Applied Mathematics Commons^™

Analysis Of Nonlinear Dispersive Model Equations, Jacob Grey

LSU Doctoral Dissertations

In this work we begin with a brief survey of the classical fluid dynamics problem of water waves, and then proceed to derive well known evolution equations via a Hamiltonian Variational approach. This method was first introduced in the seminal work of Walter Craig, et al. \cite{CG}. The distinguishing feature of this scheme is that the Dirichlet-Neumann operator of the fluid domain appears explicitly in the Hamiltonian. In the second and third chapters, we utilize the Hamiltonian perturbation theory introduced in \cite{CG} to derive the Benjamin-Bona-Mahony (BBM) and Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP)equations. Finally, we briefly review the existence theory for their corresponding …

Partial Cosine-Funk Transforms At Poles Of The Cosine-Λ Transform On Grassmann Manifolds, Christopher Adam Cross

LSU Doctoral Dissertations

The cosine-λ transform, denoted C^λ, is a family of integral transforms we can define on the sphere and on the Grassmannian manifolds of p-dimensional subspaces in Kⁿ where K is R, C or the skew field H of quaternions. We treat the Grassmannians as the symmetric spaces SO(n)/S(O(p) × O(q)), SU(n)/S(U(p) × U(q)) and Sp(n)/(Sp(p) × Sp(q)) and we work by analogy with the case of the cosine-λ transform on the sphere, which is also a symmetric space.

The family C^λ extends meromorphically in λ to the complex plane with poles at (among other values) λ …

Excluding Two Minors Of The Petersen Graph, Adam Beau Ferguson

LSU Doctoral Dissertations

In this dissertation, we begin with a brief survey of the Petersen graph and its role in graph theory. We will then develop an alternative decomposition to clique sums for 3-connected graphs, called T-sums. This decomposition will be used in Chapter 2 to completely characterize those graphs which have no P_3 minor, where P_3 is a graph with 7 vertices, 12 edges, and is isomorphic to the graph created by contracting three edges of a perfect matching of the Petersen Graph. In Chapter 3, we determine the structure of any large internally 4-connected graph which has no P_2 minor, where …

Knots, Skein Theory And Q-Series, Mustafa Hajij

LSU Doctoral Dissertations

The tail of a sequence {P_n(q)} of formal power series in Z[q^{-1}][[q]], if it exists, is the formal power series whose first $n$ coefficients agree up to a common sign with the first n coefficients of P_n. The colored Jones polynomial is link invariant that associates to every link in S^3 a sequence of Laurent polynomials. In the first part of this work we study the tail of the unreduced colored Jones polynomial of alternating links using the colored Kauffman skein relation. This gives a natural extension of a result by Kauffman, Murasugi, and Thistlethwaite regarding the highest and lowest …

Left-Orderability, Cyclic Branched Covers And Representations Of The Knot Group, Ying Hu

LSU Doctoral Dissertations

A group G is called left-orderable if one can find a total order on G, which is preserved under left multiplication. In this paper we first give a sufficient condition for the fundamental group of the nth cyclic branched cover of the three sphere over a prime knot K to be left-orderable, in terms of representations of the knot group. Then we make use of this criterion to study the left-orderability of fundamental groups of cyclic branched covers over two-bridge knots and satellite knots.

Wavelets, Coorbit Theory, And Projective Representations, Amer Hasan Darweesh

LSU Doctoral Dissertations

Banach spaces of functions, or more generally, of distributions are one of the main topics in analysis. In this thesis, we present an abstract framework for construction of invariant Banach function spaces from projective group representations. Coorbit theory gives a unified method to construct invariant Banach function spaces via representations of Lie groups. This theory was introduced by \Fch\, and \Gro\, in \cite{FG,FG1, FG2,FG3} and then extended in \cite{CO2}. We generalize this concept by constructing coorbit spaces using projective representation which is first studied by O. Christensen in \cite{O1}. This allows us to describe wider classes of function spaces as …

A New Method In Distribution Theory With A Non-Smooth Framework, Yunyun Yang

LSU Doctoral Dissertations

In this work, we present a complete treatment of the theory of thick distributions and its asymptotic expansion. We also present several applications of thick distributions in mathematical physics, function spaces, and measure theory. We also discuss regularization using different surfaces. In the last chapter we present some recent applications of distributions in clarifying the moment terms in the heat kernel expansion, and in explaining the relation between the heat kernel expansion and the cylinder kernel expansion.

Well-Quasi-Ordering By The Induced-Minor Relation, Chanun Lewchalermvongs

LSU Doctoral Dissertations

Robertson and Seymour proved Wagner's Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in …

On The Infinitesimal Theory Of Chow Groups, Benjamin F. Dribus

LSU Doctoral Dissertations

The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. Except for the case p=1, which yields an algebraic group, the Chow groups remain mysterious. This thesis explores a "linearization" approach to this problem, focusing on the infinitesimal structure of the Chow groups near their identity elements. This method was adumbrated in recent work of Mark Green and Phillip Griffiths. Similar topics have been explored by Bloch, Stienstra, Hesselholt, Van der Kallen, and others. A famous formula of Bloch expresses the Chow …

Coloring Graphs Drawn With Crossings, Daniel Allen Guillot

LSU Doctoral Dissertations

This dissertation will examine various results for graph colorings. It begins by introducing some basic graph theory concepts, focusing on those ideas relevant to graph embeddings, and by introducing terminology to allow a formal discussion of drawings of graphs. Chapter 2 focuses on results for proper colorings of graphs with good drawings, using a previous result from Král and Stacho as inspiration. Chapter 3 expands on the ideas of Chapter 2 and focuses on cyclic colorings of embedded graphs. Chapters 5 and 6 examine results for total and list colorings, respectively, of drawings of graphs. Finally, Chapter 6 introduces generalized …

Ito Formula And Girsanov Theorem On A New Ito Integral, Yun Peng

LSU Doctoral Dissertations

The celebrated Ito theory of stochastic integration deals with stochastic integrals of adapted stochastic processes. The Ito formula and Girsanov theorem in this theory are fundamental results which are used in many applied fields, in particular, the finance and the stock markets, e.g. the Black-Scholes model. In chapter 1 we will briefly review the Ito theory. In recent years, there have been several extension of the Ito integral to stochastic integrals of non-adapted stochastic processes. In this dissertation we will study an extension initiated by Ayed and Kuo in 2008. In Chapter 2 we review this new stochastic integral and …

Damage Evolution In Pressurized Domain: A Gradient Based Variational Approach, Navid Mozaffari

LSU Master's Theses

Construction of appropriate models through mathematical analysis for materials in order to find their main properties and ingredients and enhance the numerical simulations to predict their behavior under specific conditions is in interest even in mathematics departments rather than material science and engineering branches. Among these models, gradient damage models have reached to the specific stage because of their ability to bring the effects of micro cracks propagation into conventional continuum mechanics formulation and approximate brittle fracture as one of the most phenomena in the area of material behavior simulation. This thesis includes the application and extension of a previously …

Effects Of Focused Instruction Process (Fip)On Student End-Of-Course Test (Predicting End-Of-Course Test Using Teacher-Made Test), Roland Damasco Dante

LSU Master's Theses

This study took place at a medium-sized suburban high school. It was designed to determine the usefulness of certain teacher-made tests in predicting students' end-of-course (EOC) tests. The teacher taught the students the skills in which their performance was weakest on the previous state test. The students were tested after each skill on a four-point quiz (teacher-made test). Students who scored 3—4 moved on to the next lesson or enrichment, while those who scored 0—2 were re-taught and re-tested. The procedure was repeated throughout the school year. At the end of the course, students took the state-mandated end-of-course test. The …

Developing Auxiliary Resource Materials To Support The Engageny Geometry Curriculum, Joanne Griffin

LSU Master's Theses

With the advent of current education reform, and the introduction of the Common Core State Standards for Mathematics, the present offerings of the geometry curriculum have become dated. One contribution to remedy this situation is a project by the state of New York called EngageNY. EngageNY is a common core aligned mathematics curriculum across all grades. The EngageNY Geometry Curriculum Module 1 is the basis from which this thesis was developed. It is the purpose of this thesis to present a supplement to the EngageNY Geometry Module 1 Curriculum and to describe why it is advantageous to have such a …

Explicit Equations Of Non-Hyperelliptic Genus 3 Curves With Real Multiplication By Q(ζ7+ζ7-1), Dun Liang

LSU Doctoral Dissertations

This thesis is devoted to proving the following:

For all (u₁, u₂, u₃, u₄) in a Zariski dense open subset of C⁴ there is a genus 3 curve X(u₁, u₂, u₃, u₄) with the following properties:

1. X(u₁, u₂, u₃, u₄) is not hyperelliptic.
2. End(Jac((X(u₁, u₂, u₃, u₄))) ⊗Q contains the real cubic field Q(ζ₇+ζ₇^-1) where ζ₇ is …

On Matroid And Polymatroid Connectivity, Dennis Wayne Hall Ii

LSU Doctoral Dissertations

Matroids were introduced in 1935 by Hassler Whitney to provide a way to abstractly capture the dependence properties common to graphs and matrices. One important class of matroids arises by taking as objects some finite collection of one-dimensional subspaces of a vector space. If, instead, one takes as objects some finite collection of subspaces of dimensions at most k in a vector space, one gets an example of a k-polymatroid.

Connectivity is a pivotal topic of study in the endeavor to understand the structure of matroids and polymatroids. In this dissertation, we study the notion of connectivity from several …

Obstructions To Embedding Genus-1 Tangles In Links, Susan Marie Abernathy

LSU Doctoral Dissertations

Given a compact, oriented 3-manifold M in S3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S3 if T can be completed to L by adding a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We focus on the case of (S_1 x D_2, 2)-tangles, also called genus-1 tangles, and consider the following question: given a genus-1 tangle G and a link L, how can we tell if L is a closure of …

Constructive Aspects Of Kochen's Theorem On P-Adic Closures, Evan Michael Eakins

LSU Doctoral Dissertations

In this work we begin with a brief survey of set theory and arithmetic to provide background for a logical procedure to `cleanse' the Axiom of Choice from a proof of a theorem of Kochen's. We accomplish this in the following chapters. We then discuss certain theorems involving definable Skolem functions. These theorems are used in Chapter 5 to give a construction of a p-adic closure of a p-valued field. Certain further considerations and open questions are addressed in the _x000C_final chapter.

Invariants Of Legendrian Products, Peter Lambert-Cole

LSU Doctoral Dissertations

This thesis investigates a construction in contact topology of Legendrian submanifolds called the Legendrian product. We investigate and compute invariants for these Legendrian submanifolds, including the Thurston-Bennequin invariant and Maslov class; Legendrian contact homology for the product of two Legendrian knots; and generating family homology.

Selected Problems On Matroid Minors, Jesse Taylor

LSU Doctoral Dissertations

This dissertation begins with an introduction to matroids and graphs. In the first chapter, we develop matroid and graph theory definitions and preliminary results sufficient to discuss the problems presented in the later chapters. These topics include duality, connectivity, matroid minors, and Cunningham and Edmonds's tree decomposition for connected matroids. One of the most well-known excluded-minor results in matroid theory is Tutte's characterization of binary matroids. The class of binary matroids is one of the most widely studied classes of matroids, and its members have many attractive qualities. This motivates the study of matroid classes that are close to being …

Robust Preconditioners For The High-Contrast Elliptic Partial Differential Equations, Zuhal Unlu

LSU Doctoral Dissertations

In this thesis, we discuss a robust preconditioner (the AGKS preconditioner) for solving linear systems arising from approximations of partial differential equations (PDEs) with high-contrast coefficients. The problems considered here include the standard second and higher order elliptic PDEs such as high-contrast diffusion equation, Stokes' equation and biharmonic-plate equation. The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated to treat large configurations. The construction of the preconditioner consists of two phases. The first one is an algebraic phase which partitions the degrees of freedom into high and low permeability regions which …

The Gaussian Radon Transform For Banach Spaces, Irina Holmes

LSU Doctoral Dissertations

The classical Radon transform can be thought of as a way to obtain the density of an n-dimensional object from its (n-1)-dimensional sections in diff_x001B_erent directions. A generalization of this transform to infi_x001C_nite-dimensional spaces has the potential to allow one to obtain a function de_x001C_fined on an infi_x001C_nite-dimensional space from its conditional expectations. We work within a standard framework in in_x001C_finite-dimensional analysis, that of abstract Wiener spaces, developed by L. Gross. The main obstacle in infinite dimensions is the absence of a useful version of Lebesgue measure. To overcome this, we work with Gaussian measures. Specifically, we construct Gaussian measures …

Conical Representations For Direct Limits Of Riemannian Symmetric Spaces., Matthew Glenn Dawson

LSU Doctoral Dissertations

We extend the definition of conical representations for Riemannian symmetric space to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical …

Reformulations For Control Systems And Optimization Problems With Impulses, Jacob Blanton

LSU Doctoral Dissertations

This dissertation studies two different techniques for analyzing control systems whose dynamics include impulses, or more specifically, are measure-driven. In such systems, the state trajectories will have discontinuities corresponding to the atoms of the Borel measure driving the dynamics, and these discontinuities require further definition in order for the control system to be treated with the broad range of results available to non-impulsive systems. Both techniques considered involve a reparameterization of the system variables including state, time, and controls. The first method is that of the graph completion, which provides an explicit reparameterization of the time and state variables. The …

Problem Solving Strategies And Metacognitive Skills For Gifted Students In Middle School, Lorena Aguelo Java

LSU Master's Theses

This study is conducted to investigate if the designed four-step method strategy (GEAR strategy adapted from Polya, 1973) in solving math problems has improved students’ performance scores and enhanced the metacognitive skills of gifted students. The respondents of this study include middle school gifted students who took math eight course in the school year 2013-2014 at Westdale Middle School in East Baton Rouge Parish School System. There are four classes of math eight gifted students who participated in the study. The classes were chosen randomly for experimental and controlled group and were equalized on the basis of the pre-test results …

Extremal Problems In Matroid Connectivity, John Tyler Moss

LSU Doctoral Dissertations

Matroid k-connectivity is typically defined in terms of a connectivity function. We can also say that a matroid is 2-connected if and only if for each pair of elements, there is a circuit containing both elements. Equivalently, a matroid is 2-connected if and only if each pair of elements is in a certain 2-element minor that is 2-connected. Similar results for higher connectivity had not been known. We determine a characterization of 3-connectivity that is based on the containment of small subsets in 3-connected minors from a given list of 3-connected matroids. Bixby’s Lemma is a well-known inductive tool in …

Local Conjugations Of Groups And Applications To Number Fields, Bir B. Kafle

LSU Doctoral Dissertations

This dissertation studies pairs of subgroups H, H' of a finite group G together with a bijective map Φ H −> H' that is a local conjugation, meaning that each element h in H is conjugate in G to its image Φ(h). The map Φ is not required to take products to products. The motivation for studying such pairs comes from a paper of F. Gassmann in 1926, in which he formulated an equivalent but different-sounding condition now known as Gassmann’s condition. There are now at least ten equivalent reformulations of Gassmann’s condition, of which local conjugation is perhaps the …

Exponentially Convergent Generalized Finite Element Method For Multi-Scale Problems, Xu Huang

LSU Doctoral Dissertations

The overall approach I take in the thesis falls into the category of multiscale finite element methods(MsFEM). I work to identify a new class of local approximation spaces with good approximation properties. This is carried out for the equilibrium problem of linear elasticity. The choice of local approximation spaces is motivated by the kolmogorov n-width. Part of my thesis work develops an estimate to show that it is possible to achieve a local approximation error of ô with respect to the energy norm using at most ln^d+1 &frac1ô local basis functions. The global approximation error ôis controlled by the …

Combinatorial Minimal Free Resolutions Of Ideals With Monomial And Binomial Generators, Trevor Mcguire

LSU Doctoral Dissertations

In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. If Lambda is an integer lattice in the n-dimensional integers satisfying some mild conditions, S is the polynomial ring with n variables and R is the group algebra of S[Lambda], then the first result is resolutions of Lambda-invariant submodules of the Laurent polynomial ring in n variables as R-modules. A consequence will be the ability to …

Multiplicity Formulas For Perverse Coherent Sheaves On The Nilpotent Cone, Myron Minn-Thu-Aye

LSU Doctoral Dissertations

Arinkin and Bezrukavnikov have given the construction of the category of equivariant perverse coherent sheaves on the nilpotent cone of a complex reductive algebraic group. Bezrukavnikov has shown that this category is in fact weakly quasi-hereditary with Andersen--Jantzen sheaves playing a role analogous to that of Verma modules in category O for a semi-simple Lie algebra. Our goal is to show that the category of perverse coherent sheaves possesses the added structure of a properly stratified category, and to use this structure to give an effective algorithm to compute multiplicities of simple objects in perverse coherent sheaves. The algorithm is …