Physical Sciences and Mathematics Commons^™

Hierarchical Analysis Of A Distributed Evaluator, Robert M. Keller, Gary Lindstrom

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We outline the analysis of a distributed evaluator for an applicative language FGL (Function Graph Language). Our goal is to show that the least fixed point semantics of FGL are faithfully implemented by the hardware evaluator envisioned in the Applicative Multi-Processor System AMPS. Included in the analysis are a formalization of demand-driven computation , the introduction of an intermediate graphic language IGL to aid in our proofs, and discussion of pragmatic issues involved in the AMPS machine language design.

A New Lower Bound For The Number Of Switches In Rearrangeable Networks, Nicholas Pippenger

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For the commonest model of rearrangeable networks with $n$ inputs and $n$ outputs, it is shown that such a network must contain at least $6n \log _6 n + O( n )$ switches. Similar lower bounds for other models are also presented.

Liverpool University Expedition To Jamaica, Donald A. Mcfarlane

WM Keck Science Faculty Papers

During summer 1977, five members of the Liverpool University Potholing Club spent six weeks working and exploring in the caves of Jamaica. The team consisted of Don McFarlane, John Dye, Malcolm Macduff, Mike Roger and Barry Williams, all of whom contributed to this report. The expedition base was at Troy, where the villagers are owed a debt of gratitude for their hospitality. This placed the expedition in the heart of the cave region, and a number of new caves and shafts were discovered and explored. The main discovery was the Still Waters Cave, located near Accompong, where 11,800 feet of …

Comparative Schematology And Pebbling With Auxiliary Pushdowns, Nicholas J. Pippenger

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This paper has three claims to interest. First, it combines comparative schematology with complexity theory. This combination is capable of distinguishing among Strong's “languages of maximal power,” a distinction not possible when comparative schematology is based on computability considerations alone, and it is capable of establishing exponential disparities in running times, a capability not currently possessed by complexity theory alone. Secondly, this paper inaugurates the study of pebbling with auxiliary pushdowns, which bears to plain pebbling the same relationship as Cook's study of space-bounded machines with auxiliary pushdowns bears to plain space-bounded machines. This extension of pebbling serves as the …

Závisí Matematická Pravda Od Času?, Judith V. Grabiner

Pitzer Faculty Publications and Research

This is a Slovak translation of Judith Grabiner's "Is Mathematical Truth Time-Dependent?," published in Volume 81 of American Mathematical Monthly (April 1974).

Two Point Boundary Value Problem With Jumping Nonlinearities, Alfonso Castro

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We prove that a certain two point BVP with jumping nonlinearities has a solution. Our result generalizes that of [2]. We use variational methods which permit giving a minimax characterization of the solution. Our proof exposes the similarities between the variational behavior of this problem and that of other semilinear problems with noninvertible linear part (see [5]).

James Eights, Albany Naturalist: New Evidence, Char Miller, Naomi Goldsmith

Pomona Faculty Publications and Research

Eights's contributions to scientific study and to the popularization of science have been understated and misunderstood.

On The Evaluation Of Powers And Monomials, Nicholas Pippenger

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Let $y_1 , \cdots ,y_p $ be monomials over the indeterminates $x_1 , \cdots ,x_q $. For every $y = (y_1 , \cdots ,y_p )$ there is some minimum number $L(y)$ of multiplications sufficient to compute $y_1 , \cdots ,y_p $ from $x_1 , \cdots ,x_q $ and the identity 1. Let $L(p,q,N)$ denote the maximum of $L(y)$ over all $y$ for which the exponent of any indeterminate in any monomial is at most $N$. We show that if $p = (N + 1^{o(q)} )$ and $q = (N + 1^{o(p)} )$, then $L(p,q,N) = \min \{ p,q\} \log N …

Critical Point Theory And The Number Of Solutions Of A Nonlinear Dirichlet Problem, Alfonso Castro, A. C. Lazer

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No abstract provided.

Optimal 2,3-Trees, Nicholas J. Pippenger, Raymond E. Miller, Arnold L. Rosenberg, Lawrence Snyder

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The 2,3-trees that are optimal in the sense of having minimal expected number of nodes visited per access are characterized in terms of their “profiles”. The characterization leads directly to a linear-time algorithm for constructing a K-key optimal 2,3-tree for a sorted list of K keys. A number of results are derived that demonstrate how different in structure these optimal 2,3-trees are from their “average” cousins.

A Survey Of Ligand Effects Upon The Reaction Entropies Of Some Transition Metal Redox Couples, Edmund L. Yee, Robert J. Cave, Kendall L. Guyer, Paul D. Tyma, Michael J. Weaver

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The reaction entropies ∆S°_rc of a number of transition metal redox couples of the form M(III)/(II) in aqueous solution have been determined using nonisothermal electrochemical cells in order to explore the effect of varying the ligand structure upon the nature of the ion-solvent interactions. Examinations of six aquo couples of the form M(OH_2)_n&3+/2+ with varying metal M yielded ∆S°_rc values in the range 36-49 eu. In order to scrutinize the effect of replacing aquo with ammine and simple anionic ligands, Ru(III)/(II) couples were employed since the relative substituion inertness of both oxidation states allowed ∆S …

A Semilinear Dirichlet Problem, Alfonso Castro

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Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:

Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω

u(x) = 0 ; x є ∂Ω

where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rⁿ⁺¹ → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

On The Application Of Coding Theory To Hashing, Nicholas Pippenger

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Quick proofs are given for the characterization (due to Schay, Raver, Hanan, and Palermo) of the collision distance of a linear hashing function and for a dual function (called the restriction distance), which relates to the accessibility of addresses by sets of keys and the uniform distribution of sets of keys over addresses.

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

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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 …

The Kinetics Of Solvent Reorientation In Hydroxylated Solvents From The Exciting-Wavelength Dependence Of Chromophore Emission Spectra, John Milton, Robert M. Purkey, William C. Galley

WM Keck Science Faculty Papers

The disappearance of the exciting-wavelength dependence of the phosphorescence spectra of polar, aromatic chromophores in supercooled glycol–water mixtures is utilized to monitor the kinetics of solvent reorientation. Reorientation times in the nanosecond to second range are obtained for (3:2 v/v) glycerol–water and (1:1 v/v) ethylene glycol–water at 140–240 °K. The results suggest that the process is one involving a cluster of solvent molecules and in which the chromophore plays a relatively passive role. Steady-state data and direct measurements of phosphorescence shifts as a function of time indicate that the solvent reorientation process is nonexponential in nature. The decay function derived …

Hammerstein Integral Equations With Indefinite Kernel, Alfonso Castro

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This paper deals with the problem of finding solutions of the Hammerstein integral equation. It is shown that this problem can be reduced to the study of the critical points of certain functional defined on L₂(Ω). Existence of a solution of the Hammersteln integral equation is proved. Some other related results of interest are obtained.

Generalized Connectors, Nicholas Pippenger

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An $n$-connector is an acyclic directed graph having $n$ inputs and $n$ outputs and satisfying the following condition: given any one-to-one correspondence between inputs and distinct outputs, there exists a set of vertex-disjoint paths that join each input to the corresponding output. It is known that the minimum possible number of edges in an $n$-connector lies between lower and upper bounds that are asymptotic to $3n\log _3 n$ and $6n\log _3 n$ respectively. A generalized $n$-connector satisfies the following stronger condition: given any one-to-many correspondence between inputs and disjoint sets of outputs, there exists a set of vertex-disjoint trees that …

Applying Mathematics Without A License, Melvin Henriksen

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A recent educational experience made me realize the extent to which the mathematical community has become fragmented and how this has served to inhibit communication both with others and ourselves.

An Algebraic Characterization Of The Freudenthal Compactification For A Class Of Rimcompact Spaces, Melvin Henriksen

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Throughout C(X) will denote the ring of all continuous real-valued functions on a Tychonoff space X, and C*(X) will denote the subring of bounded elements of C(X). The real line is denoted by R, and N denotes the (discrete) subspace of positive integers. A subset S of X such that the map f → f|_s is an epimorphism of C(X) (resp. C*(X)) is said to be C-embedded (resp. C*-embedded) in X. As is well-known, every f Є C*(X) has a unique continuous extension βf over its Stone-Čech compactification βX [GJ, Chapter 6]. That is, X is …

Some Sufficient Conditions For The Jacobson Radical Of A Commutative Ring With Identity To Contain A Prime Ideal, Melvin Henriksen

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Throughout, the word "ring" will abbreviate the phrase "commutative ring with identity element 1" unless the contrary is stated explicitly. An ideal I of a ring R is called pseudoprime if ab = 0 implies a or b is in I. This term was introduced by C. Kohls and L. Gillman who observed that if I contains a prime ideal, then I is pseudoprime, but, in general, the converse need not hold. In [9 p. 233], M. Larsen, W. Lewis, and R. Shores ask if whenever the Jacobson radical J(R) of an arithmetical ring is pseudoprime, it follows that J(R) …

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.

Superconcentrators, Nicholas Pippenger

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An $n$-superconcentrator is an acyclic directed graph with $n$ inputs and $n$ outputs for which, for every $r \leqq n$, every set of $r$ inputs, and every set of $r$ outputs, there exists an $r$-flow (a set of $r$ vertex-disjoint directed paths) from the given inputs to the given outputs. We show that there exist $n$-superconcentrators with $39n + O(\log n)$ (in fact, at most $40n$) edges, depth $O(\log n)$, and maximum degree (in-degree plus out-degree) 16.

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

All HMC Faculty Publications and Research

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. …

Quadrupole Couplings Of N12 And B12 Implanted In Metal Single-Crystals, Richard C. Haskell, Francis David Corell, Leon Madansky

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Measurements have been made of the quadrupole couplings of 12N implanted in single crystals of Be and Mg and of 12B implanted in a single crystal of Zn. A comparison of the 12N couplings in Be and Mg suggests that (i) the final stopping sites of the implanted 12N ions are substitutional sites, i.e., the 12N ions occupy metal-ion lattice positions, and (ii) the 12N ions implanted in Be and Mg have the same charge state and quadrupole shielding factor. A procedure is outlined for deducing Q(12N). The 12B couplings in Zn imply the existence of two inequivalent stopping sites. …

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.