Applied Mathematics Commons^™

A C0 Interior Penalty Method For The Von Kármán Equations, Armin Karl Reiser

LSU Doctoral Dissertations

In this dissertation we develop a C⁰ interior penalty method for the von Kármán equations for nonlinear elastic plates. We begin with a brief survey on frequently used finite element methods for the von Kármán equations. After addressing some topics from functional analysis in the preliminaries, we present existence, uniqueness and regularity results for the solutions of the von Kármán equations in Chapter 3. In the next chapter we review the C⁰ interior penalty method for the biharmonic problem. Motivated by these results, we propose a C⁰ interior penalty method for the linearized von Kármán equations in …

Analysis Of Hdg Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, Francisco-Javier Sayas

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree $ k$ for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the …

Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We present a family of mixed methods for linear elasticity that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces, solve a symmetric positive-definite system. The design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices. These new degrees of freedom are also used to give a new simple calculation of the …

Modeling, Analysis, And Simulation Of Discrete-Continuum Models Of Step-Flow Epitaxy: Bunching Instabilities And Continuum Limits, Nicholas O. Kirby

University of Kentucky Doctoral Dissertations

Vicinal surfaces consist of terraces separated by atomic steps. In the step-flow regime, deposited atoms (adatoms) diffuse on terraces, eventually reaching steps where they attach to the crystal, thereby causing the steps to move. There are two main objectives of this work. First, we analyze rigorously the differences in qualitative behavior between vicinal surfaces consisting of infinitely many steps and nanowires whose top surface consists of a small number of steps bounded by a reflecting wall. Second, we derive the continuum model that describes the macroscopic behavior of vicinal surfaces from detailed microscopic models of step dynamics.

We use the …

Variables Affecting Crime Rate, Jaime Zeigler

Honors Program Theses

The Federal Bureau of Investigation releases data about crime statistics. They publish the raw data and provide factors that influence the crime rate. They provide the general public with a few variables that they deemed most influential, but do not provide the statistical analysis to support. After performing a statistical analysis, I support or disprove their claims. Another way to question the validity of regression is comparing the current regression to a previous regression performed two years ago by myself with the same variables from the same source. An additional aspect that this paper explores the history of factors affecting …

Paley-Wiener Theorems With Respect To The Spectral Parameter, Susanna Dann

LSU Doctoral Dissertations

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then harmonic analysis on M is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations of G. R^n can be realized as the quotient R^n=E(n)/SO(n), where E(n) is the orientation preserving Euclidean motion group. The pair (E(n), SO(n)) is a Gelfand pair. Hence this realization of R^n comes with its own natural …

Guided Modes And Resonant Transmission In Periodic Structures, Hairui Tu

LSU Doctoral Dissertations

We analyze resonant scattering phenomena of scalar fields in periodic slab and pillar structures that are related to the interaction between guided modes of the structure and plane waves emanating from the exterior. The mechanism for the resonance is the nonrobust nature of the guided modes with respect to perturbations of the wavenumber, which reflects the fact that the frequency of the mode is embedded in the continuous spectrum of the pseudo-periodic Helmholtz equation. We extend previous complex perturbation analysis of transmission anomalies to structures whose coefficients are only required to be measurable and bounded from above and below, and …

A New Theory Of Stochastic Integration, Anuwat Sae-Tang

LSU Doctoral Dissertations

In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula …

Capturing Elements In Matroid Minors, Deborah Chun

LSU Doctoral Dissertations

In this dissertation, we begin with an introduction to a matroid as the natural generalization of independence arising in three different fields of mathematics. In the first chapter, we develop graph theory and matroid theory terminology necessary to the topic of this dissertation. In Chapter 2 and Chapter 3, we prove two main results. A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n exceeding two, there is an integer f(n) so that if |E(M)| exceeds f(n), then M has a minor isomorphic to the rank-n wheel or whirl, …

Parallel-Sparse Symmetrical/Unsymmetrical Finite Element Domain Decomposition Solver With Multi-Point Constraints For Structural/Acoustic Analysis, Siroj Tungkahotara, Willie R. Watson, Duc T. Nguyen, Subramaniam D. Rajan

Civil & Environmental Engineering Faculty Publications

Details of parallel-sparse Domain Decomposition (DD) with multi-point constraints (MPC) formulation are explained. Major computational components of the DD formulation are identified. Critical roles of parallel (direct) sparse and iterative solvers with MPC are discussed within the framework of DD formulation. Both symmetrical and unsymmetrical system of simultaneous linear equations (SLE) can be handled by the developed DD formulation. For symmetrical SLE, option for imposing MPC equations is also provided.

Large-scale (up to 25 million unknowns involving complex numbers) structural and acoustic Finite Element (FE) analysis are used to evaluate the parallel computational performance of the proposed DD implementation using …

Hamiltonicity And Connectivity In Distance-Colored Graphs, Kyle C. Kolasinski

Dissertations

Abstract attached as separate file.

Currency Trading Using The Fractal Market Hypothesis, Jonathan Blackledge, Kieran Murphy

Articles

We report on a research and development programme in financial modelling and economic security undertaken in the Information and Communications Security Research Group (ICSRG, 2011) which has led to the launch of a new company - Currency Traders Ireland Limited - funded by Enterprise Ireland. Currency Traders Ireland Limited (CTI, 2011) has a fifty year exclusive license to develop a new set of indicators for analysing currency exchange rates (Forex trading). We consider the background to the approach taken and present examples of the results obtained to date.

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

Articles

This paper presents a generalized model for simulating wave fields 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 …

Cryptography Using Steganography: New Algorithms And Applications, Jonathan Blackledge

Articles

Developing methods for ensuring the secure exchange of information is one of the oldest occupations in history. With the revolution in Information Technology, the need for securing information and the variety of methods that have been developed to do it has expanded rapidly. Much of the technology that forms the basis for many of the techniques used today was originally conceived for use in military communications and has since found a place in a wide range of industrial and commercial sectors. This has led to the development of certain industry standards that are compounded in specific data processing algorithms together …

The Sir Model When S(T) Is A Multi-Exponential Function., Teshome Mogessie Balkew

Electronic Theses and Dissertations

The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑ⁿ_k=1 r_ke^-σ_kt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+e^wt, then …

A Sequel To “A Space Topologized By Functions From Omega To Omega”, Tetsuya Ishiu, Akira Iwasa

Computer Science and Mathematics Faculty Publications

We consider a topological space ⟨𝑋, 𝜏 (ℱ)⟩, where 𝑋 = {𝑝 ∗} ∪ [𝜔 Å~ 𝜔] and ℱ ⊆ 𝜔𝜔. Each point in 𝜔 Å~ 𝜔 is isolated and a neighborhood of 𝑝∗ has the form {𝑝∗}∪{⟨𝑖, 𝑗⟩ : 𝑖 ≥ 𝑛, 𝑗 ≥ 𝑓(𝑖)} for some 𝑛 ∈ 𝜔 and 𝑓 ∈ ℱ. We show that there are subsets ℱ and 𝒢 of 𝜔𝜔 such that ℱ is not bounded, 𝒢 is bounded, yet ⟨𝑋, 𝜏 (ℱ)⟩ and ⟨𝑋, 𝜏 (𝒢)⟩ are homeomorphic. This answers a question of the second author posed in A space topologized by functions …

Quantitative Stability And Optimality Conditions In Convex Semi-Infinite And Infinite Programming, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra

Mathematics Research Reports

This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T. Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is loo(T). Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this …

Solving A Generalized Heron Problem By Means Of Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr

Mathematics Research Reports

The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of IR!, find a point such that the sum of the distances from that point to n given nonempty …

Explicit Level Lowering Of 2-Dimensional Modular Galois Representations, Rodney Keaton

All Theses

Let f be a normalized eigenform of level Npα for some positive integer α and some odd prime p satisfying gcd(p,N)=1. A construction of Deligne, Shimura, et. al., attaches a p-adic continuous two-dimensional Galois representation to f. The Refined Conjecture of Serre states that such a representation should in fact arise from a normalized eigenform of level prime to p.
In this presentation we present a proof of Ribet which allows us to 'strip' these powers of p from the level while still retaining the original Galois representation, i.e., the residual of our new representation arising from level N will …

The Möbius Geometry Of Hypersurfaces, Ii, Michael Bolt

University Faculty Publications and Creative Works

No abstract provided.