Mathematics

Hudson Colloquium Series - 2009 and Earlier

Fall 2009

November 11
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.

October 28
Dr. Jared Schlieper

Financial Mathematics
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.)

October 7
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.

September 2
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.

Spring 2009

April 15
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.

April 1
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.

March 11
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.

February 19
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.

February 4
Nova Films

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.

Fall 2008

November 12
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.

October 29
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.

October 8
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.

September 17
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.

Spring 2006

April 5

Elijah Allen
Prime Constellations
Consider a k-tuple of prime numbers in ascending order. Such a k-tuple is considered inadmissible if there are no other k-tuples of prime numbers that match its intervals between successive primes exactly. Thus, admissible k-tuples of primes establish a pattern that is repeatable with other k-tuples of primes. An admissible k-tuple with the smallest possible difference between the last and the first terms is defined to be a prime constellation with k terms. The prime k -tuple conjecture states that every admissible pattern for a prime constellation occurs infinitely often. This research looks into this still-open question and gives results so far.

March 22
Dr. Robert L. Taylor, Clemson University

Fun and Opportunities in Probability and Statistics
Probability and statistics problems have intrigued and puzzled people for many years. Dr. Taylor will analyze some of these problems to determine logical solutions and to illustrate facetious approaches to solutions. He will present Monty Hall's "Let's Make a Deal" puzzler as one example of illogical and logical solutions. In addition, Dr. Taylor will discuss career opportunities for students in the mathematical sciences, especially probability and statistics.

March 8
Dr. Jim Brawner, Jeremiah Eisenmenger, Duc Huynh

Reflections on the 2005 Putnam Exam
The Armstrong student team for the 2005 Putman Exam in Mathematics will present a synopsis of their experiences in taking this challenging national examination. Each of the three team members will discuss a solution for one of the problems on the examination.

March 1
Dr. Ray R. Hashemi

A Signature-Based Predictive System for Liver Cancer
Dr. Hashemi will present a hybrid predictive system that improves the prediction of liver cancer caused by a group of chemical agents. The system employs both SOM net and Hopfield net. The SOM net performs the clustering of the training set and delivers a signature for each cluster. Hopfield net treats each signature as an exemplar made up of 2,717 × 2,717 digits and then learns the exemplars. Each record of the test set is also converted into a vector of 2,717 elements and is considered a corrupted signature. The Hopfield net tries to un-corrupt the test record through several iterations using its associative memory property and then attempts to map it to one of the signatures and consequently to the prediction value associated with the mapped signature.

February 22
Amy Chambers, University of Colorado at Boulder

Cuntz Algebras
If E is a directed graph, the graph C*-algebra C*(E) is the universal C*-algebra generated by families of partial isometries and projections corresponding to the edges and vertices of the graph E satisfying certain relations that form a Cuntz-Krieger E-system. Graph C*-algebras have been much studied in the last ten years by D. Pask, A. Kumjian, and I. Raeburn and have proved useful in the general structure theory of C*-algebras. In this talk we will examine the question of the existence of a conditional expectation from the tensor product of two graph C*-algebras, C*(E1) ⊗ C*(E2 ), to the subalgebra B = span{SmSv* ⊗ Sa Sb* : m and v are paths in E1 with the same source, a and b are paths in E2 with the same source, and |m| - |v| = |a| - |b|}. Using an action of the unit circle T on C*(E1) ⊗ C*(E 2), we will show that there always exists a conditional expectation from C*(E1) ⊗ C*(E2) onto B. We will then define a directed graph e derived from the graphs E1 and E2 and examine two examples. In our first example, the conditional expectation maps O d1 ⊗ Od2 , the tensor product of two Cuntz algebras, onto B = C*(e) = Od1d2. The second example we give exhibits a case in which C*(E) does not equal B. Finally, with these two examples in mind, we will make precise the requirements necessary for C*(E) to be equal to our subalgebra B.

February 15
Dr. Charles W. Champ, Georgia Southern University

Using Multiple Characteristics In Quality Assessments -Properties of Multivariate Control Charts with Estimated Parameters.
In this presentation, Dr. Champ will discuss his and co-author L. Allison Jones-Farmer’s research into Hotelling's T², multivariate exponentially weighted moving average (MEWMA), and several multivariate cumulative sum (MCUSUM) charts. Traditionally, these types of charts track varying attributes of a product or service over time. He will present two descriptions of each chart, with estimated parameters for monitoring the mean of a vector of quality measurements. For each chart, one description explains how the chart can be applied with estimated parameters in practice and the other description is useful for analyzing the run length performance of the chart. Run lengths are important in quality control because they offer information about the expected time until a “false alarm” (i.e., a stop-the-manufacturing-line signal that is erroneous). Dr. Champ demonstrates that, if the covariance matrix is “in control”, the run length distribution of most of these charts depends only on the distributional parameters through the size of the process shift in terms of statistical distance. Simulation is used to provide performance analyses and comparisons of these charts. Dr. Champ presents an example to illustrate the MCUSUM and MEWMA charts when parameters are estimated.

February 8
Jim Brawner

The Marriage Problem
As Valentine’s Day approaches, you may be wondering about a strategy for finding the spouse of your dreams. (Then again, you may consider advice on dating from a mathematician to be about as helpful as an ethics seminar conducted by Jack Abramoff. ) In this talk, Dr. Brawner will discuss the problem of finding an optimal strategy for pairing men and women into stable marriages based on their preferences for the members of the opposite sex. In addition to offering at least one genuine piece of advice for marriage seekers, Dr. Brawner will discuss why this problem might be of particular interest to pre-med, pre-law, and economics majors.

February 1
Tim Ellis

The Complete Dummy's Guide to the Greatest Unsolved Problem in Mathematics
In 1859, Bernhard Riemann was appointed a corresponding member of the Berlin Academy, based on his 1851 doctoral dissertation and his 1857 work on abelian functions. In response to this honor, he submitted a paper entitled "On the Number of Prime Numbers Less Than a Given Quantity". In this paper, he presented an educated guess (since known as the Riemann Hypothesis), which is arguably the greatest unsolved problem in all of mathematics. The purpose of this presentation is to explore the background of the Riemann Hypothesis, to shed some light on its meaning, to delve into the history of attempts to prove or disprove it, and to describe the current prognosis of a solution. This presentation will be fully understandable by anyone possessing a passing familiarity with complex numbers and Calculus I.

January 18
Selwyn Hollis

Nuts and Bolts of Nonlinear Optimization
In the latter half of the 20th century, advances in computing technology spurred numerous scientific and mathematical fields. Among them is the field of optimization, which in its broadest sense overlaps significantly with operations research, numerical analysis, the calculus of variations, and optimal control theory. However, the field known to today's applied mathematics community as optimization is essentially a subfield of