Physical Sciences and Mathematics Commons^™

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.

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, …

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 …

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...

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 …

Estimation Of Parameters Of Normal Populations From Truncated Samples, Kuo-Chung Liang

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

In statistical experiments, if a random sample of items drawn from a population is tested until all items fail, conventional statistical techniques may be employed but normal frequency distributions may not be satisfied. Failure of the data to satisfy the assumption of normality may lead to an invalid result. Some statistical results, however, have been shown to be robust to the failure of the data to meet normality. When using conventional experimental methods, considerable time and money are required to produce failure of all samples. To be economical, many experiments are concluded before all test items fail. A sample obtained …

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 …

Numerical Approximations To The Cumulative Chi-Square Distribution, The Cumulative T-Distribution And The Cumulative F-Distribution For Digital Computers, Grace Yuan-Chuen Wang

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

There are good tables of the frequently used cumulative frequency distributions. These tables have some limitations with respect to the number of percentage points that are available. The main drawback in computer usage of these tables is that large amounts of storage and elaborate search and interpolation techniques are necessary for their use.

It is the purpose of this study to present associated numerical methods for digital computer which are satisfactorily accurate and which are reasonably economical in both time and machine memory capacity. To carry out this objective the following procedures were used:

1. A review of literature on …

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.

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 …

Tests Of Independence In Contingency Tables, Su-Feng Wongbhan

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

This report is a survey of the literature for a combination of different tests for both two-way and multi-way tests of independence in contingency tables. The derivation of the commonly used chi-square statistic for the tests will be shown immediately below, then followed by the summary of the different tests which will be the contents of this report.

Reduction And Summarization Of Forage Production Data From Pasture And Range Lands, Julian Leigh Wilkinson Jr.

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

In an attempt to control and properly manage our vast pasture and range lands there have been many problems that the scientist has been confronted with. Among these has been the problem of reducing the great quantity of collected data from the ranges and pasture lands into a form suitable for scientific analysis. In the past this reduction process has been accomplished by slow, painstaking hand calculation methods, thus making it virtually impossible for the scientist to analyze and propose corrective measures at the time they are needed. However, with the advent of high speed computing equipment this need no …

Review Of Reliability Techniques, Eugene Richard Doherty

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In the development of any product to perform a specific function the first concern of the engineer is to design for satisfactory operation. Engineers originally approached the reliability problem by using excessive safety factors to be assured the structure or material would withstand the calculated loads and stresses. The engineer also learned from operating or testing the equipment until failures occurred and then redesigning as mistakes became apparent. These methods were time consuming and often resulted in bulky over designed products. These approaches became impractical with the advent of new technological advancements. The accelerated industrial development of aircraft, missiles, and …

A Logistic System Simulation Model Encompassing Poisson Processes And Normal Or Weibull Life, Willard A. Hansen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This thesis describes a computer simulation model for determining effective spares stock levels for recoverable items at Air Force bases and depots. The simulation model is based on the following fundamental inventory theory; whenever a demand arises, it is satisfied from stock on hand, and the quantity equal to that demand is recorded immediately; when a demand exceeds stock on hand, the excess demand is backordered immediately and when item life expires procurement action is initiated at depot level. The resulting product of the model cam be used as a guide for the optimum distribution of available spares or as …

Bayesian Inference, Sing-Chou Wu

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

Bayes' original paper "Essay Towards Solving a Problem in the Doctrine of Change" was published in Philosophical Transactions of the Royal Society, 1763. Over 200 years, Bayes' Concepts have survived numerous critical onslaughts. Even though Bayesian Inference is still regarded as being somewhat unorthodox, it is becoming more generally accepted each year by statisticians and other scientists.

It is the purpose of this paper to provide a bird's-eye view of Bayesian Inference with emphasis on the comparison of Bayesian approaches and conventional approaches. This paper is written for those who have had about one year's background in mathematical statistics.

Some …

Fredholm Solution Of Linear Integral Equations, Dewey F. Furness

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

An integral equation of the form

ζ(x) - λ∫ K(x, s) ζ(s) ds = f(x) a ≤ x ≤ b

is called a Fredholm equation. By the method of successive approximations a solution can be obtained if the parameter λ is sufficiently small. If the kernel, K (x, s), is degenerate then a solution can be obtained by reducing it to a system of linear algebraic equations. In the general case the kernel is represented as an infinite Fourier Series. With this representation the solution is obtained by combining the two methods mentioned. The solution is the solution of two …

Formulation Of Error Structures Under Non-Orthogonal Situations, Justus Frandsen Seely

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

To gain an appreciation or understanding for the title of this study we must first understand what the phrases "non-orthogonal" and "error structure" mean. With an understanding of these terms the title of this study will become clear.

To obtain an understanding of the term non-orthogonal, consider an experiment where differing treatments are applied to groups of experi­mental units in order to observe the differential treatment responses. If an equal number of experimental units are in each group, then we say we have an orthogonal situation. This means that when equal numbers exist among the experimental units, that the variability …

Boolean Space, Tzeng-Hsiang Sun

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

M. H. A. Stone showed in 1937 and subsequently that many interesting and important results of general topology involve latices and Boolean rings. This type of result forms the substance of this thesis.

Theorem 4, page 11, states that for any r ≠ 0 in a Boolean ring, there exists a homomorphism h into I₂ , (the field of integers modulo 2), such that h(r) = 1.

Theorem 3, page 6, states that any subring of a characteristic ring of a Boolean space X is the whole ring if it has the two points property (that is, given x, …

Barypact Topological Spaces, Bradley Y. Maughan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Recently, Kimber [3] has discovered a general class of topological spaces, the members of which are termed barypact spaces, that includes the compact topological spaces. This class is distinct from the set of all compact topological spaces, but its members possess many of the useful properties associates with compactness. As a consequence, several standard compactness theorems become special cases of corresponding theorems in a more general setting and the techniques of proof applied to these extensions provide new, and sometimes remarkably simple, proofs of the very theorems they generalize. The purpose of this paper is to extend to this …

Comparison Of Methods To Select A Probability Model, Howard Lyndal Colburn

Masters Theses

"A comparison is made between the estimates of the parameters in a gamma distribution obtained by the method of moments with those obtained by a numerical approximation to the maximum likelihood estimates. The estimates obtained by the numerical approximation had a smaller mean squared error from the true value than the estimates obtained by the method of moments. Modifications to tests of fit are made in order to develop methods to select a distribution from a set of possible distributions for a population with an unknown distribution. These selection methods are compared in their ability to make correct selections. Although …

Linear And Quadratic Programming With More Than One Objective Function, William John Lodholz

Masters Theses

"A computational procedure is presented for determining optimal solutions to the linear and quadratic programming problem when there is more than one objective function subject to linear constraints. In general a unique solution does not exist and a set of "best" or "efficient" points is determined and presented in graphical or tabular form. To solve the mathematical programming problems the simplex method is used for linear objective functions and Wolfe's method is used for quadratic objective functions"--Abstract, page ii.

Tschebyscheff Fitting With Polynomials And Nonlinear Functions, George F. Luffel

Masters Theses

"It is the purpose of this study to survey the properties of the Tschebyscheff polynomials with particular reference to how they are employed as approximants and interpolants. The survey is extended to include the process known as "Tschebyscheff Approximation" or "Tschebyscheff Fitting" of a function by functions other than polynomials. One such fitting technique which will be of particular interest in this study is that of fitting f(x) by the function ab^x + c where a, b, c are real and b ≠ 1. In addition to a survey of the properties of the Tschebyscheff polynomials and of Tschebyscheff …

A Study Of Certain Conservative Sets For Parameters In The Linear Statistical Model, Roger Alan Chapin

Masters Theses

"In the case of the linear statistical model, it has been shown that under certain conditions the confidence intervals obtained by considering the parameters one at a time are conservative when used as a joint confidence region, using the product of confidences as the confidence. However, nothing had been known of how conservative they are. This research provides accurate estimates of the true confidence for these cases.

Also, it has not yet been proved that they are indeed conservative for all cases. It is thought that they are, and the results of this research support this conjecture"--Abstract, page ii.

Latent Class Analysis And Information Retrieval, George Loyd Jensen

Masters Theses

"Information retrieval may be defined roughly as a procedure to either locate or physically retrieve a document or documents containing information on a given topic with a high degree of reliability. Information retrieval is one of the newest fields in computing science. Being so new there are many unexplored areas. The research that has been done has not been standardized beyond the point of the effort which has been slanted toward solving the "Library Problem" ... Several different information retrieval methods have been suggested and some research done on them to try and solve the "Library Problem". Of these methods …

A Development Of The Number System, Janet R. Olsen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers.

Estimation And Tabulation Of Bias Coefficients For Regression Analysis In Incompletely Specified Linear Models., Harry Kerry Edwards

Masters Theses

"There is often some uncertainty as to the exact number of predictors to include in the specification of the linear model when the theory of regression analysis is applied to specific data. To aid the experimenter in defining the correct number of predictors, a bias function that estimates the deviation of a linear model from the true model has been defined.

The writer has derived an accurate, economical, numerical method for calculating the bias function associated with an incompletely specified linear model and has tabulated bias function coefficients for a reasonable number of doubtful predictors"--Abstract, page ii.

The Effect Of Matrix Condition In The Solution Of A System Of Linear Algebraic Equations., Herbert R. Alcorn

Masters Theses

"The solution of a system of linear non-homogeneous equations may contain errors which originate from many sources. A system of linear equations in which small changes in the coefficients cause large changes in the solution is unstable and the coefficient matrix is ill- conditioned .

The purpose of this study is to define several measures of matrix condition and to test them by correlation with a measure of the actual errors introduced into a system of equations.

The study indicates that three of the five measures of condition tested were reliable indices of the magnitude of error to expect in …

A Method To Give The Best Linear Combination Of Order Statistics To Estimate The Mean Of Any Symmetric Population, Robert M. Smith

Masters Theses

“This paper gives a distribution for the ratio of two non-independent normally distributed random variables. The random variables involved are jointly distributed as the bivariate normal with means zero and variance- covariance matrix V.

The sample mean and one-half the sum of the smallest and largest sample observations from the symmetrically truncated Cauchy density function are compared as estimates of the mean of this density function. The reason for this comparison is to determine which of these two estimators gives the best estimate of the mean.

A method to empirically determine the best linear combination of order statistics for any …