last Theorem was completed in 1995 by Sir Andrew Wiles et al proving a conjecture about elliptic curves, called the Taniyama-Shimura-Weil Conjecture. One of the most prominent problems to solve in the theory of elliptic curves today is the Birch-and-Swinnerton-Dyer Conjecture. This conjecture asserts that some analytic complex-valued generating function L(s), called an L-function, associated with an elliptic curve reveals a pack of arithmetic information about the elliptic curve in its Taylor expansion at s=1. Many number theorists in this area are interested in proving a probabilistic implication of this conjecture on a certain family of elliptic curves, called the quadratic twists of elliptic curves, and Dr. Chang will introduce the literature and his work in this area.
Dr. Paul Hadavas
Operations Research: The Time of Your Life
Abstract: Operations Research(OR) has officially been a field of study in mathematics for almost 60 years. But what is it and how does it affect your daily life? In this talk, we'll dig a little deeper into Operations Research, recently dubbed "the science of better", and look at four examples from every day life where OR techniques can be applied. These examples include:
- brewing beer
- electing a president
- getting home the quickest
- spending $700 billion in the most efficient way
In addition, we'll examine how different mathematical formulations for these problems can lead to optimal solutions within seconds instead of hours.
Dr. Selwyn Hollis
Turing Instability and the Leopard's Spots
Abstract: In a 1952 paper, The Chemical Basis of Morphogenesis, Alan Turing explained that spatial patterns of chemical concentration can be generated by simultaneous reaction and diffusion processes, suggesting that this behavior may account for the development of some animal pigmentation patterns such as a leopard's spots. In this talk, Dr Hollis will present an introduction to reaction-diffusion equations and outline the mathematical basis for Turing's theory of pattern formation, which has become known as Turing (or diffusion-driven) instability. Several Mathematica-generated animations will provide illustration.
Dr. Jim Brawner
Playing with Polyhedra
Abstract: What do Plato, Archimedes, Johannes Kepler, and Norman Johnson have in common? They each have a class of polyhedra named after them. We will survey a variety of polyhedra, some better known than others, and explore some interesting properties and relationships among them.