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Calculus And The Computer: A Conservative Approach, Melvin Henriksen
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.
A Theoretical Model For Gas Separation In A Glow Discharge: Cataphoresis, Fredrick H. Shair, Donald S. Remer
A Theoretical Model For Gas Separation In A Glow Discharge: Cataphoresis, Fredrick H. Shair, Donald S. Remer
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A theoretical model for transient and steady-state cataphoresis is developed starting with the macroscopic equations of continuity. After a brief breakdown period, the impurity ions are assumed to be closely coupled with their neutral counterparts. The basic assumptions in the model are that after breakdown, the level of ionization of the impurity, and the axial electric field remain constant; it is demonstrated that under these conditions a system involving rapid ionization-recombination reactions is equivalent to a system in which no reaction occurs, but in which the "effective'' ion mobility is a product of the true ion mobility and the fraction …
Problems Encountered With Control Networks In Highly-Restructurable Digital Systems, Donald F. Wann, Robert A. Ellis, Mishell J. Stucki, Robert M. Keller
Problems Encountered With Control Networks In Highly-Restructurable Digital Systems, Donald F. Wann, Robert A. Ellis, Mishell J. Stucki, Robert M. Keller
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This paper discusses problems encountered with control networks in highly restructurable digital systems. In particular the treatment of implementation errors is covered with emphasis on concurrent processing. The implementation of concurrent processing networks may result in errors which will be quite complex to detect and systematic methods are warranted. Four meta control elements are employed in obtaining convenient concurrent structures. We analyze several error detecting schemes and conclude that the arc-node method with node partitioning appears to be the most realistic approach at this time.
The Space Of Minimal Prime Ideals Of A Commutative Ring, Melvin Henriksen, Meyer Jerison
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
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
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 xm), then every regular matrix sums all f in C*(X) (to f(xm)).
The main result of this paper is that if a regular matrix sums all f in C*(X) then it sums f …
On The Structure Of A Class Of Archimedean Lattice-Ordered Algebras, Melvin Henriksen, D. G. Johnson
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 …
Lattice-Ordered Rings And Function Rings, Melvin Henriksen, John R. Isbell
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.
Some Properties Of Compactifications, Melvin Henriksen, John R. Isbell
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
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
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
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
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
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
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
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 …
An Isomorphism Theorem For Real-Closed Fields, P. Erdös, L. Gillman, Melvin Henriksen
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.
Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen
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).
Concerning Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen
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 …
On A Theorem Of Gelfand And Kolmogoroff Concerning Maximal Ideals In Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen, Meyer Jerison
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
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 Rings Of Entire Functions Of Finite Order, Melvin Henriksen
On Rings Of Entire Functions Of Finite Order, Melvin Henriksen
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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 Rλ 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
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 …
On The Ideal Structure Of The Ring Of Entire Functions, Melvin Henriksen
On The Ideal Structure 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. 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 …