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Coordinates Arising From Affine Fibrations, Andrew Lewis

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We study the question of whether residual coordinates arising from affine fibrations are coordinates. We show that the second Venereau polynomial is a coordinate, and introduce a related class of residual coordinates, called Venereau-type polynomials, and show many of them to be coordinates. We give some partial results towards the Dolgachev-Weisfeiler conjecture in the case of tame strongly residual coordinates.

Quotients Of Subgroup Lattices Of Finite Abelian P-Groups, Marina Dombrovskaya

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Let G be a finite abelian p-group of type λ. It is well-known that the lattice L(p) of subgroups of G is the order-theoretic p-analogue of the chain product [0, λ]. However, any surjection φ : L(p) → [0, λ] with order analogue properties does not respect group automorphisms. We are interested in L, the quotient lattice of L(p) under the action of a Sylow p-subgroup of the automorphism group of G. This quotient lattice is particularly interesting since it respects group automorphisms, has the property that the size of an orbit of the action is a power of p, …

Comparison Theorems In Elliptic Partial Differential Equations With Neumann Boundary Conditions, Jeffrey Langford

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In this thesis, we study how the solution of a PDE changes when the data are rearranged. Specifically, we prove comparison theorems on spherical shells, spheres, and hemispheres, showing that under rearrangement of the data, the solution's convex mean increases. We also obtain similar weighted comparison results in balls.

Castelnuovo-Mumford Regularity Of General Rational Curves On Hypersurfaces., Sara Gharahbeigi

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We show that for a general smooth rational curve on a general hypersurface of degree $d\leq N$ in $\mathbb{P}^N$, $N\geq 4$, the restriction map of global sections is of maximal rank, and therefore the regularity index of such curves is as small as possible.

Billiard Markov Operators And Second-Order Differential Operators, Jasmine Ng

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We will consider a class of Markov operators that arise from billiard dynamical systems. In addition to discussing results about their convergence to second-order differential operators, we will approximate the spectrum of one in terms of the other.

Composite Multi Resolution Analysis Wavelets, Benjamin Manning

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Composite dilation wavelets are a class of wavelets that include additional dilations from a countable subgroup of the invertible matrices. We consider the case when these additional dilation matrices form a finite group. A theory of MRA wavelets is established in this setting along with a theory of shift invariant subspaces. We examine accuracy of this class of MRA wavelets and produce several examples of compactly support composite MRA wavelets.

Using Dirichlet Process Priors For Bayesian Mixture Clustering, Xiao Huang

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We describe a non-parametric Bayesian model using genotype data to classify individuals among populations where the total number of populations is unknown. The model assumes that a population is characterized by a set of allele frequencies that follow multinomial distributions. The Dirichlet Process is applied as the prior distribution. The method estimates the number of populations together with the allele frequencies and the ancestry coefficients of each individual. Distance matrices and bootstrap support numbers based on MCMC runs are generated to create a phylogeny of the ancestral populations.

The Boundary Behavior Of Holomorphic Functions, Baili Min

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In the theory of several complex variables, the Fatou type problems, the Lindel\"{o}f principle, and inner functions have been well studied for strongly pseudoconvex domains. In this thesis, we are going to study more generalized domains, those of finite type. In Chapter 2 we show that there is no Fatou's theorem for approach regions complex tangentially broader than admissible ones, in domains of finite type. In Chapter 3 discussing the Lindel\"{o}f principle, we provide some conditions which yield admissible convergence. In Chapter 4 we construct inner functions for a type of domains more general than strongly pseudoconvex ones. Discussion is …

Markov Chains Derived From Lagrangian Mechanical Systems, Scott Cook

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The theory of Markov chains with countable state spaces is a greatly developed and successful area of probability theory and statistics. There is much interest in continuing to develop the theory of Markov chains beyond countable state spaces. One needs good and well motivated model systems in this effort. In this thesis, we propose to produce such systems by introducing randomness into familiar deterministic systems so that we can draw upon the existing: deterministic) results to aid the analysis of our Markov chains. We will focus most heavily on models drawn from Lagrangian mechanical systems with collisions: billiards).

Computable Performance Analysis Of Recovering Signals With Low-Dimensional Structures, Gongguo Tang

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The last decade witnessed the burgeoning development in the reconstruction of signals by exploiting their low-dimensional structures, particularly, the sparsity, the block-sparsity, the low-rankness, and the low-dimensional manifold structures of general nonlinear data sets. The reconstruction performance of these signals relies heavily on the structure of the sensing matrix/operator. In many applications, there is a flexibility to select the optimal sensing matrix among a class of them. A prerequisite for optimal sensing matrix design is the computability of the performance for different recovery algorithms. I present a computational framework for analyzing the recovery performance of signals with low-dimensional structures. I …

Physical Models In Community Detection With Applications To Identifying Structure In Complex Amorphous Systems, Peter Ronhovde

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We present an exceptionally accurate spin-glass-type Potts model for the graph theoretic problem of community detection. With a simple algorithm, we find that our approach is exceptionally accurate, robust to the effects of noise, and competitive with the best currently available algorithms in terms of speed and the size of solvable systems. Being a "local" measure of community structure, our Potts model is free from a "resolution limit" that hinders community solutions for some popular community detection models. It further remains a local measure on weighted and directed graphs. We apply our community detection method to accurately and quantitatively evaluate …

Modeling Aerial Refueling Operations, Allen Mccoy

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Aerial Refueling: AR) is the act of offloading fuel from one aircraft: the tanker) to another aircraft: the receiver) in mid flight. Meetings between tanker and receiver aircraft are referred to as AR events and are scheduled to: escort one or more receivers across a large body of water; refuel one or more receivers; or train receiver pilots, tanker pilots, and boom operators. In order to efficiently execute the Aerial Refueling Mission, the Air Mobility Command: AMC) of the United States Air Force: USAF) depends on computer models to help it make tanker basing decisions, plan tanker sorties, schedule aircraft, …

Properties Of Truncated Toeplitz Operators, Nicholas Sedlock

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We discuss the multiplication of truncated Toeplitz operators: or TTOs) on backward shift invariant subspaces of the Hardy space of the unit disc. Specifically we discuss when the product of two TTOs is itself a TTO, finding an equivalent to a similar result of Brown and Halmos for ordinary Toeplitz operators. This leads us to investigate the commutants of certain rank-one perturbations of the compressed shift operator, deriving a symbol calculus for TTOs, as well as several other results.

Equivariant Deformations Of Horospherical Surfaces, Michael Deutsch

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The classical Goursat transform for minimal surfaces is interpreted as conformal transformation of the Gauss map, allowing us to "bend" these surfaces for certain geometric purposes. A simple analogue of this deformation is defined for CMC1 surfaces which makes the Goursat transform equivariant with respect to the Lawson correspondence, thereby increasing the number of explicitly computable examples of minimal/CMC1 cousin pairs. We then indicate how the Goursat transformation law and integrability conditions for the "spin curve" of a horospherical surface are analogous to the Lorentz transformation law and equations of motion for the wavefunction of a massless fermion.

The Nonexistence Of Shearlet-Like Scaling Multifunctions That Satisfy Certain Minimally Desirable Properties And Characterizations Of The Reproducing Properties Of The Integer Lattice Translations Of A Countable Collection Of Square Integrable Functions, Robert Houska

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In Chapter 1, we introduce three varieties of reproducing systems—Bessel systems, frames, and Riesz bases—within the Hilbert space context and prove a number of elementary results, including qualitative characterizations of each and several results regarding the combination and partitioning of reproducing systems.

In Chapter 2, we characterize when the integer lattice translations of a countable collection of square integrable functions forms a Bessel system, a frame, and a Riesz basis.

In Chapter 3, we introduce composite wavelet systems and generalize several well-known classical wavelet system results—including those regarding pointwise values of the Fourier transform of the wavelet and scaling function …

Filling Essential Laminations, Michael Hamm

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Thurston and, later, Calegari-Dunfield found superlaminations in certain laminated 3-manifolds, the existence of which implies inclusions into Homeo S1 of the fundamental groups of those manifolds. The present paper extends the construction of the superlamination, and finds an infinite class of manifolds to which the extension does not yield such an inclusion of groups. Specifically, Calegari and Dunfield's proof of the existence of such an inclusion into Homeo S1 depended on their filling lemma, which states that essential laminations with solid torus guts can have leaves added to them to yield essential laminations with solid torus complementary regions.: Roughly, a …

Connections Between Floer-Type Invariants And Morse-Type Invariants Of Legendrian Knots., Michael Henry

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We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\sK$ with generic front projection $\sfront$, we define a combinatorial/algebraic object on $\sfront$ called a \emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\sNgres$ of the front $\sfront$. …