Digital Commons Network^™

Looking Beyond The Curriculum In Jamaica, Jon T. Jacobsen, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

In August 2004, we had the opportunity to travel to Jamaica to lead a pilot workshop for Jamaican high school math teachers. The workshop focused on the importance of mathematical context in the teaching of mathematics. It was sponsored by the Gibraltar Institute, a Jamaica-based nongovernmental organization led by Trevor Campbell (Pomona College) and Reginald Nugent (Cal State Pomona), Jamaica’s College of Agriculture, Science and Education, and Harvey Mudd College.

Pythagorean Primes And Palindromic Continued Fractions, Arthur T. Benjamin, Doron Zeilberger

All HMC Faculty Publications and Research

In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued fractions. It uses an idea by Henry J. S. Smith ([3], [5], and [6]) most recently described in [4] (which provides a new proof of uniqueness and reprints Smith's paper in the original Latin). Smith's proof makes heavy use of nontrivial properties of determinants. Our purely combinatorial proof is self-contained and elementary.

Recounting The Odds Of An Even Derangement, Arthur T. Benjamin, Curtis D. Bennet, Florence Newberger

All HMC Faculty Publications and Research

No abstract provided in this article.

Fibonacci Determinants — A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn

All HMC Faculty Publications and Research

In this paper, we provide combinatorial interpretations for some determinantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to generalize and discover new ones.

Grid-Enabling A Vibroacoustic Analysis Application, Brian Bentow, Jon Dodge, Aaron Homer, Christopher D. Moore, Robert M. Keller, Matthew T. Presley, Robert Davis, Jorge Seidel, Craig Lee, Joseph Betser

All HMC Faculty Publications and Research

This paper describes the process of grid-enabling a vibroacoustic analysis application using the Globus Toolkit 3.2.1. This is the first step in a project intended to grid-enable a suite of tools being developed as a service-oriented architecture for spacecraft telemetry analysis. Many of the applications in the suite are compute intensive and would benefit from significantly improved performance. In this paper we show the advantage of using Globus to grid-enable a single tool in a vibroacoustic analysis flow, with the result that using as few as eleven nodes, that tool’s runtime improved by a factor of eight. While communication overhead …

Using Ultrasonic Atomization To Produce An Aerosol Of Micron-Scale Particles, Thomas D. Donnelly, J. Hogan '03, A. Mugler '04, M. Schubmehl '02, N. Schommer '04, Andrew J. Bernoff, S. Dasnurkar, T. Ditmire

All HMC Faculty Publications and Research

A device that uses ultrasonic atomization of a liquid to produce an aerosol of micron-scale droplets is described. This device represents a new approach to producing targets relevant to laser-driven fusion studies, and to rare studies of nonlinear optics in which wavelength-scale targets are irradiated. The device has also made possible tests of fluid dynamics models in a novel phase space. The distribution of droplet sizes produced by the device and the threshold power required for droplet production are shown to follow scaling laws predicted by fluid dynamics.

Proof Without Words: Alternating Sums Of Odd Numbers, Arthur T. Benjamin

All HMC Faculty Publications and Research

Proof for alternating sums of odd numbers in two figures.

Internet Accessibility And Usage Among Urban Adolescents In Southern California: Implications For Web-Based Health Research, C. Anderson Johnson, Ping Sun, Jennifer B. Unger, Paula Palmer, Peggy E. Gallaher, Chih-Ping Chou, Steve Sussman, Lourdes Baezconde-Garbanati

CGU Faculty Publications and Research

The World Wide Web (WWW) poses a distinct capability to offer interventions tailored to the individual’s characteristics. To fine tune the tailoring process, studies are needed to explore how Internet accessibility and usage are related to demographic, psychosocial, behavioral, and other health related characteristics. This study was based on a cross-sectional survey conducted on 2373 7th grade students of various ethnic groups in Southern California. Measures of Internet use included Internet use at school or at home, Email use, chat-room use, and Internet favoring. Logistic regressions were conducted to assess the associations between Internet uses with selected demographic, psychosocial, behavioral …

Q.954 And A.954, Quickie Problem And Solution, Arthur T. Benjamin, Michel Bataille

All HMC Faculty Publications and Research

Problem and proof proposed by authors.

Another proof, using lattice paths, can be found in Robert A. Sulanke's article, Objects Counted by the Central Delannoy Numbers, The Journal of Integer Sequences, Vol 6, 2003. A proof by polynomials is in Michel Bataille's paper Some Identities about an Old Combinatorial Sum, The Mathematical Gazette, March 2003, pp. 144-8. A slight change in the above proof leads to m ≥ n, a generalization proved by Li Zhou using lattice paths in The Mathematical Gazette.

Getting Grounded In The Post Hometown World, Hedley Burrell, David E. Drew

CGU Faculty Publications and Research

Remember when Americans had hometowns? "Where are you from?" we'd ask one another.

And the answer would come back: New York City. St. Joseph, Mo. Atlanta. Santa Barbara, Calif. Chattanooga, Tenn.

But odds are that now we'd get a more complicated response. It'd go something like this: "Well, I was born in Atlanta but we moved to Baltimore when I was 11 and in my junior year of high school, we went out to L.A. I've been in Chicago for a year."

And even this might not be quite accurate. The speaker may have been born in an Atlanta exurb …

A Constructive Proof Of Ky Fan's Generalization Of Tucker's Lemma, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.

Upper Estimates For The Energy Of Solutions Of Nonhomogeneous Boundary Value Problems, Alfonso Castro, Mónica Clapp

All HMC Faculty Publications and Research

We establish upper bounds for the energy of critical levels of the functional associated to a perturbed superlinear elliptic boundary value problem. We show that the perturbed problem satisfies the estimates obtained by Bahri and Lions (1988) for the symmetric problem. We use these estimates to prove the existence of nonradial solutions to a radial elliptic boundary value problem. Our results fill a gap in an earlier paper by Aduén and Castro.

Understanding The Rise And Transformation Of Business Collective Action In India, Aseema Sinha

CMC Faculty Publications and Research

Scholars of business associations have recently learned a great deal about how associations contribute to development, but much less about the origins of such developmental associations. This essay introduces and assesses a new political explanation for the origins of ‘developmental associations.’ Conventional wisdom holds that developmental associations must be able to rise above political and collusive pressures and establish autonomy from states. Yet, I argue that these associations’ developmental capacities emerge as a result of active state support by key actors, and in response to challenges and threats posed by competitive business organizations. Developmental associations emerge and acquire their capacities …

Effects Of Information And Machine Learning Algorithms On Word Sense Disambiguation With Small Datasets, Gondy Leroy, Thomas C. Rindflesch

CGU Faculty Publications and Research

Current approaches to word sense disambiguation use (and often combine) various machine learning techniques. Most refer to characteristics of the ambiguity and its surrounding words and are based on thousands of examples. Unfortunately, developing large training sets is burdensome, and in response to this challenge, we investigate the use of symbolic knowledge for small datasets. A naïve Bayes classifier was trained for 15 words with 100 examples for each. Unified Medical Language System (UMLS) semantic types assigned to concepts found in the sentence and relationships between these semantic types form the knowledge base. The most frequent sense of a word …

Two-Dimensional Self-Assembly In Diblock Copolymers, Anette E. Hosoi, Dmitriy Kogan '03, Caitlin E. Devereaux '02, Andrew J. Bernoff, Shenda M. Baker

All HMC Faculty Publications and Research

Diblock copolymers confined to a two-dimensional surface may produce uniform features of macromolecular dimensions (∼10–100  nm). We present a mathematical model for nanoscale pattern formation in such polymers that captures the dynamic evolution of a solution of poly(styrene)-b-poly(ethylene oxide), PS-b-PEO, in solvent at an air-water interface. The model has no fitting parameters and incorporates the effects of surface tension gradients, entanglement or vitrification, and diffusion. The resultant morphologies are quantitatively compared with experimental data.

Harmonic Generation In Thin Films And Multilayers, William S. Kolthammer '04, Dustin Barnard '03, Nicole Carson, Aaron D. Edens '00, Nathan A. Miller '01, Peter N. Saeta

All HMC Faculty Publications and Research

A general method for computing harmonic generation in reflection and transmission from planar nonmagnetic multilayer structures is described. The method assumes plane waves and treats harmonic generation in the parametric approximation. The method is applied in studying the second- and third-harmonic generation properties of thin crystal silicon layers surrounded by thermal oxide. Most independent components of the nonlinear susceptibility tensor have unique signatures with silicon layer thickness d, allowing their strength to be determined in principle by measuring harmonic generation as a function of d. Surface and bulk contributions to third-harmonic generation are cleanly distinguished, with the bulk signal dominating. …

Some Effective Diophantine Results Over Q-Bar, Lenny Fukshansky

CMC Faculty Publications and Research

In his 1999 paper D. W. Masser talks about effective search bounds for polynomial equations over integers and rationals. This discussion can also be extended over number fields. Unfortunately, as illustrated by Matiasevich's negative answer to Hilbert's 10-th problem, search bounds in general probably do not exist. Some special cases are understood, but in general very little is known. I will talk about effective search bounds for solutions of polynomial equations over Q-bar with some additional arithmetic conditions. This discussion also naturaly ties into the realm of "absolute" diophantine results, like Siegel's lemma of Roy and Thunder. I will try …

Book Review: Across The Board: The Mathematics Of Chessboard Problems By John J. Watkins, Arthur T. Benjamin

All HMC Faculty Publications and Research

I think I became a mathematician because I loved to play games as a child. I learned about probability and expectation by playing games like backgammon, bridge, and Risk. But I experienced the greater thrill of careful deductive reasoning through games like Mastermind and chess. In fact, for many years I took the game of chess quite seriously and played in many tournaments. But I gave up the game when I started college and turned my attention to more serious pursuits, like learning real mathematics.

Counting On Determinants, Arthur T. Benjamin, Naiomi T. Cameron

All HMC Faculty Publications and Research

No abstract provided in this article.

Review: Moira R. Rogers, Newtonianism For The Ladies And Other Uneducated Souls: The Popularization Of Science In Leipzig, 1687-1750 (New York, 2003), Andre Wakefield

Pitzer Faculty Publications and Research

Reviewed work: Moira R. Rogers. Newtonianism for the Ladies and Other Uneducated Souls: The Popularization of Science in Leipzig, 1687-1750. New York and Bern: Peter Lang, 2003. xiii + 181 pp. $61.95 (cloth), ISBN 978-0-8204-5029-2.

X-Ray Generation From Metal Targets Coated With Wavelength-Scale Spheres, D. R. Symes, H. A. Sumeruk, I. V. Churina, Thomas D. Donnelly, J. Landry, T. Ditmire

All HMC Faculty Publications and Research

X-ray yield measurements from targets coated with wavelength-scale spheres are compared with measurements from polished targets. Evidence for a hotter resonant electron temperature due to field enhancements from Mie resonances in the spheres is investigated.

The Electro-Acoustic Music Of Frederick Lesemann, Bill Alves

All HMC Faculty Publications and Research

A survey of the electro-acoustic music compositions of Frederick Lesemann from 1970-2004.

Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su

All HMC Faculty Publications and Research

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their …

A Mathematical Model For Treatment-Resistant Mutations Of Hiv, Helen Moore, Weiqing Gu

All HMC Faculty Publications and Research

In this paper, we propose and analyze a mathematical model, in the form of a system of ordinary differential equations, governing mutated strains of human immunodeficiency virus (HIV) and their interactions with the immune system and treatments. Our model incorporates two types of resistant mutations: strains that are not responsive to protease inhibitors, and strains that are not responsive to reverse transcriptase inhibitors. It also includes strains that do not have either of these two types of resistance (wild-type virus) and strains that have both types. We perform our analysis by changing the system of ordinary differential equations (ODEs) to …

Review: Bettina Wahrig And Werner Sohn, Eds. Zwischen Aufklärung, Policey Und Verwaltung. Zur Genese Des Medizinalwesens, 1750-1850 (Wiesbaden, 2003), Andre Wakefield

Pitzer Faculty Publications and Research

Reviewed work: Bettina Wahrig; Werner Sohn (Editors). Zwischen Aufklärung, Policey, und Verwaltung Zur Genese des Medizinalwesens, 1750–1850. 212 pp., index. Wiesbaden: Harrassowitz Verlag, 2003. €59.

Combinatorial Proofs Of Fermat's, Lucas's, And Wilson's Theorems, Peter G. Anderson, Arthur T. Benjamin, Jeremy A. Rouse

All HMC Faculty Publications and Research

No abstract provided in this article.

Selectron Mass Reconstruction And The Resolution Of The Linear Collider Detector, Sharon J. Gerbode, Heath Holguin, Troy Lau, Paul Mooser, Adam Pearlstein, Joe Rose, Bruce Schumm

All HMC Faculty Publications and Research

We have used ISAJET and the JAS LCD fast simulation to explore the precision of Snowmass Point SPS1a selectron mass reconstruction for the Silicon Detector concept. Simulating collisions at E_cm = 1 TeV, we have found that most of the information constraining the selectron mass is carried in the forward (|cos(theta)| > 0.8) region. We have also found that, for a beam energy spread of 1% (conventional RF design), detector resolution limitations compromise the selectron mass reconstruction only in the forward region. However, for a beam energy spread of less than 0.2% (superconducting RF design), the detector resolution compromises the selectron …

Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005.

At The Creation: The National Forest Commission Of 1896-97, Gerald W. Williams, Char Miller

Pomona Faculty Publications and Research

Among the central forces in the creation of the legislation necessary to establish federal forestry was the National Forest Commission. Its members included some of the leading conservationists of the 1890s, including Charles Sprague Sargent and Gifford Pinchot; John Muir was an unofficial member. Its final report advocated the establishment of a national forest system and served as the basis for the so-called Organic Act, which cleared the way for active management on federal forests and grasslands. Unlike the other articles, this one contains several excerpted documents interspersed with exposition.

The 19th Century Excavation Of Kent's Cavern, England, Donald A. Mcfarlane, Joyce Lundberg

WM Keck Science Faculty Papers

Between 1858 and 1880, William Pengelly developed revolutionary new techniques for the archeological and paleontological excavation of cave deposits. His work at Brixham Cave and Kent’s Cavern, England, yielded tens of thousands of specimens from the mid-Pleistocene to the Holocene, settled the intellectual debate over the co-existence of humans and extinct mammals, and accumulated an unparalleled resource for continued study. Although the Brixham Cave work was thoroughly summarized in print, Pengelly never published the plans of his much more thorough and extensive excavations at Kent’s Cavern. Here we present a reconstructed plan of the Pengelly excavations that we hope will …