Thesis Abstracts

Regression With Missing Data:  An Investigation of the Case with Uniform Predictors and Missingness Related to the Response Variable

Jack T. Ervasti

Missing data is a very important problem in many fields, including the social, behavioral and medicinal sciences. As a result, a number of techniques for analyzing data sets with missing values have been developed and refined in the last few decades. There has also been a significant amount of research done on the bias introduced with different types of missing data when these techniques are performed.

In this paper, I investigate how various types of missingness affect the bias of regression parameters under imputation and complete case analysis. Using simulated data sets, I examine cases with normally and uniformly distributed predictor variables and different types of simulated missingness. I find that uniformly distributed predictors cause bias under different circumstances than normally distributed predictors when missing values are imputed. In particular, I find that if the predictors are uniformly distributed, regression parameters are biased when missingness is related to the response variable and are approximately unbiased when missingness is related to missing values. These results indicate a lack of investigation into missing data with uniformly distributed variables and missingness that is conditional on the response variable. Based on these findings I perform an experiment to gain a deeper understanding of the relationship between types of missingness and the bias of regression parameters in the case with uniform predictor variables. 

A Trajectory Smoothing and Clustering Method for the Identification of Potent shRNAs

Alexander H. Greaves-Tunnell

RNA interference (RNAi) is a potent and specific mechanism of gene silencing with extensive applications to research, biotechnology, and medicine. Recently, there has been considerable interest in short hairpin RNAs (shRNAs) as triggers for “programmable” RNAi, due in part to the fact that they enable stable and heritable gene silencing. However, the experimental identification of potent shRNAs is costly and inefficient, and prediction of potent shRNAs for novel targets remains a major challenge. In this paper, we introduce a smoothing and clustering method for data collected from the Sensor assay, the first massively parallel biological procedure for the identification of potent shRNAs. This method is based on a novel treatment of the data as fundamentally longitudinal in nature. We identify a set of roughly 300 top performing shRNAs for the given targets, and conduct preliminary validation based on three sequence and thermodynamic features of known potent shRNAs.S

Benford’s Law and Stick Fragmentation

Joy Jing

Many datasets and real-life functions exhibit a leading digit bias, where the first digit base 10 of a number equals 1 not 11% of the time as we would expect if all digits were equally likely, but closer to 30% of the time. This phenomenon is known as Benford’s Law, and has applications ranging from the detection of tax fraud to analyzing the Fibonacci sequence. It is especially applicable in today’s world of ‘Big Data’ and can be used for fraud detection to test data integrity, as most people are unaware of the phenomenon.

The cardinal goal is often determining which datasets follow Benford’s Law. We know that the decomposition of a finite stick based on a reiterative cutting pattern determined by a ‘nice’ probability density function will tend toward Benford’s Law. We extend these previous results to show that this is also true when the cuts are determined by a finite set of nice probability density functions. We further conjecture that when we apply the same exact cut at every level, as long as that cut is not equal to 0.5, the distribution of lengths will still follow Benford’s Law.

Perimeter-Minimizing Tilings by Convex and Non-Convex Pentagons

Zane K. Martin

We study the presumably unnecessary convexity hypothesis in the theorem of Chung et al. on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for further research.

Clustering Time Dependent PITCHf/x Data

Christopher P. Picardo

In this paper I extend the powerful model based clustering framework to data that incorporates an entire time period, specifically single seasons from the PITCHf/x database. Traditional clustering methods are reviewed and described in detail in order to motivate the introduction of model based clustering. In order to apply model based clustering to the time indexed data, a cluster consistency algorithm is proposed that treats the cluster selection problem as equivalent a model selection problem from the supervised learning literature. Finally, the cluster consistency procedure is applied to the PITCHf/x dataset to select the appropriate number of clusters for several pitchers over an entire season. The PITCHf/x season data for two starting pitchers is then analyzed using the cluster movements for the entire season.  

Generalizing Nondeterminism for Algebraic Computation Machines

Scott Sanderson

In this thesis we present an introduction to the BSS Machine model, which serves as a generalization of the Turing Machine model of computation. Motivated by the classical equivalence of nondeterministic computation and deterministic verifiability, we develop an extension to the BSS Machine model that preserves important structural features of nondeterministic Turing Machines. We use our machines to develop a new family of relativized complexity classes, and we prove some containment relations between these and the BSS Machine generalizations of P and NP.

The Forest Through the Trees in Multilabel Classification

Benjamin Bradbury Seiler

Traditional machine learning classification algorithms are not suited for statistical classification problems in which an instance can simultaneously belong to more than one class. Such multilabel classification problems have prompted significant research in recent years including a concerted effort to bridge the gap between established classification techniques and this nonstandard framework. Based on such works as recently as Tsoumakas and Katakis [2007] and Vogrincic and Bosnic [2011], the vast majority of novel multilabel classification algorithms are compared to baseline problem transformation techniques using only support vector machines or linear models. In this study, we broaden the pool of potential base learners for problem transformation techniques and discover significant evidence to suggest the superiority of partition tree based methods in many cases, thereby, raising the bar for baseline competitiveness.

Formal Fibers of Height-n Primes and Completions of Complete Intersection Domains

Philip D. Tosteson

            Of interest in commutative algebra is the relationship between a Noetherian local ring and its completion. This thesis investigates the relationship between a complete Noetherian local ring (T,M), and Notherian local subrings R of T that have I as their completion. In particular, given an ideal  I of T and a countable collection of prime ideals C of T, we ask whether there exists a subring R, with completion T, such that (I intersect R) is prime, and the formal fiber of R at (I intersect R) has maximal elements precisely C. This question quickly relates to the construction of complete intersection domains whose completions are complete intersection rings and which have specified generic formal fiber. We study this question in several specific special cases, and further discuss progress and a method of attack on a more general case.

Dynamics, Information, and Energy of Morris-Lecar Neurons

Ji Won Ahn

We studied the Morris-Lecar model, which is a mathematical model of a motor neuron.  In particular, we studied the mutual information, metabolic energy cost, and energy efficiency of unidirectionally connected Morris-Lecar neurons, and compared our result to the work of Moujahid et al., who studied the mutual information, metabolic energy cost, and energy efficiency of unidirectionally coupled Hodgkin-Huxley neurons.

We found that unidirectionally coupled Morris-Lecar models behave differently from unidirectionally coupled Hodgkin-Huxley neurons in both information transfer and energy efficiency.  Unlike Moujahid et al., we found that among groups of one, five, ten, and twenty postsynaptic neurons, the single Morris-Lecar neuron synchronizes with the presynaptic neuron the best and is the most energy efficient.

On Multiply Recurrent and Manifold Mixing Properties on Infinite Measure Preserving Transformations

Praphruetpong Athiwaratkun

We show an example of an infinite measure preserving transformation such that it is not 2-recurrent and not power weakly mixing. This example demonstrates the striking difference between measure preserving transformations in a finite and sigma-finite measure spaces.

Totally Knotted and Semi-Free Seifert Surfaces 

Thomas N. Crawford

In 2005 Osamu Kakimizu determined the Kakimizu Complex, a simplicial complex whose vertices correspond to isotopy classes of Seifert surfaces of a given knot, for all knots with 10 crossings or fewer. We investigate a few properties the surfaces themselves. Specifically we show various combinations of semi-free and totally knotted surfaces, can be embedded in the same knot complement. We restrict ourselves to hyperbolic knots allowing us to also look at the maximal cusp diagrams of the resultant manifold.

Monkemeyer Map Analogues to Stern’s Diatomic Sequence

Noah N. Goldberg

Stern’s Diatomic Sequence is a well-studied sequence of integers which stems from continued fractions.  The Monkemeyer Map is a type of multidimensional continued fraction.  We will examine an analogue of Stern’s Diatomic Sequence for the Monkemeyer Map.

Ergodic Properties of TRIP Maps:  A Family of Multidimensional Continued Fractions

Stephanie Jensen

We study the ergodic properties of several of the most relevant TRIP maps, a family of multidimensional continued fractions that encompasses many well-known algorithms. As a first step, we show these maps converge almost everywhere.  From there, we are able to prove ergodicity.

Spaces of Planar Polygons

Brian Li

We introduce the space of convex planar polygons with different side lengths. We then consider the side lengths that produce valid linkages as well as the relation of this space to the associahedra and M_0,n.

Choose to Play:  A New Take on the Spatial Prisoner’s Dilemma

Connor McKean Stern

The Prisoner’s dilemma is one of the most important models we have to study the evolution of cooperation in a world of self-interested individuals.  Defecting is the only evolutionarily stable strategy, but from previous studies we know that in repeated games and in games with spatial effects cooperating becomes not only possible, but also preferable under certain conditions.  In this paper we explore a new repeated model of the spatial prisoner’s dilemma game where a player can select which opponents to continue interacting with.  By giving players this option we are rejecting the key condition of the repeated game that players cannot avoid interaction, yet we find that this new model shares the same underlying structure of the traditional spatial prisoner’s dilemma.

Stochastic Calculus and Applications to Mathematical Finance

Gregory White

In this paper, we review fundamental probability theory, the theory of stochastic processes, and Ito calculus.  We also study an application of Ito calculus in mathematical finance: the Black-Scholes option pricing model for the European call option. We study the development of the model and the assumptions necessary to arrive at the Black-Scholes no arbitrage rational price for a European call option.

We supplement the simple Black-Scholes model by relaxing the assumption that trading can be performed continuously in time, and studying the deviation the Black-Scholes replicating portfolio exhibits from the self-financing characteristic of the continuous-time portfolio.  We term this deviation the cumulative correction of the portfolio and explain in detail its construction.  We study the cumulative correction of Black-Scholes portfolios by performing a numerical analysis of the cumulative correction for outcomes of the stock price stochastic process. While finding a closed form probability distribution representing the cumulative correction proves difficult and we do not pursue that route in this paper, the numerical analysis indicates that the second central moment of the distribution of cumulative corrections decreases as the number of discrete time steps at which the portfolio is rebalanced increases.  Additionally, we analyze the cumulative correction required to replicate the European call option for the historical stock price data series of certain actual stocks, finding examples of a stock that would have required a positive cumulative correction and a stock that would have required a negative cumulative correction.

Spectral Theory for Matrix Orthogonal Polynomials on the Unit Circle

Liyang Zhang

In this thesis, we first introduce the classical theory of orthogonal polynomials on the unit circle and its corresponding matrix representations – the GGT representation and the CMV representation.  We briefly discuss the Sturm oscillation theory for the CMV representation.  Motivated by Schulz-Baldes’ development of Sturm oscillation theory for matrix orthogonal polynomials on the real line, we study matrix orthogonal polynomials on the unit circle.  We prove a connection between spectral properties of GGT representation with matrix entries, CMV representation with matrix entries with intersection of Lagrangian planes.  We use this connection and Bott’s theory on intersection of Lagrangian planes to develop a Sturm oscillation theory for GGT representation with matrix entries and CMV representation with matrix entries.

A Study of Hitting Times for Random Walks on Finite, Undirected Graphs

Ariel Joseph Binder

This thesis applies algebraic graph theory to random walks.  Using the concept of a graph’s fundamental matrix and the method of spectral decomposition, we derive a formula that calculates expected hitting times for discrete-time random walks on finite, undirected, strongly connected graphs.  We arrive at this formula independently of existing literature, and do so in a clearer and more explicit manner than previous works.  Additionally we apply primitive roots of unity to the calculation of expected hitting times for random walks on circulant graphs.  The thesis ends by discussing the difficulty of generalizing these results to higher moments of hitting time distributions, and using a different approach that makes use of the Catalan numbers to investigate hitting time probabilities for random walks on the integer number line.

n-Level Densities of the Low-Lying Zeroes of Quadratic Dirichlet L-Functions

Jake Levinson

The statistical distributions of zeros of L-functions can be used to study prime numbers, elliptic curves and even the ideal class groups of number fields. L-functions have been studied in connection with random matrix theory, which provides easier methods of computing these distributions.  One statistic, the n-level density of low-lying zeros for a family of L-functions, measures the distribution of zeros near the central point s = 1/2. The Density Conjecture of Katz and Sarnak states that the n-level density for an L-function family depends on a classical compact group associated to the family.  We extend previous work by Gao on the n-level densities of quadratic Dirichlet L-functions. Our main result is to confirm up to n = 6 that, for test functions of suitable support, the density is as predicted by random matrix theory.  We also consider a (conjectural) combinatorial identity for certain Fourier transforms of the test functions which, if true, would help in extending the result to all n.

Chains of Rings with Local Formal Fibers

Sean Carlos Pegado

Let R be a local (Noetherian) commutative ring with unity. If R is complete, its structure is understood; however, less is known if R is not complete, and thus the relationship between a ring and its completion is a subject of current research. To this end, previous work has begun to investigate the relationship between prime ideals of a ring and the prime ideals of its completion. We generalize these results to chains of rings that share the same completion.

Optimal Control of the Generalized Moving Point Mass Dynamic

Thuy Vinh Pham

We study the generalized time-optimal control problem where the underlying dynamic is a moving point mass under Newtonian mechanics with acceleration and velocity constraints. The optimal control of this control problem coincides with the viscosity solution of a specific partial differential equation of Hamilton-Jacobi type. Using the dynamic programming approach, we derive the associated Hamilton-Jacobi-Bellman equation and obtain its numerical solution with a semi-Lagrangian discretization scheme.

Robust Regression Boosting

Ville Satopaa

In 2010 Long and Servedio suggested that boosting algorithms that are based on convex loss functions are flawed in a sense that they cannot tolerate outliers.  Inspired by Long and Servedio’s observation, this undergraduate thesis introduces a novel regression boosting algorithm that is based on a non-convex loss function.  First, several properties of this algorithm are stated and proven.  Second, experimental evidence showing that this algorithm is highly robust in the presence of outliers is given.

Generic Formal Fibers

Philip Vu

Let T be a complete local ring.  We present necessary and sufficient conditions for which there exists a local integral domain A, a subring of T, whose completion is T with a generic formal fiber that has countably many maximal elements.  We also present results on the elements we can adjoin to this integral domain A.

Geometric Degree of 2-Bridge Knots

Jacob Wagner

In 1987, Kuiper introduced geometric degree alongside superbridge index, but degree has been studied far less than superbridge. In this thesis, we calculate degree for all 2-bridge, 4-superbridge knots. Then, we modify the definitions of degree and a related invariant, thin position, to generate new measures.

Sturm-Liouville Oscillation Theory for Differential Equations and Applications to Functional Analysis

Zhaoning Wang

We study the connection between second-order differential equations and their corresponding difference equations.  With this connection in mind, we investigate quantitative and qualitative properties of the zeros of the solutions of differential/difference equations and of the eigenvalues of the associated Jacobi matrices.  In particular, we study various applications of the Sturm-Liouville Oscillation Theory to differential equations and spectral theory.

The Limiting Spectral Measure for the Ensemble of Generalized Real Symmetric Block m-Circulant Matrices

Wentao Xiong

Given an ensemble of N x N random matrices with independent entries chosen from a nice probability distribution, a natural question is whether the empirical spectral measures of typical matrices converge to some limiting measure as N tends to infinity. It has been shown that the limiting spectral distribution for the ensemble of real symmetric matrices is a semi-circle, and that the distribution for real symmetric circulant matrices is a Gaussian. As a transition from the general real symmetric matrices to the highly structured circulant matrices, the ensemble of block m-circulant matrices with toroidal diagonals of period m exhibits an eigenvalue density as the product of a Gaussian and a certain even polynomial of degree 2m-2. This paper generalizes the m-circulant pattern and shows that the limiting spectral distribution is determined by the pattern of the i.i.d.r.v. elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged. For an arbitrary pattern, the empirical spectral measures converge to some nice probability distribution as N tends to infinity.

Evolutionary Dynamics on Weighted Edge Graphs with Structural Balance Conditions:  A Generalized Model of Social Networks

 

Christophe Dorsey-Guillaumin

We present a generalized model of social networks using a weighted-edge graph with dynamics.  Specifically, each edge in this model evolves in accordance with its membership in one or more triads, or edge triples; the stability of these triads will be defined by a dynamical interpretation of a variation of Balance Theory. We analyze this system in the single triad and general case, find several types of  fixed points in the system, and point to directions for further study.

Non-Orientable Heegaard Splittings

Andrew Scott Lee

Certain decompositions of 3-manifolds are called Heegaard splittings.  Starting from the figure eight knot, we exhibit an infinite class of hyperbolic examples in the non-orientable case derived from knot complements and describe some splittings of surface bundles over the circle.

From Doodles to Diagrams

Noel F. MacNaughton

Consider a diagram as a four-valent graph on a sphere, that is, a graph where every vertex is adjacent to exactly four edges. We look at the faces of these diagrams as being m-gons when the face has exactly m edges. We consider what types of diagrams can be drawn that have their only faces be 2-gons or n-gons for some n≥5, how many of these diagrams exist, and how many components they can have. We then consider the same information for the case when all faces are either 3-gons or n-gons for n≥5.

Modeling Convolutions of L-Functions

Ralph Elliott Morrison

A number of mathematical methods have been shown to model the zeroes of L-functions with remarkable success, including the Ratios Conjecture and Random Matrix Theory. In order to understand the structure of convolutions of families of L-functions, we investigate how well these methods model the zeros of such functions.  Our primary focus is the convolution of the L-function associated to Ramanujan’s tau function with the family of quadratic Dirichlet L-functions, for which J.B. Conrey and N.C. Snaith computed the Ratios Conjecture’s prediction. Our main result is performing the number theory calculations and verifying these predictions for the one-level density up to square-root error term. Unlike Random Matrix Theory, which only predicts the main term, the Ratios Conjecture detects the arithmetic of the family and makes detailed predictions about their dependence in the lower order terms. Interestingly, while Random Matrix Theory is frequently used to model behavior of L-functions (or at least the main terms), there has been little if any work on the analogue of convolving families of L-functions by convolving random matrix ensembles. We explore one possibility by considering Kronecker products; unfortunately, it appears that this is not the correct random matrix analogue to convolving families.

Chains of Excellent Reduced Local Rings

Bolor Turmunkh

Let (T,M) be a complete local ring with dimension at least one which contains the rationals, C a  finite set of incomparable non-maximal prime ideals of T.  We find sufficient conditions for T to be the completion of an excellent integral domain B0 with semilocal generic formal fiber ring with maximal ideals the elements of C, and excellent reduced local rings B1, B2 … Bk such that B0 is contained in B1 which is contained in B2 and etc.  We also require that B1,…Bk have semilocal formal fiber rings, whose maximal ideals we can prescribe.  In other words, we find a relatively weak sufficient conditions such that for a given complete local ring T we have an excellent integral domain B0 and a chain of excellent reduced local rings B1,…,Bk such that all of them complete to T and we have a containment as well as the properties concerning the formal fiber rings.

Semilocal Formal Fibers

Domenico Aiello

Let (T,M) be a complete local Noetherian ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p ϵ T a nonzero element.  We find necessary and sufficient conditions for T to be the completion of integral domains A and B where A ⊆ B, the generic formal fiber of A is semilocal with maximal ideals the elements of C, and pB is a height one prime ideal of B whose formal fiber is semilocal with maximal ideals the elements of C.  We also show that given a complete local ring of the form T = k[[x1, x2,…, xn]]/I, where after proper reordering of the indeterminants, I ⊆ (xk+1, xk+2,…, xn)T with k < n and given a prime ideal Q = (x1, x2,…, xl)of T, k ≤ l < n, there exists a domain A such that  = T and (x1, x2,…, xk)A is a height k prime ideal of A whose formal fiber is local with maximal ideal Q.

Unknotting Tunnels and Geodesic Heegaard Splittings of Hyperbolic 3-Manifolds

Karin Knudson

Using the geometric structure associated with the complement of a hyperbolic knot, we present several conditions that are sufficient to ensure that a given arc in the knot complement is an unknotting tunnel.  Then, we apply similar techniques to determine when a closed geodesic in a closed hyperbolic 3-manifold can be used to generate a Heegaard splitting of that manifold.

The Soap Bubble Problem on the Sphere

Edward Souder Newkirk

We consider the soap bubble problem on the sphere S2, which seeks a perimeter-minimizing partition into n regions of given areas.  For n = 4, it is conjectured that a tetrahedral partition is minimizing.  We prove that there exists a unique tetrahedral partition into given areas, and that this partition has less perimeter than any other partition dividing the sphere into the same four connected areas.

Class Number Divisibility in Quadratic Fields

Natee Pitiwan

Number fields and function fields are finite algebraic extensions of the field of rational numbers and the quotient field of polynomials over finite fields, respectively.  To each number field and function field we associate the class group and class number, which contain information on how close the ring of integers of the field is to being a unique factorization domain.  It is known that infinitely many number fields and function fields have class number divisible by a given integer.  The Reflection Theorem by Scholz shows that there are infinitely many corresponding pairs of real and imaginary quadratic number fields with class number divisible by 3.  Based on Komatsu’s generalization of this result, we prove an analogue for function fields.  Another related question on class groups is the n-rank of the group.  It has been shown that there are infinitely many quadratic number fields with 3-rank at least 2, but less is known about other n-ranks.  We consider the case of 5-rank and show a partial result on class number divisibility.

Isoperimetric Regions on a Weighted 2-Dimensional Lattice

Deividas Seferis

In this thesis we investigate isoperimetric regions in the 1st quadrant of the two-dimensional lattice, where each point is weighted by the sum of its coordinates.  We analyze the isoperimetric properties of five types of regions located in the first quadrant of the Cartesian plane:  squares, rectangles, quarter circles, diamonds, crosses and triangles.  To compute volume and perimeter of each region we use summation and integration methods which give comparable but not identical results.  Among our candidates we find that the diamond has the least perimeter for given volume.

A Compactification of the Configuration Space of Particles on a Graph

Rahul Shah

We generalize the compactification of configuration spaces and tilings from points on one-manifolds to points on graphs.  The compactification of the real moduli space, M_0n(R), is combinatorially equivalent to the compactification of the configuration space of n – 3 particles on a circle with three marked points, and this equivalence provides a tiling of [M]_0n(R) by associahedra.  We find a tiling of the compactification of the configuration space of n particles on an arbitrary graph by polytopes such as associahedra and cyclohedra.

Extensions of Extremal Graph Theory to Grids

Bret Thacher

We determine an upper and a lower bound for the number of edges that a grid graph with no rectangles can have.

On Panti’s Generalization of the n-Dimensional Minkowski Question-Mark Function

Amy Steele

A real number x is a quadratic irrational if and only if it has an eventually periodic continued fraction expansion. This property led Hermann Minkowski to construct a function that can be seen as the confrontation of regular continued fractions and the alternated dyadic system within [0,1]. The function has zero derivative almost everywhere, and is continuous and strictly increasing. In this this, we discuss the n-dimensional analogue of Minkowski’s function as defined by Giovanni Panti.

The Number of Summands in the Ostrowski Alpha Numeration

Wasin Vipismakul

One of the beautiful facts in number theory is that every natural number can be expressed uniquely as a sum of non-consecutive Fibonacci numbers. In fact, we can generalize this result to a more general sequence, and we call such sum the Ostrowski alpha-numeration, where alphia is a root of some quadratic polynomial. A natural question to ask is “How many non-zero terms, in average, are required in the sum?” We will define what it means to be “average” and compute it for some class of alpha.

On Equivalence Relations on Sequence Spaces

Paul Alexander Woodard

Given a sequence space S, we can define an equivalence relation ~_x on S by (x_n ~_x(y_n) for (x_n),(y_n) in S if and only if (y_n-x_n) is in X, where X is a subspace of S, such as 1₁, the space of absolutely summable sequences, or c₀, the space of sequences converging to 0. The quotient space S/~_x is also a vector space, so we can study the linear functionals which act on it. To this end, we examine infinite matrices whose rows, as elements of the dual space of X converge weak* to (0).

A Numerical Analysis of the Spectrum of the Almost Mathieu Operator

Sunmi Yang

In 1981, Marc Kac offered ten martinis to anyone who could prove that the spectrum of the almost Mathieu operator is a Cantor Set. This problem, which became known as the Ten Martini Problem, remained unsolved until 2005, when Avila and Jitomirskaya published their solution. Although the theoretical solution now exists, it is difficult to develop an intuitive understanding of these results. In this study we present a numerical analysis of the spectrum of the almost Mathieu operator, using the software Mathematica, in an attempt to better understand the implications of these results.

The Spectrum of the Random Schrodinger Operator

Irina Yurieva Zhecheva

Random Schrodinger operators have important applications in physics. We use results from ergodic theory, probability, and functional analysis to find about the spectrum of the random Schrodinger operator. Specifically, we show why the spectrum of the random Schrodinger operator is deterministic.

Slicing Polyhedra: Searching for Convex Cross-Sections

Katherine Baldiga

We develop a method for determining whether or not it is possible to slice a polyhedron and produce only convex cross-sections. This slicing takes the form of rotating and translating a slicing plane over the polyhedron in a continuous sweep. Then, we address whether this can be done using a slicing plane with a fixed normal direction. Our methods use three-dimensional dualization techniques, where solutions appear in the form of paths through the dual. We improve upon previous methods used to slice polygons into one-dimensional cross-sections by incorporating more geometric information into the dual. Finally, we reveal how these improvements yield more insightful solutions not only to the convex cross-section question but also to other types of decomposition questions.

Weighted Blow-Ups of the Braid Arrangement

Colin D. Carroll

We use compactifications of the braid arrangement as a motivation to weight points on a line and define a way to use these weights to produce building sets. We define two operations on bracketings on a path with /n/ nodes which describe the poset structure of truncated simplices by weighting points in the configuration space. We provide both global and local descriptions of the spectrum of blow-ups of the braid arrangement.

Growth and Combinatorial Properties of the Triangle Sequence

Shea Daniel Chen

Triangle sequences are a type of multi-dimensional continued fraction. We investigate growth rates of the denominators in triangle sequences, in analog to the growth rates of the denominators in continued fractions. In particular we look at the analog of the Euler totient function for triangle sequences, which gives us the number of points in the triangle sequence given a denominator. We also study the distribution of denominators for special sets of triangle sequences. Finally, we present a combinatorial representation for triangle sequences.

Alpha-Regular Stick Knots

Diana Davis

A stick knot is a closed chain of line segments attached end to end. An alpha-regular stick knot has unit-length segments where the angle at each vertex is the same, some angle that we call alpha. If we have found an example of a stick knot that is very nearly alpha-regular, with sticks that are very close to unit length and angles that are very close to alpha, we would like to say that a stick knot exists of the same knot type, where the sticks are exactly unit length and the angles are exactly alpha. Previous work has proved this result for regular stick knots (with unit-length sticks but different angles) and for very specific cases of alpha-regular stick knots. We prove this result in full generality, with one small caveat. We also provide some new results for the trivial knot and some general discussion of alpha-regular knots.

Simultaneous Confidence Interval Estimation for Multivariate Binary Data

Douglas Robert Hammond

We first consider the different methods which are currently used to form confidence intervals for the true proportion of univariate binary distributions. Then, we consider the methods which are currently used to form simultaneous confidence intervals for the true proportions of multivariate binary distributions. Next we compare the relative performance of these latter methods over a range of marginal probabilities and correlation structures. Finally, we evaluate the problems of Peter Westfall’s iterative bootstrap method for forming simultaneous confidence intervals, provide suggestions about how to deal with these problems and propose and evaluate a slightly different form of Westfall’s method.

On Sensitivity in Topological Dynamics

Jennifer Elizabeth James

Sensitive dependence on initial conditions captures the notion that small differences between initial states result in great distinctions between eventual behaviors. In this work I discuss several topological properties and distinct concepts of chaos. I also examine results that imply sensitivity on compact spaces and prove that various properties imply the sensitivity of continuous maps on locally compact spaces.

A Classification of Spanning Surfaces for Alternating Links

Thomas Kindred

A surface spans a link if it has boundary equal to the link. We present a new construction that gives spanning surfaces for any link, and we prove that this construction produces all possible spanning surfaces for alternating links, up to a certain equivalence. As corollaries, we present easy methods for determining the cross-cap number and overall (orientable or non-orientable) genus for any alternating link.

Partition Congruences and Modular Forms

Ross Daniel Kravitz

The partition function of a positive integer n counts the number of different ways of writing n as a sum of positive integers. It is a purely combinatorial object. Modular forms are holomorphic functions on the upper half plane satisfying a certain growth condition and functional equation, and their theory is part of complex analysis. We’ll look at how the theory of modular forms can be used to study congruence properties of the partition function, an area of research initiated by Srinivasa Ramanujan. In particular, we’ll look at congruence properties modulo 2 and 3, which strangely seem to be the most difficult primes to get a handle on.

Descriptive Dynamics of Borel Endomorphisms and Group Actions

Kathryn Anne Lindsey

This thesis explores the dynamical properties of Borel endomorphisms and group actions on Polish spaces equipped with their -algebra of Borel sets, and obtains descriptive versions of key results from measurable dynamics. Sets in WT, the ideal consisting of all countable unions of wandering sets, are considered “trivial,” and most results are proven to hold modulo a set in WT. Original results presented here include descriptive analogues of the Poincare Recurrence lemma, Rohlin lemma, and Birkhoff Ergodic Theorem for Borel endomorphisms, as well as generalizing the notions of the Shelah-Weiss ideal, saturation, compressability, decomposability, and the Hopf ideal to the case of Borel endomorphisms. A similar theory is developed for Borel actions of countable groups.

Chains of Rings with Local Formal Fibers at Principal Prime Ideals

Myron Minn-Thu-Aye

Given a local ring R, we can define a metric on R and complete the ring with respect to this metric. While it is difficult to determine the properties of rings in general, we know much more about the structure of complete local rings. Therefore, we can study a local ring by considering its relationship to its completion. In particular, suppose we are given a complete local ring (T,M). Let P1, P2, …, Pn be a chain of nonmaximal prime ideals of T with Pi contained in Pj for all i < j. Let p be a regular element of T contained in P1. We give necessary and sufficient conditions for there to exist a chain of local domains Bn, B(n-1), …, B1 where Bi is contained in Bj for all i > j such that p is contained in Bn, each Bi completes to T and the formal fiber of each Bi at pBi is local with maximal ideal Pi.

Least-Perimeter Partitions of the Sphere

Conor Quinn

We consider generalizations of the honeycomb problem to the sphere S2 and seek the perimeter-minimizing partition into n regions of equal area. We provide a new proof of Masters’ result that three great semicircles meeting at the poles at 120 degrees minimize perimeter among partitions into three equal areas. We also treat the case of four equal areas, and we prove under various hypotheses that the tetrahedral arrangement of four equilateral triangles meeting at 120 degrees minimizes perimeter among partitions into four equal areas.

Spectral Properties of Random Unitary Band Matrices

Brian Zachary Simanek

Verblunsky’s Theorem states that there is a one to one correspondence between sequences of complex numbers in the unit disc and nontrivial measures on the unit circle. Given any nontrivial measure on the unit circle, we can obtain a sequence of orthogonal polynomials that obey a recurrence relation involving a sequence of numbers in the unit disc, the so-called Verblunsky coefficients. In 2005, Stoiciu proved that if the Verblunsky coefficients are i.i.d. random variables distributed uniformly on the disc of radius less than 1, then the asymptotic distribution of the eigenvalues of the corresponding CMV matrix is almost surely Poisson. The first part of this thesis is devoted to proving the same result, but with the Verblunsky coefficients coming from different distribution that is more concentrated at the origin. The second part of this thesis studies a different type of unitary band matrix that we call a “Joye Matrix.” Using known methods, we can obtain much information about the spectral properties of these matrices. We conclude with a proof of Aizenman’s Theorem for Joye Matrices when the distribution of the phases is any one of a very large class of distributions. This is a major step towards proving that the asymptotic distribution of the eigenvalues of these matrices is also Poisson.

A New Multi-Strategy Approach to Ensemble Classification

Lindsey Wu

Although classification techniques can be very powerful on their own, they perform even better when combined. An aggregated model of multiple classifiers is called an ensemble, and current research focuses on finding ensemble methods that are effective over a wide variety of classification problems. For an ensemble to be effective, its base classifiers must be accurate and diverse, but it is difficult to increase one of these properties without decreasing the other. In this thesis, we explore how a heterogeneous ensemble, one that combines a variety of different classification strategies, might allow both diversity and accuracy to increase, resulting in a higher-performing ensemble. We examine a new hybrid model which we call 7-Stack, and find that its predictions are consistently more accurate than the predictions of other ensemble methods.

Implementation of Stochasticity in Differential Equation Models with Applications to Modeling Hematological Diseases

Christina Brakken-Thal

Biological models have been criticized for not being able to take into account stochasticity found in biological data, particularly variations in time delays. In this paper, I show how to implement stochasticity, using the Naor process, into time delays in discrete differential equation models. The Naor process was implemented in the red blood cell model of cyclic hemolytic anemia proposed by Mahaffy, Belair, and Mackey in 1998. The introduction of stochasticity into the red blood cell model significantly decreases the size of the red blood cell cycles and the size of the period of the cycles. The introduction of stochasticity also has a minor effect on the bifurcation point of where the cycles start to occur. These results indicate that stochasticity is an important consideration when trying to model the size and the period of cyclic diseases.

Excluded Blocks in Cellular Automata

James Clayton Kingsbery, Jr.

Cellular Automata (CA) are systems that have locally defined behavior that are capable of exhibiting complex global behavior. In this work, we find very very tight bounds for the shortest excluded blocks of one particular type of CA, which is believed to have the longest such block possible. This bound is drastically tighter than that found in any previous work. We go own to look at consequences of this result.

Isoperimetric Regions in Spaces

Michelle D. Lee

We examine the least-perimeter way to enclose given area in various spaces including some spaces with density.

The Honeycomb Problem on Hyperbolic Surfaces

Vojislav S. Sesum

Assuming a certain conjectured Polygonal Isoperimetric Unequality, we prove that a valence three tiling of a compact hyperbolic manifold by regular N-gons is parameter minimizing. We prove the Polygonal Isoperimetric Inequality for some special cases and give some negative computational evidence for other cases.

The Stick Number of Torus Knots

Todd Brooks Shayler

What is the least number of sticks glued end-to-end needed to construct a given knot K? What is the least number of sticks in any projection of K? These invariants are known as the stick number and projection stick number, respectively. Are there embeddings of stick knots realizing the stick number such that we can project into some plane, causing half of the stick to disappear? We find such embeddings of (p,2p+1)-torus knots where one less than half of the sticks are parallel!

Flat Folding with Thick Paper

Tomio Ueda

Computational origami has thus far concerned itself only with paper that was infinitely thin. We explored new issues and possibilities when a thickness is assigned to the paper, such as folding models, combinatorics regarding the diameter of the half-circles around folds, and the phenomenon known as creeping in both the 1D and 2D cases with thickness.

Simultaneous Interval Estimation for Multivariate Normal and Binary Data

Ya Xu

We first look at different methods to construct simultaneous confidence intervals for the mean values of multivariate normal distributions. We propose a computer intensive numerical method that produces shorter intervals than the traditional analytical methods. We then extend these methodologies to multivariate binary data. Based on the binomial probability function, we again propose a numerical method to produce shorter intervals.

Diophantine Approximation through Nonsimple Continued Fractions and Planar Curves

Nicholas Sasowski Yates

Here we introduce an explicit function whose graph is a smooth curve that spirals in to the golden ratio phi and crosses the x-axis at precisely the best rational approximates to phi. We then analyze the structure of this Golden Diophantine Spiral. In particular, we determine its limiting proportions, through which we discover a connection between our curve and the well-known Golden Rectangle. We extend our results and define Diophantine Spirals for a large class of real quadratic irrational numbers. We then examine two relatively-unexplored continued fraction representation systems, focusing especially on the expansions of real quadratic irrationals. It is well-known that a number is a real quadratic irrational if and only if its simple continued fraction is eventually periodic. Here we show that, with a fixed integer numerator, all quadratic irrationals can be written periodically with a period of length one. We also explore Diophantine approximation issues within the context of these new expansions. We further investigate whether a similar period-one expansion holds for a system of non-simple continued fractions in which each numerator depends on the previous denominator. Using the dynamics of a related map to study this system, we offer preliminary results and conjectures in this direction, and place these in context with what is currently known.

Helical Structures

Stephen Savinar Moseley

We explore the structural properties of a class of stable structures resembling triple helices. We assume a simplified physics model, and observe an ideal system as it settles. By changing the relative sizes of elements and varying the properties of the rules that define the system’s dynamics, we identify which systems assume regular, stable configurations. We further test stability by applying Brownian perturbations and stretching settled configurations to observe how quickly they resettle. Given the final range of variables that yield stable systems, we compare our structures to the ideal physical characteristics of the collagen protein (which forms a regular triple helix) and hypothesize how the differences between the rules of our simulation and those in nature cause our stable systems to differ from collagen.

Juggling Braids, Links, and Artin Groups

John Mugno

We study the SITESWAP notation used by jugglers and mathematicians for denoting juggling patterns. We construct a map from the space of juggling patterns to links, and prove that this map is onto. In other words, all links can be juggled. We extend this to other juggling patterns that arise from alternate Artin groups.

On Diophantine Approximation Along Algebraic Curves

Ashok Pillai

Building on the previous work of Carsten Elsner from 2001, here we discover a method for approximating almost all positive real numbers by integer points that lie on homogeneous algebraic curves of degree two. We first examine circles and ellipses as special cases before generalizing our work to produce a result for all symmetric homogeneous quadratic curves. Next we extend this generalization to all homogeneous quadratic curves. Finally we employ our methods to approximate certain U-numbers by rational points on singular cubic curves.

Two-Cycles in Three-Dimensional Space

Jordan Rodu

Two-Cycles are approximations of stationary trajectories of flows under probabilistic control, formed when two flows are anti-parallel at a particular point. We know what these two-cycles look like in two dimensional space. In this paper, we will investigate the structure and conditions of two-cycles in three dimensional space. Specifically, we show that locally the locus of points in which flows are anti-parallel is a curve, and that two cycles that approximate these points form a two parameter family of curves.

Class Groups of Function Fields and the Decomposition of Irreducibles in Field Extensions

Matthew P. Spencer

Let n be an integer greater than 2 and suppose S, T and U are pairwise disjoint finite sets of monic irreducible polynomials in Fq (T). We construct infinitely many quadratic function fields K of degree m such that n divides the size of the class group of K, and such that polynomials in S split completely, polynomials in T remain inert, and polynomials in U are totally ramified in K. We present further results concerning higher degree extensions and class groups of high n-rank.

Triangle Sequence Revisited: An In-Depth Look at Triangle Iterations

Christopher Stine Calfee

Purely periodic triangle sequences correspond to cubic irrationals alpha and beta. We will show a variety of methods for finding the irreducible cubic polynomials corresponding to both alpha and beta. Finally, we will explore some of the interesting polynomials which emerge from the sequences that are purely periodic of periodicity length one.

Minimal Blow-ups of Spherical Coxeter Complexes and their Homotopy

Eric Hershel Engler

The goal of my thesis is to find a presentation for the fundamental group of projective spherical Coxeter complexes with minimal blow-ups. It is based on work by Davis, Januszkiewicz and Scott (DJS), who prove that the fundamental group is the kernel of a map p from a group OW that acts on the universal cover of the space onto the underlying group W. DJS prove this result for abstract systems, and thus translating their work is non-trivial, in fact very difficult. We translate their work into the language of graph-associaheda developed by the SMALL 2004 configuration spaces group and specifically compute OW and p. Given these computations, we calculate the fundamental group of these spaces (through dimension six) using java code and a computational algebra package called GAP. From these results, we develop a conjecture for the first homology group.

Double Bubbles in S3, H3, and Gauss Space

Neil Reardon Hoffman

This thesis is the near completion of work done by the 2001-2003 Geometry Groups to prove the double bubble conjecture in the three-sphere S3 and hyperbolic three-space H3 in the cases where we can apply Hutchings theory: in S3, both enclosed volumes and the complement occupy at least 10% of the volume of S3; in H3, the smaller volume is at least 85% that of the larger; And in Gauss space Gm for three-equal-volume double bubbles. A balancing argument and asymptotic analysis reduce the problem in S3 and H3 to some computer checking. The computer analysis has been designed and fully implemented in S3. In H3, it has been only partially implemented.

Completions of UFDs with Semi-Local Formal Fibers

David Jensen

Let (T,M) be a complete local ring such that |T/M| = |T|. Given a finite set of incomparable non-maximal prime ideals C of T, we provide necessary and sufficient conditions for T to be the completion of a local UFD A with semi-local generic formal fiber with maximal ideals the elements of C. We also prove an extension of this result where A contains a height one prime ideal with semi-local formal fiber with maximal ideals the elements of C. In addition, we discuss the possibility of forcing our UFD A to be excellent.

Identifying Best Rational Approximations Through Sharp Diophantine Inequalities

Kari Frazer Lock

Using the theory of continued fractions, we produce a new sharp Diophantine inequality involving an irrational number and a rational approximation to that number, such that the only solutions are precisely all the best rational approximates to the given irrational number; that is, the complete list of its convergents. This work generalizes and extends previously known results appearing in the literature. We also identify the best rational approximates when simultaneously approximating a finite number of generalized golden ratios in the same quadratic field.

Rotating Linkages in a Normed Plane

Jonathan Lovett

In this paper we examine the implications of rotating linkages in generalized norms. We prove that fully rotating a rhombus with both diagonals implies that the norm is linearly equivalent to Euclidean or that the triangle has a certain exceptional property. We also demonstrate that the same is implied by full rotation of some non-exceptional isosceles triangle with median or right triangle with median. In addition, we prove that all triangles can be fully rotated in any norm, and that that rotation is continuous if the norm is strictly convex.

Totally Geodesic Seifert Surfaces in Hyperbolic Complements of Knots in 3-Manifolds

Aaron Daniel Magid

A rich class of hyperbolic 3-manifolds can be represented as the complement of a knot or link in a closed orientable 3-manifold. For these cusped manifolds, we are interested in finding totally geodesic Seifert surfaces, surfaces whose boundary is the knot or link. We consider knot complements for knots embedded in Euclidean 3-manifolds, spherical 3-manifolds, and S2 x S1. We show that all of the closed Euclidean 3-manifolds contain a hyperbolic knot with totally geodesic Seifert surface. Additionally, we show that S2 x S1 and all lens spaces L(p,q) contain a hyperbolic knot with totally geodesic Seifert surface. Also, we give examples of some immersed totally geodesic surfaces in knot complements in the 3-sphere.

The Farey-Bary Map Revisited

Andrew Noah Marder

Two generalizations of the Minkowski ?(x) function are given. As ?(x) maps quadratic irrationals to rational numbers, it is shown that both generalizations send natural classes of pairs of cubic irrational numbers in the same cubic number field to pairs of rational numbers. It is also shown that these functions satisfy an analog to the fact that ?(x), while continuous and increasing, has derivative zero almost everywhere. Both extend earlier work of Beaver-Garrity on the Farey-Bary map.

A Triangle Sequence Pell Equation

Michael T. Baiocchi

Using triangle sequences, a multi-dimensional continued fraction algorithm, this paper develops a higher-dimensional version of the Pell Equation. The set of solutions to this Pell-Analog has the same structure as the solution set to the original Pell Equation. Further, this paper explores the connection both Pells share with the units of particular fields.

Characterization of Completions of Domains with Semi-Local Generic Formal Fiber

Philippa L. Charters

In this paper, we prove the following characterization of the completion of a domain with given generic formal fiber: Let (T,M) be a complete local ring, G Í SpecT such that G is nonempty and the number of maximal elements in G is finite. Then there exists a local domain A such that the completion of A is T and the generic formal fiber of A is exactly G if and only if T is a field (and G = {(0)}) or the following conditions hold: 1. M Ï G, and G contains all the associated primes of T

2. If Q Ï G and P Î SpecT such that P Í Q then P Î G

3. If Q Î G then Q Ç prime subring of T = (0)

From this theorem and its proof, we will also derive some more specific theorems, including a characterization of completions of excellent domains with semi-local generic formal fiber in the characteristic zero case.

On Completion and Tight Closure

Brian P. Katz

Tight closure is one of the most active areas in current algebra research. It is conjectured that tight closure and completion will commute for excellent rings, finally providing ring theorists with a sufficiently strong condition to study the relationship between a ring and its completion. I constructed two local rings, a unique factorization domain and an “almost excellent” domain (all fibers are geometrically regular except the generic one), for which tight closure and completion do not commute.

Phase Transitions of Multidimensional Generalizations of the Knauf Number-Theoretical Chain Model

Edvard Major

This thesis briefly reviews basic concepts of statistical mechanics. A detailed exposition of the Farey Number-Theoretical Chain (FNTC) model is provided. Critical phenomenon of this statistical-mechanic model is further discussed. The Knauf Number-Theoretical Chain (KNTC) model is revisited, and an elegant new proof of exact phase transition location is provided.

A couple of new two-dimensional number sequence models that assume Knauf-like, denominator interactions are proposed. The first one is based on a triangle sequence introduced by Von Rudolf Monkemeyer and D. Grabiner. The existence of the model’s phase transition is verified. To construct the remaining models, a couple of new continued fraction Re2 algorithm-generalizations are proposed, and their properties analyzed. The existence of respective phase transitions is proved.

Generalized Continued Fractions and the Units of Cubic Fields

Mark P. Rothlisberger

Every real number a has a continued fraction expansion which can be developed in several ways. We will examine some of the properties of continued fractions in order to work on generalizing them. Contained in Sections 1, and 3, this work is well known. An introduction to the Geometry of Numbers developed by Minkowski can be found in [4], while a slightly different, but still geometric approach to continued fractions is presented in [5]. Section 2 is also an introduction to well-known background material. Continued fractions are closely tied to distinguishing quadratic irrationals and determining properties of the algebraic fields that they determine. The generalized continued fractions we develop and investigate will follow the approach of Minkowski by using convex bodies in R3, namely parallelepipeds, to approximate certain vectors and planes. These methods will resemble the geometric development of continued fractions from Section 3, and we will demonstrate that some of the results from continued fractions generalize as a result of this method. Two approaches will be given: the first, contained in Section 4, is not original; the same approach is contained in [2] under the title A Criterion for Algebraic Numbers. The second generalization, in Section 5, is original in the choice of parallelepipeds, but employs methods from The Theory of Continued Fractions in [2]. We will also examine the connection between the two generalizations.

Augmentations of Knot and Link Complements

Eric M. Schoenfeld

It is conjectured that the meridian length for any alternating knot complement is bounded above by 2, though the best known upper bound approaches 3 for high crossing knots. We show that the bound of 2 is held for “almost all” alternating knots, and indeed almost all alternating links as well. Moreover, we show that any knot complement, and indeed any link complement, can be realized as Dehn surgery on a special type of link with meridian length exactly 2.

Singular Maps of Surfaces into Hyperbolic 3-Manifolds

Eric Michael Katerman

We construct singular maps of surfaces into hyperbolic 3-manifolds in order to find upper bounds for meridian length, longitude length, and maximal cusp volume of those manifolds. We also provide ample background and history of hyperbolic geometry and 3-manifold theory for this exposition to be accessible to undergraduate mathematics majors. Generalizations and attempts to strengthen our results are also included for completeness.

Spatially Explicit Biological Population Models

Jonathan A. Othmer

This thesis presents a spatially explicit hybrid system population model. Populations are assumed to exist in discrete patches, which we approximate using a hexagonal tiling of the plane. Dynamics within one patch are controlled by a system of differential equations while intra-patch dynamics are controlled via a set of transition functions and threshold values. We explore a variety of behaviors of this model, filling the plane, reaching static equilibrium, and reaching dynamic equilibrium. We also present and explore a spatially attracting, self-synchronizing cycle that arises out of the model.

On Solution to the Generalized Pell Equation with Applications to Diophantine Approximation

Charles Samuels

Suppose pn/qn are the convergents of c, where c is a positive integer not a perfect square. We show that NewtonÕs method applied to F(x) = x2 Ð c with initial approximation pmlÐ1/qmlÐ1, for any natural number m, generates the sequence {p2nmlÐ1/p2nmlÐ1}, n = 0,1,2,É. Subsequently, we generalize these results to all functions of the form F(x) = x2 Ð bx Ð c, where b > 0, c > 0 are integers such that b2 + 4c is not a perfect square. We finally explore the dynamics of some polynomial functions in the p-adic numbers.

Calculus: Its History, Teaching, and Pedagogy

Camille S. Burnett

There are three components to this study – the first, a historical and analytical survey of the calculus; the second, a comparison of teaching methods and approaches across two cultures, the United States and Jamaica; the third, a section on course module development. In the history of the calculus, we examine how calculus developed, the motivation of the theory and the major problems encountered. We present an overview of contributions by early mathematicians, a more in-depth look at the work of Newton and Leibniz, and discussed how calculus was made rigorous in the 1800s.

Rank One Mixing and Dynamical Sequences

Darren Creutz

Rank one transformations are a class of ergodic transformations constructed using a cutting and stacking method. We show that a class of rank one transformations characterized by adding spacer levels that have restricted growth but also tending toward a uniform type of distribution are indeed mixing transformations. All previously known mixing rank one transformations, including staircase transformations satisfying the restricted growth condition, fall into our class.

Four-Manifolds and Related Topological Investigations

Richard Haynes

In this thesis I investigate high dimensional manifolds through the lens of four-dimensional topology. In this vein, I use invariants of four-dimensional spaces to specify related properties of larger ambient spaces. This relationship provides restrictions on the possible structures of these larger spaces.

Power Weak Mixing and Recurrence in Infinite Measure

Abhaya N. Menon

In this thesis, we explore the idea of Power Weak Mixing and demonstrate the existence of a family of transformations exhibiting this property. We then investigate the recurrence properties of this family of transformations.

An Improvement on Legendre’s Theorem from Diophantine Approximation

Rungporn Roengpitya

In this thesis, we explore two questions from Diophantine analysis. First, we improve Legendre’s Theorem by finding the best possible constants for j, the golden ratio, and the generalized golden ratio j2 and j3. Then, we explore the nature of the function ;x;where x is an irrational number in one and two dimensions.

Uniqueness in mimensional Triangle Sequences

Tegan Cheslack-Postava

In the generalization of continued fractions introduced by Garrity, each point in an m-dimensional simplex is represented by a sequence of nonnegative integers. After introducing the algorithm for generating these sequences, we show that the representation map is in general not injective. We use the notions of partition simplices and associated dimension to investigate the set of points identified by an m-triangle sequence.

Strict Minimality of Alternating Knots in S x I

Thomas Fleming

In the late 1800′s Tait conjectured that for knots that lie in the plane, a reduced alternating projection has the smallest possible number of crossings for any projection of that knot, and that any non-alternating projection must have more crossings. This fact was proven in 1984 by Kaufmann, Murasugi, and Thistlethwaite. In the summer of 1999, the Colin Adams directed Knot Theory SMALL group of Fleming, Levin and Turner was able to prove that if the knot projection lies on a surface (such as a torus) and the knot lies in a layer around that surface (the surface cross an interval), then a reduced alternating projection has the smallest possible number of crossings for any projection of that knot. We will extend this work to prove that for a knot in a surface cross an interval, a reduced alternating projection of the knot must have strictly fewer crossings than a non-alternating projection. We will use arguments based on a generalized Kauffman bracket polynomial, Menasco-type geometric arguments, and covering space techniques.

Applying a Bayesian Hierarchical Model to a Data Set Consisting of Hospital Mortality Rates

Cory Heilmann

Bayesian hierarchical modeling is often applicable to data sets where the data originate from many different entities, each of which measures a similar quantity. Examples of these data sets are students’ test scores from different schools and mortality rates from different hospitals. This sort of modeling is particularly useful when we wish to estimate means and variances of each entity, but some of the entities have low numbers of observations, and thus the maximum likelihood estimator is unreliable. This thesis uses a Bayesian hierarchical model on a data set consisting of the mortality rates from organ transplants in 131 hospitals. We will rank the hospitals according to their predicted mortality rate, and also decide whether the mortality rates of small hospitals appear to be larger than the mortality rates of large hospitals.

A Structural Analysis of the Triangle Iteration

Adam Schuyler

Classically, it is know that the continued fraction sequence for a real number a is eventually periodic if and only if a is a quadratic irrational. In response to this, Hermite posed the general question which asks for ways of representing numbers that reflect special algebraic properties. Specifically, he was inquiring about possible generalizations of the continued fraction. In this paper we will study the triangle iteration, a two-dimensional analogue of the continued fraction. We will take a primarily geometric approach and look at the probabilities of the occurrences of certain sequences.

Relationships and Syzygies in Classical Invariant Theory for Vector-Valued Bilinear Forms

Zachary J. Grossman

The goal of invariant theory is to describe the algebra of invariants for a vector space under a given group action. After introducing invariant theory and its two main problems, we will prove the Second Fundamental Theorem for vector-valued bilinear forms, which describes the basis relations between invariants of vector-valued bilinear forms.

Bend Minimization for Hexagonal Graph Drawing

Davina Kunvipusilkul

In this thesis, we give an overview of some of the optimization problems that arise in computing orthogonal and hexagonal drawings of graphs. We then employ the concepts of spine and spirality to develop a polynomial-time algorithm that, given a biconnected, 6-planar, series parallel graph, computes a hexagonal drawing having the minimum number of bends over all possible embeddings. The algorithm runs in O(n^8) time. This work extends similar results by Di Battista, Liotta, and Vargiu on bend minimization for orthogonal graph drawings.

Supercrossing Number of Knots

Sang Pahk

One of the oldest invariants utilized for the study of knots is the crossing number of a knot, which is the least number of crossings in any projection of the knot. In this thesis, the supercrossing number of knots, a variation on crossing number, is investigated. It is proved that the supercrossing number is always at least 3 greater than the crossing number. The trefoil knot is shown to have supercrossing number 6 or 7. The crossing map is then investigated as a tool to understand the supercrossing number.

The Cusped Hyperbolic Three-Manifolds of 2nd Smallest Volume

Scott B. Reynolds

A hyperbolic 3-manifold is defined as the quotient of hyperbolic 3-space by a discrete group of fixed point-free isometries. It is known that the set of volumes of all noncompact (cusped) hyperbolic 3-manifolds is well-ordered, and in 1987, Professor Adams proved that the hyperbolic manifold of smallest volume (V=1.0149…) is the Gieseking manifold. Working in the upper-half space model of hyperbolic 3-space, the Gieseking manifold is obtained by taking a regular ideal (vertices at infinity) tetrahedron and identifying its edges with each other. This paper provides background on these types of problems and then proves that, for a large class of cusped hyperbolic 3-manifolds, the manifold of second-smallest volume is the non-orientable one obtained by gluing two 45-45-90 (angles between vertical faces, measured in degrees) ideal tetrahedra together. This manifold has volume approximately equal to 1.83.

Generic Formal Fibers of Excellent Local Polynomial Rings

Aaron D. Weinberg

Let (T, M) be a complete regular local ring of dimension at least two containing the rationals, such that the cardinality of the residue field T/M is at least the cardinality of the real numbers. Suppose p is a nonmaximal prime ideal of T and L is a set of prime ideals of T[[X₁, ..., X_n]] (where X₁, …, X_n are indeterminates) such that the cardinality of L is strictly less than the cardinality of T/M, Q intersected with T is a subset of p for each Q in L, and if Pi is the prime subring of T, then Pi[X₁, ..., X_n] intersected with Q is the zero ideal for each Q in L. Then there exists an excellent regular local ring A such that the completion of A is T, the generic formal fiber of A is local (this means that the ring T \otimes_A K is a local ring where K is the quotient field of A) with p \otimes_A K its maximal ideal, and Q intersected with A[X₁, ..., X_n] is the zero ideal for each Q in L.

On Spheres and Smooth Structures of Four-Dimensional Manifolds

Craig C. Westerland

For smooth four-dimensional manifolds M we explore the representation of classes in H2(M) as smooth embeddings of two-spheres into M. For simply connected manifolds, it is known that such a representation is always possible for continuous embeddings, but the smooth case is in general mostly unknown. Given a class that can be represented in this manner, we determine several bounds on the self-intersection of the class. Additionally, we demonstrate a relationship between the occurrence of certain types of these homology classes that can be represented as smooth spheres in a manifold and the smooth structure on that manifold. Finally, we present an abortive attempt to determine a class of manifolds whose second homology contains no non-characteristic classes that are representable as smooth spheres. To introduce the appropriate background to complete the work herein, we also include several chapters on differential topology, algebraic topology, and four-manifold theory.

Smooth 2-Spheres in Some Compact, Orientable, Simply Connected 4-Manifolds

Alexandre Wolfe

This thesis discusses techniques and results in the study of necessary conditions for representability of second homology classes of compact, orientable, simply connected 4-manifolds by smooth 2-spheres.

Comparison of Ordinary Differential Equation Models of HIV Infection

Laura Louise Christensen

Many ordinary differential equation models have been bused in research on HIV to te et al and Hraba et al) are compared with and without treatment incorporated. The conclusions of this comparison are as follows. The McLean model, which is very similar to the Perelson model, but designed to model dynamics during treatment, is robust only for initial conditions which reflect an infected steady-state and not for infection dynamics from initial inflection or after the completion of a treatment course. Treatment dynamics are compared between the Perelson and McLean models. Particular treatments were implemented on both the Hraba and Perelson models yielding the conclusion that, though the models are quite different, they give similar predictions of relative treatment effectiveness.

Cost Minimizing Networks Separating Immiscible Fluids in R²

Brian Elieson

Cost minimizing networks model certain behavior of immiscible fluids in the plane. This paper proves the existences of minimizers of straight lines with an upper bound on the number of nodes, closely following the work of Alfaro. It provides some basic examples of minimizers. The paper also gives sufficient conditions for an upper bound on the number of regions meeting around a point.

Characterization of Completions of Integral Domains

Deborah L. Greilsheimer

Christer Lech characterized the complete local (Noetherian) rings that are completions of domains. We reprove Lech’s result showing that a complete local ring T is the completion of a local domain if and only if no integer of T is a zero divisor, and, unless equal to (0), the maximal ideal of T does not belong to (0) as an associate prime ideal. Moreover, suppose p (does not equal) M is a prime ideal of T such that Q is an associated prime of T implies Q à p, and suppose that |{q Î Spec T | q Ë p}| £ | T/M²| and p intersected with the prime subring of T is the zero ideal. In this case, we construct a local domain A such that  = T and the generic formal fiber ring of A is local with p^*_AK the maximal ideal where K is the quotient field of A.

Weak Mixing, III0 Staircase Zd Actions

Erich Muehlegger

This thesis presents two new examples of staircase Z^d actions, functions mapping points from Z^d x R to R. Expanding on the work of Silva/Adams and Touloumtzis, the first construction is shown to be an infinite measure preserving action with weakly mixing basis transformations. The second and more interesting example is a type III₀ action with weakly mixing basis transformations. In addition, the several concepts and criteria applying to Z² actions are generalized to their Z^d counterpart.

Bayesian Prediction Intervals for Symmetric Shrinking Linear Smoothers

Jason Ross Schweinsberg

In many statistical problems, it is important to estimate the relationship between a dependent variable y and some independent variables, so that given values for the independent variables, a 95 percent prediction interval for y can be computed. Here we describe how Bayesian methods can be used to calculate prediction intervals for a class of modeling methods called symmetric shrinking linear smoothers. We show that how well these “Bayesian prediction intervals” perform can be related to simple properties of the smoother. When one “smoothing parameter” is chosen optimally, we show that Bayesian prediction intervals perform well asymptotically for generalized ridge regression. For other smoothers, the asymptotic performance of Bayesian prediction intervals is conjectured to depend on the eigenvalues of a linear operator called the covariance operator.

Outer-crossing Numbers: A New Parameter for Graphs

Alexander Woo

Outer-crossing numbers of graphs are defined and basic properties are given. The outer-crossing numbers of complete bipartite graphs are found. Some conjectures and a generalization to surfaces other than the plane are discussed.

Finding Incompressible Surfaces

Jeffrey Bevelander

In an attempt to supplement the machinery already developed to identify knots and links, this thesis presents an algorithm which will detect the presence of incompressible surfaces, which can assist in the process of knot and link identification and analysis. Examining the link in its ideal triangulated form, the algorithm produces the simpler surfaces that can be expressed “nicely” by interconnecting triangles and quadrilaterals. Expressing a surface “nicely” basically means that each triangle and each quadrilateral used in the representation lies entirely in a single tetrahedron, and that none of the polygons intersect each other in a given tetrahedron. These two restrictions create a combinatorial requirement for a surface that sits “nicely” in the tetrahedra. Utilization of this requirement allows for the identification of the sets of triangles and quadrilaterals that lie “nicely” within the tetrahedra, and upon associating these sets of polygons with the surfaces they form, it is then possible to identify the surfaces that can be placed “nicely” into a link’s ideal triangulated form. Since, incompressible surfaces always have a “nice” representation in the ideal triangulation, once we have identified our “nice” surfaces, we will have found all the simple incompressible surfaces in a knot complement, and gathered information that will further our efforts towards identifying the knot or link.

Analysis of Manifolds Using Morse-Smale Homology

David DelaCruz

A topological object can be defined as a set of points in a space. A particularly beautiful topological object to examine is the manifold.

Given two n-manifolds, it is interesting to know if they are topologically equivalent — that is, whether we can bend, twist, and stretch one to get the other. The topological invariants of two manifolds must be the same if two manifolds are the same under topology, one such invariant is homology groups.

Singular homology theory describes homology groups for manifolds in the abstract. Morse-Smale homology allows us to capture homology groups algebraically as well.

The thesis is an exposition of singular homology theory, Morse theory, and Morse-Smale homology, followed by an algorithm that can be used to approximate Morse Smale gradient flows on algebraically-defined manifolds.

Computing the Intersection Homology Groups of a Complex Algebraic Variety

Christopher French

An algorithm is given to compute the intersection homology groups for a complex algebraic variety. Two previously developed algorithms, the Collin’s cad algorithm and Prill’s Adjacency algorithm, are presented and used.

One Approach to Factoring Multivariate Rational Polynomials Over the Complex Numbers

Dimitry Korsunsky

Factoring a given multivariate polynomial is an important task in symbolic computation. Potential uses for an efficient solution to this problem could be found in various branches of applied mathematics, such as computer-aided design and theorem proving. Several algorithms giving different methods for factoring multivariate polynomials had been created over the years (Noether 1922, Davenport and Trager 1981, Christov and Grigoryev 1983 etc.). The theoretical basis for the algorithm, implementation of which is the subject of this paper had been put forward in Bajaj et al [4]. There it is proved that the suggested approach when implemented in parallel will execute in shorter time as compared to earlier solutions. Although a sequential solution which was implemented achieves lower efficiency there is a significant advantage in that it can be used in a large variety of settings. The program had been written using the Mathematica software package.

Bumper Drawings: A New Type of Proximity Drawing

Michael Pelsmajer

Open and closed ß-bumper drawings are defined exactly as ß-drawings, except that proximity regions contain no vertices and edges. Open and closed [[infinity]]-bumper graphs are completely classified. Maximal outerplanar drawings are defined for 1-bumper drawings, and partial results and methods are detailed.

Hyperbolic 3-Orbifolds

Edward Welsh

Suppose we have a hyperbolic 3-orbifold with discrete fundamental group G generated by two elliptic transformations, alpha and beta. This thesis finds restrictions on the possible distances between the axes of alpha and beta under various conditions. Introductions to both hyperbolic geometry and algebraic topology are included.