Physical Sciences and Mathematics Commons^™

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 …

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.

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 …

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 …

A Conditioned Gaussian-Poisson Model For Default Phenomena, Tyler Brannan

LSU Doctoral Dissertations

We introduce a new model to study the behavior of a portfolio of defaultable assets. We refer to this model as the Gaussian-Poisson model. It builds upon one-factor Gaussian copula models and Poisson models (specifically Cox processes). Our model utilizes a random variable Y along with probability measures ℙ^• and ℙ^†. The measures ℙ^• and ℙ^† will act as market pricing measures and are obtained via conditioning. The random variable Y will act as a default descriptor.

We provide the distribution of Y under both ℙ^• and ℙ^†. We use a conditional …

Fractal Shapes Generated By Iterated Function Systems, Mary Catherine Mckinley

LSU Master's Theses

This thesis explores the construction of shapes and, in particular, fractal-type shapes as fixed points of contractive iterated function systems as discussed in Michael Barnsley's 1988 book ``Fractals Everywhere." The purpose of the thesis is to serve as a resource for an undergraduate-level introduction to the beauty and core ideas of fractal geometry, especially with regard to visualizations of basic concepts and algorithms.

Option Volatility & Arbitrage Opportunities, Mikael Boffetti

LSU Master's Theses

This paper develops several methods to estimate a future volatility of a stock in order to correctly price corresponding stock options. The pricing model known as Black-Scholes-Merton is presented with a constant volatility parameter and compares it to stochastic volatility models. It mathematically describes the probability distribution of the underlying stock price changes implied by the models and the consequences. Arbitrage opportunities between stock options of various maturities or strike prices are explained from the volatility smile and volatility term structure.

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 …

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 …

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 …

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 …

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 …

Riemann-Hilbert Formalism In The Study Of Crack Propagation In Domains With A Boundary, Aleksandr Smirnov

LSU Doctoral Dissertations

The Wiener-Hopf technique is a powerful tool for constructing analytic solutions for a wide range of problems in physics and engineering. The key step in its application is solution of the Riemann-Hilbert problem, which consists of finding a piece-wise analytic (vector-) function in the complex plane for a specified behavior of its discontinuities. In this dissertation, the applied theory of vector Riemann-Hilbert problems is reviewed. The analytical solution representing the problem on a Riemann surface, and a numerical solution that reduces the problem to singular integral equations, are considered, as well as a combination of the numerical and analytical techniques …

On Properties Of Matroid Connectivity, Simon Pfeil

LSU Doctoral Dissertations

Highly connected matroids are consistently useful in the analysis of matroid structure. Round matroids, in particular, were instrumental in the proof of Rota's conjecture. Chapter 2 concerns a class of matroids with similar properties to those of round matroids. We provide many useful characterizations of these matroids, and determine explicitly their regular members. Tutte proved that a 3-connected matroid with every element in a 3-element circuit and a 3-element cocircuit is either a whirl or the cycle matroid of a wheel. This result led to the proof of the 3-connected splitter theorem. More recently, Miller proved that matroids of sufficient …

Towards Theory And Applications Of Generalized Categories To Areas Of Type Theory And Categorical Logic, Lucius Traylor Schoenbaum

LSU Doctoral Dissertations

Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to higher category theory, we undertake a detailed study of a new mathematical abstraction, the generalized category. It is a partially defined monoid equipped with endomorphism maps defining sources and targets on arbitrary elements, possibly allowing a proximal behavior with respect to composition. We first present a formal introduction to the theory of generalized categories. We describe functors, equivalences, natural transformations, adjoints, and limits in the generalized …

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 Study Of Mathematical Equivalence: The Importance Of The Equal Sign, Christy De'sha Duncan

LSU Master's Theses

The purpose of this study was to investigate students’ understanding and knowledge of the equal sign, so that instructional resources could be identified to improve student’s conceptual understanding about mathematical equivalence. A test, consisting of a combination of items taken from previous studies, as well as items developed by the researchers, was designed to gauge students’ understanding of the equality symbol. The test was administered to 54 seventh-graders in Spring 2015. The results of the test indicated a significant number of students in our district have a limited understanding of mathematical equivalence. This papers ends with some suggested activities recommended …

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 …

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 …

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 …

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 …

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.

Shape Optimization For Drag Minimization Using The Navier-Stokes Equation, Chukwudi Paul Chukwudozie

LSU Master's Theses

Fluid drag is a force that opposes relative motion between fluid layers or between solids and surrounding fluids. For a stationary solid in a moving fluid, it is the amount of force necessary to keep the object stationary in the moving fluid. In addition to fluid and flow conditions, pressure drag on a solid object is dependent on the size and shape of the object. The aim of this project is to compute the shape of a stationary 2D object of size 3.5 m2 that minimizes drag for different Reynolds numbers. We solve the problem in the context of shape …

Increasing Student Engagement In The Secondary Math Classroom, Chantell Holloway Walker

LSU Master's Theses

This thesis reports on a professional development package developed by the author to help three teachers increase the level of student engagement in their math classrooms. There were three phases: 1) initial presentation of strategies and sample lessons, 2) classroom implementation, 3) reflection and evaluation. As a result of the professional development, the Louisiana Compass Teacher Evaluation Rubric scores of the teachers improved in the area of student engagement. This thesis can be used as a guide for principals or instructional specialists who wish to provide professional development for small groups of teachers, with a focus on increasing student engagement.

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 …

Exploring Rational Numbers In Middle School, Robyn Jasmin Boudoin

LSU Master's Theses

The move by the state of Louisiana to fully implement the Common Core State Standards (CCSS) from 2013 -2014 school year on and to align all state mandated tests to the CCSS has caused teachers to change the way they teach and how they deliver content. The overall most crucial new part of the CCSS in Mathematics is the emphasis on the “Standards for Mathematical Practice”. In order to illustrate the meaning of the Mathematical Practice Standards, non routine problems must be used that allow students and teachers to “dig deeper” and practice their mathematical habits of mind. Rational numbers …

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 …

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.

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.