Physical Sciences and Mathematics Commons^™

Dramathizing Functions: Building Connections Between Mathematics And Arts, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on connections between mathematics and performance arts (drama). More specifically we offer an exposition of a segment of college algebra mathematics (an introduction to functions), with an approach primarily emphasizing the aesthetic aspects of mathematical learning, teaching, and performing.

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z², as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum …

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at Institut de Mathématiques in Bordeaux, France, June 2008.

A Summary Of Results On Order-Cauchy Completions Of Rings And Vector Lattices Of Continuous Functions, Melvin Henriksen

All HMC Faculty Publications and Research

This paper is a summary of joint research by F. Dashiell, A. Hager and the present author. Proofs are largely omitted. A complete version will appear in the Canadian Journal of Mathematics. It is devoted to a study of sequential order-Cauchy convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified for compact spaces X in …

Averages Of Continuous Functions On Countable Spaces, Melvin Henriksen, John R. Isbell

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Let X = {x1, x2, ...} be a countably infinite topological space; then the space C*(X) of all bounded real-valued continuous functions f may be regarded as a space of sequences (f(x1), f(x2), ...). It is well known [7, p. 54] that no regular (Toeplitz) matrix can sum all bounded sequences. On the other hand, if (x1, x2, ...) converges in X (to x_m), then every regular matrix sums all f in C*(X) (to f(x_m)).

The main result of this paper is that if a regular matrix sums all f in C*(X) then it sums f …

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.

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

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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

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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 …

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

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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

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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 Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen

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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 …