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. …

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. …

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 Brief Consideration Of The Different Methods Of Integration, Donald Golden

Master's Theses or Doctor of Nursing Practice

Any student of mathematics is familiar with the importance of the process of integration. Integration is as fundamental to analysis as the basic principles of the number theory are to arithmetical calculation. Some of the common applications of integration are, finding the distance a falling body has travelled during a particular interval of time; to determine the equation of a curve, given different conditions (such as slope of the curve equal numerically to one-half the abscissa, or some similar problem); motion of a projectile; motion in a resisting medium; finding areas and volumes of revolution; length of a curve; areas …

The Theory Of Determinants: Some Special Forms And Applications, William Ralph Eikelberger

Master's Theses or Doctor of Nursing Practice

The theory of determinants is a relatively new branch of mathematics; it was not firmly established until a brief 200 years ago. It was not until the middle of the 18th century that Cramer, one of the independent discoverers of the fundamental idea, was fortunate enough to attract attention to the theory of determinants. Much of the literature is written on the assumption that its reader has a good background in the subject. Then other authors just give the fundamental properties of determinants without proof of their statements. These latter authors are not interested in the mathematical theory of determinants, …

Student Algebra Notebook, Jack Thorington, Joseph S. Winter

Winter Letterbooks

Jack Thorington's Algebra notebook. Contains practice algebra problems.

Vollständige Anleitung Zur Algebra, Erster Theil, Leonhard Euler

All Works by Eneström Number

No abstract provided.

Vollständige Anleitung Zur Algebra, Zweyter Theil, Leonhard Euler

All Works by Eneström Number

No abstract provided.