Benford's Law

Students: Kelly K. Horan, Ryan M. Galgon, Jonathan R. Bradley, Dan P. Pike
Adviser: Dr. David Farnsworth

Benford’s Law says that many naturally occurring sets of observations follow a certain logarithmic law. Relative frequencies of the first significant digits k are log(1 + 1/k) for k = 1, 2, ..., 9, where the base of the logarithm is ten. Financial and other auditors routinely check data sets against this law in order to investigate for fraud.

A Network Theoretic Approach to Hyperspectral Image Classification
Student: Ryan Lewis (2008-9)
Adviser: Dr. Anthony Harkin

Abstract: A hyperpectral image has n pixels with k>100 spectral bands. Hyperspectral imaging has a variety of applications, for exampe: geological research, welands mapping, and plant and mineral identification. We are working on a novel technique to classify the pixels of a hyperspectral image into spectrally similar groups. Our method represents the image data as a subset of R^k, and is based on Newman's Method of Optimal Modularity in Social Networks.

Minimal k-Rankings of Prism Graphs
Student: Andrew Zemke (2008-9)
Adviser: Dr. Darren A. Narayan

Abstract: A k-ranking of a graph is an assignment of integers 1,2,...,k such that for every pair of vertices with the same rank there is a vertex of larger rank. The smallest k for which G has a k-ranking is called the rank number of G. A minimal ranking is a ranking where the reduction of any label violates the ranking property. We investigate the rank number for prism graphs.

Evolutionary Environmental Graph Theory
Student: Gregory Puleo (2008-9)

Abstract: We consider a simple spatial model of competition between two species. The environment is represented by a graph with red and blue vertices, which offer different levels of reproductive fitness to the two species. In general, the process appears to be difficult to analyze. However, in the case where the coloring of the vertices is a "proper" two-coloring, we show that the graphs are fair: neither species has an overall advantage.

Abstract: Einstein's theory of general relativity has radically altered the way in which we perceive the universe. Einstein's breakthrough was to realize that the fabric of space is deformable in the presence of mass, and that space and time are linked into a continuum. Much evidence has been gathered in support of general relativity over the decades. Some of the indirect evidence for GR includes the phenomenon of gravitational lensing, the anomalous perihelion of mercury, and the gravitational redshift. One of the most striking predictions of GR, that has not yet been confirmed, is the existence of gravitational waves. The primary source of gravitational waves in the universe is thought to be produced during the merger of binary black hole systems, or by binary neutron stars. The starting point for computer simulations of black hole mergers requires highly accurate initial data for the space-time metric and for the curvature. The equations describing the initial space-time around the black hole(s) are non-linear, elliptic partial differential equations (PDE). In this work, we use a pseudo-spectral (collocation) method to calculate initial puncture data corresponding to both single and binary black hole systems.

Abstract: If an object bounces on a frictionless horizontal surface, its center of mass does not move horizontally. If we let (H_n, T_n) be the height and angular orientation of a 2-dimensional object on the n_th bounce off such a surface, we have the mapping (H_(n+1) , T_(n+1)) = F(H_n, T_n) that defines a discrete dynamical system in the plane. The dynamics of such a system can be complex. The system can have periodic trajectories and it can have chaotic regions. The details of the dynamics depend on the shape of the bouncing object. In this project, in collaboration with Profs. Ross, Thurston, and Franklin, we developed a numerical method for solving a system of nonlinear ordinary differential equations that define shapes tailored to produce desired dynamics.

Abstract: There are many theories about the collapse of the Classic Mayan Empire. These theories include drought, depleting to the soil too quickly, war, lack of food to support the population, etc. In the summer of 2005, Hye Yon and Dr. Bill Basener adapted population-resource differential equations that have been used to model the collapse of the population on Easter Island in order to account for the two main resources the Mayans had available (maze and breadnut) . They also formulated the harvesting rate (which represents the percentage harvested to feed the people from one year to the next) as an equation instead of a constant. The standard discrete logistic equation, with the addition of the change in harvesting equation, created a model that lead to the appropriate collapsing results, which demonstrates that the collapse of a civilization may happen because of overuse of resources

Abstract: Kenn spent the summer of 2005 investigating the Theory of Distributions, the Fourier Transform, and their importance to the attempt to solve linear systems that describe cancerous growths in skin tissue. With the help of Dr. Joe DeLorenzo, Kenn began work toward describing the way that back-scattering can be used to characterize tumors

Abstract: Chlamydia is caused by a parasitic microorganism, known as Chlamydia trachomatis, that can only survive in a host cell. Spread through close sexual contact, Chlamydia infections are currently the most commonly reported and most rapidly growing of the sexually transmitted diseases. In 2000, all 50 states and the District of Columbia were required to report Chlamydia cases to the Centers for Disease Control for the first time. However, because most Chlamydial infections are asymptomatic, there is clearly under detection and underreporting of cases. From the data collected by the Centers for Disease Control, an SIS epidemiology model was developed to characterize the transmission of the disease. From 1990-2003, the average number of secondary infections caused by each infective was 1.040 per year, with an average infective in sufficient contact with 1.223 persons. Utilizing the computed model, a carrying capacity of 3.84% of the US population was calculated, with an estimate of 272 years until this value is reached. The purpose of creating this model is to try and predict future cases, and learn more about the inherent nature of the transmission of this disease. Several limitations of the model, such as underreporting of cases, lack of knowledge of the characteristics of the disease, and lack of sufficient data, are currently being considered for future testing. Further examination, including corrections and improvements to the model, are ongoing.

Abstract: A k-ranking of a graph is a labeling of the vertices with integers such that for any pair of vertices with the same label contains a vertex with a larger label. A k-ranking is minimal if reducing any label larger than 1 violates the described ranking property. The rank number of G is the smallest k such that G has a minimal k-ranking. Early studies involving the rank number of a graph were sparked by its numerous applications including designs for very large scale integration (VLSI) layouts and Cholesky factorizations associated with parallel processing. An interesting relation involves the rank number of a path and the solution to the Towers of Hanoi problem. For a set of disks listed in increasing size, instructions for which disk to move next can be found by reading the labels in a minimum ranking of a path. A label of i in the ranking would indicate to move the i-th smallest disk from one stack to another. A recent result by Kratochvíl and Tuza showed that the rank number of an oriented tree is bounded by one plus the rank number of its longest directed path. In the summer of 2005, Patrick and Dr. Darren Narayan proved that no such bound holds for undirected graphs. They determined new results involving rank numbers for a family of trees containing t copies of a path on t vertices.

Abstract: Both Principle Component Analysis (PCA) and Independent Component Analysis (ICA) seek to conceptualize underlying patterns or structures of observed variables. PCA and ICA can be viewed as a rotation between data spaces where, in effect, one is trying to reduce the dimensionality of the data to summarize the most important (i.e. defining) parts while simultaneously filtering out noise. Devin worked with Professor Joseph Delorenzo of the Department of Mathematics and Statistics in the summer of 2005 using PCA and ICA to speperate the Electroencephalogram (EEG) signals into mutually independent components to gain insight into cerebral activity.

Abstract: Semiconductor lasers are extremely sensitive to optical feedback, which results from undesired reflections from optical elements and detectors. A small amount of feedback is sufficient to produce chaotic instabilities. In 1980, Lang and Kobayashi formulated a model consisting of two ordinary delay differential equations for the complex electrical field and the carrier number. Delay differential equations are difficult to explore analytically or even numerically. Working with Drs. Tamas Wiandt and Vassilios Kovanis in the summer of 2005, Emma used cutting edge numerical softwares to investigate the Lang-Kobayashi equations for short external cavity.

Abstract: Nathan is a student in the Applied Mathematics program who studied the application of Topological knot theory to DNA supercoiling. The helix axis of DNA is referred to the central line running through helix of DNA. The backbone of DNA is the two boundary strands that twist around the helix axis. A DNA molecule that twists in a right handed-helix is in a relaxed state if there is one complete turn for every 10.5 base pairs. Sometimes there is an excessive number of twists in the DNA molecule. When this happens the helix begins to twist around itself in order to reduce torsional stress of the molecule, and becomes coiled up, so-called "supercoling." This conformation of DNA is said to be plectonemic or interwound. Another supercoiling can occur is for the DNA to wrap itself around a protein molecule. Nathan worked with Drs. William Basener (Mathematics) and George Thurston (Physics) during the summer of 2005.

Abstract: Problem B-3 on the 1991 William Lowell Putnam Examination asked "Does there exist a natural number L such that if m and n are integers greater than L, then an m x n rectangle may be expressed as a union of 4x6 and 5x7 rectangles, any two intersect at most along their boundaries?" It is known that all rectangles with length and width at least 34 can be partitioned into 4x6 and 5x7 rectangles. However the case involving rectangles with a dimension less than or equal to 33 is still unsolved. Rachell, Frances, Carol, and Aisosa worked with Dr. Darren Narayan during the summer of 2005 on the general problem involving any m and n and seek to determine a definitive list of which rectangles can be tiled and which rectangles cannot. This research was supported through a program of the Mathematical Association of America and is funded by the National Science Foundation, the Moody's Foundation, and the National Security Agency.