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Articles 121 - 150 of 188
Full-Text Articles in Applied Mathematics
Algorithms Related To Subgroups Of The Modular Group, Constantin Cristian Caranica
Algorithms Related To Subgroups Of The Modular Group, Constantin Cristian Caranica
LSU Doctoral Dissertations
Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose of this thesis is to find and characterize small-index noncongruence subgroups of the modular group PSL_2{Z}. We use the concept of Farey symbol to describe the subgroups of PSL_2{Z}. The first part contains results concerning the geometry of subgroups of PSL_2{Z}. The second part describes a graph-theoretical approach to finding all subgroups of a given …
The Structure Of 4-Separations In 4-Connected Matroids, Jeremy M. Aikin
The Structure Of 4-Separations In 4-Connected Matroids, Jeremy M. Aikin
LSU Doctoral Dissertations
Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we …
White Noise Methods For Anticipating Stochastic Differential Equations, Julius Esunge
White Noise Methods For Anticipating Stochastic Differential Equations, Julius Esunge
LSU Doctoral Dissertations
This dissertation focuses on linear stochastic differential equations of anticipating type. Owing to the lack of a theory of differentiation for random processes, the said differential equations are appropriately understood and studied as anticipating stochastic integral equations. The unfolding work considers equations in which anticipation arises either from the initial condition or the integrand. In this regard, the techniques of white noise analysis are applied to such equations. In particular, by using the Hitsuda-Skorokhod integral which nicely extends the It integral to anticipating integrands, we then apply the S-transform from white noise analysis to study this new equation.
Function Spaces, Wavelets And Representation Theory, Jens Gerlach Christensen
Function Spaces, Wavelets And Representation Theory, Jens Gerlach Christensen
LSU Doctoral Dissertations
This dissertation is concerned with the interplay between the theory of Banach spaces and representations of groups. The wavelet transform has proven to be a useful tool in characterizing and constructing Banach spaces, and we investigate a generalization of an already known technique due to H.G. Feichtinger and K. Gröchenig. This generalization is presented in Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be described using the theory. The first example clears up a question regarding a wavelet characterization of Bergman spaces related to a non-integrable representation. The second example is a wavelet …
Unavoidable Minors In Graphs And Matroids, Carolyn Barlow Chun
Unavoidable Minors In Graphs And Matroids, Carolyn Barlow Chun
LSU Doctoral Dissertations
It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges …
A Regularization Technique In Dynamic Optimization, Alvaro Guevara
A Regularization Technique In Dynamic Optimization, Alvaro Guevara
LSU Doctoral Dissertations
In this dissertation we discuss certain aspects of a parametric regularization technique which is based on recent work by R. Goebel. For proper, lower semicontinuous, and convex functions, this regularization is self-dual with respect to convex conjugation, and a simple extension of this smoothing exhibits the same feature when applied to proper, closed, and saddle functions. In Chapter 1 we give a introduction to convex and saddle function theory, which includes new results on the convergence of saddle function values that were not previously available in the form presented. In Chapter 2, we define the regularization and extend some of …
The Extended Picture Group, With Applications To Line Arrangement Complements, Charles Richard Egedy
The Extended Picture Group, With Applications To Line Arrangement Complements, Charles Richard Egedy
LSU Doctoral Dissertations
We obtain the picture group as the quotient with a torsion subgroup, of an extended picture group, which is isomorphic to the kernel of a precrossed module homomorphism. In addition to expanding the notion of a picture group, the new formulation gives a natural way to construct homomorphisms between picture groups by describing deformations of one-vertex subpictures. The extended picture group thus provides a convenient way to describe generators for the second homotopy group of line arrangement complements as well as homomorphisms between these groups. In particular, we show that the homomorphisms relate to a lattice structure corresponding roughly to …
Some Results On Cubic Graphs, Evan Morgan
Some Results On Cubic Graphs, Evan Morgan
LSU Doctoral Dissertations
Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley's question in general remains unanswered, our investigations led to two graph operations (Chapters 2 and 4) which are of independent interest. We present some partial results toward Oxley's question in Chapter 3. The central results of the dissertation involve an operation on cubic graphs called the switch; in the literature, a similar operation is known as the edge slide. In Chapter 2, the author proves that we can transform, …
Homological Width And Turaev Genus, Adam Lowrance
Homological Width And Turaev Genus, Adam Lowrance
LSU Doctoral Dissertations
Khovanov homology and knot Floer homology are generalizations of the Jones polynomial and the Alexander polynomial respectively. They are bigraded Z-modules, and their underlying polynomials are recovered by taking the graded Euler characteristic. The two homologies share many characteristics, however their relationship has yet to be fully understood. In both Khovanov homology and knot Floer homology, the two gradings can be combined into a single diagonal grading. Homological width is a measure of the support of the homology with respect to the diagonal grading. In this thesis, we show that the homological width of Khovanov homology and knot Floer homology …
A Discrete Model Of Guided Modes And Anomalous Scattering In Periodic Structures, Natalia Grigoryevna Ptitsyna
A Discrete Model Of Guided Modes And Anomalous Scattering In Periodic Structures, Natalia Grigoryevna Ptitsyna
LSU Doctoral Dissertations
We study a discrete prototype of anomalous scattering associated with the interaction of guided modes of a periodic scatterer and plane waves incident upon the scatterer. The transmission anomalies arise because of the non-robustness of a guided mode, a mode that exists only at a specific frequency and wave number pair. The simplicity of the discrete prototype allows one to make certain explicit calculations and proofs, and to examine details of important resonant phenomena of the open wave guides. The main results are (1) a formula for transmission anomalies near a non-robust guided mode with rigorous error estimates that extends …
Stochastic Navier-Stokes Equations With Fractional Brownian Motions, Liqun Fang
Stochastic Navier-Stokes Equations With Fractional Brownian Motions, Liqun Fang
LSU Doctoral Dissertations
The aim of this dissertation is to study stochastic Navier-Stokes equations with a fractional Brownian motion noise. The second chapter will introduce the background results on fractional Brownian motions and some of their properties. The third chapter will focus on the Stokes operator and the semigroup generated by this operator. The Navier-Stokes equations and the evolution equation setup will be described in the next chapter. The main goal is to prove the existence and uniqueness of solutions for the stochastic Navier-Stokes equations with a fractional Brownian motion noise under suitable conditions. The proof is given with full details for two …
Local Behavior Of Distributions And Applications, Jasson Vindas
Local Behavior Of Distributions And Applications, Jasson Vindas
LSU Doctoral Dissertations
This dissertation studies local and asymptotic properties of distributions (generalized functions) in connection to several problems in harmonic analysis, approximation theory, classical real and complex function theory, tauberian theory, summability of divergent series and integrals, and number theory. In Chapter 2 we give two new proofs of the Prime Number Theory based on ideas from asymptotic analysis on spaces of distributions. Several inverse problems in Fourier analysis and summability theory are studied in detail. Chapter 3 provides a complete characterization of point values of tempered distributions and functions in terms of a generalized pointwise Fourier inversion formula. The relation of …
The Segal-Bargmann Transform On Inductive Limits Of Compact Symmetric Spaces, Keng Wiboonton
The Segal-Bargmann Transform On Inductive Limits Of Compact Symmetric Spaces, Keng Wiboonton
LSU Doctoral Dissertations
We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\{L^2(M_n)^{K_n}\}_n$ where $\{M_n = U_n/K_n\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \ n = 1,2,...$. For each $n$, let $\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\widehat{U_n/K_n} \subseteq \widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\widehat{U_n/K_n}$ …
Convolution Semigroups, Kevin W. Zito
Convolution Semigroups, Kevin W. Zito
LSU Doctoral Dissertations
In this dissertation we investigate, compute, and approximate convolution powers of functions (often probability densities) with compact support in the positive real numbers. Extending results of Ursula Westphal from 1974 concerning the characteristic function on the interval $[0,1]$, it is shown that positive, decreasing step functions with compact support can be embedded in a convolution semigroup in $L^1(0,infty)$ and that any decreasing, positive function $pin L^1(0,infty)$ can be embedded in a convolution semigroup of distributions. As an application to the study of evolution equations, we consider an evolutionary system that is described by a bounded, strongly continuous semigroup ${T(t)}_{tgeq0}$ in …
Impulsive Control Systems, Wei Cai
Impulsive Control Systems, Wei Cai
LSU Doctoral Dissertations
Impulsive control systems arose from classical control systems described by differential equations where the control functions could be unbounded. Passing to the limit of trajectories whose velocities are changing very rapidly leads to the state vector to "jump", or exhibit impulsive behavior. The mathematical model in this thesis uses a differential inclusion and a measure-driven control, and it becomes possible to deal with the discontinuity of movements happening over a small interval. We adopt the formulism of impulsive systems in which the velocities are decomposed by the slow and fast ones. The fast time velocity is expressed as the multiplication …
Multiscale Analysis Of Heterogeneous Media For Local And Nonlocal Continuum Theories, Bacim Alali
Multiscale Analysis Of Heterogeneous Media For Local And Nonlocal Continuum Theories, Bacim Alali
LSU Doctoral Dissertations
The dissertation provides new multiscale methods for the analysis of heterogeneous media. The first part of the dissertation treats heterogeneous media using the theory of linear elasticity. In this context, a methodology is presented for bounding the higher order moments of the local stress and strain fields inside random elastic media. Optimal lower bounds that are given in terms of the applied loading and the volume (area) fractions for random two-phase composites are presented. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. The second part of …
Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
LSU Doctoral Dissertations
This dissertation is concerned with providing a description of certain symmetric bilinear forms, called trace forms, associated with finite normal extensions N/K of an algebraic number field K, with abelian Galois group Gal(N/K). These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). Complete descriptions are obtained when the base field K has exactly one dyadic prime and either no real embeddings or one real embedding. For these fields K, the set of abelian trace forms is closed under multiplication in the Witt ring W(K).
Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara
Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara
LSU Doctoral Dissertations
In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes …
Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger
Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger
LSU Doctoral Dissertations
The main categories of study in this thesis are the categories of diffeological and Fr\"olicher spaces. They form concrete cartesian closed categories. In Chapter 1 we provide relevant background from category theory and differentiation theory in locally convex spaces. In Chapter 2 we define a class of categories whose objects are sets with a structure determined by functions into the set. Fr\"olicher's $M$-spaces, Chen's differentiable spaces and Souriau's diffeological spaces fall into this class of categories. We prove cartesian closedness of the two main categories, and show that they have all limits and colimits. We exhibit an adjunction between the …
Surgery Description Of Colored Knots, Steven Daniel Wallace
Surgery Description Of Colored Knots, Steven Daniel Wallace
LSU Doctoral Dissertations
By a knot, or link, we mean a circle, or a collection of circles, embedded in the three-sphere S3. The study of knots is a very rich subject and plays a key role in the area of low-dimensional topology. In fact, a theorem of W.B.R. Lickorish and A.D. Wallace states that any three-dimensional manifold may be described by Dehn surgery along a link which is the process of removing the link from S3 and then gluing it back in a way that possibly changes the resulting manifold. In this dissertation, we will be interested in the pair (K, ρ) consisting …
Stochastic And Copula Models For Credit Derivatives, Chao Meng
Stochastic And Copula Models For Credit Derivatives, Chao Meng
LSU Doctoral Dissertations
We prove results relating to the exit time of a stochastic process from a region in N-dimensional space. We compute certain stochastic integrals involving the exit time. Taking a Gaussian copula model for the hitting time behavior, we prove several results on the sensitivity of quantities connected with the hitting times to parameters of the model, as well as the large-N behavior. We discuss the relationship of these results to certain credit derivative instruments. Relevant simulations are presented.
Laplace Transform Inversion And Time-Discretization Methods For Evolution Equations, Koray Ozer
Laplace Transform Inversion And Time-Discretization Methods For Evolution Equations, Koray Ozer
LSU Doctoral Dissertations
In this dissertation, we introduce Post-Widder-type inversion methods for the Laplace transform based on A-stable rational approximations of the exponential function. Since the results hold for Banach-space-valued functions, they yield efficient time-discretization methods for evolution equations of convolution type; e.g., linear first and higher order abstract Cauchy problems, inhomogeneous Cauchy problems, delay equations, Volterra and integro-differential equations, and problems that can be re-written as an abstract Cauchy problem on an appropriate state space.
Fast Marching Methods - Parallel Implementation And Analysis, Maria Cristina Tugurlan
Fast Marching Methods - Parallel Implementation And Analysis, Maria Cristina Tugurlan
LSU Doctoral Dissertations
Fast Marching represents a very efficient technique for solving front propagation problems, which can be formulated as partial differential equations with Dirichlet boundary conditions, called Eikonal equation: $F(x)|\nabla T(x)|=1$, for $x \in \Omega$ and $T(x)=0$ for $x \in \Gamma$, where $\Omega$ is a domain in $\mathbb{R}^n$, $\Gamma$ is the initial position of a curve evolving with normal velocity F>0. Fast Marching Methods are a necessary step in Level Set Methods, which are widely used today in scientific computing. The classical Fast Marching Methods, based on finite differences, are typically sequential. Parallelizing Fast Marching Methods is a step forward for …
Subrepresentation Semirings And An Analogue Of 6j-Symbols, Nam Hee Kwon
Subrepresentation Semirings And An Analogue Of 6j-Symbols, Nam Hee Kwon
LSU Doctoral Dissertations
Let G be a quasi simply reducible group, and let V be a representation of G over the complex numbers $mathbb{C}$. In this thesis, we introduce the twisted 6j-symbols over G which have their origin to Wigner's 6j-symbols over the group SU(2) to study the structure constants of the subrepresentation semiring S_{G}(End(V)), and we study the representation theory of a quasi simply reducible group G laying emphasis on our new G-module objects. We also investigate properties of our twisted 6j-symbols by establishing the link between the twisted 6j-symbols and Wigner's 3j-symbols over the group G.
Backward Stochastic Navier-Stokes Equations In Two Dimensions, Hong Yin
Backward Stochastic Navier-Stokes Equations In Two Dimensions, Hong Yin
LSU Doctoral Dissertations
There are two parts in this dissertation. The backward stochastic Lorenz system is studied in the first part. Suitable a priori estimates for adapted solutions of the backward stochastic Lorenz system are obtained. The existence and uniqueness of solutions is shown by the use of suitable truncations and approximations. The continuity of the adapted solutions with respect to the terminal data is also established. The backward stochastic Navier-Stokes equations (BSNSEs, for short) corresponding to incompressible fluid flow in a bounded domain $G$ are studied in the second part. Suitable a priori estimates for adapted solutions of the BSNSEs are obtained …
Comparison Of Kp And Bbm-Kp Models, Gideon Pyelshak Daspan
Comparison Of Kp And Bbm-Kp Models, Gideon Pyelshak Daspan
LSU Doctoral Dissertations
In this dissertation we show that the solution of the pure initial-value problems for the KP and regularize KP equations are the same, to within the order of accuracy attributable to either, on the time scale from zero to epsilon to negative three halves power, during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.
Sign Ambiguities Of Gaussian Sums, Heon Kim
Sign Ambiguities Of Gaussian Sums, Heon Kim
LSU Doctoral Dissertations
In 1934, two kinds of multiplicative relations, extit{norm and Davenport-Hasse} relations, between Gaussian sums, were known. In 1964, H. Hasse conjectured that the norm and Davenport-Hasse relations are the only multiplicative relations connecting the Gaussian sums over $mathbb F_p$. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture when Gaussian sums are considered as numbers. This counterexample was a new type of multiplicative relation, called a {it sign ambiguity} (see Definition ef{defi:of_sign_ambi}), involving a $pm$ sign not connected to elementary properties of Gauss sums. In Chapter $5$, we provide an explicit product formula giving an infinite class …
Multiplicative Renormalization Method For Orthogonal Polynomials, Suat Namli
Multiplicative Renormalization Method For Orthogonal Polynomials, Suat Namli
LSU Doctoral Dissertations
To study the orthogonal polynomials, Asai, Kubo and Kuo recently have developed the multiplicative renormalization method. Motivated by infinite dimensional white noise analysis, it is an alternative to the computational part of the classical Gram-Schmidt process to find the orthogonal polynomials for a given measure. Instead of finding the orthogonal polynomials recursively as described in the Gram-Schmidt process, one analyzes different types of generating functions systematically in order to obtain polynomials after power series expansion. This work also produces the Jacobi-Szego parameters easily and paves the way for the study of one-mode interacting Fock spaces related to these parameters. They …
An Inverse Homogenization Design Method For Stress Control In Composites, Michael Stuebner
An Inverse Homogenization Design Method For Stress Control In Composites, Michael Stuebner
LSU Doctoral Dissertations
This thesis addresses the problem of optimal design of microstructure in composite materials. The work involves new developments in homogenization theory and numerical analysis. A computational design method for grading the microstructure in composite materials for the control of local stress in the vicinity of stress concentrations is developed. The method is based upon new rigorous multiscale stress criteria connecting the macroscopic or homogenized stress to local stress fluctuations at the scale of the microstructure. These methods are applied to three different types of design problems. The first treats the problem of optimal distribution of fibers with circular cross section …
Integral Cohomology Of The Siegel Modular Variety Of Degree Two And Level Three, Mustafa Arslan
Integral Cohomology Of The Siegel Modular Variety Of Degree Two And Level Three, Mustafa Arslan
LSU Doctoral Dissertations
In this thesis work Deligne's spectral sequence Ep,qr with integer coefficients for the embedding of the Siegel modular variety of degree two and level three, A2(3) into its Igusa compactification, A2(3)*, is investigated. It is shown that E3 = E∞ and this information is applied to compute the cohomology groups of A2(3) over the integers.