Strong Orbit Equivalence And Residuality, 2010 University of Denver

#### Strong Orbit Equivalence And Residuality, Brett M. Werner

*Electronic Theses and Dissertations*

In this dissertation, we consider notions of equivalence between minimal Cantor systems, in particular strong orbit equivalence. By constructing the systems, we show that there exist two nonisomorphic substitution systems that are both Kakutani equivalent and strongly orbit equivalent. We go on to define a metric on a strong orbit equivalence class of minimal Cantor systems and prove several properties about the metric space. If the strong orbit equivalence class contains a finite rank system, we show that the set of finite rank systems is residual in the metric space. The last result shown is that set of systems with ...

A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, 2010 University of Texas at Austin

#### A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, Leszek Demkowicz, Jay Gopalakrishnan

*Mathematics and Statistics Faculty Publications and Presentations*

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated ...

A Projection-Based Error Analysis Of Hdg Methods, 2010 Portland State University

#### A Projection-Based Error Analysis Of Hdg Methods, Jay Gopalakrishnan, Bernardo Cockburn, Francisco-Javier Sayas

*Mathematics and Statistics Faculty Publications and Presentations*

We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG ...

A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, 2010 University of Texas at Austin

#### A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, Leszek Demkowicz, Jay Gopalakrishnan

*Mathematics and Statistics Faculty Publications and Presentations*

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size *h* and polynomial degree *p*, and outperforms the standard upwind DG method.

Symmetry And Stability Of Homogeneous Flocks, 2010 Portland State University

#### Symmetry And Stability Of Homogeneous Flocks, J. J. P. Veerman

*Mathematics and Statistics Faculty Publications and Presentations*

The study of the movement of flocks, whether biological or technological, is motivated by the desire to understand the capability of coherent motion of a large number of agents that only receive very limited information. In a biological flock a large group of animals seek their course while moving in a more or less fixed formation. It seems reasonable that the immediate course is determined by leaders at the boundary of the flock. The others follow: what is their algorithm? The most popular technological application consists of cars on a one-lane road. The light turns green and the lead car ...

Spectral Methods For The Hamiltonian Systems, 2010 Wayne State University

#### Spectral Methods For The Hamiltonian Systems, Nairat Kanyamee

*Wayne State University Dissertations*

We conduct a systematic comparison of spectral methods with some

symplectic methods in solving Hamiltonian dynamical systems. Our

main emphasis is on the non-linear problems. Numerical evidence has

demonstrated that the proposed spectral collocation method preserves

both energy and symplectic structure up to the machine error in each

time (large) step, and therefore has a better long time behavior.

Financial Securities Under Nonlinear Diffusion Asset Pricing Model, 2010 Wilfrid Laurier University

#### Financial Securities Under Nonlinear Diffusion Asset Pricing Model, Andrey Vasilyev

*Theses and Dissertations (Comprehensive)*

In this thesis we investigate two pricing models for valuing financial derivatives. Both models are diffusion processes with a linear drift and nonlinear diffusion coefficient. The forward price process of these models is a martingale under an assumed risk-neutral measure and the transition probability densities are given in analytically closed form. Specifically, we study and calibrate two different families of models that are constructed based on a so-called diffusion canonical transformation. One family follows from the Ornstein-Uhlenbeck diffusion (the UOU family) and the other—from the Cox-Ingersoll-Ross process (the Confluent-U family).

The first part of the thesis considers single-asset and ...

Traveling Wave Solutions For A Nonlocal Reaction-Diffusion Model Of Influenza A Drift, 2010 Wright State University - Main Campus

#### Traveling Wave Solutions For A Nonlocal Reaction-Diffusion Model Of Influenza A Drift, Joaquin Riviera, Yi Li

*Mathematics and Statistics Faculty Publications*

In this paper we discuss the existence of traveling wave solutions for a nonlocal reaction-diffusion model of Influenza A proposed in Lin et. al. (2003). The proof for the existence of the traveling wave takes advantage of the different time scales between the evolution of the disease and the progress of the disease in the population. Under this framework we are able to use the techniques from geometric singular perturbation theory to prove the existence of the traveling wave.

Methods Of Variational Analysis In Pessimistic Bilevel Programming, 2010 Wayne State University

#### Methods Of Variational Analysis In Pessimistic Bilevel Programming, Samarathunga M. Dassanayaka

*Wayne State University Dissertations*

Bilevel programming problems are of growing interest both from theoretical and practical points of view. These models are used in various applications, such as economic planning, network design, and so on. The purpose of this dissertation is to study the pessimistic (or strong) version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation developed by B. Mordukhovich.

This dissertation begins with analyzing pessimistic bilevel programs, formulation of the problems, literature review, practical application, existence of ...

Primes Of The Form X² + Ny² In Function Fields, 2010 Louisiana State University and Agricultural and Mechanical College

#### Primes Of The Form X² + Ny² In Function Fields, Piotr Maciak

*LSU Doctoral Dissertations*

Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits ...

Koszul Duality For Multigraded Algebras, 2010 Louisiana State University and Agricultural and Mechanical College

#### Koszul Duality For Multigraded Algebras, Fareed Hawwa

*LSU Doctoral Dissertations*

Classical Koszul duality sets up an adjoint pair of functors establishing an equivalence of categories. The equivalence is between the bounded derived category of complexes of graded modules over a graded algebra and the bounded derived category of complexes of graded modules over the quadratic dual graded algebra. This duality can be extended in many ways. We consider here two extensions: first we wish to allow a multigraded algebra, meaning that the algebra can be graded by any abelian group (not just the integers). Second, we will allow filtered algebras. In fact we are considering filtered quadratic algebras with an ...

Multigrid Methods For Maxwell's Equations, 2010 Louisiana State University and Agricultural and Mechanical College

#### Multigrid Methods For Maxwell's Equations, Jintao Cui

*LSU Doctoral Dissertations*

In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second ...

Dimer Models For Knot Polynomials, 2010 Louisiana State University and Agricultural and Mechanical College

#### Dimer Models For Knot Polynomials, Moshe Cohen

*LSU Doctoral Dissertations*

A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph. This work ...

Multiresolution Inverse Wavelet Reconstruction From A Fourier Partial Sum, 2010 CUNY Kingsborough Community College

#### Multiresolution Inverse Wavelet Reconstruction From A Fourier Partial Sum, Nataniel Greene

*Publications and Research*

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities.In previous work [11,12] we described a numerical procedure for overcoming the Gibbs phenomenon called the Inverse Wavelet Reconstruction method (IWR). The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series. However, we only described the method standard wavelet ...

Subgroups Of The Torelli Group, 2010 Louisiana State University and Agricultural and Mechanical College

#### Subgroups Of The Torelli Group, Leah R. Childers

*LSU Doctoral Dissertations*

Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation:

Mod(Sg) ---> Sp(2g, Z).

The kernel of this representation is called the Torelli group, I(Sg).

We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of ...

Perverse Poisson Sheaves On The Nilpotent Cone, 2010 Louisiana State University and Agricultural and Mechanical College

#### Perverse Poisson Sheaves On The Nilpotent Cone, Jared Lee Culbertson

*LSU Doctoral Dissertations*

For a reductive complex algebraic group, the associated nilpotent cone is the variety of nilpotent elements in the corresponding Lie algebra. Understanding the nilpotent cone is of central importance in representation theory. For example, the nilpotent cone plays a prominent role in classifying the representations of finite groups of Lie type. More recently, the nilpotent cone has been shown to have a close connection with the affine flag variety and this has been exploited in the Geometric Langlands Program. We make use of the following important fact. The nilpotent cone is invariant under the coadjoint action of G on the ...

Method Of Riemann Surfaces In Modelling Of Cavitating Flow, 2010 Louisiana State University and Agricultural and Mechanical College

#### Method Of Riemann Surfaces In Modelling Of Cavitating Flow, Anna Zemlyanova

*LSU Doctoral Dissertations*

This dissertation is concerned with the applications of the Riemann-Hilbert problem on a hyperelliptic Riemann surface to problems on supercavitating flows of a liquid around objects. For a two-dimensional steady irrotational flow of liquid it is possible to introduce a complex potential w(z) which allows to apply the powerful methods of complex analysis to the solution of fluid mechanics problems. In this work problems on supercavitating flows of a liquid around one or two wedges have been stated. The Tulin single-spiral-vortex model is employed as a cavity closure condition. The flow domain is transformed into an auxiliary domain with ...

Power Series Expansions For Waves In High-Contrast Plasmonic Crystals, 2010 Louisiana State University and Agricultural and Mechanical College

#### Power Series Expansions For Waves In High-Contrast Plasmonic Crystals, Santiago Prado Parentes Fortes

*LSU Doctoral Dissertations*

In this thesis, a method is developed for obtaining convergent power series expansions for dispersion relations in two-dimensional periodic media with frequency dependent constitutive relations. The method is based on high-contrast expansions in the parameter _x0011_ = 2_x0019_d=_x0015_, where d is the period of the crystal cell and _x0015_ is the wavelength. The radii of convergence obtained are not too small, on the order of _x0011_ _x0019_ 102. That the method applies to frequency dependent media is an important fact, since the majority of the methods available in the literature are restricted to frequency independent constitutive relations. The convergent ...

Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, 2009 University of Wollongong

#### Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, Tim Marchant

*Tim Marchant*

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of ...

Multiple Decrement Modeling In The Presence Of Interval Censoring And Masking, 2009 Laurentian University

#### Multiple Decrement Modeling In The Presence Of Interval Censoring And Masking, Peter Adamic, Stephanie Dixon, Daniel Gillis

*Stephanie Dixon*

A self-consistent algorithm will be proposed to non-parametrically estimate the cause-specific cumulative incidence functions (CIFs) in an interval censored, multiple decrement context. More specifically, the censoring mechanism will be assumed to be a mixture of case 2 interval-censored data with the additional possibility of exact observations. The proposed algorithm is a generalization of the classical univariate algorithms of Efron and Turnbull. However, unlike any previous non-parametric models proposed in the literature to date, the algorithm will explicitly allow for the possibility of any combination of masked modes of failure, where failure is known only to occur due to a subset ...