Physical Sciences and Mathematics Commons^™

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. …

The Characteristic Roots Of Certain Real Symmetric Matrices, Joseph Frederick Elliott

Masters Theses

The main purpose of this thesis is to collect and coordinate some known results in the study of characteristic roots of certain real symmetric matrices. Specifically, most of the matrices considered have their elements a_ij = 0 except for I i - J I < 1 and are known as Jacobi matrices. The thesis fills in the details of two papers by D. E. Rutherford [12; 13]¹ , with some translation of his results for determinants to the problem of finding characteristic roots of matrices. Some of the results obtained are briefly discussed relative to certain theorems on bounds for characteristic roots.

The first part of the paper is an attempt to show some of the physical sources of the problem …

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 …}

On Rings Of Entire Functions Of Finite Order, Melvin Henriksen

All HMC Faculty Publications and Research

In an earlier paper, the author showed that if M is any maximal ideal of R, the residue class field R/M is isomorphic with the complex field K. In this paper, under some restrictions, this theorem is extended to the ring R_λ of all entire functions of order no greater than λ, and hence to R*.

On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let R be the ring of entire functions, and let K be the complex field. In an earlier paper [6], the author investigated the ideal structure of R, particular attention being paid to the maximal ideals. In 1946, Schilling [9, Lemma 5] stated that every prime ideal of R is maximal. Recently, I. Kaplansky pointed out to the author (in conversation) that this statement is false, and constructed a non maximal prime ideal of R (see Theorem 1(a), below). The purpose of the present paper is to investigate these nonmaximal prime ideals and their residue class fields. The author is …

Convexity And Starlikeness Of Analytic Functions, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

The Spieker Circle And Certain Related Configurations, Berthola Elmo Lumzy

Electronic Thesis and Dissertation

It was the purpose of the present study 1) to present the major known theorems concerning the Spieker circle; 2) to show the remarkable analogy between the Spieker circle and the nine point circle; and 3) to extend theorems of the Spieker circle and the definition of the Spierker circle to related configuration that are located outside the given triangle.

The Solution Of Boundary Value Problems By Use Of The Laplace Transformation As Compared With Classical Methods, Dan W. Stoddard

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this paper is to present a study of different methods of solving certain boundary value problems. In particular it will be concerned with solutions by classical methods and by operational methods. Of the various operational methods that may be considered, the Laplace transformation appears to be the best⁽¹⁾ and will be used in this paper.

In the 1951 Encyclopedia Americana Annual is this report on the activities in applied mathematics for the previous year:

Progress was made on the general problem of finding the eigenvalues of matrices and systems of differential equations. Considerable effort was also …

What Is A Riemmian Manifold?, Frederick Griffin

Journal of the Arkansas Academy of Science

No abstract provided.

Equivalent Functions Of Strategies In The Theory Of Games, Harlan D. Mills

The Harlan D. Mills Collection

No abstract provided.

The Expansion Of A Single-Valued Function Of Several Variables Having For Its Rangeadenumerable Set Of Real Numbers And For Its Domain Of Definitionaset Of Real Numbers, Nicholas Christ Scholomiti

Master's Theses

No abstract provided.

The Spieker Circle And Certain Related Configurations, Berthola E. Lumzy

Electronic Thesis and Dissertation

No abstract provided.

On The Ideal Structure Of The Ring Of Entire Functions, Melvin Henriksen

All HMC Faculty Publications and Research

Let R be the ring of entire functions, and let K be the complex field. The ring R consists of all functions from K to K differentiable everywhere (in the usual sense).

The algebraic structure of the ring of entire functions seems to have been investigated extensively first by O. Helmer [1].

The ideals of R are herein classified as in [2]: an ideal I is called fixed if every function in it vanishes at at least one common point; otherwise, I is called free. The structure of the fixed ideals was determined in [1]. The structure of the …

A Critical Analysis Of Several Product And Factoring Formulas, Herbert Wills

Masters Theses

No abstract provided.

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.

The Diophantine Equation, Frederic C. Barnett

Mathematics & Statistics ETDs

In the first part of Chapter II of this thesis a study is made of these surfaces F_n. In the second part of Chapter II, several differential geometric properties of F_n are adduced, and again, an attack based on certain of these is shown to lead to the very theorem we seek to prove. Finally, we obtain three geometrical implications of the assumption that an integral triad exists on F_n suggesting new lines of attack on the theorem of Fermat.

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 Galois Group Of A Polynomial Equation With Coefficients In A Finite Field, Charles Leston Bradshaw

Masters Theses

Introduction [Abbreviated]:

The primary purpose of this thesis is to demonstrate a method for the determination of the Galois group of a polynomial equation with coefficients in a finite field. The problem of finding the Galois group of an arbitrary equation, where the coefficient field is either finite or infinite, is neither new nor unsolved.

Prime Ideals In Semigroups, Helen Bradley Grimble

Masters Theses

The concept of prime ideal, which arises in the theory of rings as a generalization of the concept of prime number in the ring of integers, plays a highly important role in that theory, as might be expected from the central position occupied by the primes in arithmetic. In the present paper, the concept is defined for ideals in semigroups, the simplest of the algebraic systems of single composition, and some analogies and differences between the ring and semigroup theories are brought out. We make only occasional references to ring theory, however; a reader acquainted with that theory will …

Constant Rank Matrices, Harlan D. Mills

The Harlan D. Mills Collection

A matrix is a rectangular array of quantities which, as an array, obeys certain rules when combined with other matrices by the operations of addition and multiplication, or when combined with scalar quantities by the operation of multiplication. These operations have meaning if and only if the matric quantities and scalar quantities are elements of a ring. A minor of a matrix is a certain function of a square sub-array of the matrix, and has a unique meaning if and only if the elements of the sub-array are commutative. The rank of a matrix is a function of all possible …

Endomorphisms And Translations Of Semigroups, Eldon E. Posey

Masters Theses

This thesis is a study of some of the properties of mappings or semigroups into semigroups. In Section 1, definitions and basic concepts are given, with brief mention of some well known properties of semigroups and of mappings. Section 2 is devoted to a few theroems which are useful in the process of finding the semigroup of all endomorphisms of a given semigroup. When theorems on endomorphisms are valid in the more general situation in which a semigroup is mapped homomorphically on another semigroup, we state and prove the theorems in the more general setting. Section 3 consists of some …

Semigroup Ideals, Kenneth Scott Carmen

Masters Theses

In the literature of ideals a left ideal L of a system S has usually been defined by the inclusion of SL ⊆ L and a right ideal R by the inclusion RS ⊆ R. On the other hand a left zero element z is usually defined by the equation za = z and a right zero element by the equation az = z ; similarly a left or right identity element e is defined by the equation ea = a or ae = a respectively. The author has in this paper made the terms left and right when applied …

Direct And Semidirect Products Of Semigroups, John Charles Harden

Masters Theses

The purpose of this thesis is to exhibit the direct products and the semidirect products of semigroups of orders two, three, and four. Section I consists of the definitions necessary for the proofs and computations which appear in the first five sections. In Section 2 theorems on direct products of semigroups of any order are proved. A method of writing by inspection the multiplication table of the direct product of two finite semigroups is presented in Section 3. Section 4 consists of tables listing the direct products of semigroups of order two with semigroups of orders two, three and four …

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 Analytic Study Of Eight-Grade Pupil Performance As Revealed Through The Stanford Achievement Test In Arithmetic, Dominic J. Brungardt

Master's Theses

The immediate purpose of the study were: (a) to determine how closely the grade equivalents of two-hundred and sixty pupils follow the normal curve ; (b) to plot the results of the one-step problems on a graph with an analytic explanation of the graph ; (c) to place the results of the two-step problems on a graph with an analytic explanation of the graph; to determine what types of error were most prevalent; to discover apparent causes of error ; (d) to make suggestions helpful to teachers for improving pupil achievement.

On Matrices With Elements In A Principal Ideal Ring, William Leavitt, George Whaples

Department of Mathematics: Faculty Publications

We prove the following theorem.
THEOREM 1. Let D be any commutative principal ideal ring without divisors of zero, and A any matrix with elements in D whose characteristic equation factors into linear factors in D. Then there exists a unimodular matrix T, with elements in D, such that T^-1 AT has zeros below the main diagonal.

The Application Of The Law Of Virtual Work In The Solution Of Civil Engineering Structures, Martin Trester Dyke

University of the Pacific Theses and Dissertations

The purpose of this thesis is to explore the civil engineering application of the law of virtual work to the determination of defelections, shears, bending stresses in truss structures, beams, structures subjected to both direct stress and bending, and indeterminate structures, and further to give examples of their mathematical solutions.

Arithmetical Vocabulary : A Factor In Verbal Problem Solving In Sixth Grade Arithmetic, Ernest A. Hoopes

Master's Theses

During the writer's experience of teaching in elementary and junior high schools in Kansas he had excellent opportunity through supervision and classroom teaching to note a more-than-ordinary difficulty experienced by most children in the subject of arithmetic. Not only was the dull student baffled by the subject but many times the average and good students were "lost" when certain problems were presented. The exceptional. troublesomeness of this subject led the writer to ponder on the possible cause or causes of incorrect problem solving. One thing was apparent from observation and this was that many children seemed to lack a readiness …