Graduate Students

Carrie Nixon (2010)

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Cyran Marek (2010)

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Ryan Lewis (2010)

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Nicholas Battista (2010)

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Grant Dietert (2010)

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Jacqieline McClive (2010)

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Aaron Kaufer (2009)

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Gabriella Jaramillo (2009)

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Karthik Bathena (2009)

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Michael Margitus (2009)

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Aaron Kaufer (2009)

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Timothy Doster (2009)

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Edgar Amaca (2008)

Abstract: The existence of an equivalence subset of rational functions with Fibonacci numbers as coefficents and the Golden Ratio as a fixed point is proven. The proof is based on two theorems establishing basic relationships underlying the Fibonacci Sequence, Pascal's Triangle and the Golden Ratio. Equaions from the two theorems are related to each other and seen to generate the equivalence subset of rational functions. Proof by induction on these equations constitutes the proof of the existence of this subset of rational functions. It is found that this subset of rational functions possesses interesting mathematical properties, particularly that of convergence to the Golden Ratio at the limit. Further investigation shows that this subset of rational functions possesses algebraic structures that would take us into the realms of abstract algebra and complex analysis.

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HyeYon Yi (2007)

Abstract: With the increasing threat of biological warfare, and the fear of an epidemic outbreak of deadly influenza, the field of epidemic modeling is becoming increasing important in science. This work focuses on a model for the spread of disease, and the qualitative changes in that spread that occur from changing parameters. The model is linearly stable when diffusion does not exist within the population. As diffusion is incorporated, Turing instabilities occur.

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Halyna Romanyuk (2007)

Abstract: It is our goal to investigate the global character of the positive solutions of a particular rational difference equation with initial conditions. We examine the global stability, periodic nature, boundedness and monotonicity of the positive solutions.

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Anne Marino (2007)

Abstract: The so-called "Inspection Paradox" refers to the fact that in a renewal process, the length of hte interarrival period which contains a fixed time t is stochastically larger than the length of an ordinary interarrival period. In this work, we use conditioning arguments to derive explicity formulas for the distributions of the lengths of interarrival periods in the continuous time case where the renewal process is a time-homogeneous Poisson Process, and in a discrete time where the renewal process is a Bernoulli Process.

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Chris Kerbert (2007)

Abstract: We investigate the periodic character and the boundedness nature of positive solutions of max-type difference equations whose coefficients are periodic sequences of positive real numbers.

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Ben Zindle (2007)

Abstract: We use the principle of inclusion-Exclusion and rook polynomials to study certain configurations of rooks on two and three dimensional boards. Some even-odd problems and certain Lottery and mystery cases are solved using rook polynomials.

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Jeremy Nieman (2007)

Abstract: Consider a cube ¡oating in space: how many unique ways can we color that cube with two colors uniquely? If the cube were øxed, the problem would simplify to two color choices for each face of the cube, with six faces. In other words, 2 x 2 x 2 x 2 x 2 x 2 = 26 = 64. A floating cube becomes a more di±cult problem due to the cube's geometric symmetries. We can develop a formula based on these symmetries to count the distinct colorings of a floating figure. With only a little more work, we can also obtain a generating function for the pattern inventory of the figure in question.

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Nathan Reff (2007)

Abstract: The Birkhoff-von Neumann Theorem tells us that a square matrix over the real numbers is doubly stochastic if and only if it is a convex linear combination of permutation matrices. In this work, we extended this theorem to a more general case involving hypermatrices of dimension 3. Further we investigate systems of distinct representatives (SDRs), lower bounds for hyperpermanents, applications to Latin squares and finally Birkhoff Polytopes.

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Rachel Marilley (2007)

Abstract: In the thesis we examine the dynamics of semiconductor lasers subject to optical injection. We study typical bifurcation scenarios, namely the saddle-node and Hopf bifurcations associated with this system. First, the system is investigated analytically, then numerical simulations verify our analytical conclusions.

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Deanna Olles (2006)

Abstract: In this work; three specific dynamical systems models, the Basic, Maki- Thompson, and Daley-Kendall, are used to model rumor transmission on social networks. Rumor flow is a measure of the time it takes for the rumor to completely pass through a specified network. Comparisons between random social networks and a small world social networks yield the faster transmission of a rumor over a small world network. Using unique adjacency matrices that define our random networks, observations of some characteristics of the random networks will be made that are specific to this type of graph. Differences in the constructs of the two networks will be illustrated by comparing these properties to those of the small world networks (created by a certain rewiring scheme of a k-regular network). Interesting comparisons are to be made about the networksÕ defining characteristics include average clustering coefficients, centrality measures, and average path lengths. The flow of a rumor through each type of network reveals the characteristics of the network. A rumor will clearly flow through a small world network faster than in a random network, mainly due to higher density, increased clustering, and better defined centrality.

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Mark Bellavia (2006)

Abstract: It is our goal to investigate the long term behavior of the solutions to difference equations whose coefficients are periodic sequences of positive real numbers. We will examine how the different periods of the sequence and the relationship of the terms of the sequence affect the long term behavior of the solutions.

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Amir Barghi (2006)

Abstract: By briefly introducing chromatic polynomials of finite graphs and their properties, we compute chromatic polynomials of some sequences of graphs. In addition to that, we introduce the ring of ordinomials which are utilized to define chromatic ordinomials of infinite graphs.

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Matt Ford (2006)

Abstract: Linear Programming has its origins in the 1940's, when complex planning problems needed to be solved to contribute to the wartime operations. The Diet Problem was one of those problems. My project deals with the formulation of my own diet problem tailored to my personal tastes where I minimize the cost of eating for one week while maintaining a healthy diet. I gathered data for different meals as well as nutritional information such as calorie count and number of servings. The formulation was eventually modified to introduce variety into the diet.

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Carol Panepinto (2005)

Abstract: A photoconductorÕs mobility is a measure of the speed at which electrons migrate through the material under the influence of an electric field. The mobility determines how long a packet of charge takes to go through the photoconductor. It also determines how much and in what manner the E-field changes during the packet transit. The problem in which we are interested is inferring the mobility of a photoconductor from time-of-flight measurements, that is, from measurements of the current produced per unit time by a known charge packet. Mathematically, the problem is an initial-boundary value problem for a nonlinear, non-local, hyperbolic conservation law that characterizes the E-field in the photoconductor. In this paper, we discuss the mathematical formulation of this problem, its solution using the method of characteristics, and the application of the solution to the problem of inferring mobilities.

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David Fraser (2005)

Abstract: The Necklace Illustrator Java Applet, the necklace problem is a well known problem from combinatorics and a variety of problems can be formulated as necklace counting problems. A Java Applet is presented which finds the number of unique necklaces. It does so by computing the cycle index and pattern inventory for the given input. It also draws each of these necklaces by comparing their string representations considering rotations and reflections.

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Gillian Gallle (2005)

Abstract: A ranking is when the vertices of a graph, G, are labeled with numbers such that every path between any two vertices labeled with the same number contains a vertex labeled with a higher number. A ranking is called minimal if decreasing any label greater than 1 violates the above ranking property. The arank number of a graph, denoted Pk(G), is defined to be the largest k such that G has a minimal k-ranking. While much work has been done concerning the arank of stars and paths, this paper explores arank trends over all the nonisomorphic graphs on 6 vertices or less paying special attention to furthering the work of Ghoshal, Laskar and Pillone. These results are followed by an investigation of the potential correlation of reduction classes to the arank number. Finally, the arank number of a specific family of trees is examined.

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Olin Stratton (2005)

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Chris Cappon (2005)

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Kevin Gonzales (2005)

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Joseph Rhoads (2005)

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