Physical Sciences and Mathematics Commons^™

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.

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 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 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 …

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.

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 …

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.

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 …

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.

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.

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.

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) …

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

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No abstract provided.

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.

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.

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. …

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 …

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

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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 …

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

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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.

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.

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.