Applied Mathematics Commons^™

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 …

A Survey Of The Applications Of Difference Equations, Roberta Lanice Harkey

Mathematics & Statistics ETDs

In asserting that people in other fields often tend to be afraid to use mathematics, F.K. Mechta, an economist, expressed an uncertainty which contributes to the hesitancy to use mathematics. “Mathematics is tricky, it maintains silence, does its work quietly; and when we do not understand its ways and misinterpret its message, it just smiles. It never loses its temper, never laughs; we can observe a suppressed smile on its lips. Such is mathematics.” It is the purpose of this paper to conduct a brief survey of the applications of difference equations. The use of these equations is often rather …

Degeneration Of The Solutions Of Certain Well Posed Systems Of Partial Differential Equations Depending On A Small Parameter, Larry Bobisud

Mathematics & Statistics ETDs

Let be a system of N partial differential equations, where α is a multi­index, p is a positive number, the B_α are N x N matrices of constants, A(ϵ) is an N X N diagonal matrix with N-m ϵ's followed by m ones (1 ≤ m ≤ N-1), and For each ϵ > 0 V_ϵ is to satisfy V(0, x) = f(x). Under sufficiently strong assumptions on (1), this problem can be solved for each sufficiently small ϵ > 0 provided f has a sufficient number of L₁ and continuous derivatives. Let M₁ denote the upper left (N …

On The Optimal Search Problem, Wallace E. Franck Jr.

Mathematics & Statistics ETDs

Suppose a searcher is at a given point on a line and wishes to locate an object which is known to be somewhere on the line. Suppose further that there is a probabilistic law which governs the location of the object on the line. The searcher can only locate the object by traveling to the point where it is. He desires to formulate a search plan which will minimize the average distance traveled before locating the object. The only initial choice he has is as to which direction to go. Then he must decide how far to go before turning …

Some Properties Of Certain Sets Of Coprime Integers, Roger C. Entringer

Mathematics & Statistics ETDs

The set P(n) of all primes equal to or less than n has the obvious property that it contains exactly one multiple of each prime equal to or less than n. We use this partial description of P(n) as a basis for the following

Definition 1.1. An increasing sequence {a₁,...,a_k} of integers greater than 1 is a coprime chain if it contains exactly one multiple of each prime equal to or less than a_k.

Convergence Functions And Their Related Topologies, Darrell C. Kent

Mathematics & Statistics ETDs

A convergence function is a correspondence between the filters on a given set S and the subsets of S which specifies which filters converge to which points of S. This concept is defined to include types of convergence which are more general than that defined by specifying a topology on S. Thus a convergence function may be regarded as a generalization of a topology.

On The Initial Value Problem For The Quasi-Linear Parabolic Partial Differential Equation, Henry Hermes

Mathematics & Statistics ETDs

In this paper we consider second order parabolic partial differential equations on the infinite strip. We confine our attention to the quasi-linear equations, and in particular, the problem

(0-1) u͙(t,x) = f(u)u_xx(t.x) , (t,x) ε(0,T)x(-∞,∞) ≡ D_{T, T>0}

u(0,x) = u₀(x) , x ε (-∞,∞).

Existence, uniqueness and properties of the solution are discussed. These results can be immediately generalized to problems where the differential equation is of the form u͙(t,x) = f(t,x,u)u _xx(t,x).

A Numerical Treatment Of One-Dimensional Non-Steady Compressible Flows, Paul J. Kazek

Mathematics & Statistics ETDs

The problem that will be considered here will be the flow in a cylinder filled with a homogeneous ideal gas, bounded on one end by a vacuum and/or a rigid wall with the motion initiated by a piston on the other end. First, the basic Lagrangian differential equations will be derived along with other necessary thermodynamical relations. A scheme of difference equations will be presented and the viscosity term discussed as well as a method for insuring stability of the difference equations during the calculations

A Statistical Technique For Predicting A Two Dimensional Vector With Application, Richard E. Vogel

Mathematics & Statistics ETDs

The problem of multiple regression analysis where the dependent and independent variables are components of a two dimensional vector is discussed, and a complete statistical development of the solution of estimators for the parameters in the model given. The theory regarding predictions and confidence statements about such predictions is also developed. A computer code was written for the IBM 704 computer which solves the above problem and a description of the code appears in the appendix.

The statistical model was applied to a meteorological problem in wind forecasting at the Eniwetok Proving Ground, and prediction equations were developed and evaluated.

Beta And Gamma Distributions, Calvin Rogers

Mathematics & Statistics ETDs

The purpose of this paper is to exhibit the main properties of Gamma and Beta distributions and show their relation to certain well known distributions.

In chapter II the Gamma and Beta distributions are defined in terms of Gamma and Beta functions. The moments of these distributions are calculated, and the moment generating function and cumulant generating function for the Gamma distribution are obtained. The curves are classified with respect to parameter values and the curves are graphically illustrated in Figures 1, 2, and 3. The exponential distribution, as a special case of interest, is shown to be a Gamma …

The Use Of Kamke's Transformation In Approximating The Zeros Of Orthogonal Polynomials, Robert L. Daniels

Mathematics & Statistics ETDs

The importance of the classical orthogonal polynomials has long been acknowledged. It has not been possible, however, to represent them in such a way that all of their important properties are immediately evident. In particular, the location of the zeros of these polynomials is of considerable interest.

This thesis is primarily concerned with a different technique in which Kamke's transformation is applied to the differential equations frequently used to define these polynomials. The resulting trigonometric differential equations cannot be explicitly solved either, but certain characteristics of these solutions facilitate the derivation of approximations to the zeroes of the solutions.

The Numerical Treatment Of A Simple Hydrodynamical Shock By The Von Neumann-Richtmyer Method, Charles F. Sprague Iii

Mathematics & Statistics ETDs

When the differential equations describing the flow of compressible fluids are derived under the assumptions that (1) forces in the fluid are due only to variations in pressure and (2) that the entropy of any volume element remains constant, it can be shown mathematically that these differential equations cannot have continuous solutions under all circumstances. If we adopt the notion of “shock discontinuities” in these solutions, the differential equations describing the flow in continuous regions together with conditions expressing the laws of conservation across the discontinuities suffice to completely determine the flow. An alternative procedure is to use a method, …

Estimates For The Zeros Of Ultraspherical Polynomials, Frank B. Correia

Mathematics & Statistics ETDs

In this work an attempt is made to develop a new method for estimating the zeros of the Ultraspherical Polynomials. This method presupposes knowledge of the differential equation that these polynomials satisfy. However the method is applicable to a much wider class of functions since it may also be applied to estimating the zeros of the solutions of any homogeneous linear differential equation of the second order which satisfies certain initial conditions.

The Variation Problems Of Weierstrass-Bliss And Radon, Douglas M. Gragg

Mathematics & Statistics ETDs

Problems of the Calculus of Variations are generalizations of the familiar minimum problems treated in the differential calculus. The relationships between the ordinary minimum problems of the calculus and the generalizations dealt with in the Calculus of Variations is possibly best seen by examining the general Hilbert-Moore minimum problem, and the special examples of such problems formulated in the table below.

Inverse Problems Of Hamel-Type., Robert G. Schrandt

Mathematics & Statistics ETDs

The formulation and discussion of the simplest (fixed) end point direct problem of the calculus of Variation is a necessary preliminary to attack on the inverse problems considered in Chapters II and III of this thesis. Since the plane problem is already comprehensively treated in the literature, only enough of its theory is developed here to render intelligible to the reader the inverse problems studied in the sequel.

The Darboux Inverse Problem In The Calculus Of Variations, Frank O. Lane

Mathematics & Statistics ETDs

The simplest non-parametric problem of the calculus of variation, the so-called direct problem of the plane, is the problem of finding that arc C_o of a family of admissible arcs y=y (x) joining two fixed pointed (x₁ , y₁ ), (x_1, y₂) in the x,y-plane such that along the C_o the integral takes on a minimum value.

An Investigation Of The Meaning Of Α3 As A Measure Of Skewness, John W. Coy

Mathematics & Statistics ETDs

The purpose of this study is the interpretation of α₃ by means of a relatively simple formula which will predict the amount of shift in the effective limits of the Type III curve for a given change in the skewness. The methods used are chiefly empirical.

An Investigation Of The Nature Of The Coefficients Of Entire Function, Marie Ann Philips

Mathematics & Statistics ETDs

The purpose of this study is to investigate the nature of the coefficients of certain power series. In particular, it is desired to know what characteristics the coefficients must possess in order that the series shall represent an entire function.

A Study Of Certain Substitution Groups, Merle Mitchell

Mathematics & Statistics ETDs

This thesis purposes to study a certain group of movements which can be expressed as substitutions. The groups of movements which send a square into itself is to be studied as a group of eight substitutions on the vertices for the purpose of leading up to the real problem of this paper. From the octic group, it is natural to proceed to a study of the movements which send a cube into itself. In particular, it is the aim of this thesis to discover the group of the cube and to analyze some of its properties. There are twenty-eight rotations …