Parallel session abstracts

Brittany Ambeau, Rose-Hulman Institute of Technology and Rochester Institute of Technology
Harris Enniss, Rose-Hulman Institute of Technology and Harvey Mudd College
Stefan Schnake, Rose-Hulman Institute of Technology and Murray State University
Nondestructive electrothermal detection of corrosion 

Nondestructive testing and imaging plays an important role in many industries, e.g., the monitoring and maintenance of aircraft. The general technique is to input energy in some form into an object, observe the object's response, and from this input-output information determine internal structure. New techniques are always being explored, and recently there has been much interest in methods that use multiple forms of energy. In this vein, we examine a new technique for imaging corrosion or material loss in an object by combining electrical and thermal measurements on some accessible portion of the object's outer boundary. The flow of electrical and thermal energy through the object is modeled using certain partial differential equations, and imaging the corrosion leads to a mathematical``inverse problem'' We examine the limits and stability of this type of imaging, and develop an effective numerical algorithm for solving certain of these types of problems.

Hanna Astephan, Indiana Univerity and University of Michigan
Negatively curved surfaces in R^3

We explore Vaigant's surface, the only known negatively curved surface with four cuspidal ends. Building on his work, we hope to modify this surface in a variety of ways. The first modification entails adding a genus to a surface and exploring the topological ramifications. The second modification transforms Vaigant's surface to a six-ended surface. We are investigating ways to preserve negative curvature after these modifications. As a part of this, we are looking at the indices of planar slice curves and relating them to the Euler characteristics of these new surfaces. We are also considering gluing methods to replace non-positively curved regions.

Clark Butler, Indiana University and Ohio State University
Negatively curved slab surfaces 

There are very few known examples of complete surfaces of negative curvature which are embedded in a slab in Euclidean 3-space, the region between two parallel planes. All of these examples are homeomorphic to a plane or an annulus, and have no more than two ends. We construct new examples of these topological types and analyze whether or not these slab surfaces can possess genus or more than two ends, using the behavior of the Gauss map at infinity as well as the line fields of principle and asymptotic directions.

JingJing Chen, Rose-Hulman Institute of Technology and Pomona College
Mark Lotts, Rose-Hulman Institute of Technology and Randolph-Macon College
Structure and randomness of the discrete Lambert map 

We investigate the structure and cryptographic applications of the Discrete Lambert Map (DLM), the mapping x -> xg^x mod p, for p a prime and some fixed g in (Z/pZ)*. The mapping is closely related to the Discrete Log Problem (DLP), but has received far less attention since it is considered to be a more complicated map that is likely even harder to invert. However, this mapping is quite important because it underlies the security of the ElGamal Digital Signature Algorithm. Using functional graphs induced by this mapping, we were able to find non-random properties that could potentially be used to exploit the ElGamal DSA.

Michael Dairyko, Valparaiso University and Pomona College
Samantha Tyner, Valparaiso University and Augustana College
Casey Wynn, Valparaiso University and Hendrix College

In this talk we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by modifying a known algorithm that counts binary trees avoiding a single contiguous tree pattern. Next, we use this algorithm to prove several theorems about the generating function whose nth coefficient gives the number of n-leaf trees avoiding a pattern. In addition, we investigate and structurally explain the recurrences that arise from these generating functions. Finally, we examine the enumeration of binary trees avoiding multiple tree patterns.

Lisa Dion, Valparaiso University and Providence College
Jeremy Jank, Valparaiso University and Concordia University -- NE
Nicole Rutt, Valparaiso University and Concordia College -- Moorhead, MN 
Computer monitored problem solving dialogues 

This project``looks over the shoulder'' at students collaboratively engaged in a math problem-solving activity. One task we looked at was mechanically classifying the students' current activity or knowledge state, of which we have identified 15 different categories. We have produced an automatic classifier that examines student sentences and is 55% accurate in identifying utterances as containing certain bits of knowledge or evidence of certain activities. The classifier was built from a corpus of student-written reports. Treating each sentence as a bag of words, we built vector space models of the word co-occurrence matrix using both non-negative matrix factorization (NMF) and latent semantic analysis (LSA). Classification was achieved by comparing new, unknown, sentences with pre-built bundles of manually tagged sentences, one bundle for each classification. Our categories are specific to the problem being solved, particular bits of knowledge needed to understand a two-person game called Poison. We have also been characterizing the dialogues with problem-independent categories: a math collaborative dimension and a problem-solving dimension. This will enable us to classify utterances with regard to in what ways students are participating in the dialogue and the problem-solving process. The context of this work is a quantitative problem-solving course in which students work in small groups. Our goal is for the computer to notice some of the same aspects of the activity that a teacher walking around the classroom might observe, such as what realizations a group has achieved and how students are collaborating. This type of computer-mediated collaborative problem solving exposes student thinking, providing opportunities to gain insights about student learning.

Nick Edelen, Indiana University and University of Edinburgh
Characterizing helicoidal surfaces of constant mean curvature 

We explore a kinetic condition for constant mean curvature due to Perdomo, who uses a rolling construction on the generating curve to characterize CMC helicoidal surfaces. We give a more direct proof through parameterizing by angle of normal, and provide a conservation interpretation of the condition.

Christopher J. Evans, Rose-Hulman Institute of Technology and University of Arkansas
The elliptic curve discrete logarithm and functional graphs

The Elliptic Curve Discrete Logarithm Problem (ECDLP) is an open question in number theory with important applications to cryptography and computer security. We will focus on a graph-theoretic approach in an attempt to find underlying structure in the ECDLP. Any indications of underlying structure could suggest a method of quickly solving the ECDLP, and undermine the security of certain cryptographic systems. In real-world applications, this could allow the theft of personal or corporate information, along with significant financial losses.

Jordan Fitch, Rose-Hulman Institute of Technology and Centre College
Bread foams and periodic structures 

The shape of a slice of French bread is examined with particular regard to the inner structure of gaseous cells, which can be modeled as a two-dimensional foam under gravity. Using the calculus of variations, the total energy of the cell wall configuration can be minimized; this total energy is comprised of the surface energy and the gravitational potential energy. After establishing a graph structure for small configurations of cells, or bread foams, the process of evolution to a minimized, optimal state is explored. A catalog of small bread foams suggests that the behavior of larger, more realistic bread foams could be generalized into a periodic hexagonal structure when observed at its fully-evolved, optimal energy configuration.

Ewain Gwynne, Indiana University and Northwestern University
A Quaternionic analogue of the cross ratio 

In the complex plane, a fractional linear transformation (az + b)/(cz+d) is completely determined by the images of any three points: the image of a fourth point can be found using the cross ratio of the images of the first three. By contrast, fractional linear transformations over the quaternions are not uniquely determined by their values at three or even four points. Nevertheless, we derive an analogue of the cross ratio for quaternion fractional linear transformations and give necessary and sufficient conditions for such a transformation to map four or five given points to four or five other given points. In doing so, we also provide geometric descriptions of the set of possible images of a given quaternion under a fractional linear transformation and show that many of the classical properties of the complex cross ratio also hold for our quaternionic one.

Andrew Harris, Rose-Hulman Institute of Technology
Investigating the baking process 

A mathematical model for the baking process will be presented. This model is primarily for baking a cake or bread. The model incorporates the temperature distribution, the moisture content, the vapor content, and the deformation induced during the baking process. The resulting model is a coupled system of nonlinear partial differential equations that are solved using numerical methods. We start the model with an oversimplified situation and add complexity, using basic partial differential equations (heat/diffusion equation), some thermodynamics (evaporation/enthalpy/pressure), numerical methods, and some continuum mechanics.

Chaim Hodges, Goshen College
Fair division with budget constraints 

Divisible goods are to be allocated fairly among two or more players. For example, Antonio and Beth are to share the gift of a chocolate bar. Different players value the goods differently. Perhaps Antonio thinks the chocolate bar is worth $3.00 while Beth thinks it is only worth $1.90. Players are willing to exchange money to obtain a better allocation: Antonio taking the whole chocolate bar and giving Beth $1.20 is more valuable to both than simply splitting the chocolate bar in half. Nonetheless, players have budget constraints: Antonio may be unwilling to spend more than $0.50. In this situation, what is a ``fair'' allocation? We investigate allocations that are efficient (there is no way to simultaneously improve the allocation for everyone), envy-free (no one would want to trade portions with another), or satisfy other fairness properties.

Rebecca McCarthy, Indiana University and Rose-Hulman Institute of Technology
Properties and applications of the Ising model

The Ising Model arises in the study of classical statistical mechanics. Using this model, we can determine certain physical properties of a system based on its components. Through the use of Linear Algebra techniques, the problem can be simplified for further analysis and application to other areas of physics. This talk will include information about the aforementioned techniques and applications as well as their implications about the behavior of the system at extreme temperatures.

Matthew Mizuhara, Indiana University and Bucknell University
Distance in the Farey graph

The Farey graph is defined as follows: as vertices, take the rational numbers together with the point 1/0. Given p/q and r/s in lowest terms, join them by an edge if |ps-rq| = 1. A metric can be induced on the graph so that every edge has length one, so the natural question arises in finding the distance between arbitrary vertices in the graph. We will prove a deterministic method of calculating a geodesic path in the Farey graph requiring only elementary methods.

Rachel Moger-Reischer, Indiana University and Bucknell University
Combinatorial expansion factors 

Given a hexagon $Q_0$ and a rule for subdividing the hexagon into finitely many pieces, each of which is identified with $Q_0$, we can examine the combinatorial distance $D(n)$ across the resulting complex, where $n$ is the number of times the subdivision rule has been applied. In particular, we will discuss the behavior of this distance as $n$ approaches infinity and look at the limit $\lim_{n\to\infty} (D(n))^{1/n}$.

Nathan Rehfuss, Rose-Hulman Institute of Technology and Iowa State University
Small bread foams under gravity 

A model for the interior structure of bread is a cluster of bubbles that are shaped by the force of gravity. This talk focuses on the topological structure, the legal and illegal configurations of a bread foam, and methods for describing and classifying them.

Frederick Robinson, Indiana University and Northwestern University
Maximum parsimony and phylogenetics 

We will consider the following conjecture in phylogenetics by Victor Albert, modified by Mike Steel, further modified as follows: Consider a binary tree whose nodes transmit a binary (0 or 1) signal along edges. If a node is 0 then an adjacent node is 0 with probability (1-p) but it switches to 1 with probability p. Similarly, if a node is 1 then an adjacent node is 1 with probability (1-p) but it switches to 0 with probability p. You should think of p as being small but fixed while you should think of n , the number of nodes, as growing large. Now suppose you don't know the tree structure. You only see the signal on the leaf nodes. In fact, you can see as many signal sets as you would like. Can you recover the tree simply by minimizing the number of changes that occur in the signals that you see?

Maple So, Rose-Hulman Institute of Technology and Arizona State University
Cloaking against thermal energy 

There has been a lot of recent interest in cloaking and invisibility in the mathematics and science communities, and in fact physically plausible mechanisms have been proposed (some built) for cloaking an object against detection using a variety of electromagnetic methods. The ideas are very general, however, and should allow one to design cloaks that work against other forms of imaging. In this talk we examine the possibility of cloaking an object to make it invisible to an observer using thermal energy (heat) as the imaging tool.

Ben Thompson, Rose-Hulman Institute of Technology and Cornell College
Weighted wet foams: Decorating dry foams 

A dry foam consists of thin liquid walls separating gas cells like in soap suds. A wet foam is where the liquid in the cell walls pool at the intersections of cell walls. It is well understood how to ``decorate'' the intersection of a weightless dry foam to produce a weightless wet foam in its equilibrium state. This talk will concern how to decorate weighted foams, where the weight of the liquid is not negligible, and the weight of the liquid pooling at the intersection will play a role in the equilibrium structure of the foam.

Alex Wood, Rose-Hulman Institute of Technology and DePaul University
The square discrete exponentiation map 

The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. It maps numbers modulo a prime number p. This presentation focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this talk focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p=2q+1, where q is also prime.