Physical Sciences and Mathematics Commons^™

Topological Groupoids, Ronson J. Warne

Doctoral Dissertations

A groupoid is a set G in which a single valued product ab is defined for every pair of elements a, b ε G. If G is a groupoid and at the same time a Hausdorff topological space, and, moreover, the multiplication in the groupoid G is continuous in the topological space G, then G is called a topological groupoid. Our aim in this dissertation is two-fold: (1) to study topological groupoids for their own sake; (2) to investigate the relation of certain topological properties to associativity. We note, in relation to the first motif, that many authors have dealt …

A Mathematical Theory Of Retail Space Management, Harlan D. Mills

The Harlan D. Mills Collection

No abstract provided.

Studies On The Sampling Methodology Of Peas For Yield And Quality, Pratapsinha Chintamani Pendse

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Pea¹ growers have much at stake in getting high yields of peas of prime quality. The income accruing from a pea crop grown for processors is determined by the yield as well as quality. Therefore the farmers' efforts are directed toward growing such a crop.

Research workers are interested in knowing the yield of peas with known tenderometer values which will indicate the quality of peas. Present methods of field harvesting are costly and time consuming which tend to limit the number of varieties that can be satisfactorily evaluated for trial.

A comparison of sampling techniques with present harvesting …

On Linear Associative Algebras Corresponding To Association Schemes Of Partially Balanced Designs, R. C. Bose, Dale M. Mesner

Department of Mathematics: Faculty Publications

Given v objects 1, 2, .. , v, a relation satisfying the following conditions is said to be an association scheme with m classes:
(a) Any two objects are either 1st, or 2nd, . . ,or mth associates, the relation of association being symmetrical, i.e., if the object α is the ith associate of the object β, then β is the ith associate of α.
(b) Each object a has n_i ith associates, the number n_i being independent of α.
(c) If any two objects α and β are ith associates, then …

Stochastic Properties Of Elementary Logistic Components, Harlan D. Mills

The Harlan D. Mills Collection

No abstract provided.

Structures In Logistic Operations, Harlan D. Mills

The Harlan D. Mills Collection

No abstract provided.

On An Explicit Method For The Solution Of A Stefan Problem, William F. Trench

William F. Trench

No abstract provided.

Spectral Theory Of Self-Adjoint Ordinary Differential Operators, Charles C. Oehring

Doctoral Dissertations

Introduction: Many of the properties of the ordinary Fourier series expansion of a given function are shared by the orthogonal expansion in terms of eigenfunctions of a second order ordinary differential operator. Let p = p(x) and q = q(x) be real-valued functions such that p, p', and q are continuous, and p(x) > 0, on a finite interval a ≤ x ≤ b. Let λ be a complex parameter. The classical Strum-Liouville theory [9, section 27; 4, Chapter 7; 21, Chapter 1]¹ is concerned with solutions of the differential equation -(py') + qy = λ, which satisfy certain real …

On A Theorem Of Hölder, Donald W. Miller

Department of Mathematics: Faculty Publications

A well-known result, due to Hölder [1], is the following: The symmetric group S_n, has outer automorphisms if and only if n = 6. The classical proof of the existence of a class of outer automorphisms of S₆, as formulated by Burnside [2], rests in part on the theory of primitive groups and entails extensive computation. In this note we offer a direct method for constructing such automorphisms. The author is grateful to Professor R. H. Bruck for raising this problem and for subsequent helpful remarks.

Some Properties Of Compactifications, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

A compactification of a topological space X is a compact (Hausdorff) space containing a dense subspace homeomorphic with X. Since only completely regular spaces have compactifications, all spaces mentioned here will be completely regular unless the contrary is assumed explicitly. This paper is a study of properties of the sets of points which may be added to a space in compactifying it.

Local Connectedness In The Stone-Cech Compactification, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

This is a study of when and where the Stone-Čech compactification of a completely regular space may be locally connected. As to when, Banaschewski [1] has given strong necessary conditions for βX to be locally connected, and Wallace [19] has given necessary and sufficient conditions in case X is normal. We show below that Banaschewski's necessary conditions are also sufficient and may be restated as follows: βX is locally connected if and only if X is locally connected and pseudo-compact (Corollary 2.5). Moreover, the requirement that βX be locally connected is so strong that it implies that every completely regular …

Tame, Finite Complexes In Three-Space, P. H. Doyle Iii

Doctoral Dissertations

Introduction: A fundamental problem in topology of Euclidean n-space is to determine under what conditions two homeomorphic subsets of an n-space are strongly homeomorphic; that is, to determine when one can be carried onto the other by homeomorphism of space onto itself. This problem is of particular importance when one of the two subsets is a polyhedron.

Approximately Finite Geometries And Their Coordinate Rings, James Howard Alexander

Doctoral Dissertations

Introduction: Von Newmann [8]¹ has discovered a class of geometries in which the undefined elements are subspaces and the undefined operations are meets and joins of subspaces. His axioms require that the system consisting of the set ℒ of subspaces together with the two operations form an irreducible, complete, complemented, modular lattice satisfying an additional dual pair of continuity of conditions.

Some Remarks On A Paper Of Aronszajn And Panitchpakdi, Melvin Henriksen

All HMC Faculty Publications and Research

In the paper of the title [1], a number of problems are posed. Negative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self contained.

On Minimal Completely Regular Spaces Associated With A Given Ring Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let C(X) denote the ring of all continuous real-valued functions on a completely regular space X. If X and Y are completely regular spaces such that one is dense in the other, say X is dense in Y, and every f ε C(X) has a (unique) extension f E C(Y), then C(X) and C(Y) are said to be strictly isomorphic. In a recent paper [2], L. J. Heider asks if it is possible to associate with the completely regular space X a dense subspace μX minimal with respect to the property that C(μX) and C(X) are strictly isomorphic.

A Comparison Of The Accuracy Of Finite Difference Methods And Variational Methods In The Numerical Solution Of Partial Differential Equations, Robert F. Brodsky

Mathematics & Statistics ETDs

The purpose of this thesis is to investigate some of the numerical methods available to solve partial differential equations intractable to analytic solutions with a view towards: 1. Indicating where particular methods appear most applicable, from the standpoints of accuracy, easy of solution, minimum expenditure of effort; and 2. Indicating the accuracy to be expected in the use of two of the more powerful methods investigated as applied to a problem of elasticity.

On The Equivalence Of The Ring, Lattice, And Semigroup Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

A large number of papers have been published that are devoted to showing that certain algebraic objects obtained by defining operations on the set of all continuous real-valued functions on a suitably restricted topological space determine the space. We mention but a few of them in this article.

A Historical Survey Of Methods Of Solving Cubic Equations, Minna Burgess Connor

Master's Theses

It has been said that the labor-saving devices ot this modern age have been made possible by the untiring efforts of lazy men. While working with cubic equations, solving them according to the standard methods appearing in modern text-books on the theory of equations, it became apparent, that in many cases, the finding of solutions was a long and tedious process involving numerical calculations into which numerous errors could creep. Confessing to laziness, and having been told at an impressionable age that "any fool can do it the hard way but it takes a genius to find the easy …

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.

On Rings Of Bounded Continuous Functions With Values In A Division Ring, Ellen Correl, Melvin Henriksen

All HMC Faculty Publications and Research

Let C*(X, A) denote the ring of bounded continuous functions on a (Hausdorff) topological space X with values in a topological division ring A. If, for every maximal (two-sided) ideal M of C*(X, A), we have C*(X, A)/M is isomorphic with A, we say that Stone's theorem holds for C*(X, A). It is well known [9; 6] that Stone's theorem holds for C*(X, A) if A is locally compact and connected, or a finite field. In giving a partial answer to a question of Kaplansky [7], Goldhaber and Wolk showed in [5] that, with restriction on X, and if A …

Some Remarks About Elementary Divisor Rings, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that S be an Hermite ring (i.e., every matrix over S can be reduced to triangular form). This is an improvement, since, in any case, it is necessary that S be an Hermite ring, while, on the other hand, it is not necessary that all zero-divisors be in the radical. In fact, we show that every regular commutative ring with identity is adequate. However, the condition that S be adequate is not necessary either.

We succeed in obtaining a necessary …

Rings Of Continuous Functions In Which Every Finitely Generated Ideal Is Principal, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

The outline of our present paper is as follows. In §1, we collect some preliminary definitions and results. §2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring).

The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, σ-compact space (e.g., the reals), then βX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are:

(i) for every f …

An Introduction To The Derivative Of A Polynomial, Floyd A. Miller

Masters Theses

No abstract provided.

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, …

Certain Equivalence Relations In Transformation Semigroups, Carol G. Doss

Masters Theses

The general object of this thesis is to study certain equivalence relations defined on a semigroup, in particular, to study certain equivalence relations defined on semigroups of single-valued transformations. We are interested in semigroups of transformations partly because every semigroup has as homomorphic image a semigroup of transformations (and hence a subsemigroup of a transformation semigroup of degree n for some n). This is a well-known fact, analogous to the Cayley Theorem on abstract groups, but we shall give a brief proof in Section 1. Section 1 is devoted to definitions and basic concepts. In Section 2 we prove some …

An Isomorphism Theorem For Real-Closed Fields, P. Erdös, L. Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

A classical theorem of Steinitz states that the characteristic of an algebraically closed fields, together with its absolute degree of transcendency, uniquely determine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the absolute transcendence degree are needed in order characterize a real-closed field.

Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen

All HMC Faculty Publications and Research

A commutative ring S with identity element 1 is called an elementary divisor ring (resp. Hermite ring) if for every matrix A over S there exist nonsingular matrices P, Q such that PAQ (resp. AQ) is a diagonal matrix (resp. triangular matrix). It is clear that every elementary divisor ring is an Hermite ring, and that every Hermite ring is an F-ring (that is, a commutative ring with identity in which all finitely generated ideals are principal).

Mathematicians And Royalty: A Historical Survey, Loren W. Pixley

Masters Theses

No abstract provided.

Concerning Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen

All HMC Faculty Publications and Research

The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X.

The first of these, treated in §§1-7, is the study of what we call P-spaces -- those spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in §1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in …