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Articles 1 - 15 of 15
Full-Text Articles in Physical Sciences and Mathematics
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 …
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 …
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 $\widehat{U_n/K_n}$ …
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 …
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 …
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 …
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 …
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, …
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 …
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 …
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 …
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 …
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 …
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.
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 …