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 ...

A Characterization Of Near Outer-Planar Graphs, 2010 Louisiana State University and Agricultural and Mechanical College

#### A Characterization Of Near Outer-Planar Graphs, Tanya Allen Lueder

*LSU Master's Theses*

This thesis focuses on graphs containing an edge whose removal results in an outer-planar graph. We present partial results towards the larger goal of describing the class of all such graphs in terms of a finite list of excluded graphs. Specifically, we give a complete description of those members of this list that are not 2-connected or do not contain a subdivision of a three-spoke wheel. We also show that no members of the list contain a five-spoke wheel.

Correlation Of Defaults In Complex Portfolios Using Copula Techniques, 2010 Louisiana State University and Agricultural and Mechanical College

#### Correlation Of Defaults In Complex Portfolios Using Copula Techniques, Adam Lodygowski

*LSU Master's Theses*

This work, dealing with the correlation between subportfolios in more complex portfolios, begins with a brief survey of the necessary theoretical background. The basic statistical and probabilistic concepts are reviewed. The notion of copulas is introduced along with the fundamental theorem of Sklar. After this background a numerical procedure and code are developed for correlated defaults in multiple correlated portfolio. Further on, interesting results regarding the impact of changes in correlation on the portfolio performance are investigated in the simulations. The most valuable observations regarding the expected default ratios of two subportfolios considered jointly are presented and explained with particular ...

Hamilton-Jacobi Theory For Optimal Control Problems On Stratified Domains, 2010 Louisiana State University and Agricultural and Mechanical College

#### Hamilton-Jacobi Theory For Optimal Control Problems On Stratified Domains, Richard Charles Barnard

*LSU Doctoral Dissertations*

This thesis studies optimal control problems on stratified domains. We first establish a known proximal Hamilton-Jacobi characterization of the value function for problems with Lipschitz dynamics. This background gives the motivation for our results for systems over stratified domains, which is a system with non-Lipschitz dynamics that were introduced by Bressan and Hong. We provide an example that shows their attempt to derive a Hamilton-Jacobi characterization of the value function is incorrect, and discuss the nature of their error. A new construction of a multifunction is introduced that possesses properties similar to those of a Lipschitz multifunction, and is used ...

Optimal Control And Nonlinear Programming, 2010 Louisiana State University and Agricultural and Mechanical College

#### Optimal Control And Nonlinear Programming, Qingxia Li

*LSU Doctoral Dissertations*

In this thesis, we have two distinct but related subjects: optimal control and nonlinear programming. In the first part of this thesis, we prove that the value function, propagated from initial or terminal costs, and constraints, in the form of a differential equation, satisfy a subgradient form of the Hamilton-Jacobi equation in which the Hamiltonian is measurable with respect to time. In the second part of this thesis, we first construct a concrete example to demonstrate conjugate duality theory in vector optimization as developed by Tanino. We also define the normal cones corresponding to Tanino's concept of the subgradient ...

Orthogonal Grassmannians And Hermitian K-Theory In A¹-Homotopy Theory Of Schemes, 2010 Louisiana State University and Agricultural and Mechanical College

#### Orthogonal Grassmannians And Hermitian K-Theory In A¹-Homotopy Theory Of Schemes, Girja Shanker Tripathi

*LSU Doctoral Dissertations*

In this work we prove that the hermitian K-theory is geometrically representable in the A^1 -homotopy category of smooth schemes over a field. We also study in detail a realization functor from the A^1 -homotopy category of smooth schemes over the field R of real numbers to the category of topological spaces. This functor is determined by taking the real points of a smooth R-scheme. There is another realization functor induced by taking the complex points with a similar description although we have not discussed this other functor in this dissertation. Using these realization functors we have concluded ...

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 ...

A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves, 2010 Dublin Institute of Technology

#### A Generalized Nonlinear Model For The Evolution Of Low Frequency Freak Waves, Jonathan Blackledge

*Articles*

This paper presents a generalized model for simulating wavefields associated with the sea surface. This includes the case when `freak waves' may occur through an effect compounded in the nonlinear (cubic) Schrodinger equation. After providing brief introductions to linear sea wave models, `freak waves' and the linear and nonlinear Schrodinger equations, we present a unified model that provides for a piecewise continuous transition from a linear to a nonlinear state. This is based on introducing a fractional time derivative to develop a fractional nonlinear partial differential equation with a stochastic source function. In order to explore the characteristics of this ...

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 ...

Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, 2010 Portland State University

#### Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, Dylan M. Copeland, Jay Gopalakrishnan, Minah Oh

*Mathematics and Statistics Faculty Publications and Presentations*

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex then the multigrid Vcycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for ...

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 ...

Quantitative Modelling Approaches For Ascorbic Acid Degradation And Non-Enzymatic Browning Of Orange Juice During Ultrasound Processing, 2010 Technological University Dublin

#### Quantitative Modelling Approaches For Ascorbic Acid Degradation And Non-Enzymatic Browning Of Orange Juice During Ultrasound Processing, Vasilis Valdramidis, Patrick Cullen, Brijesh Tiwari, Colm O’Donnell

*Articles*

The objective of this study was to develop a deterministic modelling approach for non-enzymatic browning (NEB) and ascorbic acid (AA) degradation in orange juice during ultrasound processing. Freshly squeezed orange juice was sonicated using a 1,500 W ultrasonic processor at a constant frequency of 20 kHz and processing variables of amplitude level (24.4 – 61.0 μm), temperature (5 – 30 oC) and time (0 – 10 min). The rate constants of the NEB and AA were estimated by a primary model (zero and first order) while their relationship with respect to the processing factors was tested for a number of ...

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 ...

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 ...

A Tangent-Plane, Marker-Particle Method For The Computation Of Three-Dimensional Solid Surfaces Evolving By Surface Diffusion On A Substrate, 2009 Western Kentucky University

#### A Tangent-Plane, Marker-Particle Method For The Computation Of Three-Dimensional Solid Surfaces Evolving By Surface Diffusion On A Substrate, Ping Du, Mikhail Khenner, Harris Wong

*Mikhail Khenner*

We introduce a marker-particle method for the computation of three-dimensional solid surface morphologies evolving by surface diffusion. The method does not use gridding of surfaces or numerical differentiation, and applies to surfaces with finite slopes and overhangs. We demonstrate the method by computing the evolution of perturbed cylindrical wires on a substrate. We show that computed growth rates at early times agree with those predicted by the linear stability analysis. Furthermore, when the marker particles are redistributed periodically to maintain even spacing, the method can follow breakup of the wire.

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, 2009 Western Kentucky University

#### Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

*Mikhail Khenner*

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=O#1;#3;Bi#2;, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

Thickness-Dependent Spontaneous Dewetting Morphology Of Ultrathin Ag Films, 2009 Washington University in St Louis

#### Thickness-Dependent Spontaneous Dewetting Morphology Of Ultrathin Ag Films, H Krishna, R Sachan, J Strader, C Favazza, Mikhail Khenner, Ramki Kalyanaraman

*Mikhail Khenner*

We show here that the morphological pathway of spontaneous dewetting of ultrathin Ag films on SiO2 under nanosecond laser melting is found to be film thickness dependent. For films with thickness h between 2<=h<=9.5 nm, the intermediate stages of the morphology consisted of bicontinuous structures. For films 11.5<=h<=20 nm, the intermediate stages consisted of regularly-sized holes. Measurement of the characteristic length scales for different stages of dewetting as a function of film thickness showed a systematic increase, which is consistent with the spinodal dewetting instability over the entire thickness range investigated. This change in morphology with thickness is consistent with observations made previously for polymer films [A. Shama et al, Phys. Rev. Lett., v81, pp3463 (1998); R. Seemann et al, J. Phys. Cond. Matt., v13, pp4925, (2001)]. Based on the behavior of free energy curvature that incorporates intermolecular forces, we have estimated the morphological transition thickness for Ag on SiO2. The theory predictions agree well with observations for Ag. These results show that it is possible to form a variety of complex Ag nanomorphologies in a consistent manner, which could be useful in optical applications of Ag surfaces, such as in surface enhanced Raman sensing.

Symmetries Of The Central Vestibular System: Forming Movements For Gravity And A Three-Dimensional World, 2009 Portland State University

#### Symmetries Of The Central Vestibular System: Forming Movements For Gravity And A Three-Dimensional World, Gin Mccollum, Douglas A. Hanes

*Gin McCollum*

Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, inertial, somatosensory, auditory) and body movements (leg, trunk, head/neck, eye). They are also unified by geometric conditions. Symmetry groups have been found to structure experimentally-recorded pathways of the central vestibular system. When related to geometric conditions in three-dimensional physical space, these symmetry groups make sense as a logical foundation for sensorimotor coordination.