Mathematics Commons^™

Heat Trace And Heat Content Asymptotics For Schrodinger Operators Of Stable Processes/Fractional Laplacians, Luis Guillermo Acuna Valverde

Open Access Dissertations

Let V be a bounded and integrable potential over R^d and 0 < α ≤ 2. We show the existence of an asymptotic expansion by means of Fourier Transform techniques and probabilistic methods for the following quantities [special characters omitted] and [special characters omitted] as t ↓ 0. These quantities are called the heat trace and heat content in R^d with respect to V, respectively. Here, p((α)/ t)(x, y) and p( ^H_V/t)(x, y) denote, respectively, the heat kernels of the heat semigroups with infinitesimal generators given by (-Δ)(α/2) and H_V = (-Δ)(α/2) + V. The former operator is known as the fractional Laplacian whereas the latter one is known as the fractional Schrödinger Operator. ^ The study …

Orderability And Rigidity In Contact Geometry, Peter Weigel

Open Access Dissertations

We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold (&Sgr;, ξ = ker(α)). Previous results (see [1]) show that a large class of Liouville-fillable contact manifolds admit contractible positive loops. In contrast, we show that for any Liouville-fillable (&Sgr;, α) with dim(&Sgr;) ≥ 7, there exists a Liouville-fillable contact structure ξ' on &Sgr; which admits no positive loop at all. Further, ξ' can be chosen to agree with ξ' on the complement of a Darboux ball. We then define a relative version of orderability for a Legendrian submanifold, and discuss the relationship between the two …

Permutohedra, Configuration Spaces And Spineless Cacti, Yongheng Zhang

Open Access Dissertations

It has been known that the configuration space F(R², n) of n distinct ordered points in R² deformation retracts to a regular CW complex with n!permutohedra P_n as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the spaceCact(n) of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between F (R², n) and Cact(n) are then given. It is well known that the little 2-discs space D₂(n) is homotopy equivalent toF(R², n). …

Uncertainty Quantification And Calibration Of Physical Models, Xian He

Open Access Dissertations

An ecosystem model is a representation of a real complex ecological system, and is usually described by sophisticated mathematical models. Terrestrial Ecosystem Model (TEM) is one of the ecosystem models, that describes the dynamics of car- bon, nitrogen, water and other vegetation related variables. There are uncertainties in the TEM which are attributed to inaccurate input data, insufficient knowledge of the parameters, inherent randomness and simplification of the physical model. Quantification of uncertainty of such an ecosystem model is computationally very heavy. Bayesian calibration method has been used as an efficient way to calibrate and quantify uncertainties of the computer …

Parallel Symmetric Eigenvalue Problem Solvers, Alicia Marie Klinvex

Open Access Dissertations

Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, we describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, we discuss alternative methods for solving the saddle point …

Investigating Synergy: Mathematical Models For The Coupled Dynamics Of Hiv And Hsv-2 And Other Endemic Diseases, Christina M Alvey

Open Access Dissertations

This dissertation presents epidemiological models that investigate synergy: synergy between HIV and HSV-2 or between humans and mosquitoes in a malaria study. Each of the three coupled disease models addresses different epidemiological questions with regard to gender or disease structure in the context of sexually-transmitted diseases (STDs), while the malaria model focuses on age-structure of the human population. ^ Mounting evidence indicates that HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the …

Applications Of Microlocal Analysis To Some Hyperbolic Inverse Problems, Andrew J Homan

Open Access Dissertations

This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when …

Functional Inequalities And The Curvature Dimension Inequality On Totally Geodesic Foliations, Bumsik Kim

Open Access Dissertations

We discover following analytic / geometric properties on Riemannian foliations with bundle-like metric and totally geodesic leaves, or shortly, totally geodesic foliations. Under a certain curvature condition, we obtain (1) Sobolev-isoperimetric inequalities, global Poincar\'e inqualities, and a lower bound for Cheeger's isoperimetric constant, (2) Poincar\'e inequalities on balls and uniqueness of positive(or $L^p,p\geq 1$) solutions for the subelliptic heat equation, (3) A lower bound for the first non-zero eigenvalue of sub-Laplacians (Lichnerowicz theorem), and Obata's sphere theorem. In this context, the curvature condition is a sub-Riemannian analogue of lower bounds for Ricci curvature tensor. Earlier, it is given by Baudoin-Garofalo's …

A Pure-Jump Market-Making Model For High-Frequency Trading, Chi Wai Law

Open Access Dissertations

We propose a new market-making model which incorporates a number of realistic features relevant for high-frequency trading. In particular, we model the dependency structure of prices and order arrivals with novel self- and cross-exciting point processes. Furthermore, instead of assuming the bid and ask prices can be adjusted continuously by the market maker, we formulate the market maker's decisions as an optimal switching problem. Moreover, the risk of overtrading has been taken into consideration by allowing each order to have different size, and the market maker can make use of market orders, which are treated as impulse control, to get …

G-Frobenius Manifolds, Byeongho Lee

Open Access Dissertations

The goal of this dissertation is to introduce the notion of G-Frobenius manifolds for any finite group G. This work is motivated by the fact that any G-Frobenius algebra yields an ordinary Frobenius algebra by taking its G-invariants. We generalize this on the level of Frobenius manifolds. To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative "rings of functions", generalizing the commutative rings of power series on ordinary vector spaces. As the genus zero part of any ordinary …

A P-Adic Spectral Triple, Sumedha Hemamalee Rathnayake

Open Access Dissertations

We construct a spectral triple for the C*-algebra of continuous functions on the space of p-adic integers. On the technical level we utilize a weighted rooted tree obtained from a coarse grained approximation of the space combined with the forward derivative D on the tree. Our spectral triple satisfies the properties of a compact spectral metric space and the metric on the space of p-adic integers induced by the spectral triple is equivalent to the usual p-adic metric. Furthermore, we show that the spectrum of the operator D*D is closely related to the roots of a certain …

Alg Calculus Ii (Lecture Slides), Lake Ritter, Shangrong Deng

Mathematics Ancillary Materials

Authors' Description:

Upon completing this course students should be able to:

1. Find derivatives and integrals of transcendental functions.
2. Apply techniques to evaluate integrals.
3. Use tests to determine series convergence.
4. Determine Taylor series for common functions.
5. Describe curves in parametric form and polar coordinates.

Important Note:

All are welcome to use and modify these slides for nonprofit educational activities. They are intended to be used with smart board/smart podium or touch screen technology and cannot serve as a primary text. They were produced with LaTeX and converted to .ppt.

The original Beamer slides are superior quality but require several additional figure …

On Products Of Generalized Geometries, Ralph R. Gomez, Janet Talvacchia

Mathematics & Statistics Faculty Works

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimensional space that admits a generalized almost contact structure and an even dimensional space that admits a generalized almost complex structure. We also draw attention to the relationship of the Courant bracket to the classical notion of normality for almost contact structures.

Exploring The Unknown: Electrophysiological And Behavioural Measures Of Visuospatial Learning, Brendan Quinlivan, John Butler, Abdur R. Ridwan, Ines Beiser, Laura Williams, Eavan Mcgovern, Sean O'Riordan, Michael Hutchinson, Richard B. Reilly

Articles

Visuospatial memory describes our ability to temporarily store and manipulate visual and spatial information, and is employed for a wide variety of complex cognitive tasks. Here, a visuospatial learning task requiring fine motor control is employed to investigate visuospatial learning in a group of typically developing adults. Electrophysiological and behavioural data are collected during a target location task under two experimental conditions: Target Learning and Target Cued. Movement times (MTs) are employed as a behavioural metric of performance, while dynamic P3b amplitudes and power in the alpha band (approximately 10 Hz) are explored as electrophysiological metrics during visuospatial learning. Results …

Sensitivity Of Mixed Models To Computational Algorithms Of Time Series Data, Gunaime Nevine

Doctoral Dissertations

Statistical analysis is influenced by implementation of the algorithms used to execute the computations associated with various statistical techniques. Over many years; very important criteria for model comparison has been studied and examined, and two algorithms on a single dataset have been performed numerous times. The goal of this research is not comparing two or more models on one dataset, but comparing models with numerical algorithms that have been used to solve them on the same dataset.

In this research, different models have been broadly applied in modeling and their contrasting which are affected by the numerical algorithms in different …

Root Cover Pebbling On Graphs, Claire A. Sonneborn

Honors Theses

Consider a graph, G, with pebbles on its vertices. A pebbling move is defined to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The cover pebbling number of a graph, γ(G), is the minimum number of pebbles such that, given any configuration of γ(G) pebbles on the vertices of G, pebbling moves can be used to place one pebble on each vertex of G. We define the root vertex of a graph and fix an initial configuration of pebbles on G where we place all pebbles on the root …

Hurwitz's Theorem, Marianna Malek

Masters Essays

No abstract provided.

An Analysis Of Lake Erie’S Near Shore Habitats With The Benthic Diatom Metric, Kenjon Yeong

Masters Essays

No abstract provided.

The Atiyah Class Of A Dg-Vector Bundle, Rajan Amit Mehta, Mathieu Stiénon, Ping Xu

Mathematics Sciences: Faculty Publications

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields X(M) on a dg-manifold M with homological vector field Q admits a structure of L_∞[1]-algebra with the Lie derivative L_Q as unary bracket λ₁, and the Atiyah cocycle At_M corresponding to a torsion-free affine connection as binary bracket λ₂.

On The Fattening Of Lines In P3, Mike Janssen

Faculty Work Comprehensive List

We follow the lead of Bocci and Chiantini and show how differences in the invariant alpha can be used to classify certain classes of subschemes of P^3. Specifically, we will seek to classify arithmetically Cohen-Macaulay codimension 2 subschemes of P^3 in the manner Bocci and Chiantini classified points in P^2. The first section will seek to motivate our consideration of the invariant alpha by relating it to the Hilbert function and gamma, following the work of Bocci and Chiantini, and Dumnicki, et. al. The second section will contain our results classifying arithmetically Cohen-Macaulay codimension 2 subschemes of P^3. This work …