Physical Sciences and Mathematics Commons^™

Excluding A Weakly 4-Connected Minor, Kimberly Sevin D'Souza

LSU Doctoral Dissertations

A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\Delta …

Properties Of Polynomial Identity Quantized Weyl Algebras, Jesse S. F. Levitt

LSU Doctoral Dissertations

In this work on Polynomial Identity (PI) quantized Weyl algebras we begin with a brief survey of Poisson geometry and quantum cluster algebras, before using these as tools to classify the possible centers of such algebras in two different ways. In doing so we explicitly calculate the formulas of the discriminants of these algebras in terms of a general class of central polynomial subalgebras. From this we can classify all members of this family of algebras free over their centers while proving that their discriminants have the properties of effectiveness and local domination. Applying these results to the family of …

Method Of The Riemann-Hilbert Problem For The Solution Of The Helmholtz Equation In A Semi-Infinite Strip, Ashar Ghulam

LSU Doctoral Dissertations

In this dissertation, a new method is developed to study BVPs of the modified Helmholtz and Helmholtz equations in a semi-infinite strip subject to the Poincare type, impedance and higher order boundary conditions. The main machinery used here is the theory of Riemann Hilbert problems, the residue theory of complex variables and the theory of integral transforms. A special kind of interconnected Laplace transforms are introduced whose parameters are related through branch of a multi-valued function. In the chapter 1 a brief review of the unified transform method used to solve BVPs of linear and non-linear integrable PDEs in convex …

Dynamic Resonant Scattering Of Near-Monochromatic Fields, Gayan Shanaka Abeynanda

LSU Doctoral Dissertations

Certain universal features of photonic resonant scattering systems are encapsulated in a simple model which is a resonant modification of the famous Lamb Model for free vibrations of a nucleus in an extended medium. We analyze this "resonant Lamb model" to garner information on dynamic resonant scattering of near-monochromatic fields when an extended system is weakly coupled to a resonator. The transmitted field in a resonant scattering process consists of two distinct pathways: an initial pulse (direct transmission) and a tail of slow decay (resonant transmission). The resonant Lamb model incorporates a two-part scatterer attached to an infinite string with …

Spectral Properties Of Photonic Crystals: Bloch Waves And Band Gaps, Robert Paul Viator Jr

LSU Doctoral Dissertations

The author of this dissertation studies the spectral properties of high-contrast photonic crystals, i.e. periodic electromagnetic waveguides made of two materials (a connected phase and included phase) whose electromagnetic material properties are in large contrast. A spectral analysis of 2nd-order divergence-form partial differential operators (with a coupling constant k) is provided. A result of this analysis is a uniformly convergent power series representation of Bloch-wave eigenvalues in terms of the coupling constant k in the high-contrast limit k -> infinity. An explicit radius of convergence for this power series is obtained, and can be written explicitly in terms of the …

Cluster Algebras And Maximal Green Sequences For Closed Surfaces, Eric Bucher

LSU Doctoral Dissertations

Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work …

Derived Geometric Satake Equivalence, Springer Correspondence, And Small Representations, Jacob Paul Matherne

LSU Doctoral Dissertations

It is known that the geometric Satake equivalence is intimately related to the Springer correspondence when restricting to small representations of the Langlands dual group (see a paper by Achar and Henderson and one by Achar, Henderson, and Riche). This dissertation relates the derived geometric Satake equivalence of Bezrukavnikov and Finkelberg and the derived Springer correspondence of Rider when we restrict to small representations of the Langlands dual group under consideration. The main theorem of the before-mentioned paper of Achar, Henderson, and Riche sits inside this derived relationship as its degree zero piece.

Global A Priori Estimates And Sharp Existence Results For Quasilinear Equations On Nonsmooth Domains., Karthik Adimurthi

LSU Doctoral Dissertations

This thesis deals obtaining global a priori estimates for quasilinear elliptic equations and sharp existence results for Quasilinear equations with gradient nonlinearity on the right. The main results are contained in Chapters 3, 4, 5 and 6. In Chapters 3 and 4, we obtain global unweighted a priori estimates for very weak solutions below the natural exponent and weighted estimates at the natural exponent. The weights we consider are the well studied Muckenhoupt weights. Using the results obtained in Chapter 4, we obtain sharp existence result for quasilinear operators with gradient type nonlinearity on the right. We characterize the function …

Evolution Semigroups For Well-Posed, Non-Autonomous Evolution Families, Austin Keith Scirratt

LSU Doctoral Dissertations

The goal of this dissertation is to expand Berhard Koopman's operator theoretic global linearization approach to the study of nonautonomous flows. Given a system with states x in a set \Omega (the state space), a map t\to \gamma(t,s,x) (t\geq s \geq 0) is called a global flow if it describes the time evolution of a system with the initial state x \in \Omega at time t \geq s \geq 0. Koopman's approach to the study of flows is to look at the dynamics of the observables of the states instead of studying the dynamics of the states directly. To do …

Twisted Reflection Positivity, Mostafa Ahmad Hayajneh

LSU Doctoral Dissertations

Reflection positivity has several applications in both mathematics and physics. For example, reflection positivity induces a duality between group representations. In this thesis, we coin a new definition for a new kind of reflection positivity, namely, twisted reflection positive representation on a vector space. We show that all of the non-compactly causal symmetric spaces give rise to twisted reflection positive representations. We discover examples of twisted reflection positive representations on the sphere and on the Grassmannian manifold which are not unitary, namely, the generalized principle series with the Cosine transform as an intertwining operator. We give a direct proof for …

Beyond The Tails Of The Colored Jones Polynomial, Jun Peng

LSU Doctoral Dissertations

In [2] Armond showed that the heads and tails of the colored Jones polynomial exist for adequate links. This was also shown independently by Garoufalidis and Le for alternating links in [8]. Here we study coefficients of the "difference quotient" of the colored Jones polynomial. We begin with the fundamentals of knot theory. A brief introduction to skein theory is also included to illustrate those necessary tools. In Chapter 3 we give an explicit expression for the first coefficient of the relative difference. In Chapter 4 we develop a formula of t_2, the number of regions with exactly 2 crossings …

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 …

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.

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 …

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 …

Topological Dynamics On Compact Phase Spaces, Lieth Abdalateef Majed

LSU Doctoral Dissertations

Our main focus will be to investigate the various facets of what are commonly called dynamical systems or flows, which are triples $(S,X,\pi)$, where $X$ is a compact Hausdorff space and $\pi:S \times X \longrightarrow X$ is a separately continuous action of a semigroup $S$ on $X$. Historically, as was introduced by R.Ellis 1960, the enveloping semigroup, which is a closure of the set of continuous functions on a compact space $X$, was discovered to be an important tool to study dynamical systems. Soon, a realization of the existence of a universal compactification of a phase semigroup with an extended …

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 …

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.

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 …

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 …

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 …

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.

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 …

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 …

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.

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 …