Student Talk Abstracts

Nicholas Brubaker, Stephen Carter, Millersville University of PA

Title: Double Bubble Experiments in the 3-Torus

Abstract: During the spring of 2007, Nicholas Brubaker, Stephen Carter, Sean Evans, Daniel Kravatz, Sherry Linn, Stephen Peurifoy, and Ryan Walker demonstrated the existence of at least ten distinct topological types of doubble bubbles in a 3-torus by physically constructing a soap film representation of each type in a plexiglass box. This talk describes their work and the techniques they used to create their physical representations.

Foram Dave, Penn State - Lehigh Valley

Title: Dirchlet to Neumann Operators

Abstract: We discuss the boundary value operator known as the Dirichlet to Neumann Operator. We look at the specific example of the Dirichlet problem on a half-plane, and discuss the operator under perturbations.

James Davis, Rutgers Camden

Title: Simplicity

Abstract: Using basic tools from mathematical logic a calculus of simplicity is developed. The calculus assigns integer values to axiomatic systems based on how simple they are. Emphasis will be placed on how the calculus fits into the study of axiomatic systems.

Danielle Dombrowski, Seton Hall University

Title: Component Order Edge Connectivity of the Harary Graphs

Abstract: The component order edge connectivity of a graph is the fewest number of edges that must be deleted from the graph to produce a subgraph with all components of order less than some predetermined threshold value k. No formula is known to compute this parameter for an arbitrary graph. We consider a particular class of graphs, the Harary graphs, and find a formula that will derive the component edge connectivity of any graph in this class.

Daniel B Fagerburg, West Chester University

Title: Recreational Mathematics: Modular origami

Abstract: I will explain how math can be used to create modular origami models. I will derive each of the two "families" of models I discovered. I will explain how each family has a set of equations which it is governed by.

Daniel Fillman, Ben Mizack, Moravian College

Title: The Effects of Minimum Wage on Labor Force Networks

Abstract: During our talk, we will discuss the effects of a minimum wage being implemented on a labor force. To model this phenomenon, we will use a social network model where vertices represent laborers and edges represent acquaintances between two laborers. The model will show how job offer information is passed among laborers, how this transfer of information changes once a minimum wage takes effect, and what the limiting behavior of the model will be.

Joanna Gillen, Seton Hall University

Title: Spanning Trees of Bipartite Graphs That Contains Specified Edges

Abstract: In this talk we develop formulas for the number of spanning trees of the bipartite graph K_r,r+1 that contain a specified two-edge forest. These formulas are useful in answering a question about spanning tree edge densities and dependencies.

Michael Governale, Kutztown University

Title: Math Programming takes the fun out of SUDOKU Puzzles

Abstract: Although Sudoku puzzles first appeared in 1979, wide-spread interest in these puzzles did not take off until 2005. Since then these puzzles, which boast "No math required!", appear almost everywhere from newspapers, books, on web sites, even on cell phones. A Sudoku puzzle consists of a 9 by 9 grid with some of the 81 boxes containing the digits 1 through 9. To solve a Sudoku puzzle requires filling in the blank boxes so that each of the nine rows, each of the nine columns, and each of the nine 3 by 3 sections all contain every digit from 1 to 9. In this paper we show how mathematics can be used to solve Sudoku puzzles. Specifically, we present two distinct mathematical programming formulations, both of which can be used to solve Sudoku puzzles. Also, the coding of these formulations in the software optimization package LINDO, using the modeling language LINGO, will be discussed. We will illustrate the efficiency and effectiveness of these formulations by solving a variety of Sudoku puzzles ranging from easy to hard to challenging in at most a few seconds on a 2 GHz PC. Finally, we will use the two math formulations to solve the infamous "Qassim Hamza" Sudoku puzzle.

Rebecca Greenwood, Elizabethtown College

Title: Cryptanalysis of Classical Ciphers Applied to Sound

Abstract: Classical ciphers have historically been applied to written text, and the corresponding cryptanalytic techniques developed over the centuries take advantage of the statistical properties of text. However, applying those same ciphers to digital sound files requires corresponding changes in cryptanalysis. I will discuss new methods and provide examples of the decryption of classically enciphered sound files without knowing the secret key.

Susanna Holloway, Seton Hall University

Title: Realizing Degree Sequences for Multigraphs

Abstract: Previously, characterizations of multigraphs having degree sequences (0, 1, 2,�, n-1), (0,1,2,�,n-2,n), (1, 2,�, n) and (1,2,�,n-1,n+1), all of which are not realizable for simple graphs, were investigated, and algorithms were developed to realize certain criteria arranged in order of priority. In this talk, we present algorithms for constructing a multigraph with a different ordering of these criteria, namely, 1. The fewest components; 2. The smallest maximum multiplicity of any multiedge; 3. The fewest number of places in the multigraph where multiple edges occur.

Emily Ibanez, Penn State University, Harrisburg

Title: Probability: Insurance's Root

Abstract: I will be speaking on the history of probability that led to the creation of the field of insurance, the fundamental usages of risk reduction as relating to commerce as far back as 3000 BC and I will expand the beginnings of formal probability beyond Fermat and Pascal to Pacioli and Cardano. Basic probability concepts will be briefly discussed (i.e. Law of Large Numbers, Bayes Rule, ...). Finally, the insurance principle of reserve requirements and reinsurance will also be covered.

Rachel Irby, Elizabethtown College

Title: Time Discretization of Markov Chains

Abstract: Although many real life processes are more conducive to be modeled using continuous-time Markov chains, it would be much simpler to model with a discrete-time chain. Therefore, we propose a method for constructing, for a given continuous-time Markov chain, an approximating discrete-time Markov chain. The order of approximation is studied. Stationary and limiting distributions are also addressed.

Antonio Jimenez, Penn State - Lehigh Valley

Title: A Tribute to Katherine Okikiolu, Spectral Zeta Functions and

Abstract: We look at K. Okikiolu's contributions to the study of geometry. In particular, we look at the Laplacian operator and its eigenvalues and associated spectral zeta function. We shall also mention applications including geometric properties and physics.

Timothy Mills, Moravian College

Title: Spheriosity - A (partial) Computer Emulation of Spherical Geometry

Abstract: Ever wished you had some better computer tools to work with spherical geometry? Well I sure did, and thus Spheriosity was born as a small project for my Higher Geometry class. You will hear about some of the mathematical challenges I had as well as see a demo of what I have done so far on the application.

James Mulligan, Penn State Harrisburg

Title: The Riemann Hypothesis

Abstract: Ever since Euclid's proof of the infinitude of primes, mathematicians have wondered about the distribution of the prime numbers. An approximate answer was obtained in the form of the prime number theorem. But is there a more complete picture of this distribution? I will discuss the Riemann Hypothesis, which states that the non-trivial zeros of the zeta function have real part one-half. I will also discuss the connection between these non-trivial zeros and the prime numbers. If the hypothesis should be proven, it will lead to another way of thinking about the distribution of the prime numbers.

Rachael Todd, Moravian College

Title: Extending Evan's Conjecture to Latin Cubes

Abstract: A latin square is an n ×n array in which every row and column contains n distinct symbols, each of which occur exactly once. Any partial latin square containing n-1 elements is able to be completed, according to Evan's Conjecture. Is this same conjecture true for latin cubes? This talk focuses on constructions which can be used to complete partial latin cubes originally containing n-1 elements.

Veronica Tripaldi, Abby Barrett, Alex Liobis, The University of Scranton

Title: The Lucas-Lehmer Test: Pushing the Primes to New Limits

Abstract: For centuries, the prime numbers have been one of fascination for mathematicians. In fact, Civilizations as far back as the ancient Greeks were aware that every integer greater than one could be written uniquely as the product of primes. This talk will present the background and mathematics of the Lucas- Lehmer Test, a convenient algorithm for testing the primality of large Mersenne numbers. It will also include a brief portrait of the mathematicians who made it possible and a discussion on how the test is used in the Great Internet Mersenne Prime Search.

Geometric Construction Workshops Abstracts

George W. Hart will be presenting the following workshops today. Since space is limited, we ask that if you are interested you sign up for these workshops at the registration desk.

"Slide-Together" Geometric Paper Constructions

11:20-12:00, room 302

Abstract: This activity consists of several attractive constructions which are fun and relatively easy to make because one simply cuts out paper pieces and slides them together. A number of mathematical skills are developed, concerning geometric structure, coloring patterns, and concrete and mental visualization. I have found these to be good classroom activities for middle-school, high school, and college students. Furthermore, as team-building projects, these work well if assembled in groups of two or three students. That encourages collaboration and mathematical communication.

Each "slide-together" is made from identical copies of a single type of regular polygon (e.g., just squares or just triangles) with slits cut at the proper locations. I make them from colored card stock, simply photocopying the templates onto the sheets. In most cases, glue or tape is not needed if you use a stiff stock. But you might want to use a small dot of glue at the corners or bit of scotch tape on the interior to fasten the components together. Having the corners meet crisply is the key to producing a neat geometric impression.

Information about this project can be found at: http://www.georgehart.com/slide-togethers/slide-togethers.html

Paper Polylinks

1:40-2:20, room 302

Abstract: Instructions are given for making paper examples of "regular polylinks" or "orderly tangles." These are models of forms first presented briefly in a 1971 book by Alan Holden. He then developed the idea into a book-length exposition in 1983. I have explored his idea further and have written software that can be used to generate many related forms. The equipment required is card stock (i.e., heavy paper), a copier or computer printer to print the templates, scissors to cut them out, and a small amount of clear tape to assemble the parts. I recommend proceeding in sequence from simplest to hardest: (A) Four Triangles (related to a tetrahedron); (B) Six Squares (related to a cube); (C) Six Pentagons (related to a dodecahedron).

I have used these as individual and group activities in a classroom. They develop geometric ideas, visualization skills, and mathematical communication. But be warned: these constructions are trickier puzzles than they might first appear.

Information about this project can be found at http://www.georgehart.com/orderly-tangles-workshop/paper-polylinks.html