Michael W. Daub
One of the main objects of study in algebraic number theory is the class group of a number field K. The integral closure of Z in K is a Dedekind domain, and we can define an equivalence relation on its nonzero ideals. The class group is the finite abelian group formed by the equivalence classes of these ideals. The class number, or size of the class group, is 1 if and only if the ring is a unique factorization domain, and in fact, the class group gives even more information about the structure of the ring and its field of fractions. One tool used to study the class groups of K is using a special field extension of K, called the Hilbert class field of K. In particular, if the Hilbert class field of K has class number 1, then we get some nice results about the class numbers of fields related to K.
Anna Elizabeth Ferguson
Diversity of form and function are hallmarks of the animal kingdom, despite being surprisingly similar. How are the similarities and differences in traits seen in animals translated in the genome? Rearrangement of the genome yields either a change or conservation in protein function. As lineages separate, each function of a multifunctional protein may separate as well, resulting in similar, but different proteins, called homologues. Since proteins are the mechanism by which traits are realized, the study of homologues may reveal more about the how such great diversity is achieved from just one source.
I study hunchback, a member of the C2H2 zinc finger class of proteins, known in protostomes (a group of animals), and Ikaros, a C2H2 zinc finger protein in deuterostomes (another group of animals) that highly resembles hunchback structurally. Since simple search processes based on sequence identity shared do not find a deuterostome hunchback homologue, many researchers have concluded that the hunchback protein has been lost in the deuterostome lineage. However, the conservation of zinc finger structure suggests that there may be a common origin between Ikaros and hunchback. Throughout my research, I used comparative analyses using structural elements in the hunchback class of proteins as well as molecular phylogenetic data to try and differentiate the two groups. Together, these two independent units of comparative data suggest that the two classes of zinc finger proteins are separate and that Ikaros must not be the hunchback homologue in deuterostomes. Based on these results, I provide two possible evolutionary histories of the two classes of proteins.
Son L. Ho
Our main goal is to build a machinery to work with manifolds of higher codimension in Rn+k. We take a look at the map from the manifold M to a polynomial space, defined by using second fundamental form matrices. the multiplicity of roots of polynomials in (M) are preserved under various changes of coordinates. This leads us to believe that the map captures interesting geometric information of M. And in the last section we prove a result which indicates that it is the case for a certain type of codimension 2 manifolds.
Haydee Lindo
Matthew D. Simonson
We solve the isoperimetric problem, the least-perimeter way to enclose a given area, on various Euclidean, spherical, and hyperbolic surfaces, sometimes with cusps or free boundary. On hyperbolic two-holed tori, Adams and Morgan characterized the four possible types of isoperimetric regions. We show that on every hyperbolic two-holed torus, one of the solutions must be an annulus.
In a Euclidean planar annulus bounded by two concentric circles, we show that the least-perimeter way to enclose a given area is an arc against the outer boundary or a pair of spokes. We generalize this result to regions on the sphere or hyperbolic plane bounded by circles, horocycles, and other constant-curvature curves. In one case the solution alternates back and forth between two types, a phenomenon we have yet to see in the literature.
We also examine non-orientable surfaces such as Mobius bands and twisted chimney spaces.
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.
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.
Paul Alexander Woodard
Given a sequence space S, we can define an equivalence relation ~x on S by (xn ~x(yn) for (xn),(yn) in S if and only if (yn-xn) is in X, where X is a subspace of S, such as 11, the space of absolutely summable sequences, or c0, the space of sequences converging to 0. The quotient space S/~x is also a vector space, so we can study the linear functionals which act on it. To this end, we examine infinite matrices whose rows, as elements of the dual space of X converge weak* to (0).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Michelle D. Lee
We examine the least-perimeter way to enclose given area in various spaces including some spaces with density.
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Aaron D. Weinberg
Let (T, M) be a complete regular local ring of dimension at least two containing the rationals, such that the cardinality of the residue field T/M is at least the cardinality of the real numbers. Suppose p is a nonmaximal prime ideal of T and L is a set of prime ideals of T[[X1, ..., Xn]] (where X1, ..., Xn are indeterminates) such that the cardinality of L is strictly less than the cardinality of T/M, Q intersected with T is a subset of p for each Q in L, and if Pi is the prime subring of T, then Pi[X1, ..., Xn] intersected with Q is the zero ideal for each Q in L. Then there exists an excellent regular local ring A such that the completion of A is T, the generic formal fiber of A is local (this means that the ring T \otimes_A K is a local ring where K is the quotient field of A) with p \otimes_A K its maximal ideal, and Q intersected with A[X1, ..., Xn] is the zero ideal for each Q in L.
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.
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.
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.
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.
Deborah L. Greilsheimer
Christer Lech characterized the complete local (Noetherian) rings that are completions of domains. We reprove Lech's result showing that a complete local ring T is the completion of a local domain if and only if no integer of T is a zero divisor, and, unless equal to (0), the maximal ideal of T does not belong to (0) as an associate prime ideal. Moreover, suppose p (does not equal) M is a prime ideal of T such that Q is an associated prime of T implies Q à p, and suppose that |{q Î Spec T | q Ë p}| £ | T/M2| and p intersected with the prime subring of T is the zero ideal. In this case, we construct a local domain A such that  = T and the generic formal fiber ring of A is local with p*AK the maximal ideal where K is the quotient field of A.
Erich Muehlegger
This thesis presents two new examples of staircase Zd actions, functions mapping points from Zd x R to R. Expanding on the work of Silva/Adams and Touloumtzis, the first construction is shown to be an infinite measure preserving action with weakly mixing basis transformations. The second and more interesting example is a type III0 action with weakly mixing basis transformations. In addition, the several concepts and criteria applying to Z2 actions are generalized to their Zd counterpart.
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.
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.
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.
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.
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.
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.
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.
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.