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Articles 151 - 180 of 232
Full-Text Articles in Applied Mathematics
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 …
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 …
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 …
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 …
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 …
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 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 $\widehat{U_n/K_n}$ …
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).
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 …
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.
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 …
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 …
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 …
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.
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 …
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.
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 …
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 …
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 …
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.
Index Future Pricing Under Imperfect Market And Stochastic Volatility, Wei-Hsien Li
Index Future Pricing Under Imperfect Market And Stochastic Volatility, Wei-Hsien Li
LSU Master's Theses
Financial markets in emerging countries are volatile and imperfect, so pricing model under traditional perfect-market frameset may not give reliable price of financial derivatives. The most famous pricing model for stock index future is the cost of carry model. The mis-pricing of cost of carry model inspires lots of following researches. Even transaction costs, dividends, stochastic interest rate, stochastic volatility, market imperfection, and other factors are considered, we still do not obtain a model price consistently better than cost of carry model. But these researches offer important insights, for example, the market needs time to mature and the more complex …
Using Elimination To Describe Maxwell Curves, Lucas P. Beverlin
Using Elimination To Describe Maxwell Curves, Lucas P. Beverlin
LSU Master's Theses
Cartesian ovals are curves in the plane that have been studied for hundreds of years. A Cartesian oval is the set of points whose distances from two fixed points called foci satisfy the property that a linear combination of these distances is a fixed constant. These ovals are a special case of what we call Maxwell curves. A Maxwell curve is the set of points with the property that a specific weighted sum of the distances to n foci is constant. We shall describe these curves geometrically. We will then examine Maxwell curves with two foci and a special case …
Characterization Of The Dependency Across Foreign Exchange Markets Using Copulas, Ryan Coelho
Characterization Of The Dependency Across Foreign Exchange Markets Using Copulas, Ryan Coelho
LSU Master's Theses
Though Pearson's correlation coefficient provides a convenient approach to measuring the dependency between two variables, in the last few years, there has been a significant amount of literature cautioning against the use of Pearson's correlation coefficient, as it does not remain invariant under monotone transformations of the underlying distribution functions. Since we are interested in examining the dependency pattern observed by the return on the Sterling Pound with that of the Japanese Yen, we will use the notion of a copula to approximate the joint density function between the daily returns on the Sterling Pound and the Japanese Yen. In …
Optimal Binary Trees With Height Restrictions On Left And Right Branches, Song Ding
Optimal Binary Trees With Height Restrictions On Left And Right Branches, Song Ding
LSU Master's Theses
We begin with background definitions on binary trees. Then we review known algorithms for finding optimal binary search trees. Knuth's famous algorithm, presented in the second chapter, is the cornerstone for our work. It depends on two important results: the Quadrangle Lemma and the Monoticity Theorem. These enabled Knuth to achieve a time complexity of O(n2), while previous algorithms had been O(n3) (n = size of input). We present the known generalization of Knuth's algorithm to trees with a height restriction. Finally, we consider the previously unexamined case of trees with different restrictions on left and …
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 …
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 …
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 …
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 …