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Articles 151 - 180 of 188
Full-Text Articles in Applied Mathematics
Topics In Quantum Topology, Khaled Moham Qazaqzeh
Topics In Quantum Topology, Khaled Moham Qazaqzeh
LSU Doctoral Dissertations
In chapter 1, which represents joint work with Gilmer, we define an index two subcategory of a 3-dimensional cobordism category. The objects of the category are surfaces equipped with Lagrangian subspaces of their real first homology. This generalizes the result of [9] where surfaces are equipped with Lagrangian subspaces of their rational first homology. To define such subcategory, we give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over R. In chapter 2, we find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus …
Extension Of Shor's Period-Finding Algorithm To Infinite Dimensional Hilbert Spaces, Jeremy James Becnel
Extension Of Shor's Period-Finding Algorithm To Infinite Dimensional Hilbert Spaces, Jeremy James Becnel
LSU Doctoral Dissertations
Over the last decade quantum computing has become a very popular field in various disciplines, such as physics, engineering, and mathematics. Most of the attraction stemmed from the famous Shor period--finding algorithm, which leads to an efficient algorithm for factoring positive integers. Many adaptations and generalizations of this algorithm have been developed through the years, some of which have not been ripened with full mathematical rigor. In this dissertation we use concepts from white noise analysis to rigorously develop a Shor algorithm adapted to find a hidden subspace of a function with domain a real Hilbert space. After reviewing the …
Classifying Quadratic Number Fields Up To Arf Equivalence, Jeonghun Kim
Classifying Quadratic Number Fields Up To Arf Equivalence, Jeonghun Kim
LSU Doctoral Dissertations
Two number fields K and L are said to be Arf equivalent if there exists a bijection T : ΩK → ΩL of places of K and of L such that KP and LTP are locally Arf equivalent for every place P ε ΩK. That is, |K*p/K*2p| = |L*TP/L*2TP|, type[( , )P] = type[( , )TP], and Arf(rP ) = Arf(rTP ) for every place P ε ΩK, where rP is the local …
Representation Properties Of Definite Lattices In Function Fields, Jean Edouard Bureau
Representation Properties Of Definite Lattices In Function Fields, Jean Edouard Bureau
LSU Doctoral Dissertations
This work is made of two different parts. The first contains results concerning isospectral quadratic forms, and the second is about regular quadratic forms. Two quadratic forms are said to be isospectral if they have the same representation numbers. In this work, we consider binary and ternary definite integral quadratic form defined over the polynomial ring F[t], where F is a finite field of odd characteristic. We prove that the class of such a form is determined by its representation numbers. Equivalently, we prove that there is no nonequivalent definite F[t]-lattices of rank 2 or 3 having the same theta …
Limit Theorems For Weighted Stochastic Systems Of Interacting Particles, Jie Wu
Limit Theorems For Weighted Stochastic Systems Of Interacting Particles, Jie Wu
LSU Doctoral Dissertations
The goal of this dissertation is to (a) establish the weak convergence of empirical measures formed by a system of stochastic differential equations, and (b) prove a comparison result and compactness of support property for the limit measure. The stochastic system of size n has coefficients that depend on the empirical measure determined by the system. The weights for the empirical measure are determined by a further n-system of stochastic equations. There is a random choice among N types of weights. The existence and uniqueness of solutions of the interacting system, weak convergence of the empirical measures, and the identification …
Filippov's Operator And Discontinuous Differential Equations, Khalid Abdulaziz Alshammari
Filippov's Operator And Discontinuous Differential Equations, Khalid Abdulaziz Alshammari
LSU Doctoral Dissertations
The thesis is mainly concerned about properties of the so-called Filippov operator that is associated with a differential inclusion x'(t) ε F(x(t)) a.e. t ε [0,T], where F : Rn → Rn is given set-valued map. The operator F produces a new set-valued map F[F], which in effect regularizes F so that F[F] has nicer properties. After presenting its definition, we show that F[F] is always upper-semicontinuous as a map from Rn to the metric space of compact subsets of Rn endowed with the Hausdorff metric. Our main approach is to study the …
Circuits And Structure In Matroids And Graphs, Brian Daniel Beavers
Circuits And Structure In Matroids And Graphs, Brian Daniel Beavers
LSU Doctoral Dissertations
This dissertation consists of several results on matroid and graph structure and is organized into three main parts. The main goal of the first part, Chapters 1-3, is to produce a unique decomposition of 3-connected matroids into more highly connected pieces. In Chapter 1, we review the definitions and main results from the previous work of Hall, Oxley, Semple, and Whittle. In Chapter 2, we introduce operations that allow us to decompose a 3-connected matroid M into a pair of 3-connected pieces by breaking the matroid apart at a 3-separation. We also generalize a result of Akkari and Oxley. In …
On Moment Conditions For The Girsanov Theorem, See Keong Lee
On Moment Conditions For The Girsanov Theorem, See Keong Lee
LSU Doctoral Dissertations
In this dissertation, the well-known Girsanov Theorem will be proved under a set of moment conditions on exponential processes. Our conditions are motivated by the desire to avoid using the local martingale theory in the proof of the Girsanov Theorem. Namely, we will only use the martingale theory to prove the Girsanov Theorem. Many sufficient conditions for the validity of the Girsanov Theorem have been found since the publication of the result by Girsanov in 1960. We will compare our conditions with some of these conditions. As an application of the Girsanov Theorem, we will show the nonexistence of an …
Impulsive Systems, Stanislav Zabic
Impulsive Systems, Stanislav Zabic
LSU Doctoral Dissertations
Impulsive systems arise when dynamics produce discontinuous trajectories. Discontinuties occur when movements of states happen over a small interval that resembles a point-mass measure. We adopt the formalism in which the controlled dynamic inclusion is the sum of a slow and a fast time velocities belonging to two distinct vector fields. Fast time velocities are controlled by a vector valued Borel measure. The trajectory of impulsive systems is a function of bounded variation. To give a definition of solutions, a notion of graph completion of the control measure is needed. In the nonimpulsive case, a solution can be defined as …
Zeta Functions Of Finite Graphs, Debra Czarneski
Zeta Functions Of Finite Graphs, Debra Czarneski
LSU Doctoral Dissertations
Ihara introduced the zeta function of a finite graph in 1966 in the context of p-adic matrix groups. The idea was generalized to all finite graphs in 1989 by Hashimoto. We will introduce the zeta function from both perspectives and show the equivalence of both forms. We will discuss several properties of finite graphs that are determined by the zeta function and show by counterexample several properties of finite graphs that are not determined by the zeta function. We will also discuss the relationship between the zeta function of a finite graph and the spectrum of a finite graph.
Quasicontinuous Derivatives And Viscosity Functions, Rodica Cazacu
Quasicontinuous Derivatives And Viscosity Functions, Rodica Cazacu
LSU Doctoral Dissertations
In this work we demonstrate how the continuous domain theory can be applied to the theory of nonlinear optimization, particularly to the theory of viscosity solutions. We consider finding the viscosity solution for the Hamilton-Jacobi equation H(x, y) = g(x), with continuous hamiltonian, but with possibly discontinuous right-hand side. We begin by finding a new function space Q(X,L), the space of equivalence classes of quasicontinuous functions from a locally compact set X to a bicontinuous lattice L and we will define on Q(X,L) the qo-topology, which is a variant of classical order topology defined on complete lattices. On this …
Multiscale Strain Analysis, Timothy Donald Breitzman
Multiscale Strain Analysis, Timothy Donald Breitzman
LSU Doctoral Dissertations
The mathematical homogenization and corrector theory relevant to prestressed heterogeneous materials in the linear-elastic regime is discussed. A suitable corrector theory is derived to reconstruct the local strain field inside the composite. Based on this theory, we develop an inexpensive numerical method for multi scale strain analysis within a prestressed heterogeneous material. The theory also provides a characterization of the macroscopic strength domain. The strength domain places constraints on the homogenized strain field which guarantee that the actual strain in the heterogeneous material lies inside the strength domain of each material participating in the structure.
Laguerre Functions Associated To Euclidean Jordan Algebras, Michael Aristidou
Laguerre Functions Associated To Euclidean Jordan Algebras, Michael Aristidou
LSU Doctoral Dissertations
Certain differential recursion relations for the Laguerre functions, defined on a symmetric cone Ω, can be derived from the representations of a specific Lie algebra on L2(Ω,dμv). This Lie algebra is the corresponding Lie algebra of the Lie group G that acts on the tube domain T(Ω)=Ω+iV, where V is the associated Euclidean Jordan algebra of Ω. The representations involved are the highest weight representations of G on L2(Ω,dμv). To obtain these representations, we start from the highest weight representations of G on Hv(T(Ω)), the Hilbert space of holomorphic functions …
Virtual Strings For Closed Curves With Multiple Components And Filamentations For Virtual Links, William Schellhorn
Virtual Strings For Closed Curves With Multiple Components And Filamentations For Virtual Links, William Schellhorn
LSU Doctoral Dissertations
The theory of filaments on oriented chord diagrams can be used to detect some non-classical virtual knots. We extend existing filament techniques to virtual links with more than one component and give examples of virtual links that these techniques can detect as non-classical. Given a signed Gauss word underlying an oriented chord diagram, we describe how to construct a finite sequence of integers that encodes all of the filament information for the diagram. We also introduce a square array of integers called a MIN-square that summarizes the filament information about all of the signed Gauss words having a given Gauss …
Dynamical Systems With Time Delay, Norma Ortiz
Dynamical Systems With Time Delay, Norma Ortiz
LSU Doctoral Dissertations
In this dissertation, we study necessary conditions and weak invariance properties of dynamical systems with time delay. A number of results have been obtained recently that refine necessary conditions of optimal solutions for nonsmooth dynamical systems without time delay. In this dissertation, we examine the extension of some of these results to problems with time delay. In particular, we study the generalized problem of Bolza with the addition of delay in the state and velocity variables and refer to this problem as the Neutral Problem of Bolza. We consider the relationship between the generalized problem of Bolza with time delay …
Stability In Dynamical Polysystems, George Cazacu
Stability In Dynamical Polysystems, George Cazacu
LSU Doctoral Dissertations
A dynamical polysystem consists of a family of continuous dynamical systems, all acting on a given metric space. The first chapter of the present thesis shows a generalization of control systems via dynamical polysystems and establishes the equivalence of the two notions under certain lipschitz condition on the function defining the dynamics. The remaining chapters are focused on a basic theory of dynamical polysystems. Some topological properties of limit sets are described in Chapter 2. Chapters 3 and 4 provide characterizations for various notions of strong stability. Chapter 5 makes use of the theory of closed relations to study Lyapunov …
Dissipative Lipschitz Dynamics, Vinicio Rafael Rios
Dissipative Lipschitz Dynamics, Vinicio Rafael Rios
LSU Doctoral Dissertations
In this dissertation we study two related important issues in control theory: invariance of dynamical systems and Hamilton-Jacobi theory associated with optimal control theory. Given a control system modelled as a differential inclusion, we provide necessary and sufficient conditions for the strong invariance property of the system when the dynamic satisfies a dissipative Lipschitz condition. We show that when the dynamic is almost upper semicontinuous and satisfies the dissipative Lipschitz property, these conditions can be expressed in terms of approximate Hamilton-Jacobi inequalities, which subsumes the classic infinitesimal characterization of strongly invariant systems given under the Lipschitz assumtion. In the important …
Error Estimates For Stabilized Approximation Methods For Semigroups, Sarah Campbell Mcallister
Error Estimates For Stabilized Approximation Methods For Semigroups, Sarah Campbell Mcallister
LSU Doctoral Dissertations
In this work we analyze error estimates for rational approximation methods, and their stabilizations, for strongly continuous semigroups. Chapter 1 consists of a brief survey of time discretization methods for semigroups. In Chapter 2, we demonstrate a new method for obtaining convergent approximations in the absence of stability for strongly continuous semigroups with arbitrary initial data. In Section 2.2, we state the stabilization result in more general form and show that this method can be used to improve known error estimates by a magnitude of up to one half for smooth initial data. In Section 2.3, we give concrete examples …
Wavelet Sets With And Without Groups And Multiresolution Analysis, Mihaela Dobrescu
Wavelet Sets With And Without Groups And Multiresolution Analysis, Mihaela Dobrescu
LSU Doctoral Dissertations
In this dissertation we study a special kind of wavelets, the so-called minimally supported frequency wavelets and the associated wavelet sets. Most of the examples of wavelet sets are for dilation sets which are groups. In this work we construct wavelet sets for which the dilation set, D, is of the form D=MN, where the product is direct, and so D is not necessarily group. In the second part of this dissertation we construct multiwavelets associated with MRA's and we generalize the rotations in the dilation sets to Coxeter groups.
Asymptotic Laplace Transforms, Claudiu Mihai
Asymptotic Laplace Transforms, Claudiu Mihai
LSU Doctoral Dissertations
In this work we discuss certain aspects of the classical Laplace theory that are relevant for an entirely analytic approach to justify Heaviside's operational calculus methods. The approach explored here suggests an interpretation of the Heaviside operator ${cdot}$ based on the "Asymptotic Laplace Transform." The asymptotic approach presented here is based on recent work by G. Lumer and F. Neubrander on the subject. In particular, we investigate the two competing definitions of the asymptotic Laplace transform used in their works, and add a third one which we suggest is more natural and convenient than the earlier ones given. We compute …
On Qualitative Properties And Convergence Of Time-Discretization Methods For Semigroups, Mihaly Kovacs
On Qualitative Properties And Convergence Of Time-Discretization Methods For Semigroups, Mihaly Kovacs
LSU Doctoral Dissertations
In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We …
The Radon-Gauss Transform, Vochita Mihai
The Radon-Gauss Transform, Vochita Mihai
LSU Doctoral Dissertations
Gaussian measure is constructed for any given hyperplane in an infinite dimensional Hilbert space, and this is used to define a generalization of the Radon transform to the infinite dimensional setting, using Gauss measure instead of Lebesgue measure. An inversion formula is obtained and a support theorem proved.
Class Groups And Norms Of Units, Costel Ionita
Class Groups And Norms Of Units, Costel Ionita
LSU Doctoral Dissertations
Our object of study is relative quadratic extensions of algebraic number fields. In 'Class Number Parity', the authors P.E. Conner and J. Hurrelbrink study in detail the cases of real and CM-extensions. In this paper we generalize some of the results without any assumption on the type of the relative quadratic extension.
Orbit Structure On The Silov Boundary Of A Tube Domain And The Plancherel Decomposition Of A Causally Compact Symmetric Space, With Emphasis On The Rank One Case, Troels Roussau Johansen
Orbit Structure On The Silov Boundary Of A Tube Domain And The Plancherel Decomposition Of A Causally Compact Symmetric Space, With Emphasis On The Rank One Case, Troels Roussau Johansen
LSU Doctoral Dissertations
We construct a G-equivariant causal embedding of a compactly causal symmetric space G/H as an open dense subset of the Silov boundary S of the unbounded realization of a certain Hermitian symmetric space G1/K1 of tube type. Then S is an Euclidean space that is open and dense in the flag manifold G1/P', where P' denotes a certain parabolic subgroup of G1. The regular representation of G on L2(G/H) is thus realized on L2(S), and we use abelian harmonic analysis in the study thereof. In particular, …
On The Geometry And Topology Of Moduli Spaces Of Multi-Polygonal Linkages, Michael Edward Holcomb
On The Geometry And Topology Of Moduli Spaces Of Multi-Polygonal Linkages, Michael Edward Holcomb
LSU Doctoral Dissertations
The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal linkage by taking two free linkages and identifying initial and terminal vertices. This can be generalized so that one takes three free linkages and identifies initial and terminal vertices. Then one obtains a linkage which contains multiple polygons, any two of which have shared edges. The geometric and topological properties of moduli spaces of these multi-polygonal linkages are studied. These spaces turn out to be compact …
Splitter Theorems For 3- And 4-Regular Graphs, Jinko Kanno
Splitter Theorems For 3- And 4-Regular Graphs, Jinko Kanno
LSU Doctoral Dissertations
Let g be a class of graphs and ≤ be a graph containment relation. A splitter theorem for g under ≤ is a result that claims the existence of a set O of graph operations such that if G and H are in g and H≤G with G≠H, then there is a decreasing sequence of graphs from G to H, say G=G0≥G1≥G2...Gt=H, all intermediate graphs are in g, and each Gi can be obtained from Gi-1 by applying a single …
Book Embeddings Of Graphs, Robin Leigh Blankenship
Book Embeddings Of Graphs, Robin Leigh Blankenship
LSU Doctoral Dissertations
We use a structural theorem of Robertson and Seymour to show that for every minor-closed class of graphs, other than the class of all graphs, there is a number k such that every member of the class can be embedded in a book with k pages. Book embeddings of graphs with relation to surfaces, vertex extensions, clique-sums and r-rings are combined into a single book embedding of a graph in the minor-closed class. The effects of subdividing a complete graph and a complete bipartite graph with respect to book thickness are studied. We prove that if n ≥ 3, …
Equations Of Parametric Surfaces With Base Points Via Syzygies, Haohao Wang
Equations Of Parametric Surfaces With Base Points Via Syzygies, Haohao Wang
LSU Doctoral Dissertations
Suppose $S$ is a parametrized surface in complex projective 3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1 imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem is to compute an implicit equation $F=0$ of $S$ using the parametrization $phi$. An algorithm using syzygies exists for computing $F$ if $phi$ has no base points, i.e. $phi$ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. …
The Kauffman Bracket Skein Module Of The Quaternionic Manifold, John Michael Harris
The Kauffman Bracket Skein Module Of The Quaternionic Manifold, John Michael Harris
LSU Doctoral Dissertations
In this work, we study the structure of the Kauffman bracket skein module of the quaternionic manifold over the field of rational functions. We begin with a brief survey of manifolds whose Kauffman bracket skein modules are known, and proceed in Chapter 2 by recalling the facts from Temperley-Lieb recoupling theory that we use in the proofs. In Chapter 3, using recoupling theory and with Mathematica's assistance, we index an infinite presentation of the skein module, and conjecture that it is five-dimensional. In Chapter 4, using a new set of relations, we prove that the skein module is indeed spanned …
Explicit Multiplicative Relations Between Gauss Sums, Brian J. Murray
Explicit Multiplicative Relations Between Gauss Sums, Brian J. Murray
LSU Doctoral Dissertations
H.Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.