Physical Sciences and Mathematics Commons^™

Some Properties Of Positive Derivations On F-Rings, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set …

Specifying And Proving Properties Of Sentinels, Krithivasan Ramamritham, Robert M. Keller

All HMC Faculty Publications and Research

This paper presents a technique for specifying and verifying properties of "sentinels," a high-level language construct for synchronizing access to shared resources. Statements in the specification language possess formal temporal semantics. As a prelude to proving the correctness of sentinels, the semantics of constructs used in sentinels is given. The proof technique involves showing that the temporal behavior of a sentinel conforms to that defined by the specification. The methodology is illustrated by applying it to a typical synchronization problem.

The Design Of Oxygen Rebreather Equipment For Use In Foul-Air Speleology, Donald A. Mcfarlane

WM Keck Science Faculty Papers

The design of a lightweight oxygen rebreather set suitable for short duration explorations in foul-air caves is described, together with a discussion of its performance, limitations, and possible improvement.

Algebraic Complexity Theory, Nicholas Pippenger

All HMC Faculty Publications and Research

Algebraic complexity theory, the study of the minimum number of operations sufficient to perform algebraic computations, is surveyed with emphasis on the general theory of bilinear forms and two of its applications: polynomial multiplication and matrix multiplication. Though by no means exhausting algebraic complexity theory, these topics illustrate well its development and its methods, and provide examples of its most striking successes.

Bounds On The Performance Of Protocols For A Multiple-Access Broadcast Channel, Nicholas Pippenger

All HMC Faculty Publications and Research

A general model is presented for synchronous protocols that resolve conflicts among message transmissions to a multiple-access broadcast channel. An information-theoretic method is used now to show that if only finitely many types of conflicts can be distinguished by the protocol, utilization of the channel at rates approaching capacity is impossible. A random-coding argument is used to show that if the number of conflicting transmissions can be determined (which requires distinguishing infinitely many types of conflicts) then utilization of the channel at rates arbitrarily close to capacity can be achieved.

On Periodic Solutions Of Weakly Coupled Systems Of Differential Equations, Alfonso Castro, A. C. Lazer

All HMC Faculty Publications and Research

No abstract provided.

Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro

All HMC Faculty Publications and Research

In this paper, we study the existence of weak solutions of the problem

□u + ∇G(u) = f(t,x) ; (t,x) є Ω ≡ (0,π)x(0,π)

u(t,x) = 0 ; (t,x) є ∂Ω

where □ is the wave operator ∂²/∂t² - ∂²/∂x², G: Rⁿ→R is a function of class C² such that ∇G(0) = 0 and f:Ώ→R^n is a continuous function having first derivative with respect to t in (L₂,(Ω))ⁿ and satisfying

f(0,x) = f(π,x) = 0

for all x є [0,π].

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

No abstract provided.

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

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 …

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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.