Mathematics Commons^™

Operators As Elements In A Partially Ordered Linear Algebra., Edward Wayne Davenport

Mathematics & Statistics ETDs

R. DeMarr {unpublished) has begun a study of linear operators as elements in a Dedekind O-complete partially ordered linear algebra {dsc-pola). The order structure necessary to produce elements in a dsc-pola which behave as specific operators on a specific space of functions is studied.

Some Applications Of Decompositions Of The Identity To Ordered Vector Spaces., John Annulis

Mathematics & Statistics ETDs

We study decompositions of the identity (for definition see Vulikh [13]) in Dedekind a-complete vector lattices. In Chapter 1, we define and develop a notion of continuous elements with respect to a decomposition of the identity. We shall also develop a spectral theory for these elements. In Chapter 2, two Stieltjes-type type integrals are defined with respect to decompositions of the identity. The relationship of these integrals to the continuous elements is then explored. In Chapter 3, we prove two results in ordered vector spaces. The first result is that every infinite dimensional Dedekind complete vector lattice has a base …

A Maximum Principle For An Optimal Control Problem With Integral Constraints., Vernon Bakke

Mathematics & Statistics ETDs

In this problem necessary conditions are obtained for an optimal control problem ·whose state variables are given in terms of integral equations. The conditions are obtained separately for Volterra equations and Fredholm equations. The main result for each case is the maximum principle and multiplier rule. For the Volterra equations, transversality conditions are obtained.

Convergence Rates For The Central Limit Theorem For Random Sums, Christopher E. Olson

Mathematics & Statistics ETDs

Let (X_i} be a sequence of independent, identically-distributed random variables with EX²_i < ꝏ and E(X_i - EX_i)² = 1.

On Some Special Classes Of Partially Ordered Linear Algebras., Taen-Yu Dai

Mathematics & Statistics ETDs

R. DeMarr (unpublished) has begun a study of Banach algebras as subalgebras of partially ordered linear algebras which are Dedekind a -complete. In [3] he has shown that the real Banach algebra of norm-bounded linear operators (mapping a real Banach space into itself) can be made into a partially ordered linear algebra which is Dedekind a -complete. This leads us to study a more gen­eralized function algebra by using the order structure. In this paper from analytical point of view we will study some special classes of partially ordered linear algebras which are Dedekind a -complete. In chapter 1 we …

Stacking Methods And Mixing Properties Of Measure-Preserving Transformations., Sarah Lee Meyer Christiansen

Mathematics & Statistics ETDs

In this paper we study ergodic and mixing properties of transformations. Stacking methods are employed to construct the transformations used in this paper. Number theoretical methods are developed and used to study mixing properties. Two new mixing properties B-mixing and S-mixing are defined, and are shown to be between partial mixing and weak mixing. Finally a transformation is constructed which is weak mixing but not S-mixing, and hence not partially mixing. Several applications to additive number theory are given.

Random Evolutions On Diffusion Processes, Donald Quiring

Mathematics & Statistics ETDs

Let {V(t,ω), t ≥ O, ω ε Ω} be a diffusion process on the real line with infinitesimal operator 1/2σ²(⋅)D² + m(⋅)D. Markov processes {V_n, n = 1,2,....} on the real line are constructed in such a way that the paths of V_n are step functions with jump size n^-1/2 and

P_O [lim sup |V_n(s)-V(s)| = 0] =1

n∞ 0≤s≤t,

where P_O assigns probability one to paths starting at the origin at t = 0.

Let {T_V(t), t≥0, vε R} be a family of linear contraction operators …

I. Existence Of Eigenvalues For Integral Equations; Ii. A Collocation Method For Boundary Value Problems, Robert Dodd Russell

Mathematics & Statistics ETDs

I. The existence of eigenvalues is shown for certain classes of integral equations with continuous kernels. A number of interesting and useful results are thereby treated in a unified and relatively elementary way. The simplicity of these new proofs make the results accessible to introductory courses on the theory of integral equations.

II. Collocation with piecewise polynomial functions is developed as a method for solving two-point bour:rlary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of …

Goursat Problems For The Abstract Equation Urs = Lu, Walter J. Roth

Mathematics & Statistics ETDs

Transitive Subgraphs Of Undirected And Directed Graphs., Charles C. Harner

Mathematics & Statistics ETDs

A graph G is transitive if it is transitive when considered as a relation; i.e., if the edges ab and be are in G, so is ac. In this paper, several natural questions concerning transitivity and related concepts are investigated.

Because the terminology and notation of graph theory is not standard, a brief section of definitions and notation is included in Section II. Section III is a study of the transitive cut number of a graph, the minimum number of edges which must be removed from the graph so that the remaining subgraph is transitive. Undirected graphs on n points …

The Ideal And New Interval Topologies On Lattice Ordered Groups., Frieda Koster Holley

Mathematics & Statistics ETDs

In his 1967 edition of Lattice Theory, G. Birkhoff posed the question: "When is a directed group a topological group and a topological lattice in its new interval topology?'. This problem and the similar problem involving the ideal topology are investigated.

A New Characterization Of Besicovitch Almost Periodic Functions., Abdallah N. Dabboucy

Mathematics & Statistics ETDs

Besicovitch and Bohr have given an internal description or Besicovitch almost periodic functions which involves the notion of “satisfactory uniformity” and is somewhat complicated. Raouf Doss has given an alternative simpler internal description, which, however, involves an infinity of independent conditions. Theorem. Let f E Lp(a,b) for all real numbers a,b. Then ff (BP-AP}

if end only if (1) the functions fn obtained by truncating f atn=l,2, … !·ll-converge to f; (2) lim)C+{) x llf-fl! BP =O; (3a) for every f: > 0 the set (x: l!f x -fl' P < eJ is relatively dense; and (3b) for every E: > 0

for all sufficiently large T, where E = {x: l\fx-fllB < e). Here fx(y) = f(x+y), XE denotes the characteristic function of E and 1 :Sp< oo. The theorem extends to LC connected groups, compactly generated Abelian groups and a somewhat stronger version of it extends to discrete countable Abelian groups.

Right Baer Semigroups., William Christopher Hardy

Mathematics & Statistics ETDs

In his paper, "A Semigroup Approach to Lattices, 11 (Canad . Jour. Math., Vol. 18 (1966)), M. F. Janowitz showed that every lattice with 0 and 1 is isomorphic to the lattice formed by the left annihilators of a suitably chosen Baer semigroup when they are ordered under set inclusion. A Baer semigroup, s~ was then said to coordinatize a lattice, L, if Lis isomorphic to the lattice of left annihilators of S. This dissertation presents a similar semigroup approach to lattices by showing that a poset, P, with O and 1 forms a meet semilattice if and only if …

Analysis Of Heat Transfer In A Two-Layer Circular Cylinder:Constant Flux On Outer Surface And Zero Flux On Inner Surface, Mary Elena Franklin

Mathematics & Statistics ETDs

The one-dimensional time-dependent equation of heat conduction is solved analytically for an infinite two-layer circular cylinder whose core may be either hollow or solid. On the outer surface of the cylinder, which has no heat loss due to convection, a constant heat flux from an external source of heating is applied uniformly. The layers are in perfect thermal contact, and there is no heat loss at their interface. At the smaller radius of the inner layer is a perfect insulator so that the heat flux on the inner surface of the two-layer cylinder is zero. The temperature is uniform initially …

Adaptive Nonlinear Prediction And Convergence Rates., Dirk Andrew Dahlgren

Mathematics & Statistics ETDs

Let x1,…,xm be radom variables with an arbitrary (initial) distribution functions F. For i>=m, define

Xi+1=f(theta,x1,…,x1-m+1)+oZi

Where theta is an unknown parameter in Euclidean r space and {Zi}inf m is a sequence of independent, identically distributed random variables with E[Zi]=0 and E[Z 2 i]=1. This type of process will be termed a generalized autoregressive process in the sequel.

A Combinational Analysis Of Finite Minimal Uniform Covers, Gustavus J. Simmons

Mathematics & Statistics ETDs

No abstract provided.

Pseudo-Lattices With Application To Topology And Analysis., Ih-Ching Hsu

Mathematics & Statistics ETDs

In recent times, lattice theory has pervaded many branches of mathematics. Usually, however, the structures of primary interest are not lattices themselves. The lattice structure is arrived at by passing to a quotient set.

Most Stable Tolerance Limits For Log-Convex Distributions, John Richard Ellefson

Mathematics & Statistics ETDs

Necessary and sufficient conditions are given for reversing Jensen's inequality for continuous convex functions, and the result is then stated for the discrete case. The main theorem of Hanson and Koopmans' (1964) pioneering tolerance limit paper is extended to the case of any finite number of order statistics from a probability distribution F for which either -log(1 - F) or - log F is convex. In the first case upper tolerance limits and in the second case lower tolerance limits are exhibited for any content p and any confidence ɣ, 0 < p, ɣ < 1 whenever the sample size is not less than two. When F is such that both functions are convex, two-sided, p-content tolerance intervals at confidence level rare constructed for any sample size greater than one. The distributions for which both functions are convex include the Pólya distribution functions of order two, which includes the normal and exponential families. The distribution of the coverage of the tolerance limits is calculated using an extension of Renyi’s (1953) work on exponential random variables. Goodman and Madansky's (1962) most stable criterion for tolerance intervals is characterized in terms of its second moment about a point, and some of its other properties are established. The results are applied to develop a method for finding the most stable one-sided tolerance limits. The general method is applicable to the problem of finding the most stable two-sided tolerance intervals, but the required explicit calculation of the probability distribution function is lacking. Instead the two-sided tolerance interval is found in terms of two one-sided tolerance limits.

Semi-Groups And A Class Of Singular Perturbation Problems, Andrew Y. Schoene

Mathematics & Statistics ETDs

No abstract provided.

Enumeration Of Certain Binary Arrays, Douglas Jackson

Mathematics & Statistics ETDs

T.V. Narayana [9] and later L. Carlitz [3] have solved the following problem. Given positive integers n, r, L₁,…,L_n-1, a₁,…,a_n such that a_i ≥ r, i = 1,...,n and a_i + L_i ≥ a_i+l, i = 1,…,n-1, how many n x r matrices [a_ij] of positive integers are there which satisfy

1.) a_i1 = a_i i=1,...,n

2.) a_ij > a_i,j+1 i = 1,...,n j = 1,..., r-1

3.) a_ij + L_i ≥ a_i+1,j i=1,...,n-1 j=1,...,r.

Narayana gave a formula for …

The Convergence Of Numerical Solutions Of Hydrodynamic Shock Problems., Darrell Lee Hicks

Mathematics & Statistics ETDs

In 1944 von Neumann Conjectures that only averages of numerical solutions, produced by a discrete scheme he had proposed to solve hydrodynamic shock problems, would converge as the meshes were refined. The hydrodynamic shock problem is defined here as: Find a solution, allowing discontinuous solutions, to the three conservation laws, the increasing-entropy law, and the ideal-gas law when giben physically acceptable initial and boundary values. For discrete schemes similar to von Neumann’s, it is verified that the averages do converge and it appears that von Neumann was also correct in surmising that only the averages converge. A smoothing device named …

Formally Real Fields, Gail L. Carns

Mathematics & Statistics ETDs

This paper is concerned with the analysis of the set of orders on a formally real field using category and sheaf theory as tools. We topologize the set of orders in a way similar to that used in the prime spectrum of algebraic geometry and define a presheaf of groups using the notion of inverse limits. We then find necessary and sufficient conditions that these presheaves be sheaves. Next we notice that if F ⊆ Fⁱ respectively, fields, θ and θⁱ are the set of orders on F and Fⁱ respectively, and Γ and Γⁱ are …

Functional Relationships For Fredholm Integral Equations Arising From Pseudo-Transport Problems., Richard Crenshaw Allen Jr.

Mathematics & Statistics ETDs

In this dissertation we consider a class of integral equations of the general form

Ψ(z)=g(z) + ∫_y^x Y(z’)K(|z-z’|) Ψ(z’)dz’

Where K has an integral representation of the type

K(u)= ∫₀^∞ k(s’)e^-a(s’)uds’

With this integral equation, we associate the pseudo transport problem

(sgn s) ∂/∂z N(z,s) + a(s)N(z,s) = k(s)Y(z) ∫_-∞^∞ N(z,s’)ds’,

Y< =z< =x, |s| < ∞

with certain boundary conditions. (Equation (2) is a generalization or the equation for particle transport in a slab geometry). Equivalence between the problems (1) and (2) is established. Thus any results concerning the problem (2) produce corresponding results concerning the …

An Examination Of The Romberg Method For Numerical Integration, Water James Halpin

Mathematics & Statistics ETDs

INTRODUCTION

This paper will amount to an inquiry into the mathematical structure and properties of a matrix of approximate integration formulas due to Werner Romberg 1 which have found considerable favor in numerical analysis during the past decade. Like other devices for approximate integration the Romberg method has its origin in the exhaustion techniques developed by the Greeks for determining the areas of figures enclosed by lines in the plane. It is particularly closely associated with the scheme employed by Archimedes2 (circa 250 B.C.) for approximating the value of [3.14]. This involves a doubling process which is essentially the basis …

On Some Categories Of Partially Ordered Sets With Residuated Mappings, Gary D. Crown

Mathematics & Statistics ETDs

Croisot [1] was apparently first to examine some of the properties of residuated mappings. It is the purpose of this paper to show that certain constructions currently useful in homological algebra can be made in a suitable category of partially ordered sets with residuated mappings. In Section 1 we note some elementary properties of residuated mappings all of which are well known cf. Janowitz [4]. In Section 2 we notice some properties of various categories of partially ordered sets with residuated mappings. In Section 3 we prove that the category of complete lattices with residuated mappings has enough injectives and …

The Approximation Of Eigenvalues And Eigenfunctions Of Convolution Kernels, Adelbert Lee Roark

Mathematics & Statistics ETDs

No abstract provided.

Completion And Compactification Functors For Cauchy Spaces, James Francis Ramaley

Mathematics & Statistics ETDs

The subject of this thesis is topological but the approach is categorical. We consider the approach as important as the subject itself and so we try to indicate whenever we have a categorically defined concept. In fact, we are led by this approach to make definitions and constructions so that certain relationships are categorical in nature. Our coreflective completion functor of Chapter IV is one example of this approach.

We wish here to give a brief outline of the historical develop­ment of convergence theory and how category theory has come to play a role in this development...

Convergence Relations On Universal Algebras., Allan Matlock Weber Carstens

Mathematics & Statistics ETDs

In this paper we derive some first results on convergence algebras, which are universal algebras with a convergence structure. The first concept was defined some time ago and a unified study of it is available in [4]. Convergence functions were first defined in [7] and have been studied in further works by Kent and others. Kent also investigated the results of imposing a convergence structure on an algebraic structure, in the special case of convergence groups [8]. Our results on loops and quasigroups extend some of his.

Convergence Rates For The Law Of The Iterated Logarithm And For Probabilities Of Moderate Deviations., James Avery Davis

Mathematics & Statistics ETDs

Let {Xn:n=1,2,…} be a sequence of independent identically distributed random variables with common distribution function F(x). If {an} is a monotonic increasing sequence and Sn=sigmak=1 n Xk, then under certain conditions

P[ Is n I > a n J or P[ sup I> 1) k > n I 8'J< 8it

A Best Running Test For Symmetry And Distribution Free Tests For Symmetry, David Lloyd Burdick

Mathematics & Statistics ETDs

A sequential test of a statistical hypothesis H₀ versus H₁ is said to be a running test if there is a positive probability that the test will not stop if H₀ is true. Tests of this nature were introduced for testing the Bernoulli case by D. A. Darling and Herbert Robbins [1]; an earlier paper of Roger Farrell [2] deals implicitly with the asymptotic expected sample size of such tests for testing the hypothesis Ѳ = 0 in the parameterized family of generalized density functions h(Ѳ)e^ϴxdµ. Herbert Robbins, in a lecture given at the Sandia …