**A Numerical Investigation of the Traveling Wave in the Replicator-Diffusion Equation**

**Cody Cao**

The replicator equation fails to capture the population dynamics in which individuals are distributed throughout a space. One solution to this shortcoming is to incorporate a diffusion term, thereby providing a spatial structure and allowing individuals to move. We call the resulting equation the replicator-diffusion equation. In this thesis, we study these evolutionary game dynamics with a spatial component through two different reaction – diffusion systems, both in 1D and 2D, based on previous works by Hutson, Vickers, and Cressman via computational simulation. We numerically investigate the behavior of the traveling wave in both dimensions and give criteria for the direction of the traveling wave in 2D. We also explore other nuances in the 2D case, namely turning points and wave shapes.

**On Higher Distance Commuting Matrices **

**Madeleine Constantina Elyze**

In general, matrix multiplication is not commutative. However, sometimes pairs of matrices do commute. We say that two matrices A and B commute, denoted A<–>B, if AB-BA=0. The collection of pairs of non-scalar matrices that commute forms an algebraic affine variety in C^{2n2}, where C denotes the complex numbers. In the case that A and B do not commute, A <-/-> B, it is still possible that A and B commute with some common, non-scalar matrix X. Then we say A<–>X<–>B. In this way, we can construct chains of matrices of arbitrary lengths, A<–>X_1<–>…<–>X_k<–>B. We say that A is distance k+1 away from B if our chain is the shortest such path between A and B. This thesis is concerned with understanding the algebraic properties of these chains in C^{2n2} space. We show that the collection of distance less than or equal to 2 matrices forms an algebraic variety and offer intuition for proving that the collection of distance less than or equal to 3 matrices also forms a variety.** **

**The Zeckendorf Game**

**Alyssa Epstein**

A brilliant theorem of Zeckendorf proves that every positive integer $n$ can be written as the sum of unique, non-adjacent Fibonacci numbers. This theorem generalizes: Miller and Wang showed that *every* positive linear recurrence relation defines a unique legal decomposition, where legal means the recurrence relation cannot be applied again and the summands are appropriately bounded. This thesis examines the Zeckendorf game, a two player game on a specific recurrence relation. This game is defined on the Fibonacci numbers and can be meaningfully extended to all positive linear recurrence relations of the form F_{n} = {i=1}^{k} cF__{n-i}. We prove that the game always terminates at the appropriate (generalized) Zeckendorf decomposition, but that there are always games of multiple lengths. We also find upper and lower bounds on the number of moves taken in a game. For the Fibonacci Zeckendorf game, we show upper bounds on the order of *nlog*(*n*); for all other Zeckendorf games, we show an upper bound on moves that grows linearly with n. The thesis centrally contains an analysis of winning strategies, including a proof of the main theorem: for the Fibonacci Zeckendorf game, Player 2 has the winning strategy for all *n* > 2.

**Almost Excellent Unique Factorization Domains**

**Sarah M. Fleming**

Let (*T*,m) be an equidimensional complete local (Noetherian) ring such that no integer of *T* is a zero divisor in *T* > 1. In addition, suppose T contains the rationals, |*T*|=|*T*/m|, and the set of all principal height one prime ideals of *T* has the same cardinality as *T*. We construct a universally catenary unique factorization domain *A* such that \widehat{A}=*T* and such that uncountably many height one prime ideals of *T* have nonzero intersection with *A* and have geometrically regular formal fibers. If *T* is assumed to be normal, then we can ensure that the generic formal fiber is also geometrically regular. Furthermore, in the case where *T* is a normal domain, we can make *A* “close” to excellent in the following sense: we can show that the formal fibers at the prime ideals of height not equal to one are geometrically regular.

**Combinatorial Problems Related to Peaks, Pinnacles, Descents, and Derangements**

**Isabella Huang**

We say a permutation π = π_{1}π_{2} … π_{n} Î_{n} has a peak at index *i* if π_{i-1 }< π_{i} > π_{i+1}, has a pinnacle at value π_{i} if π_{i-1 }< π_{i} > \ π_{i+1}, has a descent at index *i* if π_{i} > π_{i+1}, and is a derangement if for all *i* Î [n], π_{i} \neq *i*. In this thesis, we provide formulas for the number of derangements with no peaks, one descent, and special cases of two descents, the last of which connects this work to counting diagonal staircase paths. In 2017, the work of Davis, Nelson, Petersen, and Tenner studied pinnacles of a permutation and asked whether there was a particular group action that preserves pinnacles. We consider the proposed group action introduced by Foata and Strehl and find that while it does preserve pinnacles of a permutation, the orbits do not encompass all elements having the same set of pinnacles. Thereby, we establish and measure the insufficiency of this particular group action. We end the thesis with future directions of research in combinatorial problems related to peaks, pinnacles, descents, and derangements with a focus on permutahedra.

**Combinatorial Conditions for Nonsingular Rank-One Transformations**

**Sumun Sitalakshmi Iyer**

A transformation T of a measure space (X, B, m) is measure-preserving if for any set C in B, we have that m(T^{-1}(C)=m(C). We say that T is nonsingular if for any set C in B we have that m(T^{-1}(C))=0 if and only if m(C)=0.We provide necessary combinatorial conditions for a transformation to be nonsingular rank-one. We also develop a class of nonsingular Morse sequences that we conjecture to have rank two.

**Network Topology, Pollen Movement and Models of Insect Behavior in Neighboring Flower Communities**

**Molly Riley Knoedler**

The dynamics and structure of networks of ecological interaction offer insight into stability, resilience, co-evolution, and diversification of ecosystems. Visitation networks, which represent the interaction of plants and their potential pollinators, are of particular interest due to the world-wide decline of insect populations and the importance of pollination to agriculture. In Chapter 1, I examine topological properties and pairwise visitor partitioning at two sites on a small island to better understand fine-scale variation between neighboring flower communities. These results are evidence of the possibility of multiple structures of stable organization in similar natural ecosystems. In Chapter 2, I discuss the results of an agent-based model I developed that investigates how the rules of insect behavior in a heterogeneous floral landscape dictate to pollen movement. I find that even with significantly different visitor partitioning between conspecific flowers, pollen can move over large areas and keep neighboring flower communities genetically connected. While agent-based modeling of pollen movement is not yet widely used, this tool has great potential to improve understanding of pollination in agricultural contexts and aid efforts to protect declining insect populations.

**Pedagogical Approaches to Mathematics Instruction: Finding a Balance Between Lectures and Discovery**

**Edward Lachance Lauber**

Math education is an issue of critical importance in the United States and around the world, yet there is still no scientific consensus on the best ways to teach math. One of the central debates concerns the use of discovery-based learning in comparison to more standard lecture techniques. Some theorists argue that students cannot learn without discovering material for themselves, while others contend that more structured lectures enable students to learn much more. In this study, we set up multiple classrooms with three different teaching paradigms: a lecture condition, a discovery condition, and a mixed condition. Over the course of three days, participants studied the same material on cake-cutting and fair division. Our findings revealed no significant differences between conditions, indicating that perhaps there is no single best way to teach this material.

**Assessing Critical Mass at UC-Berkeley: Creating Predictive Models for** **Affirmative Action Policies in Undergraduate Admissions in the United States**

**Daniel Paul Maes**

Affirmative action refers to hiring and recruiting practices designed to combat discrimination against members of certain demographic groups. These policies are used in settings ranging from federal contractors to local employment to public education, and can focus on race, ethnicity, or gender. Due to the broad applicability and direct social impact of affirmative action, substantial effort has gone into monitoring the necessity and effectiveness of these policies. This effort has manifested itself in frequent legal and state action over the past forty years. In this thesis, we focus on the application of race-based affirmative action policies in public undergraduate college admissions in the United States, specifically through a case study of admissions to the University of California, Berkeley. We make three primary contributions. First, we introduce a quantitative framework through which to interpret a key concept used in contemporary affirmative action litigation: “critical mass.” Historically, policies have been assessed using retroactive data studies. However, using predictive models would reduce the monetary and temporal costliness of such studies. Thus, second, we construct a predictive model of college admissions demographics using Markov Chains. Third, we bring together the two previous contributions, using our quantified version of the critical mass criterion as a benchmark for assessing the outcomes of the predictive model. The type of mathematical model we construct can be modified for use at other universities or for affirmative action applied to other axes, including gender.

**Finding a Model for the ensity of States of RNA**

**Eliza Woolworth Matt**

RNA molecules can adopt a multitude of fold configurations of varying free energy G. The Boltzmann probability suggests the average energy should be near the minimum free energy, but surprisingly we find the difference between the average energy and minimum free energy to be much larger than the product of the Boltzmann constant and the temperature. In this thesis, we show this is because the density of high-energy states grows more quickly than the Boltzmann factor suppresses. We observe this for RNAs across sequence lengths and types, both randomly generated and natural, coding and non-coding. We find that both a flattened exponential and a hyperbolic function are able to approximate well the observed density of states better than either a power law function or a standard exponential function.

**Longitudinal Regression Trees and Their Application to Body Mass Index Growth Trajectories**

**Anna Neufeld**

A growing body of scientific literature suggests that exposure to environmental pollutants during key periods of development can cause lasting changes in an individual’s metabolism. These changes can impact an individual’s body mass index (BMI) trajectory and risk of obesity later in life. Given the complex ways in which environmental pollutants interact with one another and the challenges of studying trajectories over time, nontraditional statistical models are needed to truly understand the relationship between environmental pollutants and an individual’s BMI trajectory. Our goal is to cluster individuals with similar BMI trajectories and similar chemical exposures so as to understand which environmental pollutants may cause heightened risk of obesity. We propose longitudinal regression trees as a promising approach to accomplishing this goal. While a few algorithms for longitudinal regression trees have been previously developed, they have never been applied with the goal of clustering growth trajectories. We compare several existing longitudinal regression tree algorithms in a simulation study setting, and evaluate the potential of these algorithms for tackling the BMI growth trajectory problem. Along the way, we propose modifications to existing algorithm so that we can group individuals by the change in their BMI over time, rather than by the level of their BMI. We then demonstrate the potential of the existing algorithms and the modified algorithm on real BMI trajectories from the National Longitudinal Survey of Youth (NLSY).

**Multi-Crossing Braids**

**Daishiro Nishida**

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A multi-crossing is a crossing where more than two strands meet at a single point, such that each strand bisects the crossing. In this thesis we generalize ideas in traditional braid theory to multi-crossing braids. Our main result is an extension of Alexander’s Theorem. We prove that every link can be put into an n-crossing braid form for any even n, and that every link with two or more components can be put into an n-crossing braid form for any n. We find bounds on the number of strings or crossings necessary to represent a link in an n-crossing braid. We explore different ways to obtain multi-crossing braids, and use these techniques to find new multi-crossing numbers.

**Finite Subsets of the Prime Spectra of Local UFDs**

**Alex Paul Semendinger**

In a 2017 paper, Avery et al. characterized the completions of noncatenary local (Noetherian) UFDs. This was accomplished using a construction similar to one used by Heitmann in a 1993 paper. However, we know very little about the prime ideal structures of the UFDs obtained with this construction. We adjust Avery et al.’s techniques, allowing for greater control over the prime ideal structure of the constructed UFDs. This expands our collection of examples of noncatenary local (Noetherian) UFDs and gives insight into the question of which partially ordered sets can arise as the prime spectrum of a Noetherian UFD.

**Using Data to Predict Hospital Readmissions at Berkshire Medical Center **

**Hallee Erica Wong**

Hospital readmission — whether a patient returns to the hospital within 30 days of discharge — is a popular metric in healthcare policy because frequent hospitalizations are a negative outcome for both hospitals and patients’ health. A model to identify patients with high risk of readmission would enable hospitals to optimize the allocation of resources. Analyzing a dataset of hospitalizations from Berkshire Medical Center, we fit models to predict readmission using logistic regression, decision trees, ensembles of models, and integer programming-based methods. To strategically utilize high dimensional medical code data, we consider ensemble models in which patients are stratified by primary diagnosis and separate models are fit for each group. We show that an ensemble model stratified on the existing hierarchy in the International Classification of Diseases (ICD-10) performs significantly worse than a single global model. We propose a new method, Categorical Co-Frequency Analysis (CCoFA), that uses random forests to identify relationships between categorical labels. Applying CCoFA to identify three disparate clusters of patients defined by diagnosis, we show the performance of an ensemble model stratified on CCoFA clusters is not statistically significantly different from that of a single global model. Additionally, we compare the performance of RiskSLIM (Ustun and Rudin, 2017), an integer programming-based method, to logistic regression and decision trees. Using RiskSLIM to generate a risk scoring system with integer coefficients leads to a model with lower accuracy than logistic regression with real-valued coefficients.

**An Uncountable Dimension-Two Ring With a Countable Spectrum
**

**Weitao Zhu
**

In this paper, we construct a quasi-semi-local ring with two maximal ideals M and N, where M contains only countably many prime ideals but N contains uncountably many. When we localize this ring at its maximal ideal M, we produce an uncountable dimension-two Noetherian local ring with a countable spectrum, reproducing a result of Colbert for dimension-two rings.

**Periodic Paths on the Triangle and Hexagon Billiard Tables**

**Megumi Asada**

Imagine a frictionless, air resistance-less fantasy world with completely elastic collisions. For some reason, you decide to play billiards in this world except your billiard ball contains an infinite supply of paint that it releases as it travels. You also decide to play on equilateral triangle and hexagon billiard tables, because why not? Interestingly, you notice that depending on the angle with which you release the ball, the path drawn in paint on the billiard table starts to repeat itself, fixing the path drawn in paint after some period of time. I’m interested in these types of paths, which we call periodic. Specifically, given a direction vector that you know is periodic, what is the period? How many times will the billiard ball hit the walls of the table before repeating its trajectory? We’ll resolve this case for the triangle billiard table and explore progress towards understanding the hexagon billiard table.

**Properties of Finite Systems of Fixed-Length Cranks**

**Paul Pierre Baird-Smith**

I consider the paths of circular cranks that rotate at fixed speeds in a plane. For a single crank, this is simple: move in a circle; but when one crank is attached to another, a whole new family of curves appears. This talk will focus on surprising properties of these systems in R^{2}, including commutativity and minimality of systems and their curves. Notably, we show that we can draw any polynomial on any bounded interval with these systems.

**Probabilistic Bounds On Binary Classification with Dependent Experts**

**David Burt**

Weighted expert voting is central to the areas of decision theory and statistical learning. In the classical Naive Bayes’ framework, it is assumed each expert votes independently. We investigate a model of expert voting in which votes are no longer assumed to be independent. We give an optimal decision rule for this model in the case when the dependencies between experts, as well as expert competences are known. We give error estimates for this decision rule. We also investigate the case in which the dependencies between experts are unknown. Under the model assumptions, we show the asymptotic correctness of the spectral clustering algorithm with certain weights as the number of samples tends to infinity. Additionally, we give explicit lower bounds in a small case for the probability spectral clustering succeeds.

**Simultaneous Variable and Factor Selections Via Sparse Group Lasso in Factor Analysis**

**Yuanchu Dang**

This thesis investigates applications of penalization techniques to factor analysis models. In a factor analysis model one assumes that there is a linear relationship between the manifest variables (i.e. observable variables) and the hidden factors of interest. Depending on the structure of the factor loading matrix, it can be further classified as either an Exploratory Factor Analysis (EFA) model or a Confirmatory Factor Analysis (CFA) model. Previous research (Hirose and Konishi, 2012) applies the Group Lasso method, a penalization technique with group-wise sparsity, to EFA in determining the number of observable variables. In this thesis we consider adapting the Sparse Group Lasso penalty to a factor analysis model, which is a more flexible approach that allows both group-wise and parameter-wise sparsity of the factor loadings. With the help of the Sparse Group Lasso technique, one can achieve selecting both the number of observable variables and the number of factors for each observable variable. Therefore, it can be used in CFA models. We show that the proposed method can better identify the true underlying structure of a factor analysis model. Moreover, our simulation studies show that the new method gives more accurate estimation of the model parameters than the existing method.

**Benford’s Law Beyond Independence: Copulas and Detecting Fraud**

**Rebecca Durst**

Benford’s law describes a common phenomenon among many naturally occurring data sets and many common distributions, such as the exponential distribution, in which the leading digits of the data points are distributed according to log_{B}[(d+1)/d]. It is so common, in fact, that it is often applied in practice to detect fraud in fields related to science, finance, and even politics. Thus, significant effort has been made to understand when and how distributions will display Benford behavior. Most of the previous work on Benford’s law, however, has been restricted to cases of independent variables, and very little is known about its potential application in situations involving dependence. In this paper, we employ the theory of Copula distributions to investigate the Benford behavior of a product of two random variables that may be dependent. Using this copula theory, we provide a method for quantifying and numerically approximating the Benford behavior of a product of two random variables whose joint distribution function is modeled by the copula C. We then develop a concept of distance from a Benford distribution for products in which one or more of the marginals is a Benford distribution and provide an upper bound for this distance that depends only on the copula C. We then conclude our investigations with a consideration of a concept of goodness of fit for the Benford behavior of a data set that is being fit to a particular copula model.

**A New Connection Between the P****ólya-Vinogradov and Burgess Inequalities**

**Elijah Fromm**

The Pólya-Vinogradov inequality bounds sums over Dirichlet characters without taking the length of the sum into account. For a character c with modulus *q*, this bound becomes trivial when the sum is shorter than a constant multiple of log *q*. Burgess’ bound, which takes the length of the sum into account, remains nontrivial as long as the sum is longer than some constant multiple of *q* + Î. Vinogradov conjectured that character sums of length longer than q^{Î} exhibit cancellation, for any Î > 0. Here, we show that a modest improvement of the Pólya-Vinogradov bound for primitive even quadratic characters would give nontrivial bounds implying Vinogradov’s conjecture on all primitive odd quadratic characters with prime modulus. We also show that whenever the mean of a completely multiplicative function *f *: N ® [-1,1] is large, the logarithmic mean of *f* must also be large.

### A Comparison of Bayesian and Frequentist Model Averaging in Predicting the Demographics of Voter Turnout

**Kathryn Grice**

Better performing than its constituent models, a model average can preempt the dangers of choosing a single model when model specification is uncertain. Model averaging can take place under either the frequentist or Bayesian paradigm, although the underlying assumptions and methodology are vastly different. Here, we evaluate the performance of these approaches to model averaging in the context of voter turnout in US general elections. Using county-level demographic data to predict the turnout of various demographic groups in six states, frequentist and Bayesian model averages see differing levels of success, indicating that a comparison of these approaches is more complex. Further, this study seeks to debunk myths of homogeneity in the voting habits of these demographic groups: indeed, groups that show highly varied trends in turnout between states, in particular those defined by race, are more difficult to predict using model averaging. In the context of a broadening role of statistical analysis before, during, and after elections, both approaches to model averaging offer strengths for demographic analysis and bespeak the inadequacy of generalizations based on demographic group alone.

### Investigating Central-Point Vanishing of Families of L-Functions Using the 2-Level Density

**Anand Hemmady**

In the 1970s, Dyson and Montgomery discovered an unexpected connection between the zeros of the Riemann zeta function and the eigenvalues of families of random matrices. Further work uncovered deep relationships between random matrix theory and the zeros of L-functions in general. We expand upon previous work to use this connection between Number Theory and Random Matrix Theory to study the low-lying zeros (that is, zeros near the central point) of L-functions. In particular, we use the 2-level density to find upper bounds on the average order of vanishing of families of L-functions at the central point.

**Efficient Calculation of RNA Secondary Structures with Terminal Stacking**

**Nikolaus H.R. Howe**

An RNA sequence can adopt an immense number of *secondary structures* (distinct combinations of base pairs), however is often represented by only a handful of *minimum free energy* (MFE) microstates. These few MFE microstates paint an incomplete picture of the thermal ensemble of folds of a given molecule. To predict what will be observed in experiments of RNA folding it is necessary to consider the whole thermal ensemble of microstates, not just the MFE. The properties of the thermal ensemble and probabilities of each microstate are described with a partition function sum over all the microstates or a Boltzmann-weighted sample of microstates. In this thesis we build an algorithm to compute both more efficiently.

The multiplicity of secondary structures scales exponentially with length as the often-quoted O(1.8^{N}). Additionally, however, because an unpaired base adjacent to a base pair can either stack on the pair or not, for any secondary structure there are also many possible ways of stacking unpaired bases. Here we show that including terminal stacking configurations increases the multiplicity to O(2.09^{N}).

The partition function and its multiplicity can be obtained with recursion, so we employ a dynamic programming algorithm, which allows us to sum the contributions of exponentially many states in O(N^{3}) time. Our novel formulation of the recursion relations allows us to recompute the partition function in O(N^{2}) time and to sample a microstate consistent with its Boltzmann probability in O(N) time per sample.

### Meta-Analysis of Risk Differences Under Random Effects

**Hae-Min Jung**

Meta-analysis combines the results of multiple studies to infer an overall treatment effect. Random Effects methods are popular in dealing with variance between the treatment effects estimated by each study. Of the three most common effect measures for 2×2 contingency tables, the Risk Difference remains the least understood for Random Effects. Existing popular methods are structurally awed and use assumptions inappropriate for the Risk Difference.

We propose a new, unbiased estimate of the between-study variance under minimal assumptions. This new estimate is used to construct a Wilson Score-type confidence interval for the common Risk Difference. The new method is advantageous over existing methods due to its closed-form solution, lack of distributional assumptions, and flexibility regarding the selection of weights. Simulation studies show that it performs well under both low and high between-study variance.

### The Isomorphism and Centralizer Problems for Partially Bounded Transformations

** Alexander Kastner**

Ergodic theory is the study of measure-preserving transformations which map a measure space into itself. One of the central problems in the field is the isomorphism problem, which asks whether two given transformations are isomorphic. A somewhat related problem is the centralizer problem, which asks which transformations commute with a given transformation. While it can be shown that, in a precise sense, both the isomorphism and centralizer problems are intractable in general, there is hope that a complete characterization may be possible if we restrict our attention to the generic class of rank-one transformations. We investigate these questions for the newly defined class of partially bounded transformations, which encompasses many well-known infinite rank-one transformations such as the infinite Chacón and Hajian-Kakutani transformations. In particular, we establish that the only transformations that commute with a partially bounded transformation T are the powers T^{n} for integers n. Further, we characterize exactly when a partially bounded transformation is isomorphic to its inverse. Finally, we discuss generalizations to rank-one flows.

**On Trace Ideals, Duals of Ideals, and Annihilators**

**Nina Pande**

Let M be a module over a ring R. The trace ideal of M is an ideal of R obtained from the set of homomorphisms from M to R: it is the sum of the images of each homomorphism in R. We present results that describe the trace ideals of ideals (viewing these ideals as modules over R) satisfying various conditions. We determine several sufficient conditions for an ideal to be equal to its trace ideal. The set of homomorphisms from M to R is called the dual of M. We examine duals of ideals in terms of particular maps contained in the dual, in terms of grade, and in terms of generators of ideals. In addition, we characterize dimension zero Gorenstein rings in terms of their trace ideals. Throughout we explore the rich relationship between trace ideals and annihilators.

### Entropy Computation for Measure-Preserving Transformations

**Matthew James Quinn**

Entropy provides a notion of how random partitions and measure-preserving transformations are by looking at how many bits of information we can expect to gain from an experiment that is represented by them. In this paper, we provide original proofs that lead to showing that the generalized odometer, with an arbitrary sequence of cuts, has zero entropy. We likewise consider the Hajian-Kakutani Skyscraper transformation. However, entropy is traditionally defined on a finite measure space. Therefore, in studying the Hajian-Kakutani Skyscraper transformation, we suggest a new adjustment for entropy to be defined over a space of infinite measure. We then discuss an extension of our arguments to all rank-one transformations.

**Human Performance and Aging: A Statistical Approach**

**John Robert Shuck**

Previous studies of aging do not take into account the inherent uncertainty associated with aging across the lifespan. This thesis explores the deterioration of performance times in masters athletes with a new statistical approach. Uncertainty in performance deterioration with age is accounted for by more precise modelling techniques and through construction of empirical confidence intervals. Through the statistical methods proposed in this thesis, it is possible to quantify which age ranges show similar rates of performance deterioration across different masters athletic events. Ultimately, it is found that deterioration is similar in the 55-70 age range across many different sports, indicating the existence of a universal aging mechanism that is not mediated by sport specific selection variables.

**New Classes of Orthogonal Polynomials on the Unit Circle and Bernoulli Distributed Verblunsky Coefficients**

**Andy Yu Zhu Yao**

My thesis presents the general theory for orthogonal polynomials on the unit circle (OPUC) and reviews some classic examples of OPUC. I study and characterize new examples of orthogonal polynomials on the unit circle. Furthermore, I prove properties relating Bernoulli random Verblunsky coefficients and the zeros of the orthogonal polynomials.

### Generalizations of Multidimensional Continued Fractions: Tetrahedron and k-Dimensional

**Emmanuel Howard Daring
**

The decimal and continued fraction expansions of a number are periodic if and only if the number is rational or a quadratic irrational, respectively. Multidimensional continued fractions aim to replicate this property with different types of irrational numbers, partitioning a triangle to produce a periodic sequence if the coordinates of the point the sequence describes are at worst cubic irrationals in the same number field. In this paper, we redefine an existing multidimensional continued fraction algorithm which partitions the triangle. This new definition lends itself to being generalized to higher dimensions, partitioning any infinite n-th dimensional simplex to produce periodic sequences when the coordinates of a point are algebraic in the same number field of degree n + 1. We observe this generalization in action for the case n = 3, where the 3rd dimensional simplex is a tetrahedron.

### Moves on **Übercrossing** Projections of Links

**Xixi Edelsbrunner
**

Generally knot and link projections have crossings with two strands passing through. An übercrossing projection of a link *L* is a projection with exactly one crossing with any number of strands bisecting it. The übercrossing number *ü* is the least number of loops in any projection of *L*. We develop moves with which we can travel between übercrossing projections for a fixed knot. We also present a proof for a bound on braid index in terms of übercrossing number for link projections.

### Blowing Up Toric Varieites With Multidimensional Continued Fractions

**Elizabeth Frank
**

Because toric varieties are built up from convex geometry, there is a natural connection to be made with triangle partition maps, which are multidimensional continued fraction algorithms. Our motivation to explore this connection is showing that to resolve the curve y^{p}=x^{q} we follow a path of blowups given by the continued fraction expansion of p/q. Dividing the triangle according to the Triangle Map turns out to be equivalent to blowing up an axis in C^{3}. We apply these blowups to resolving singularities of curves. We also discuss these blowups and blowdowns in terms of attracting or repelling curves toward or away from curves defined by a quadratic irrational or a pair of cubic irrationals with a periodic triangle sequence.

### Bipyramind Decompositions of Multi-Crossing Link Complements

**Gregory Kehne
**

Generalizing constructions of D. Thurston and C. Adams, we present a dual pair of decompositions of the complement of a link *L* into bipyramids, given any multi-crossing projection of *L*. When *L* is hyperbolic, this gives new upper bounds on the volume of *L* given its multi-crossing projection, which empirically approach a constant factor of the volume for typical petal and über knots. Additionally, this construction yields families of immersed surfaces in the complement of *L*, including a generalization of checkerboard surfaces.

### Examining More Sum Than Difference Sets in Multiple Lattices

**Lawrence Luo
**

The definition of a traditional MSTD set is a finite set *A **⊂* Z whose sumset, defined by *A *+ *A *= *{**a*1 + *a*2 : *a*1*, a*2 *∈* *A**}*, is larger in cardinality than its difference set *A **− **A*, defined by *A **− **A *= *{**a*1 *− **a*2 : *a*1*, a*2 *∈* *A**}*. We can view a MSTD set in two dimensions by making a polytope *P* whose lattice points form the elements from which a set *A* is constructed. The sumset and difference sets of *A* are then calculated by adding or subtracting the coordinates of the lattice points.

Thao Do, Archit Kulkarni, Steven Miller, David Moon ‘16, Jake Wellens, and James Wilcox ’13 explored MSTD sets in **R**^{D} space with lattice points in **Z**^{D}, for D > 0, generalizing previous results in **Z** by allowing the dilation of polytopes in **R**^{D}. Instead of exploring MSTD sets in **R**^{D}, we examine the frequency of such sets in the **R**^{2} space specifically.

To observe MSTD sets in two dimensions, we first take the five possible lattices in the Euclidean plane (square, rectangular, rhombic, parallelogramic, and hexagonal) and prove that every convex polytope in one lattice has a “strongly” equivalent convex polytope for each of the four other lattices that preserves its MSTD characteristics.

We investigate and discuss the importance of interior points for preserving the MSTD nature of balanced convex polytopes across lattices, and also discuss how a difference dominant convex polytope remain difference dominant when taking the boundary plus any combination of its interior points.

We also discuss efficient boundary constructions for generating larger MSTD densities, and offer conjectures on optimal boundary constructions for the square and hexagonal lattices.

We conclude with a discussion on the shortcomings of the code used to generate our experimental data.

### Generalizing the Minkowski Function Using Triangle Partition Maps

**Peter Morton McDonald, Jr.
**

In this paper, we present two previous attempts at generalizing the Minkowski Question Mark Function before presenting a framework for generalizing ?(x) to a family of 216 multidimensional continued fraction algorithms known as triangle partition (TRIP) maps. Furthermore, we place these 216 maps into 15 classes whose associated generalization of the Question Mark Function is related by a linear transformation and show for 7 of these classes that this function is singular.

### Modeling the Dynamics of Persistence and Extinction in Ecology

**Alexander Dolnick Meyer
**

Mathematical models are invaluable tools for ecologists. A well-constructed model not only holds predictive power, but also matches observations made in the field and sheds light on the mechanisms that underlie the complex interactions between organisms and their environment. Simulation allows researchers to conduct experiments *in silico* that would be impractical or unethical to conduct in the field, enabling ecologists and conservationists to forecast the outcomes of environmental management strategies prior to implementation. My investigation explores the mechanisms of species persistence and extinction in two unique examples: the lethal white-nose syndrome (WNS) epidemic decimating New England’s bat populations, and the temporally/spatially synchronized emergences of periodical cicadas. In particular, I use models to determine the efficacy of several WNS control measures and to analyze the mechanisms that allow only one brood of cicadas to persist in a geographical area at the exclusion of all others.

### Colorful Graph Associahedra

**Mia Smith
**

Given a graph G, there exists a simple convex polytope called the graph associahedron of G whose face poset is based on the connected subgraphs of G. With the additional parameter of color assigned to each subgraph, we define the analogous semi-colorful graph associahedron and show it is a simple abstract polytope. Furthermore, we provide a construction based on the classical permutohedron and various combinatorial and topological properties.

### Effective Proofs of Khovanskii’s Theorem in the Integers

**Gabriel K. Staton
**

Khovanskii’s Theorem states that for any finite set A in a commutative semigroup, the sumset* nA* has polynomial size for sufficiently large values of n, but its proof is ineffective. We give novel and effective proofs of this theorem in finite fields and in the integers, and present some computational work and derived conjectures in higher dimensions.

### Rank One Mixing on Levels and a Numerical Analysis of Chacón Type Transformations and the Pascal Transformation

**Roger Vargas, Jr.
**

The rank-one canonical Chacón transformation has been widely studied and is a well understood transformation. It is known that this transformation is Mildly Mixing. The construction of this transformation can be altered in different ways to construct other rank-one (Chacón-type) transformations. Weakly mixing and stronger notions of mixing has not been proven for many of these Chacón-type transformations and the Pascal transformation (our construction of Pascal is not rank-one but is constructed using a cutting and stacking method similar to Chacón). We study the mixing properties of these transformations numerically by employing computational tools. We also give a proof for rank-one transformations that simplifies the condition of mixing for all measurable sets.

### Analysis of Technical Stock Trading Strategies

**Thomas Andrew Beaudoin, Jr.**

Technical analysis has been used by traders for decades, but it has been difficult to examine with the same level of rigor as fundamental analysis due to its highly subjective nature where two traders could see the same data and perform two different actions. In this paper, I evaluate the effectiveness of a few technical strategies that are less subjective and thus could be used algorithmically. I apply these strategies to the 1500 largest US stocks over the period of January 1, 1998 to December 31, 2007. By comparing the gains when testing the strategy over the time period versus the gains when holding the stock from the beginning of the time period, and by comparing the success rate on a stock to stock basis, I find that over the 10 year period one strategy is successful, many unsuccessful yielding potential shorting opportunities, and others yield no difference.

### The Crossing Map of Knots

**Wyatt Bradley Boyer**

The crossing map is a sphere around a knot where each point on the sphere is labeled with the number of crossings one sees in the knot when projected in that direction. Regions of constant value on the crossing map are divided by three types of curves that correspond to the three Reidemeister moves. We explore what each curve determines about the embedding of the knot in space.

**Cusp Thicknesses of Checkerboard Surfaces for a Family of Links**

**Benjamin Demeo**

The *cusp thickness* of a surface bounded by a hyperbolic knot *K* is a natural measure of its geodesicity, and depends crucially on the structure of the surface’s limit points in the universal cover *H ^{3}*. We introduce a method for analyzing limit sets of checkerboard surfaces using polyhedral decompositions and apply it to a family of links, obtaining the limit sets and cusp thicknesses. We then apply our results to relevant areas and consider further applications.

### Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-Functions

**Jesse Benjamin Freeman**

The Riemann Zeta-Function is the most studied L-function – its zeros give information about the prime numbers. We can associate L-functions to a wide array of objects. In general, the zeros of these L-functions give information about those objects. For arbitrary L-functions, the order of vanishing at the central point is of particular importance. For example, the Birch and Swinnerton-Dyer conjecture states that the order vanishing at the central point of an elliptic curve L-function is the rank of the Mordell-Weil group of that elliptic curve.

The Katz-Sarnak Density Conjecture states that this order vanishing (and other behavior) are well-modeled by random matrices drawn from the classical compact groups. In particular, the conjecture states that an average order vanishing (over a “family” of L-functions) can bounded using only a given weight function and a chosen test function φ. The conjecture is known for many families when the test functions are suitably restricted.

It is natural to ask which test function is best for each family and for each set of natural restrictions on φ. Our main result is a reduction of an otherwise infinite-dimensional optimization to a finite-dimensional optimization problem for all families and all sets of restrictions. We explicitly solve many of these optimization problems and compute the improved bound we obtain on average rank. While we do not verify the density conjecture for these new, looser restrictions, with this project, we are able to precisely quantify the benefits of such efforts with respect to average rank. Finally, we are able to show that this bound strictly improves as we increase support.

**History Dependent Stochastic Processes and Applications to Finance**

In this paper we focus on properties of discretized random walk, the stochastic processes achieved in their limit, and applications of these processes to finance. We go through a brief foray into probability spaces and sigma-fields, discrete and continuous random walks, stochastic process and Ito calculus, and Brownian motion. We study the properties that make Brownian motion unique and how it is constructed as a limit of a discrete independent random walk. Using this understanding we propose a different kind of random walk that remembers the past. We investigate this new random walk and find properties similar to those of the symmetric random walk.

We look at a well-studied stochastic process called fractional Brownian motion, which uses the Hurst parameter to remember past performance. Using Brownian motion and fractional Brownian motion to model stock behavior, we then detail the famous Black-Scholes options formula and a fractional Black-Scholes model. We compare their performance and accuracy through the observation of twenty different stocks in the market. Finally we discuss under what circumstances is the fractional model more accurate at predicting stock price compared to the standard model and explanations for why this might occur.

### Partially Rigid Strictly Doubly Ergodic Rank One Transformations and Related Examples

**Isaac Loh**

Rank one cutting and stacking transformations are a useful source of examples in infinite measure spaces. Interestingly, the structure of these cutting and stacking transformations can be reduced to integer sequences, and characterized by combinatorial methods. Our main work is to use these characterizations to develop new classes of transformations meeting certain properties. We develop a transformation which is strictly partially rigid and strictly doubly ergodic (i.e. with non-ergodic Cartesian square), and also give some general conditions for the conservativity of products of cutting and stacking transformations. We also have examples of fully rigid, strictly doubly ergodic transformations. We further the study of power weakly mixing transformations in infinite measure spaces by showing that all (t,q)-type Chacon maps are power weakly mixing, and demonstrating that there are such maps which also have strict partial rigidity but closely bounded recurrence–a result which could not be previously obtained from arguments on genericity. Finally, we come close to answering an open question by Bergelson by showing that there is an infinite measure preserving transformation T with extreme asymmetry: all rectangles sweep out under T, and T has infinite conservative index, but TxT^(-1) is not ergodic.

**Imagining a Space of Circular Split Networks**

**Samantha Petti**

Phylogenetic trees are structures used to represent evolutionary histories. A circular split network is a generalization of a tree in which multiple parallel edges signify divergence. We introduce a space of circular split networks and explore its properties. This space, which we call CSNn, is the natural extension of Billera, Holmes, and Vogtman’s tree space to circular split networks. We introduce this space and a topologically rich subspace Cn. We describe the interesting gluing properties of the space. Further, in computing the homotopy of the space, we find a connection between Cn and the real moduli space M0,n.

** Partial Rigidity Values on the Levels of Chacón-type Transformations**

**Eric George Schneider**

We describe a variety of Chacón-Type Transformations based on the Canonical Chacón Transformation that appeared in Friedman in 1970, including finite point extensions. In our search for an alternative proof to Del Junco in 1978 that the Canonical Chacón Transformation is Mildly Mixing we discovered many partial rigidity values over the levels. We provide a method for determining if a partial rigidity value on the levels of the Chacón Transformation is possible as well as establishing an upper bound. This method is then generalized to a class of Chacón-Type Transformations.

### Explicit **Forms For and Some Functional Analysis Behind a Family of Multidimensional Continued Fractions — Triangle Partition Maps — and Their Associated Transfer Operators**

**Ilya D. Amburg**

The family of 216 multidimensional continued fractions known as known as triangle partition maps (TRIP maps for short) has been used in attempts to solve the Hermite problem [3], and is hence important in its own right. This thesis focuses on the functional analysis behind TRIP maps. We begin by finding the explicit form of all 216 TRIP maps and the corresponding inverses. We proceed to construct recurrence relations for certain classes of these maps; afterward, we present two ways of visualizing the action of each of the 216 maps. We then consider transfer operators naturally arising from each of the TRIP maps, find their explicit form, and present eigenfunctions of eigenvalue 1 for select transfer operators. We observe that the TRIP maps give rise to two classes of transfer operators, present theorems regarding the origin of these classes, and discuss the implications of these theorems; we also present related theorems on the form of transfer operators arising from compositions of TRIP maps. We then proceed to prove that the transfer operators associated with select TRIP maps are nuclear of trace class zero and have spectral gaps. We proceed to show that select TRIP maps are ergodic while also showing that certain TRIP maps never lead to convergence to unique points. We finish by deriving Gauss-Kuzmin distributions associated with select TRIP maps.

### Resampling Methods With Applications in Variance Estimation

**Shiwen Chen**

In point estimation, the true parameter of interest theta of the population distribution F is often estimated by a functional of the empirical distribution of n observations, denoted as q_{n} = q(F_{n}). In statistical practice, it is important to learn about the sampling distribution and assess the precision of a point estimator by estimating its variance. For many functionals, closed form expression of the sampling distribution is not available. Resampling schemes reproduce samples from the original set of observations. With the help of these reproduced samples, the sampling distribution of q_{n} can be estimated. This thesis explores a set of topics related to resampling schemes. We make contributions in two directions: the proposal of a general class of linearly extrapolated variance estimators as a generalization of the delete-one jackknife variance estimator, and the investigation of resampling schemes for dependent data, in particular, spatial data.

### Completions of Unique Factorization Domains With Unique Factorization Modulo a Principal Prime Ideal

**Craig Matthew Corsi**

We present new work in the theory of complete local rings. Given a complete local ring *T* with maximal ideal *M*, and given *p* Î *R*, we conjecture that a set of weak conditions is necessary and sufficient to ensure the existence of a local unique factorization domain *R* such that *p* Î *R* and *R/pR* is also a unique factorization domain. We make significant progress toward proving this claim. Then, given a complete local ring *T *with maximal ideal *M*, *C* a countable set of nonmaximal, pairwise incomparable set of prime ideals of *T*, and *p **Î* Ç_{P }_{Î}_{ C}*P*, we give necessary and sufficient conditions for *T* to be the completion of a local integral domain *A* such that *p **Î** A*, and p*A* is a prime ideal whose formal fiber has maximal elements the elements of *C*. We also give conditions under which A can be constructed to be excellent.

### A Characterization of Trees With Convex Obstacle Number 1 or 2

**Philippe Demontigny**

A convex obstacle representation of a tree T is a drawing of the vertices of T in the plane with a set of convex polygons so that two vertices are connected by an edge if and only if that edge does not intersect any of the polygons. The minimum number of obstacles required to represent a tree in this way is called the convex obstacle number of the tree. This new description of a graph has steadily gained popularity since its introduction in 2009, and is particularly interesting because of its relationship to visibility graphs, which have been studied extensively and have applications in robot motion planning and architecture. So far, it is known that a representation using only 5 convex obstacles exists for all trees, which implies that the upper bound for the convex obstacle number of any given tree is five. However, not much is known about which trees have a convex obstacle number that is less than 5. In this thesis, we begin to fill this gap by providing necessary and sufficient conditions for a tree to have convex obstacle number 1 or 2. We also provide insights into how one could approach the problem of finding all trees with convex obstacle number 3 or 4.

### Modelling and improving Pitching Strategies in Major League Baseball

**Carson Eisenach**

In Major League Baseball, the bullpen is perhaps the most poorly utilized of a teams resources. Finding better strategies for using the bullpen is very valuable. In this senior thesis, I explore a framework for analyzing pitching strategies in Major League Baseball. The main contributions of this thesis are (1) the development of an extensive set of tools to create the game state data needed to analyze pitching strategies and (2) using the data discovered to develop several models for pitching strategies as well as metrics by which to assess model fit.

### A Practical Review of Time-Series Forecasting Using A Large Number of Predictors

**Vu Le**

Time-series forecasting using a large number of predictors has received increasing attention in recent years. Stock and Watson (1999), Bai and Ng (2002, 2007), among others, have developed techniques to extract relevant information from a large set of forecasting variables with promising results. In this paper, we first study the main theory behind this topic, the diffusion index forecasting model (Stock and Watson, 1999). It estimates unobserved factors nonparametrically by principal components of existing predictors. We have confirmed that the model outperforms the benchmark alternatives in terms of mean squared forecasting error (MSFE). We then explore and present the empirical efficacy of proposed refinements to the model. These include the determination of the number of factors and predictor selection using soft and hard thresholding (Bai and Ng, 2007). Along the way, we will propose and test some potential refinements to the existing methodologies.

### Benford’s Law and Fraud Detection

**Yang Lu**

Benford’s Law describes the situation in which the frequency distribution of the first digits in a real-life data set does not follow a uniform distribution. Rather, the probability of a digit d occurring as the leading digit of a data point is the difference between d+1 and d on a logarithmic scale with base 10. This phenomenon has been regularly used by auditors as a tool to detect fraud. The bootstrap is a resampling method which can be used to estimate the sampling distribution of an estimator, first proposed by Efron in 1979.

We apply the bootstrap method to find a way to reduce the number of data points required for effective fraud detection based on Benford’s Law. Oftentimes an auditor may not have access to all the data; a method using a subset of the data for fraud detection makes the auditor’s job possible and potentially saves both human and computational resources. We have found that assuming that a data set of size 5000 or more is either free of manipulation or a result of summing the original values and some values following either a normal distribution or a uniform distribution, we only require 5% of the data to detect potential data manipulation.

### Relieving and Readjusting Pythagoras: Improving the Pythagorean Theorem

**Victor Dan Luo**

The Pythagorean expectation was invented by Bill James in the late 70’s as a way of calculating how many wins a baseball team should have by utilizing just runs scored and runs allowed. His original formula predicts a winning percentage of RS^2/(RS^2+RA^2), where RS stands for runs scored and RA stands for runs allowed. Although the simplicity of the formula is a thing of beauty, with the development of more advanced baseball statistics it should be possible to enhance the formula such that it gives a more accurate prediction of a team’s wins. Implementing statistics such as ballpark effect as well as accounting for game state factors, we will test to see if it is indeed the case that adjusting the Pythagorean expectation formula gives a statistically significantly better prediction for a team’s wins than the unadjusted formula.

In order to test these adjusted formulas, we will be culling data from the internet, specifically from http://www.retrosheet.org/gamelogs/index.html and espn.com. We will then import this data into R and use our code to manipulate the data, calculating the new adjusted Pythagorean expectation and old Pythagorean expectation by year for each team. Then, using different regression models, we will determine which expectation formula is more accurate.

In addition, it has been shown that we can use a Weibull distribution in order to model run production. The versatility of the distribution is due to the fact that it accounts for three parameters that can be varied to adjust the spread, shift, and scale of the distribution. We will explore whether a linear combination of Weibulls is able to more accurately determine a team’s run production.

### Completions of Reduced Local Rings With Prescribed Minimal Prime Ideals

**Byron J. Perpetua**

A central question in commutative algebra asks when a complete local ring T is the completion of a local subring A, subject to a given set of conditions on A. Arnosti et al. answer the question when T is assumed to contain the rationals and A is required to be a reduced local ring with a finite number of maximal elements in the formal fiber at each of its minimal prime ideals. In this thesis, we construct A so that the number of maximal elements in the formal fiber at each minimal prime ideal is countable instead of finite, and we introduce a new set of necessary and sufficient conditions for T to be the completion of such a ring A, which weaken the requirement that T contain the rationals.

### Simultaneous Inference on Margins of Binary Data

**Faraz Wasiur Rahman**

Simultaneous inference deals with testing several hypotheses that might be related. Techniques in this field control the chances of rejecting one or more of the true hypotheses at a low value like 5%, and then try to reject the remaining false hypotheses with high probability. We will develop and evaluate the performance of a closed testing procedure to analyze the margins of binary data, and explore some applications of the technique in a clinical trial scenario.

### Testing Benford’s Law

**Jirapat Samanvedhya**

Benford’s Law, a phenomenon of the first-significant bias, is often used in fraud and data integrity detection. The application of Benford’s Law uses goodness-of-fit test which typically involves the chi-square, the Kolmogorov-Smirnov, or the Kuiper’s tests. There are some issues with the comparison of these tests: 1) the latter two are designed to test continuous distributions and they are found to be too conservative for discrete distributions 2) we must compare both Type I error and power. This thesis aims to address those issues by using a simulation-based approach to normalize Type I error and bypass the complication with continuity.

We use the Monte Carlo method to compare powers of the Kolmogorov-Smirnov test, the Kuiper’s test and the power-divergence test, which is a family of test statistics that generalizes the chi-square test. We test Benford’s Law against five major discrete distributions (uniform, linear, Poisson, geometric and binomial distributions) and generalize to 2-digit Benford’s Law. We find that the Kuiper’s test is one of the most powerful tests in general. However, when the alternative distribution is very similar to the Benford distribution, the power-divergence test has more power. We then test two real data sets whose distribution of first significant digits visually exhibits Benford behavior. For the AAPL daily trading volume dataset, all tests reject the null hypothesis that it obeys Benford’s Law. For the streamflow dataset, the power-divergence tests rejects the null hypothesis at 95% confidence level, whereas the other tests do not.

### A Novel Model for White-Nose Syndrome in Little Brown Bats

**David F. Stevens**

Bats are important reservoirs for emerging human and wildlife diseases. Certain pathogens that are highly virulent to humans are able to persist in healthy bats and little is known about the mechanisms by which bat immune systems are able to cope with these diseases. White nose syndrome (WNS) is a devastating emerging infectious disease in North American bat populations. In 2006, the first incidence of bats infected with WNS was discovered in a cave near Albany, New York. It has since spread rapidly across eastern North America. WNS is caused by a newly described fungus, Geomyces destructans, that grows on the exterior of hibernating bats. The infection is thought to rouse infected bats from hibernation, depleting essential fat stores and resulting in death by starvation. This disease is forecasted to cause the regional extinction of little brown bats in the northeastern United States, with the potential for serious consequences for ecosystem integrity. In this paper we outline disease control strategies for WNS with the aim of preventing the regional extinction of *Myotis lucifugus*. For this purpose, we develop a mixed-time SEI model for WNS in *Myotis lucifugus* broken into three stages: (1) roosting, (2) swarming, and (3) hibernation.

### Maximal Bipartite Subgraphs of Special 4-Regular Planar Graphs

**Sean Sutherland**

Graph theorists have long been interested in determining the number of edges included in the largest bipartite subgraph of a given graph. We first provide a simple proof to show that every 4-regular planar graph has a maximal bipartite subgraph containing at least 2/3 of its edges. By applying Hadlock’s [3] procedure for determining bipartite subgraphs of planar graphs, we provide the exact size of the maximal bipartite subgraphs of two special classes of 4-regular planar graphs. We conclude by outlining our attempt to prove that the only graphs for which the maximal bipartite subgraph realizes the 2/3 ratio are the triangular checkerboard graphs.

### Limiting Spectral Measures of Random Matrix Ensembles With a Polynomial Link Function

**Kirk Swanson**

Given an ensemble of N by N random matrices, an interesting question to ask is: do the empirical spectral measures of typical matrices converge to some limiting measure as N tends to infinity? The limiting measures of several canonical matrix ensembles have been well-studied, such as the symmetric Wigner, Toeplitz, and Hankel matrices. It is known that in the limit, the Wigner matrices have a semicircular distribution, the Toeplitz have a near-Gaussian distribution, and the Hankel have a non-unimodal distribution. Although it is not fully understood why, these ensembles exhibit the interesting property that as more constraints are introduced to a patterned random matrix, new limiting measures other than the semicircle can arise. It is natural, then, to explore the question: to what extent will a patterned random matrix continue to have a semicircular limiting eigenvalue distribution? In the following, we explore this question by generalizing the Toeplitz and Hankel ensembles. The resulting matrix ensembles with bivariate polynomial link functions have unique limiting spectral distributions. In specific cases, we establish that when the variables in the polynomial are raised to the same power the limiting measure becomes non-semicircular, but when the variables are raised to different powers the limiting measure remains semicircular.

### Primes in Arithmetic Progressions of Polynomials

**Samuel Tripp**

Dirichlet’s Theorem on Primes in Arithmetic Progressions states that if a and b are coprime, there are infinitely many primes congruent to a modulo b. The proof, however, is quite analytic. Murty and Thain ask for which pairs of integers a and b this can be proved algebraically. They prove that if the square of a is congruent to 1 modulo b, there is an algebraic proof that there are infinitely many primes congruent to a modulo b.

Knowing that the analogue of Dirichlet’s Theorem holds in the function field case, we present results establishing an analogue of the results of Murty and Thain in the function field case as well.

### 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 Â = 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 1_{1}, the space of absolutely summable sequences, or c_{0}, 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[[X_{1}, …, X_{n}]] (where X_{1}, …, 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[X_{1}, …, 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[X_{1}, …, 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 R^{2}

**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 Ã p, and suppose that |{q Î Spec T | q Ë p}| £ | T/M^{2}| and p intersected with the prime subring of T is the zero ideal. In this case, we construct a local domain A such that Â = T and the generic formal fiber ring of A is local with p^{*}_{A}K 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 III_{0} action with weakly mixing basis transformations. In addition, the several concepts and criteria applying to Z^{2} 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.