Mathematics Commons^™

Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy

Doctoral Dissertations

Many ecological systems exhibit noisy period-2 oscillations and, when they are spatially extended, they undergo phase transition from synchrony to incoherence in the Ising universality class. Period-2 cycles have two possible phases of oscillations and can be represented as two states in the bistable systems. Understanding the dynamics of ecological systems by representing their oscillations as bistable states and developing dynamical models using the tools from statistical physics to predict their future states is the focus of this thesis. As the ecological oscillators with two-cycle behavior undergo phase transitions in the Ising universality class, many features of synchrony and equilibrium …

Real-Time Dengue Forecasting In Thailand: A Comparison Of Penalized Regression Approaches Using Internet Search Data, Caroline Kusiak

Masters Theses

Dengue fever affects over 390 million people annually worldwide and is of particu- lar concern in Southeast Asia where it is one of the leading causes of hospitalization. Modeling trends in dengue occurrence can provide valuable information to Public Health officials, however many challenges arise depending on the data available. In Thailand, reporting of dengue cases is often delayed by more than 6 weeks, and a small fraction of cases may not be reported until over 11 months after they occurred. This study shows that incorporating data on Google Search trends can improve dis- ease predictions in settings with severely …

Asymptotic Behavior Of The Random Logistic Model And Of Parallel Bayesian Logspline Density Estimators, Konstandinos Kotsiopoulos

Doctoral Dissertations

This dissertation is comprised of two separate projects. The first concerns a Markov chain called the Random Logistic Model. For r in (0,4] and x in [0,1] the logistic map f_r(x) = rx(1 - x) defines, for positive integer t, the dynamical system x_r(t + 1) = f(x_r(t)) on [0,1], where x_r(1) = x. The interplay between this dynamical system and the Markov chain x_r,N(t) defined by perturbing the logistic map by truncated Gaussian noise scaled by N^-1/2, where N -> infinity, is studied. A natural question is …

The Expected Total Curvature Of Random Polygons, Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, Clayton Shonkwiler

Robert Kusner

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.

We then consider the symmetric measure on closed polygons of fixed total length constructed by …

A Table Of Elliptic Curves Over The Cubic Field Of Discriminant −23, Steve Donnelly, Paul Gunnells, Ariah Klages-Mundt, Dan Yasaki

Paul Gunnells

Abstract. Let F be the cubic field of discriminant −23 and OF its ring of integers. Let 􀀀 be the arithmetic group GL2(OF ), and for any ideal n ⊂ OF let 􀀀0(n) be the congruence subgroup of level n. In [16], two of us (PG and DY) computed the cohomology of various 􀀀0(n), along with the action of the Hecke operators. The goal of [16] was to test the modularity of elliptic curves over F. In the present paper, we complement and extend the results of [16] in two ways. First, we tabulate more elliptic curves than were found …

Metaplectic Demazure Operators And Whittaker Functions, Gautam Chinta, Paul Gunnells, Anna Pusk´As

Paul Gunnells

Abstract. In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

Parametric Sensitivity Analysis For Biochemical Reaction Networks Based On Pathwise Information Theory, Yannis Pantazis, Markos Katsoulakis, Dionisios G. Vlachos

Markos Katsoulakis

Background: Stochastic modeling and simulation provide powerful predictive methods for the intrinsic understanding of fundamental mechanisms in complex biochemical networks. Typically, such mathematical models involve networks of coupled jump stochastic processes with a large number of parameters that need to be suitably calibrated against experimental data. In this direction, the parameter sensitivity analysis of reaction networks is an essential mathematical and computational tool, yielding information regarding the robustness and the identifiability of model parameters. However, existing sensitivity analysis approaches such as variants of the finite difference method can have an overwhelming computational cost in models with a high-dimensional parameter space. …

Data Combination From Multiple Sources Under Measurement Error, Hugo Gasca-Aragon

Open Access Dissertations

Regulatory Agencies are responsible for monitoring the performance of particular measurement communities. In order to achieve their objectives, they sponsor Intercomparison exercises between the members of these communities. The Intercomparison Exercise Program for Organic Contaminants in the Marine Environment is an ongoing NIST/NOAA program. It was started in 1986 and there have been 19 studies to date. Using this data as a motivation we review the theory and practices applied to its analysis.

It is a common practice to apply some kind of filter to the comparison study data. These filters go from outliers detection and exclusion to exclusion of …

Torus Orbits On Homogeneous Varieties And Kac Polynomials Of Quivers, Paul Gunnells, Emmanuel Letellier, Fernando Rodriguez Villegas

Paul Gunnells

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

On The Cohomology Of Linear Groups Over Imaginary Quadratic Fields, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schurmann, Mathieu Dutour Sikiric, Dan Yasaki

Paul Gunnells

Let 􀀀 be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of 􀀀 for N = 3, 4 and for a selection of discriminants: D −24 when N = 3, and D = −3,−4 when N = 4. In particular we compute the integral cohomology of 􀀀 up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for 􀀀 developed by Ash [4, Ch. II] and Koecher [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K4(OD), when D = −3,−4.

Local Torsion On Abelian Surfaces, Adam Gamzon

Open Access Dissertations

Fix an integer d > 0. In 2008, Chantal David and Tom Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This dissertation is an extension of that work, investigating the frequency with which a principally polarized abelian surface A over Q with real multiplication by Q adjoin a squared-root of 5 has a nontrivial p-torsion point defined over K. Averaging by height, the main result shows that A …

Spatial Evolutionary Game Theory: Deterministic Approximations, Decompositions, And Hierarchical Multi-Scale Models, Sung-Ha Hwang

Open Access Dissertations

Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide …

A Mathematical Growth Model Of The Viral Population In Early Hiv-1 Infections, Elena Edi Giorgi

Open Access Dissertations

In this thesis we develop a mathematical model to describe HIV-1 evolution during the first stages of infection (approximately within 40-60 days since onset), when one can assume exponential growth and random accumulation of mutations under a neutral drift. We analyze the Hamming distance (HD) distribution under different models (synchronous and asynchronous) in the absence of selection and recombination. In the second part of the thesis, we introduce recombination and develop a combinatorial approach to estimate the new HD distribution. We conclude describing a T statistic to test significance differences between the HD of two genetic samples, which we derive …

Knot Contact Homology And Open Strings, Jason Frederick Mcgibbon

Open Access Dissertations

In this thesis, we give a topological interpretation of knot contact homology, by considering intersections of a particular class of chains of open strings with the knot itself. In doing so, we provide evidence toward a differential graded algebra structure on the algebra generated by chains of open strings.

Statistical Methods For Nonlinear Dynamic Models With Measurement Error Using The Ricker Model, David Joseph Resendes

Open Access Dissertations

In ecological population management, years of animal counts are fit to nonlinear, dynamic models (e.g. the Ricker model) because the values of the parameters are of interest. The yearly counts are subject to measurement error, which inevitably leads to biased estimates and adversely affects inference if ignored. In the literature, often convenient distribution assumptions are imposed, readily available estimated measurement error variances are not utilized, or the measurement error is ignored entirely. In this thesis, ways to estimate the parameters of the Ricker model and perform inference while accounting for measurement error are investigated where distribution assumptions are minimized and …

Geometry Of Satake And Toroidal Compactifications, Patrick Michael Boland

Open Access Dissertations

In [JM02, section 14], Ji and MacPherson give new constructions of the Borel--Serre and reductive Borel--Serre compactifications [BS73, Zuc82] of a locally symmetric space. They use equivalence classes of eventually distance minimizing (EDM) rays to describe the boundaries of these compactications. The primary goal of this thesis is to construct the Satake compactifications of a locally symmetric space [Sat60a] using finer equivalence relations on EDM rays. To do this, we first construct the Satake compactifications of the global symmetric space [Sat60b] with equivalence classes of geodesics in the symmetric space. We then define equivalence relations on EDM rays using geometric …

A Hierarchical Spherical Radial Quadrature Algorithm For Multilevel Glmms, Gsmms, And Gene Pathway Analysis, Jacob A. Gagnon

Open Access Dissertations

The first part of my thesis is concerned with estimation for longitudinal data using generalized semi-parametric mixed models and multilevel generalized linear mixed models for a binary response. Likelihood based inferences are hindered by the lack of a closed form representation. Consequently, various integration approaches have been proposed. We propose a spherical radial integration based approach that takes advantage of the hierarchical structure of the data, which we call the 2 SR method. Compared to Pinheiro and Chao's multilevel Adaptive Gaussian quadrature, our proposed method has an improved time complexity with the number of functional evaluations scaling linearly in the …

Geometric And Combinatorial Aspects Of 1-Skeleta, Chris Ray Mcdaniel

Open Access Dissertations

In this thesis we investigate 1-skeleta and their associated cohomology rings. 1-skeleta arise from the 0- and 1-dimensional orbits of a certain class of manifold admitting a compact torus action and many questions that arise in the theory of 1-skeleta are rooted in the geometry and topology of these manifolds. The three main results of this work are: a lifting result for 1-skeleta (related to extending torus actions on manifolds), a classification result for certain 1-skeleta which have the Morse package (a property of 1-skeleta motivated by Morse theory for manifolds) and two constructions on 1-skeleta which we show preserve …

On The Frequency Of Finitely Anomalous Elliptic Curves, Penny Catherine Ridgdill

Open Access Dissertations

Given an elliptic curve E over Q, we can then consider E over the finite field Fp. If Np is the number of points on the curve over Fp, then we define ap(E) = p+1-Np. We say primes p for which ap(E) = 1 are anomalous. In this paper, we search for curves E so that this happens for only a finite number of primes. We call such curves finitely anomalous. This thesis deals with the frequency of their occurrence and finds several examples.

Generalized Emp And Nonlinear Schrodinger-Type Reformulations Of Some Scaler Field Cosmological Models, Jennie D'Ambroise

Open Access Dissertations

We show that Einstein’s gravitational field equations for the Friedmann- Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne- Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward …

Existence, Stability, And Dynamics Of Solitary Waves In Nonlinear Schroedinger Models With Periodic Potentials, Kody John Hoffman Law

Open Access Dissertations

The focus of this dissertation is the existence, stability, and resulting dynamical evolution of localized stationary solutions to Nonlinear Schr¨odinger (NLS) equations with periodic confining potentials in 2(+1) dimensions. I will make predictions about these properties based on a discrete lattice model of coupled ordinary differential equations with the appropriate symmetry. The latter has been justified by Wannier function expansions in a so-called tight-binding approximation in the appropriate parametric regime. Numerical results for the full 2(+1)-D continuum model will be qualitatively compared with discrete model predictions as well as with nonlinear optics experiments in optically induced photonic lattices in photorefractive …

On K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki

Paul Gunnells

Abstract. In this paper we investigate the structure of the algebraic K-groups K4(Z[i]) and K4(Z[ρ]), where i := √ −1 and ρ := (1 + √ −3)/2. We exploit the close connection between homology groups of GLn(R) for n 6 5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main results are (i) K4(Z[i]) is a finite abelian 3-group, and (ii) K4(Z[ρ]) is trivial.

Correction Methods, Approximate Biases, And Inference For Misclassified Data, Meng-Shiou Shieh

Open Access Dissertations

When categorical data are misplaced into the wrong category, we say the data is affected by misclassification. This is common for data collection. It is well-known that naive estimators of category probabilities and coefficients for regression that ignore misclassification can be biased. In this dissertation, we develop methods to provide improved estimators and confidence intervals for a proportion when only a misclassified proxy is observed, and provide improved estimators and confidence intervals for regression coefficients when only misclassified covariates are observed. Following the introduction and literature review, we develop two estimators for a proportion , one which reduces the bias, …

On Thickness And Packing Density For Knots And Links, Robert Kusner

Robert Kusner

We describe some problems, observations, and conjectures concerning density of the hexagonal packing of unit disks in R2.thickness and packing density of knots and links in S3 and R3. We prove the thickness of a nontrivial knot or link in S3 is no more than 4 , the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in R3 or S3 is generally less than √12 , the density of the hexagonal packing of unit disks in R2.

On Toric Varieties And Modular Forms, Paul Gunnells

Paul Gunnells

No abstract provided.

Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells

Paul Gunnells

A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary sub-groups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.

Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends And Genus Zero, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan

Robert Kusner

We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

The Spinor Representation Of Surfaces In Space, Robert Kusner, Nick Schmitt

Robert Kusner

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan [32], which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K = T(M). Given a conformal immersion of M into R3, the unique spin strucure on S2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s1, s2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion …

Moduli Spaces Of Embedded Constant Mean Curvature Surfaces With Few Ends And Special Symmetry, Karsten Grosse-Brauckmann, Robert Kusner

Robert Kusner

We give necessary conditions on complete embedded cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are twodimensional varieties in the moduli spaces of general cmc surfaces. We characterize fundamental domains of our cmc surfaces by associated great circle polygons in the three-sphere.