Physical Sciences and Mathematics Commons^™

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 …

Towards A Theory Of Universal Speed-Independent Modules, Robert M. Keller

All HMC Faculty Publications and Research

Of concern here are asynchronous modules, i.e., those whose activity is regulated by initiation and completion signals with no clocks being present. First a number of operating conditions are described that are deemed essential or useful in a system of asynchronous modules, while retaining an air of independence of particular hardware implementations as much as possible. Second, some results are presented concerning sets of modules that are universal with respect to these conditions. That is, from these sets any arbitrarily complex module may be constructed as a network. It is stipulated that such constructions be speed independent, i.e., independent of …

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.

Study Of Stopping Sites Of B12 Nuclei Implanted In Hexagonal Single-Crystals, Francis David Correll, Richard C. Haskell, Leon Madansky

All HMC Faculty Publications and Research

A series of experiments to study the stopping sites of 12B nuclei implanted in single crystals of Be and Mg will be discussed. Previous measurements of the quadrupole couplings of 12B in these metals indicated that two different stopping sites existed, one of which produced a negligible quadrupole coupling.

Li-8 Magnetic Dipole-Moment, Richard C. Haskell, Leon Madansky

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Polarized 8Li nuclei recoiling from the reaction 7Li(d, p)8Li were implanted in Au, Pd, and Pt foils, and the Knight-shifted values of the 8Li magnetic dipole moment in these metals were measured by a resonant depolarization technique. From measurements of the spinlattice relaxation times of 8Li implanted in Au, Pd, and Pt, the Knight shifts of 8Li in these metals were estimated. A corrected value for the magnetic dipole moment of 8Li was found to be μ(8Li)=(1.653 35±0.000 35)μN. Attempts were also made to measure the quadrupole couplings of 8Li in single crystals of Be and Mg. While the resonance …

A Novel Method Of Constructing Sorting Networks, Robert M. Keller

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The construction of sorting networks has been a topic of much recent discussion. In view of the apparent difficulty of verifying whether a reasonably large proposed sorting network actually does sort, the most useful approach for constructing large networks seems to be to devise a recursive scheme which constructs a network which is guaranteed to sort, obviating the verification phase. In this note, another such approach is presented.

On The Decomposition Of Asynchronous Systems, Robert M. Keller

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This paper reports of part of a continuing investigation of parallel computation, in particular, efforts toward understanding the nature of different types of parallel control. The first section defines an asynchronous system to be a simple type of state machine. This was arrived at in an attempt to generalize from the types of control in parallel program schemata and networks of asynchronous modules without bounded delays. Asynchronous systems with output are also defined in a familiar way. The deviation from standard work comes in the definition of a parallel decomposition of asynchronous systems. Some preliminary work on compositions of this …

Nuclear Quadrupole Coupling Of B-12 In A Single Be Crystal, R. L. Williams Jr., Richard C. Haskell, Leon Madansky

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Quadrupole resonance lines of β-unstable 12B have been distinctly resolved in a single crystal of Be, and have behaved properly under field reversal and variation of the angle θ between crystal c axis and external magnetic field.

Observation Of Quadrupole Splitting Of B-12 In A Single Crystal, R. L. Williams Jr., Richard C. Haskell, Leon Madansky

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The quadrupole coupling of B-12 implanted in Be-9 has been observed using a single crystal of Be. One sees a narrow resonance line, the location of which depends in the normal way on the orientation of the crystalline c-axis with respect to the external magnetic field direction. The coupling constant is given by e2qQ/h = 54.9(6) kHz. This is consistent with our previous measurement using a Be foil. Using the field gradient at Be-9 lattice sites, calculated by Pomerantz and Das, one finds Q(B-12) = about 34.6 mb.

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

All HMC Faculty Publications and Research

No abstract provided.

Measurement Of Nuclear Magnetic Dipole Moment Of Li-8 By Implantation In Metal Foils, Richard C. Haskell, R. L. Williams Jr., Leon Madansky

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Polarized 8Li nuclei have been produced through the 7Li(d,p) reaction using the 3.5-MeV Van de Graaff accelerator at Brookhaven National Laboratory. The observed polarization was a slowly-varying function of deuteron energy over the range 1.3-2.9 MeV, reaching a maximum of about +1.6%. The recoiling nuclei were stopped in Au, Pt and Pd foils and the effective dipole moments were measured by a resonant depolarization method. The results were 1.65362(22)µN, 1.65288(20)µN and 1.65270(30)µN respectively. These are consistent with the work of Connor, who found µ(8Li) =1.6530(8)µN in a LiF crystal. An upper limit for the 8Li quadrupole moment will also be …

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 …

On Maximally Parallel Schemata, Robert M. Keller

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A model for parallel computation called a schema is presented. This model is similar to that presented in the recent work of Karp and Miller. Section 1 presents a description of the model, and some results on the characterization of computations within it. Section 2 summarizes some results on determinacy and equivalence. Section 3 presents a formalization of the property of maximal parallelism in schemata. Several alternate characterizations are shown to be equivalent for certain classes. Section 4 presents results on the complexity of a maximally parallel schema equivalent to a given schema.

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.

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

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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

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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 …