Digital Commons Network^™

Linear Programming In High School Mathematics, Vernon Earl Cotter

Dissertations and Theses @ UNI

"Modern mathematics," a term associated with experimental and revised programs for the teaching of mathematics, should be interpreted only in part as meaning new subject matter. It is true that modern mathematics is associated with such subject matter as the theory of sets and the revision of the definitions of 'relation' and 'function.' But it also connotes a new spirit, a new method of approach, and a new organization of material and problems. Also it may retain much of the traditional subject matter. Of course the order of presentation may be changed and traditional subject matter bounds may be disregarded …

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.

A Proposal For Emphasizing Modern Mathematics In Elementary Algebra, Betty Sayre Grubb

Dissertations and Theses @ UNI

For the past fifty years educators and mathematicians have urged that we change our approach to mathematics. The need for change or modernization of teaching mathematics is at least as great now as it ever was. The body of mathematical content vital to the pursuit of both pure and applied mathematics is growing continually. At the same time the demand for people trained in mathematics, including its newer developments, is growing. The only solution would appear to be the integration of these new ideas, as extensively as possible by each teacher, into all levels of training in mathematics. This thesis …

Applications Of Eighth Grade Mathematics To The Ninth Grade General Shop Program In Iowa, Robert Vern Echelbarger

Dissertations and Theses @ UNI

As a student engages in the construction of a project in an industrial arts class, it is necessary for him to learn the uses of tools and materials. The extent of his learning is reflected in part by the tangible evidence of the completed project and increased skill in the mastery of tools. Just as a student must learn the uses of tools and materials, so must he learn the use of intangible tools such as mathematics, reading, writing, and speech. The mastery of these intangible tools is important for the general education of the student . Mathematics plays an …

A Comparative Study Of Two Approaches To The Teaching Of Area In The Junior High School, Margaret Patricia Green

Dissertations and Theses @ UNI

It was the purpose of this study to compare the amount of information learned and retained by the students through two approaches to teaching the topic of Area. One approach followed the material presented in a seventh grade arithmetic book; the other followed the material presented in a unit especially constructed for this experiment. The investigation was directed toward the question: Was there a difference between the two teaching methods as measured by the amount of information learned and retained by the students participating in this experiment?

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 …

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 Study In Predicting Success In Algebra, Leslie Milton Graham

Dissertations and Theses @ UNI

Students of Manson High School at Manson, Iowa, select either algebra or general mathematics, subject to the approval of the principal. Some of the students' achievement records are incomplete, thus handicapping the principal in guiding the students to make their choices. As a result, several students have chosen to try algebra and have failed, and sometimes they have become discouraged and have left high school. It occurred to the writer that, if a predicting device could be established to show the student his chances in mathematics, it possibly could spare the pupil's failing in algebra.

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.

A Comparative Investigation Of The Use And Non-Use Of Manipulative Devices In Teaching Seventh Grade Mathematics, George E. Immerzeel

Dissertations and Theses @ UNI

The problem with which this investigation is concerned may be stated as follows: Is the study of seventh grade mathematics more meaningful when taught with the use of certain manipulative devices or when taught without their use? This study is an attempt to determine whether students taught with the use of manipulative devices are inferior, equal, or superior in terms of standardized test results, certain test questions and certain subjective evaluative techniques.

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, …

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).

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.

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

Masters Theses

No abstract provided.

The Preparation Of A Suggested Unit Using Applications From Physics In The Teaching Of Trigonometric Functions In High School Trigonometry, Oliver Warren Eason

Dissertations and Theses @ UNI

In view of the crucial role played by science and technology in contemporary civilization, and because of the intimate relationship between science and mathematics, both pure and applied, a more adequate corre1ation in teaching can and should be effected between the fields of mathematics and science.

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 …

On A Theorem Of Gelfand And Kolmogoroff Concerning Maximal Ideals In Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen, Meyer Jerison

All HMC Faculty Publications and Research

This paper deals with a theorem of Gelfand and Kolmogoroff concerning the ring C= C(X, R) of all continuous real-valued functions on a completely regular topological space X, and the subring C* = C*(X, R) consisting of all bounded functions in C. The theorem in question yields a one-one correspondence between the maximal ideals of C and those of C*; it is stated without proof in [2]. Here we supply a proof (§2), and we apply the theorem to three problems previously considered by Hewitt in [5].

Our first result (§3) consists of two simple constructions of the Q-space vX. …

On The Continuity Of The Real Roots Of An Algebraic Equation, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

It is well known that the root of an algebraic equation is a continuous multiple-valued function of its coefficients [5, p. 3]. However, it is not necessarily true that a root can be given by a continuous single-valued function. A complete solution of this problem has long been known in the case where the coefficients are themselves polynomials in a complex variable [3, chap. V]. For most purposes the concept of the Riemann surface enables one to bypass the problem. However, in the study of the ideal structure of rings of continuous functions, the general problem must be met directly. …

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.

On The Theory Of Head Waves, Patrick Heelan

Research Resources

When a combined longitudinal and transverse disturbance, diverging from a localized source, strikes a plane boundary between two solid elastic media, several systems of head waves and second order boundary waves are generated, each associated with grazing incidence of one or the other of the reflected or refracted waves. Associated with grazing incidence of P ₁ P₂, the refracted P-wave, is the head wave system comprising P₁P₂P₁ (the "refracted wave" of seismic prospectors), and P₁P₂S₁ (a transverse head wave) in the upper medium, and P₁P_{2 …}