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Orientations Of Graphs Which Have Small Directed Graph Minors., Glenn Randolph Berman
Orientations Of Graphs Which Have Small Directed Graph Minors., Glenn Randolph Berman
LSU Historical Dissertations and Theses
Graphs are characterized by whether or not they have orientations to avoid one or more of the digraphs K&ar;3 , S&ar;3 , and P&ar;3 . K&ar;3 , S&ar;3 and P&ar;3 are created by starting with a triangle, a three point star, or a path of length three respectively, and replacing each edge with a pair of arcs in opposite directions. Conditions are described when all orientations of 3-connected and 4-connected graphs must have one or more of the above digraphs as a minor. It is shown that double wheels, and double wheels without an axle, are the only 4-connected graphs …
Densities Of 4-Ranks Of K(2) Of Rings Of Integers., Robert Burke Osburn
Densities Of 4-Ranks Of K(2) Of Rings Of Integers., Robert Burke Osburn
LSU Historical Dissertations and Theses
Conner and Hurrelbrink established a method of determining the structure of the 2-Sylow subgroup of the tame kernel K2( O ) for certain quadratic number fields. Specifically, the 4-rank for these fields was characterized in terms of positive definite binary quadratic forms. Numerical calculations led to questions concerning possible density results of the 4-rank of tame kernels. In this thesis, we succeed in giving affirmative answers to these questions.
On K-Conjugacy Classes Of Maximal Tori In Semi-Simple Algebraic Groups., Uroyoan Ramon-Emeterio Walker
On K-Conjugacy Classes Of Maximal Tori In Semi-Simple Algebraic Groups., Uroyoan Ramon-Emeterio Walker
LSU Historical Dissertations and Theses
An attempt was made to make this a self-contained reading. The first three chapters are intended to provide the necessary background. Chapter one develops the tools needed from Galois Cohomology. Chapter two is a brief description of involutions, and in chapter three we define the notion of (linear) algebraic group, we give some examples and discuss some of their properties. In chapter four, we discuss some variants of the classical Skolem-Noether theorem, requiring only that the subalgebra have a unique faithful representation of full degree over a separable closure. We verify that we can extend every isomorphism to the whole …
Linear Codes Defined From Higher -Dimensional Varieties., Gary Lynn Salazar
Linear Codes Defined From Higher -Dimensional Varieties., Gary Lynn Salazar
LSU Historical Dissertations and Theses
We establish an algebraic foundation to complement the improved geometric codes of Feng and Rao. Viewing linear codes as affine variety codes, we utilize the Feng-Rao minimum distance bound to construct codes with relatively large dimensions. We examine higher-dimensional affine hypersurfaces with properties similar to those of Hermitian curves. We determine a Grobner basis for the ideal of the variety of rational points on certain affine Fermat varieties. This result is applied to determine parameters of codes defined from Fermat surfaces.
Homflypt Skein Modules., Jianyuan Zhong
Homflypt Skein Modules., Jianyuan Zhong
LSU Historical Dissertations and Theses
Let k be a subring of the field of rational functions in x, v, s which contains x+/-1, v+/-1, s+/-1. If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) x --1L+ -- xL -- = (s -- s--1 )L0; (2) L with a positive twist = (xv--1)L; (3) L ⊔ O = u-u-1 s-s-1L where O is the unknot. We give two bases for the relative Homflypt skein module of …
Sociomathematical Norms Of Elementary School Classrooms: Crossnational Perspectives On The *Reform Of Mathematics Teaching., Jeongsuk Pang
Sociomathematical Norms Of Elementary School Classrooms: Crossnational Perspectives On The *Reform Of Mathematics Teaching., Jeongsuk Pang
LSU Historical Dissertations and Theses
Mathematics education reform in the United States has marshaled large-scale support for instructional innovation, and enlisted the participation and allegiance of large numbers of mathematics teachers. However, there is concern that many teachers have not grasped the full implications of the reform ideals. This study explored the breakdown that may occur between teachers' adoption of reform objectives and their successful incorporation of reform ideals by comparing and contrasting two reform-oriented classrooms. This study was an exploratory, qualitative, comparative case study using constant comparative analysis. Seven mathematics lessons were video-taped from each class, and intensive interviews conducted with the two teachers. …
Developing Students' Understanding Of Similar Figures: A Perceptual Approach., Danny Ray Mcnabb
Developing Students' Understanding Of Similar Figures: A Perceptual Approach., Danny Ray Mcnabb
LSU Historical Dissertations and Theses
Children encounter and recognize similar figures in their everyday experiences with such things as basketballs, soccer balls, tennis balls, ping-pong balls; or a candy bar that comes in various sizes of the same shape. Yet their school experience with the mathematics of similarity generally does not build on these perceptual intuitions. Traditional mathematics curricula bypass students' visual intuitions and their quantitative understandings, proceeding directly to set piece problems solved by formal algebraic methods. The result for many students is that the topic of similarity contributes to their evolving view of mathematics as a domain of complex procedural methods divorced from …
Classically Unstable Approximations For Linear Evolution Equations And Applications., Yu Zhuang
Classically Unstable Approximations For Linear Evolution Equations And Applications., Yu Zhuang
LSU Historical Dissertations and Theses
Temporal discretization methods for evolutionary differential equations that factorize the resolvent into a product of easily computable operators have great numerical appeal. For instance, the alternating direction implicit (ADI) method of Peaceman-Rachford for 2-D parabolic problems greatly reduces the simulation time when compared with the Crank-Nicolson scheme. However, just like many other factorized approximation methods that exhibit numerical stability, the ADI method is known to satisfy only the Von Neumann stability condition, a necessary condition that is usually surmised as sufficient in practical cases as pointed out by Lax and Richtmyer. Intensive efforts have been directed to understand the Von …
On Some Optimal Control Problems For The Centroaffine Geometry On The Plane., Angel L. Cruz Delgado
On Some Optimal Control Problems For The Centroaffine Geometry On The Plane., Angel L. Cruz Delgado
LSU Historical Dissertations and Theses
A certain parametrization of substantial planar curves yields a centroaffine arclength s and a centroaffine curvature ks that remain invariant under GL(2, R ) motions. In Chapter 4 we search for those substantial curves with predetermined position and velocity at the initial and terminal points, which minimize the total square curvature 0T k2s 2ds as k varies over all square summable functions on each interval [0, T]. These curves are called centroaffine elastic curves. Thinking of the curvature k as a control, we pose our problem as an optimal control problem over the Lie group GL(2, R ) with fixed …
A Canonical Description Of The Plancherel Measure For A General Two-Step Free Nilpotent Lie Group., Chin-Te Chu
A Canonical Description Of The Plancherel Measure For A General Two-Step Free Nilpotent Lie Group., Chin-Te Chu
LSU Historical Dissertations and Theses
In this work on g=Fn,2 , free 2-step nilpotent Lie algebra on n generators, we use the group of automorphisms to give a basis-free description of the Fourier Inversion Formula, thereby generalizing and strengthening an example discussed by Corwin & Greenleaf. In the Introduction we discuss Example 4.3.14 in Corwin & Greenleaf's book. It demonstrates how different bases of F3,2 lead to different inversion formulas. But the third "more" invariant formula describes Plancherel measure on a support expressed in terms of rotations, dilations, and translations. Actually it is not canonical since it still depends on choices of bases for F3,2 …
The Effect Of Antecedent And Consequent Strategies On Increasing Student Homework Compliance And Academic Achievement., Donna Marie Gilbertson
The Effect Of Antecedent And Consequent Strategies On Increasing Student Homework Compliance And Academic Achievement., Donna Marie Gilbertson
LSU Historical Dissertations and Theses
The primary purpose of this investigation was to evaluate the effects of homework and alternatives to homework on student math completion, accuracy and fluency for low socioeconomic students. To examine possible causes of homework problems as well as the effect of different treatments on math fluency, this study used an idiographic protocol to systematically examine the effects of antecedent and consequential strategies on different types of homework performance problems through several phases. First, a brief experimental analysis was conducted for each student to identify whether poor homework performance was due to a swill deficit or a performance deficit. Next, an …
On Stochastic Integration For White Noise Distribution Theory., Said Kalema Ngobi
On Stochastic Integration For White Noise Distribution Theory., Said Kalema Ngobi
LSU Historical Dissertations and Theses
This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite dimensional space S'R endowed with the gaussian measure mu. The Hilbert space ( L2) is defined as (L2) (L2) ≡ L2( S'R , mu) and our results are based on the Gel'fand triple ( S )beta ⊂ (L2) ⊂ S* b . The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In Chapter 3 we present a generalization …
Weierstrass Pairs And Minimum Distance Of Goppa Codes., Gretchen L. Matthews
Weierstrass Pairs And Minimum Distance Of Goppa Codes., Gretchen L. Matthews
LSU Historical Dissertations and Theses
We prove that elements of the Weierstrass gap set of a pair of points may be used to define a geometric Goppa code that has minimum distance greater than the usual lower bound. We determine the Weierstrass gap set of a pair of any two Weierstrass points on a Hermitian curve and use this to increase the lower bound on the minimum distance of certain codes defined using a linear combination of the two points. In particular, we obtain some two-point codes on a Hermitian curve that have better parameters than the one-point code on this curve with the same …
Structure And Minors In Graphs And Matroids., Galen Ellsworth Turner Iii
Structure And Minors In Graphs And Matroids., Galen Ellsworth Turner Iii
LSU Historical Dissertations and Theses
This dissertation establishes a number of theorems related to the structure of graphs and, more generally, matroids. In Chapter 2, we prove that a 3-connected graph G that has a triangle in which every single-edge contraction is 3-connected has a minor that uses the triangle and is isomorphic to K5 or the octahedron. We subsequently extend this result to the more general context of matroids. In Chapter 3, we specifically consider the triangle-rounded property that emerges in the results of Chapter 2. In particular, Asano, Nishizeki, and Seymour showed that whenever a 3-connected matroid M has a four-point-line-minor, and T …
On Harmonic Analysis For White Noise Distribution Theory., Aurel Iulian Stan
On Harmonic Analysis For White Noise Distribution Theory., Aurel Iulian Stan
LSU Historical Dissertations and Theses
This thesis is composed of two parts, each part treating a different problem from the theory of Harmonic Analysis. In the first part we present an inequality in White Noise Analysis similar to the classical Heisenberg Inequality for functions in L2Rn . To do this we replace the finite dimensional space R n and its Lebesgue measure by the infinite dimensional space E' , which is the dual of a nuclear space E , and its Gaussian measure. Choosing an arbitrary element eta in E , we may define the multiplication operator Q&d5;h , which is the sum between the …
Integral Kernel Operators In The Cochran -Kuo -Sengupta Space., John Joseph Whitaker
Integral Kernel Operators In The Cochran -Kuo -Sengupta Space., John Joseph Whitaker
LSU Historical Dissertations and Theses
This dissertation contains several results about integral kernel operators in white noise analysis. The results found here apply to the space of test functions and generalized functions that were constructed in the paper of Cochran, Kuo, and Sengupta, based on a sequence of numbers &cubl0;an&cubr0; infinityn=0. . We shall prove results about existence, restrictions, and extensions of integral kernel operators based on the conditions on &cubl0;an&cubr0; infinityn=0. contained in the paper of Kubo, Kuo, and Sengupta. Also, we shall prove an analytic property and growth condition of the symbol of a continuous operator in CKS space. Our results are similar …
Inequalities Between Pythagoras Numbers And Algebraic Ranks In Witt Rings Of Fields., Sidney Taylor Hawkins
Inequalities Between Pythagoras Numbers And Algebraic Ranks In Witt Rings Of Fields., Sidney Taylor Hawkins
LSU Historical Dissertations and Theses
This dissertation establishes new lower bounds for the algebraic ranks of certain Witt classes of quadratic forms. Let K denote a field of characteristic different from 2 and let q be a quadratic form over K. The form q is said to be algebraic when q is Witt equivalent to the trace form qL∣K of some finite algebraic field extension L∣K . When q is algebraic, the algebraic rank of q is defined to be the degree of the minimal extension L∣K whose trace form is Witt equivalent to q. It is an important, unsolved problem to find reasonable bounds …
The Applications Of The Method Of Quasi Reversibility To Some Ill-Posed Problems For The Heat Equation., Xueping Ru
The Applications Of The Method Of Quasi Reversibility To Some Ill-Posed Problems For The Heat Equation., Xueping Ru
LSU Historical Dissertations and Theses
In this work, we study the Cauchy problem for the heat equation, as well as the inverse heat conduction problem, both of which are ill-posed problems in the sense of Hadamard. The first chapter provides the background material about the previous investigations on the ill-posed Cauchy problem for the heat equation and the inverse heat conduction problem by other mathematicians. The method of Quasi-Reversibility is also introduced. In the second chapter we apply the method of Quasi-Reversibility to the Cauchy problem for the heat equation and obtain a formal approximate solution. We prove that the convergence of the approximate solution …
General Riemann Integrals And Their Computation Via Domain., Bin Lu
General Riemann Integrals And Their Computation Via Domain., Bin Lu
LSU Historical Dissertations and Theses
In this work we extend the domain-theoretic approach of the generalized Riemann integral introduced by A. Edalat in 1995. We begin by laying down a related theory of general Riemann integration for bounded real-valued functions on an arbitrary set X with a finitely additive measure on an algebra of subsets of X. Based on the theory developed we obtain a formula to calculate integral of a bounded function in terms of the regular Riemann integral. By classical extension theorems on set functions we can further extend this generalized Riemann integral to more general set functions such as valuations on lattices …
Totally Ordered Monoids., Gretchen Wilke Whipple
Totally Ordered Monoids., Gretchen Wilke Whipple
LSU Historical Dissertations and Theses
In this work, we explore properties of totally ordered commutative monoids---we call them tomonoids. We build on the work in [E]. Our goal is to obtain results that will be useful for studying totally ordered rings with nilpotents. Chapter 1 presents background information. In Chapter 2, we present some criteria for determining when a tomonoid is a quotient of a totally ordered free monoid by a convex congruence. In Chapter 3, we show that every positive tomonoid of rank 2 is a convex Rees quotient of a subtomonoid of a totally ordered abelian group. In Chapter 4, we provide a …
On The Second Stiefel-Whitney Class Of Scaled Trace Forms Of Central Simple Algebras., Rosali Brusamarello
On The Second Stiefel-Whitney Class Of Scaled Trace Forms Of Central Simple Algebras., Rosali Brusamarello
LSU Historical Dissertations and Theses
In this work we compute the second Stiefel-Whitney class $w\sb2$ of the scaled trace form $Q\sb{A,a}(x)={\bf tr}\sb{A/k}(ax\sp2),$ where A is a central simple algebra over a perfect field k of characteristic different from two, $a\in A$ is a fixed element, and ${\bf tr}\sb{A/k}$ is the reduced trace. The first three chapters provide background material about quadratic forms, central simple algebras, group cohomology, and representations of linear algebraic groups. The fourth chapter presents two known results about the second Stiefel-Whitney class of trace forms: Serre's formula for the case of etale algebras and Saltman's formula for the case of central simple …
On The Value Functions Of Some Singular Optimal Control Problems., Jesus Alberto Pascal
On The Value Functions Of Some Singular Optimal Control Problems., Jesus Alberto Pascal
LSU Historical Dissertations and Theses
Infinite horizon singular optimal control problems with control taking values in a closed cone U in Rn lead to a dynamic programming equation of the form: maxF2x,v, v',v'' ,F1x,v,v' =0, forall x∈Q, where Q , the state space of the control problem, is some nonempty connected open subset of Rn , and F1, F2 are continuous real-valued functions on QxR2 and QxR3 respectively, with the coercivity assumption for F 2, that is, the function r&rarrr;F2x,u,p,r is nonincreasing on R . A major concern is to determine how smooth the value function v is across the free boundary of the problem. …
Maximal Circuits In Matroids., Pou-Lin Wu
Maximal Circuits In Matroids., Pou-Lin Wu
LSU Historical Dissertations and Theses
Lovasz, Schrijver, and Seymour have shown that if a connected matroid M has a largest circuit of size c and a largest cocircuit of size c*, then M has at most 2c+c*-1 elements. A question of Oxley that has attracted considerable recent attention is whether there is an upper bound on the size of M that is polynomial in terms of c and c*. In particular, Bonin, McNulty, and Reid conjectured that 12cc* is such a bound. In Chapter 1, we prove this conjecture for an connected graphic and cographic matroids. In Chapter 2, we give a constructive description of …
Spikes In Matroid Theory., Zhaoyang Wu
Spikes In Matroid Theory., Zhaoyang Wu
LSU Historical Dissertations and Theses
For an integer $n\ge 3,$ a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point t such that, for all k in $(1, 2,\..., n - 1),$ the union of every set of k of these Lines has rank $k + 1.$ The point t is called the tip of the n-spike. Ding, Oporowski, Oxley, and Vertigan proved that, for all $n\ge 3,$ there is an integer $N(n)$ such that every 3-connected matroid with at least $N(n)$ elements has a minor isomorphic to a wheel or whirl of rank $n,\ M(K\sb{3,n})$ or …
Pure Framed Braids And 3-Manifolds., Jonathan Natov
Pure Framed Braids And 3-Manifolds., Jonathan Natov
LSU Historical Dissertations and Theses
This dissertation looks at representations of framed pure braids and compact orientable three manifolds. A representation, $\Phi:Z\sp{n}\oplus P\sb{n}\to \Gamma\sb{n},$ is constructed from the framed pure braid group on n strands to the mapping class group on a surface of genus n. The representation is used to obtain a presentation of the fundamental group. The representation, like that of (M-P), is compatible with Heegaard and Surgery descriptions. An algorithm is presented for transforming mapping class group elements to a stably equivalent pure framed braid, under the correspondence given by the representation. A geometric description, using the representation, is given for multiplication …
Lie Theory Of Differentiable Transformations On Banach-Type Manifolds., Amadou Belly Guisse
Lie Theory Of Differentiable Transformations On Banach-Type Manifolds., Amadou Belly Guisse
LSU Historical Dissertations and Theses
Let M be a manifold modeled on a Banach space B, and let U be an open subset of M containing the domain of some chart $\varphi$. The aim of this work is to set mathematical foundations (topological, algebraic geometrical) for the theory of pseudosemigroups of local transformations S on M and their infinitesimal generators $L\sp{r}(S,\ U).$ In the first part of this dissertation we define the topology of local uniform convergence, the most suitable in this case, similar to the compact open topology in the finite dimensional case, and show what relationship it has with different topologies. In the …
A Vector-Valued Operational Calculus And Abstract Cauchy Problems., Boris Baeumer
A Vector-Valued Operational Calculus And Abstract Cauchy Problems., Boris Baeumer
LSU Historical Dissertations and Theses
Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside's operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside's method, resulting in two competing mathematical theories: (a) Laplace transform theory for functions, distributions and other generalized functions, (b) J. Mikusinski's field of convolution quotients of continuous functions. …
Unavoidable Minors Of Graphs Of Large Type., John Joseph Dittmann Jr
Unavoidable Minors Of Graphs Of Large Type., John Joseph Dittmann Jr
LSU Historical Dissertations and Theses
In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs $G = G\sb0, G\sb1,\... , G\sb{n},$ where $G\sb{i}$ is obtained by contracting or deleting one edge from each block of $G\sb{i-1}$, and where $G\sb{n}$ is edgeless. We show that a 3-connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor …
Graphs And Number Theory., Brian Heck
Graphs And Number Theory., Brian Heck
LSU Historical Dissertations and Theses
In the 1930's, L. Redei and H. Reichardt used certain matrices to aid in the determination of the structure of ideal class groups of quadratic number fields. This is a classical number theoretic problem which in general presents diffculties. Ideal class groups are finite abelian groups, and it is a result of Gauss that allows us to determine their 2-rank, in other words the number of cyclic factors of even order. Redei and Reichardt worked on determining the 4-rank, the number of factors of order divisible by 4. Later, the classical study of circulant graphs was utilized to further help …
Weak Convergence Of Interacting Stochastic Systems., George William Paslaski
Weak Convergence Of Interacting Stochastic Systems., George William Paslaski
LSU Historical Dissertations and Theses
The aim of the dissertation is to establish the weak convergence of mean-field interacting particle systems driven by Poisson random measures and semimartingales. The limit of the stochastic systems is identified by the use of martingale problems and Picard iteration schemes. The interacting systems driven by Poisson random measures are shown to be stable with respect to the coefficients of the system as well as the driving terms. The same results can be achieved when a random interaction term independent of the driving terms is introduced into the coefficients of the system. Equations driven by semimartingales do not neccesarily possess …