Physical Sciences and Mathematics Commons^™

Reliable Computation By Formulas In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

It is shown that if formulas are used to compute Boolean functions in the presence of randomly occurring failures then: (1) there is a limit strictly less than 1/2 to the failure probability per gate that can be tolerated, and (2) formulas that tolerate failures must be deeper (and, therefore, compute more slowly) than those that do not. The heart of the proof is an information-theoretic argument that deals with computation and errors in very general terms. The strength of this argument is that it applies with equal ease no matter what types of gate are available. Its weaknesses is …

A Theoretical Investigation Of Several Low-Lying States Of Trans, Trans-1,3,5-Hexatriene, Robert J. Cave, Ernest R. Davidson

All HMC Faculty Publications and Research

Results from ab initio calculations concerning several low-lying electronic states of trans,trans-1,3,5-hexatriene are presented and compared with experimental and previous theoretical results. The lowest excited singlet state is predicted to be the ¹B_u state, having essentially valencelike π → π* character. The nominally doubly excited 2¹A_g state is found to lie approximately 0.6-0.9 eV above the 1¹B_u state. Results are also presented for several Rydberg states. The implications of the present results for current parametrizations of semiempirical π molecular orbital schemes are discussed.

Ab Initio Investigation Of Several Low-Lying States Of All-Trans Octatetraene, Robert J. Cave, Ernest R. Davidson

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The results of ab initio calculations that examine vertical and nonvertical transitions to several low-lying states of all-trans-octatetraene are presented. It is found that the lowest vertical excitation is to a valence π → π* ¹B_u state, and the nominally doubly excited 2¹A_g state occurs approximately 0.4 eV higher at the geometry of the ground state. Using estimated excited-state equilibrium geometries, we find that the 2¹A_g state is indeed the lowest singlet excited state, having a 0-0 tarnsition energy of 4.15 eV. The 0-0 transition energy for the 1¹B_u state is calculated to be 4.56 eV. The present results …

The Bisection Method: Which Root?, Arthur T. Benjamin

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No abstract provided in this article.

Quasi F-Covers Of Tychonoff Spaces, Melvin Henriksen, J. Vermeer, R. G. Woods

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A Tychonoff topological space is called a quasi F-space if each dense cozero-set of X is C*-embedded in X. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi F-cover" QF(X) of a compact space X as an inverse limit space, and identify the ring C(QF(X)) as the order-Cauchy completion of the ring C*(X). In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi F-cover of an arbitrary Tychonoff space. In this paper the minimal quasi F-cover of a compact space X is constructed as the space of ultrafilters …

Ordered Products Of Topological Groups, Melvin Henriksen, Ralph Kopperman, Frank A. Smith

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The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

Infinitely Many Radially Symmetric Solutions To A Superlinear Dirichlet Problem In A Ball, Alfonso Castro, Alexandra Kurepa

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In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

Theoretical Studies Of Electron Transfer In Metal Dimers: Xy+→X+Y, Where X, Y=Be, Mg, Ca, Zn, Cd, Robert J. Cave, David V. Baxter, William A. Goddard Iii, John D. Baldeschwieler

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The electronic matrix element responsible for electron exchange in a series of metal dimers was calculated using ab initio wave functions. The distance dependence is approximately exponential for a large range of internuclear separations. A localized description, where the two nonorthogonal structures characterizing the electron localized at the left and right sites are each obtained self‐consistently, is found to provide the best description of the electron exchange process. We find that Gaussian basis sets are capable of predicting the expected exponential decay of the electronic interactions even at quite large internuclear distances.

Teaching With A Humanist, David Meredith

Humanistic Mathematics Network Journal

This paper is a report about a team-taught course at San Francisco State University, and what the author learned having taught it over 11 years with faculty in the humanities. As part of an interdisciplinary curriculum called NEXA, which explored the boundaries between science and the humanities, the course revealed to the author several ways mathematics and humanities pedagogies can reciprocally learn from each other and emphasize similar goals.

Patterns Of Emotion Within Mathematics Problem-Solving, Frances A. Rosamond

Humanistic Mathematics Network Journal

There is often a divide between the experience of positive emotions toward math on the part of mathematical educators and negative emotions toward math on the part of students. This paper utilizes psychologist Richard Lazarus's work on the effects of positive emotions in order to highlight their benefits for mathematical pedagogy, to explain the author's experiment applying Lazarus's theory, and to suggest ways this application might support and foster positive emotions in students.

Gresham's Law: Algorithm Drives Out Thought, Sherman K. Stein

Humanistic Mathematics Network Journal

Gresham's law in economics states, "Bad money drives good money out of circulation." An application of this law in mathematical pedagogy states that "Algorithm drives out thought." While universities are ideally places where classes are meant to develop students' independence and critical thinking skills, often mathematics courses reflect this altered version of Gresham's law. This paper demonstrates the ways traditional mathematical pedagogy has held up Gresham's law and presents several suggestions for ways to change this approach to mathematical education to focus more on critical thinking without sacrificing the necessity of algorithm.

Letter From The Editor, Issue 1, 1987, Alvin White

Humanistic Mathematics Network Journal

No abstract provided.

Applied Mathematics As Social Contract, Philip J. Davis

Humanistic Mathematics Network Journal

The author takes the position that mathematical education must redefine its goals so as to create a citizenry with sufficient knowledge to provide social backpressure on future mathematizations. This can be accomplished by increasing the part of mathematical education that is devoted to the description and interpretation of the processes of mathematization and by allowing the technicalities of the formal operations within mathematics itself to be deemphasized or automated out by computer.

Appendix

Humanistic Mathematics Network Journal

No abstract provided.

Wavenumber Selection Of Convection Rolls In A Box, Wayne Arter, Andrew J. Bernoff, A. C. Newell

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The dynamics of two‐dimensional Rayleigh–Bénard convection rolls are studied in a finite layer with no‐slip, fixed temperature upper and lower boundaries and no‐slip insulating side walls. The dominant mechanism controlling the number of rolls seen in the layer is an instability concentrated near the side walls. This mechanism significantly narrows the band of stable wavenumbers although it can take a time comparable to the long (horizontal) diffusion time scale to operate.

Sorting And Selecting In Rounds, Nicholas Pippenger

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We present upper bounds for sorting and selecting the median in a fixed number of rounds. These bounds match the known lower bounds to within logarithmic factors. They also have the merit of being “explicit modulo expansion”; that is, probabilistic arguments are used only to obtain expanding graphs, and when explicit constructions for such graphs are found, explicit algorithms for sorting and selecting will follow. Using the best currently available explicit constructions for expanding graphs, we present the best currently known explicit algorithms for sorting and selecting in rounds.

The Complexity Of Computations By Networks, Nicholas Pippenger

All HMC Faculty Publications and Research

We survey the current state of knowledge concerning the computation of Boolean functions by networks, with particular emphasis on the addition and multiplication of binary numbers.

Rigid And Nonrigid Achirality, Erica Flapan

Pomona Faculty Publications and Research

In order to completely characterize a molecule it is useful to understand the symmetries of its molecular bond graph in 3-space. For many purposes the most important type of symmetry that a molecule can exhibit is mirror image symmetry. However, the question of whether a molecular graph is equivalent to its mirror image has different interpretations depending on what assumptions are made about the rigidity of the molecular structure. If there is a deformation of 3-space taking a molecular bond graph to its mirror image then the molecule is said to be topologically achiral. If a molecular graph can be …

The Gradient Model Load Balancing Method, Frank C. H. Lin, Robert M. Keller

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A dynamic load balancing method is proposed for a class of large-diameter multiprocessor systems. The method is based on the "gradient model," which entails transferring backlogged tasks to nearby idle processors according to a pressure gradient indirectly established by requests from idle processors. The algorithm is fully distributed and asynchronous. Global balance is achieved by successive refinements of many localized balances. The gradient model is formulated so as to be independent of system topology.

A Theoretical Investigation Of Some Low-Lying Singlet States Of 1,3-Butadiene, Robert J. Cave, Ernest R. Davidson

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Results are presented from extensive ab initio calculations on several low-lying singlet states of cis- and trans-1,3-butadiene. The results indicate a qualitative difference between the cis and trans isomers for the lowest π → π* transition. For the cis isomer, the first excited singlet state of the same symmetry as the ground state is found to lie above the lowest π → π* transition, and we estimate that this is also the case for the trans isomer.

Do We Really Know What Makes Educational Software Effective? A Call For Empirical Research On Effectiveness, Karen Jolicoeur, Dale E. Berger

CGU Faculty Publications and Research

Empirical information on specific factors that make educational software effective in reaching instructional objectives would be of considerable value. The authors describe the current state of evaluation research with educational software and discuss how popular software review methods fall short of meeting our need to know how well specific programs work.

Computers And The Nature Of Man: A Historian's Perspective On Controversies About Artificial Intelligence, Judith V. Grabiner

Pitzer Faculty Publications and Research

The purpose of the present paper is to provide a historical perspective on recent controversies, from Turing's time on, about artificial intelligence, and to make clear that these are in fact controversies about the nature of man. First, I shall briefly review three recent controversies about artificial intelligence, controversies over whether computers can think and over whether people are no more than information-processing machines. These three controversies were each initiated by philosophers who, irrespective of what the programs of their time actually did, viewed with alarm the argument that if a machine can think, a thinking being is just a …

Distributed Recovery In Applicative Systems, Frank C. H. Lin, Robert M. Keller

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Applicative systems are promising candidates for achieving high performance computing through aggregation of processors. This paper studies the fault recovery problems in a class of applicative systems. The concept of functional checkpointing is proposed as the nucleus of a distributed recovery mechanism. This entails incrementally building a resilient structure as the evaluation of an applicative program proceeds. A simple rollback algorithm is suggested to regenerate the corrupted structure by redoing the most effective functional checkpoints. Another algorithm, which attempts to recover intermediate results, is also presented. The parent of a faulty task reproduces a functional twin of the failed task. …

A Semiclassical Model For Orientation Effects In Electron Transfer Reactions, Robert J. Cave, Stephen J. Klippenstein, R.A. Marcus

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An approximate solution to the single‐particle Schrödinger equation with an oblate spheroidal potential well of finite depth is presented. The electronic matrix element H_BA for thermal electron transfer is calculated using these wave functions, and is compared with values of H_BA obtained using the exact solution of the same Schrödinger equation. The present method yields accurate results for H_BA, within the oblate spheroidal potential well model, and is useful for examining the orientational effects of the two centers on the rate of electron transfer.

A Prime Strongly Positive Amphicheiral Knot Which Is Not Slice, Erica Flapan

Pomona Faculty Publications and Research

We begin by giving several definitions. A knot K in S³ is said to be amphicheiral if there is an orientation-reversing diffeomorphism h of S³ which leaves K setwise invariant. Suppose, in addition, that K is given an orientation. ThenK is said to be positive amphicheiral if h preserves the orientation of K. If, in addition, the diffeomorphism h is an involution then K is strongly positive amphicheiral. Finally, we say a knot is slice if it bounds a smooth disc in B⁴. In this note we shall give a smooth example of …

Chemical Applications Of Scanning Tunneling Microscopy, Paul West, John Kramer, David V. Baxter, Robert J. Cave, John D. Baldeschwieler

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The development of a scanning tunneling microscope at the California Institute of Technology is well under way. Electron tunneling has been demonstrated, and preliminary surface images of gold films have been obtained. Additional instrumental development is required to achieve the atomic resolution which is required for the study of chemical processes on surfaces. A theoretical model is also being developed for the study of tunneling of electrons from the probe to surfaces with molecular species absorbed, and with atomic and molecular species intervening between the probe and the surface. These experimental tools and theoretical models, which are being developed concurrently, …

Alphabetic Minimax Trees Of Degree At Most T*, D. Coppersmith, Maria M. Klawe, Nicholas Pippenger

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Problems in circuit fan-out reduction motivate the study of constructing various types of weighted trees that are optimal with respect to maximum weighted path length. An upper bound on the maximum weighted path length and an efficient construction algorithm will be presented for trees of degree at most t, along with their implications for circuit fan-out reduction.

Reliable Computation In The Presence Of Noise, Nicholas Pippenger

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This talk concerns computation by systems whose components exhibit noise (that is, errors committed at random according to certain probabilistic laws). If we aspire to construct a theory of computation in the presence of noise, we must possess at the outset a satisfactory theory of computation in the absence of noise.

A theory that has received considerable attention in this context is that of the computation of Boolean functions by networks (with perhaps the strongest competition coming from the theory of cellular automata; see [G] and [GR]). The theory of computation by networks associates with any two sets Q and …

Mutual Orientation Effects On Electron-Transfer Reactions Between Porphyrins, Robert J. Cave, Paul Siders, R.A. Marcus

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Mutual orientation effects on the rate of nonadiabatic electron transfer between several diporphyrin pairs of experimental interest are examined. The electronic matrix element for electron transfer is calculated within a one-electron spheroidal model for a variety of states and orientations which are relevant to both biological and synthetic electron-transfer systems. Both the mutual orientation of the pairs and the nodal structure of the donor and acceptor orbitals can have large effects on calculated rates.

The Taman Negara Batek: A People In Transition, Paul Faulstich

Pitzer Faculty Publications and Research

Batek Negritos from the vicinity of Taman Negara National Park in West Malaysia are a hunting and gathering people presently experiencing rapid encroachment by the modern world. Under the authority of the Malaysian government, they are being encouraged to settle and to emulate Malay subsistence farming communities. Unfortunately, this strategy has had a number of adverse effects on the Batek.