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Applied Mathematics Commons™

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2010

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Articles 211 - 217 of 217

Full-Text Articles in Applied Mathematics

Homogenization Of Nonlinear Partial Differential Equations, Silvia Jiménez Jan 2010

Homogenization Of Nonlinear Partial Differential Equations, Silvia Jiménez

LSU Doctoral Dissertations

This dissertation is concerned with properties of local fields inside composites made from two materials with different power law behavior. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to optical nonlinearities in dielectric media. We provide the corrector theory for the strong approximation of fields inside composites made from two power law materials with different exponents. The correctors are used to develop bounds on the local singularity strength for gradient fields inside microstructured media. The bounds are multiscale in nature and can be used to measure the amplification of applied macroscopic fields by the …

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Dimer Models For Knot Polynomials, Moshe Cohen Jan 2010

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 …

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Multigrid Methods For Maxwell's Equations, Jintao Cui Jan 2010

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 method uses …

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High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, Yin Wang Jan 2010

High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, Yin Wang

University of Kentucky Doctoral Dissertations

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques …

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Subgroups Of The Torelli Group, Leah R. Childers Jan 2010

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 the Johnson homomorphism and Birman-Craggs-Johnson homomorphism.

Then we will …

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Primes Of The Form X² + Ny² In Function Fields, Piotr Maciak Jan 2010

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 completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the …

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Hamilton-Jacobi Theory For Optimal Control Problems On Stratified Domains, Richard Charles Barnard Jan 2010

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 …

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