Mathematics Commons^™

Tychonoff Spaces That Have A Compactification With Countable Remainder, Melvin Henriksen

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In this paper, an attempt is made to characterize spaces that are Zippin or strongly Zippin. We succeed in this goal only in small part, but we do obtain a number of conditions on a space that are either necessary or sufficient for such compactifications to exist.

Mathematics In America: The First Hundred Years, Judith V. Grabiner

Pitzer Faculty Publications and Research

There are two main questions I shall discuss in this paper. First, why was American mathematics so weak from 1776 to 1876? Second, and much more important, how did what happened from 1776-1876 produce an American mathematics respectable by international standards by the end of the nineteenth century? We will see that the "weakness" -at least as measured by the paucity of great names- co-existed with the active building both of mathematics education and of a mathematical community which reached maturity in the 1890's.

Multiplicatively Periodic Rings, Ted Chinburg, Melvin Henriksen

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We prove a generalization of Luh's result without using Dirichlet's Theorem. We then use Theorem 1 to show that the J-subrings of a periodic ring form a lattice with respect to join and intersection (the join of two subrings is the smallest subring containing both of them). After noting that every J-ring has nonzero characteristic, we determine for which positive integers n and m there exist J-rings of period n and characteristic m. This generalizes a problem posed by G. Wene.

On The Juror Utilization Problem, Melvin Henriksen, George H. Orland

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One of the authors, after hearing complaints night after night from his wife who was on jury duty, and finding it too much to bear, agreed to do something about the situation in return for peace. And so this study for the more efficient use of jurors was born. The aspects and magnitude of this problem have been discussed in many places. Basically we are concerned with achieving a better match between the number of jurors in a courthouse on a given day and those used in the judicial process.

On Graphs Which Contain All Small Trees, Ii, F. R. K. Chang, R. L. Graham, Nicholas Pippenger

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No abstract provided.

The Mathematician, The Historian, And The History Of Mathematics, Judith V. Grabiner

Pitzer Faculty Publications and Research

The historian's basic questions, whether he is a historian of mathematics or of political institutions, are: what was the past like? and how did the present come to be? The second question --how did the present come to be?-- is the central one in the history of mathematics, whether done by historian or mathematician. But the historian's view of both past and present is quite different from that of the mathematician. The historian is interested in the past in its full richness, and sees any present fact as conditioned by a complex chain of causes in an almost unlimited past. …

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

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In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Sums Of Kth Powers In The Ring Of Polynomials With Integer Coefficients, Ted Chinburg, Melvin Henriksen

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A working through of two theorems.

Suppose R is a ring with identity element and k is a positive integer. Let J(k, R) denote the subring of R generated by its kth powers. If Z denotes the ring of integers, then G(k, R) = {a ∈ Z: aR ⊂ J(k, R)} is an ideal of Z.

Is Mathematical Truth Time-Dependent?, Judith V. Grabiner

Pitzer Faculty Publications and Research

Another such mathematical revolution occurred between the eighteenth and nineteenth centuries, and was focused primarily on the calculus. This change was a rejection of the mathematics of powerful techniques and novel results in favor of the mathematics of clear definitions and rigorous proofs. Because this change, however important it may have been for mathematicians themselves, is not often discussed by historians and philosophers, its revolutionary character is not widely understood. In this paper, I shall first try to show that this major change did occur. Then, I shall investigate what brought it about. Once we have done this, we can …

Fields Medals And Nevanlinna Prize Presented At Icm-94 In Zurich, Nicholas Pippenger, J. Lindenstrauss, L.C. Evans, A. Douady, A. Shalev

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The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner.

The Global Optimization Of Incoherent-Phase Signals, Henry A. Krieger, Charles Albert Schaffner

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No abstract provided.

Toeplitz Operators On Locally Compact Abelian Groups, Henry A. Krieger, C.A. Schaffner

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The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for $N = 2$, and the optimal signal sets are determined for $M = 2,3,4,6$ and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ${\bf s}_j $, $j = 1,2, \cdots ,M$, on the unit sphere $S_1 $ in $C^N $. If $W_{jk} $ is the half space determined by ${\bf s}_j …

Calculus And The Computer: A Conservative Approach, Melvin Henriksen

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This paper describes a program for making the use of numerical methods an integral part of the freshman college course in single variable calculus.

The Eigenvalue Set Of A Class Of Equimodular Matrices, Gerald L. Bradley

CMC Faculty Publications and Research

The paper is divided into four sections, the first of which is basically introductory and contains definitions, notational conventions, and well-known results. The second section contains a fundamental theorem on multilinear polynomials which forms the crux of all our mail results, and the final two sections are devoted to discussions of the combinatorial and boundary properties of Y(A), respectively.

The Space Of Minimal Prime Ideals Of A Commutative Ring, Melvin Henriksen, Meyer Jerison

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The present paper is devoted to the space of minimal prime ideals of a more-or-less arbitrary commutative ring. Rings C(X) of continuous functions on topological spaces X appear only in §5 where they serve largely to provide significant examples.

Minimal Projective Extensions Of Compact Spaces, Melvin Henriksen, Meyer Jerison

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A compact space E is called projective if for each mapping ψ of E into a compact space X, and each continuous mapping τ of a compact space Y onto X, there is a continuous mapping φ of E into Y such that ψ = τ o φ.

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 …

Lattice-Ordered Rings And Function Rings, Melvin Henriksen, John R. Isbell

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This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings -- the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning.

On The Structure Of A Class Of Archimedean Lattice-Ordered Algebras, Melvin Henriksen, D. G. Johnson

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By a Φ-algebra A, we mean an Archimedean lattice-ordered algebra over the real field R which has an identity element 1 that is a weak order unit. The Φ-algebras constitute the class of the title. It is shown that every ф-algebra is isomorphic to an algebra of continuous functions on a compact space X into the two-point compactification of the real line R, each of which is real-valued on an (open) everywhere dense subset of X. Under more restrictive assumptions on A, ropresentations of this sort have long been known. An (incomplete) history of them …

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

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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

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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

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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

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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.

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

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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.

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

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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

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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 …

Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen

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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

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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.

Concerning Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen

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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 …