Applied Mathematics Commons^™

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 …

Measure Theory And Lebesgue Integration, Evan S. Brossard

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The Lebesgue integral is a generalization of the Riemann integral which extends the collection of functions which are integrable.

Lebesgue integration differs from Riemann integration in the way the approximations to the integral are taken. Riemann approximations use step functions which have a constant value on any given interval of the domain corres­ponding to some partition. Lebesgue approximations use what are called simple functions which, like the step functions, take on only a finite number of values. However, these values are not necessarily taken on by the function on intervals of the domain, but rather on arbitrary subsets of the …

Principal Component Factor Analysis, Herbert H. Jolliff

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Factor analysis came into being around 1900 in the field f psychology to explain theories of human ability. Several methods of factor analysis exist; but according to Harman (1967) principal component factor analysis is unique in the mathematical sense, therefore, quite often the preferred method. The centroid method is computationally easier, and it gives close approximations to the principal component method on some data sets. An example of this is shown in Appendixes B and F by comparison.

Factor analysis is being used in many fields. A few of the fields are sociology, meteorology, political science, medicine, geography, business, economics, …

Tests Of Methods That Control Round-Off Error, Dale M. Rasmuson

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Methods of controlling round-off error in one-step methods in the numerical solution of ordinary differential equations are compared. A new Algorithm called theoretical cumulative rounding is formulated. Round-off error bounds are obtained for single precision, and theoretical cumulative rounding. Limits of these bounds are obtained as the step length approaches zero. It is shown that the limit of the bound on the round-off error is unbounded for single precision and double precision, is constant for theoretical partial double precision, and is zero for theoretical cumulative rounding.

The limits of round-off bounds are not obtainable in actual practice. The round-off error …

Algebraic Properties Of Endomorphisms Of Abelian Groups And Rings, Johnnie George Slagle

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The main objective of the thesis was to extend the definition of an M-Group to what is called an M-Ring. From this extension a system called an expanded ring follows naturally. To facilitate the development of the expanded ring, chapter I is devoted to developing properties on systems that are not quite rings where many interesting examples are constructed. In chapter II the definition of an M-Ring is given and some of its properties are derived. In chapter III some of the properties of expanded rings are proved, and examples of expanded rings are given to show their existence.

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 …

Topics In Semigroups, Merlin Ray Jenson

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The semigroup theories of fundamental importance were discovered in 1928 with the publication of a paper by A. K. Suschkewitsch. The paper is included in the only book published prior to 1961 dealing primarily with the algebraic theory of semigroups. The book written by the above author and published in 1937 is entitles "The Theory of Generalized Groups."

Since 1940 the number of papers on semigroups has increased to approximately 30 per year. Thus we find ourselves in the midst of a new and expanding area of mathematics. Our interest centers around the structure of semigroups and their representation by …

Linear Coordinate Systems And Statistical Independence And Vector Spaces And Linear Statistical Models, James B. Mcdonald

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The object of this report is an investigation of statistical independence and linear coordinate systems. It is hoped that this report will give some background for an ultimate investigation of independence and non­linear coordinate systems using tensor analysis.

An additional basic objective is to obtain explicit algebraic expressions for different types of linear trans- formations.

The first concepts to be covered are arbitrary linear transformations, various ways of looking at linear transformations, and the effects of a linear transformation on a vector of normally distributed random variables.

Next orthogonal transformations to independence, and then oblique transformations to independence will be …

An Exposition On Bayesian Inference, John Laffoon

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The Bayesian approach to probability and statistics is described, a brief history of Bayesianism is related, differences between Bayesian and Frequentist schools of statistics are defined, protential applications are investigated, and a literature survey is presented in the form of a machine-sort card file.

Bayesian thought is increasing in favor among statisticians because of its ability to attack problems that are unassailable from the Frequentist approach. It should become more popular among practitioners because of the flexibility it allows experimenters and the ease with which prior knowledge can be combined with experimental data.