Physical Sciences and Mathematics Commons^™

Teaching Time Savers: The Exam Practically Wrote Itself!, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

When I first started teaching, creating an exam for my upper division courses was a genuinely exciting process. The material felt fresh and relatively unexplored (at least by me), and I remember often feeling pleasantly overwhelmed with what seemed like a vast supply of intriguing and engrossing exam-ready problems. Crafting the perfect exam, one that was noticeably inviting, exceedingly fair, and unavoidably illuminating, was a real joy.

Recounting Determinants For A Class Of Hessenberg Matrices, Arthur T. Benjamin, Mark A. Shattuck

All HMC Faculty Publications and Research

We provide combinatorial interpretations for determinants which are Fibonacci numbers of several recently introduced Hessenberg matrices. Our arguments make use of the basic definition of the determinant as a signed sum over the symmetric group.

Gravity-Driven Thin Liquid Films With Insoluble Surfactant: Smooth Traveling Waves, Rachel Levy, Michael Shearer, Thomas P. Witelski

All HMC Faculty Publications and Research

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerateparabolic, and admits traveling wave solutions in which the …

Effective Structure Theorems For Quadratic Spaces Via Height, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Second International Conference on The Algebraic and Arithmetic Theory of Quadratic Forms, December 2007.

An Integrated Social Actor And Service Oriented Architecture (Soa) Approach For Improved Electronic Health Record (Ehr) Privacy And Confidentiality In The Us National Healthcare Information Network (Nhin), Gondy Leroy, Elliot Sloane, Steven Sheetz

CGU Faculty Publications and Research

The emerging US National Healthcare Information Network (NHIN) will improve healthcare’s efficacy, efficiency, and safety. The first-generation NHIN being developed has numerous advantages and limitations. One of the most difficult aspects of today’s NHIN is ensuring privacy and confidentiality for personal health data, because family and caregivers have multiple complex legal relationships to a patient. A Social Actor framework is suggested to organize and manage these legal roles, but the Social Actor framework would be very difficult to implement in today’s NHIN. Social Actor Security Management could, however, be effectively implemented using Service Oriented Architectures (SOAs), which are rapidly becoming …

Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky

CMC Faculty Publications and Research

The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current …

Solution To Problem 1751, A Combinatorial Identity, Arthur T. Benjamin, Andrew Carman '09

All HMC Faculty Publications and Research

A combinatorial proof to Iliya Bluskov's proposed Problem 1751.

Infinitely Many Radial Solutions For A Sub-Super Critical Dirichlet Boundary Value Problem In A Ball, Alfonso Castro, John Kwon, Chee Meng Tan '07

All HMC Faculty Publications and Research

We prove the existence of infinitely many solutions to a semilinear Dirichlet boundary value problem in a ball for a nonlinearity g(u) that grows subcritically for u positive and supercritically for u negative.

A Fixed Point Theorem For The Infinite-Dimensional Simplex, Douglas Rizzolo '08, Francis E. Su

All HMC Faculty Publications and Research

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.

Greedy Signal Recovery And Uncertainty Principles, Deanna Needell, Roman Vershynin

CMC Faculty Publications and Research

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of the L1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle. In the case of inaccurate measurements and approximately sparse …

As Flat As Possible, Jon T. Jacobsen

All HMC Faculty Publications and Research

How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students …

Determination Of Interphase Line Tension In Langmuir Films, Jacob R. Wintersmith '06, Lu Zou, Andrew J. Bernoff, James C. Alexander, J. Adin Mann Jr.

All HMC Faculty Publications and Research

A Langmuir film is a molecularly thin film on the surface of a fluid; we study the evolution of a Langmuir film with two coexisting fluid phases driven by an interphase line tension and damped by the viscous drag of the underlying subfluid. Experimentally, we study a 4^′-8-alkyl[1,1^′-biphenyl]-4-carbonitrile (8CB) Langmuir film via digitally imaged Brewster angle microscopy in a four-roll mill setup which applies a transient strain and images the response. When a compact domain is stretched by the imposed strain, it first assumes a bola shape with two tear-drop shaped reservoirs connected by a thin …

Nonlinear Dynamics In Combinatorial Games: Renormalizing Chomp, Eric J. Friedman, Adam S. Landsberg

WM Keck Science Faculty Papers

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to …

The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennett

All HMC Faculty Publications and Research

Euclid does integers

Positive Solutions For Classes Of Multiparameter Elliptic Semipositone Problems, Scott Caldwell, Alfonso Castro, Ratnasingham Shivaji, Sumalee Unsurangsie

All HMC Faculty Publications and Research

We study positive solutions to multiparameter boundary-value problems of the form

-Δu = λg(u)+μf(u) in Ω

u = 0 on ∂Ω

where λ>0, μ>0, Ω⊆Rⁿ; n≥2 is a smooth bounded domain with ∂Ω in class C² and Δ is the Laplacian operator. In particular, we assume g(0)>0 and superlinear while f(0)

The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennet

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No abstract provided in this article.

Hot Electron And X-Ray Production From Intense Laser Irradiation Of Wavelength-Scale Polystyrene Spheres, H. A. Sumeruk, S. Kneip, D. R. Symes, I. V. Churina, A. V. Belolipetski, G. Dyer, J. Landry, G. Bansal, A. C. Bernstein, Thomas D. Donnelly, A. Karmakar, A. Pukhov, T. Ditmire

All HMC Faculty Publications and Research

Hot electron and x-ray production from solid targets coated with polystyrene-spheres which are irradiated with high-contrast, 100 fs, 400 nm light pulses at intensity up to 2×10¹⁷ W/cm² have been studied. The peak hard x-ray signal from uncoated fused silica targets is an order of magnitude smaller than the signal from targets coated with submicron sized spheres. The temperature of the x-rays in the case of sphere-coated targets is twice as hot as that of uncoated glass. A sphere-size scan of the x-ray yield and observation of a peak in both the x-ray production and temperature at a …

Determining The Agn Fraction Of Galaxy Groups, Rachel Paterno-Mahler

Pomona Senior Theses

Using the Chandra X-ray Observatory, Martini et al. (2006) found that the AGN fraction of galaxy clusters was five times higher than previous optical studies suggested. Using visual observations only, Dressler et al. (1985) estimated the AGN fraction of field galaxies to be 5%, while that of clusters was thought to be 1%. To understand the role that the environment plays in AGN fueling, the author studied a variety of environments, ranging from the field to groups to clusters. Will the AGN fraction of groups also be higher than that of the field? The author demonstrates how the AGN fraction …

Fuzzy Blackholes, Anand Murugan

Pomona Senior Theses

The fuzzball model of a black hole is an attempt to resolve the many paradoxes and puzzles of black hole physics that have revealed themselves over the last century. These badly behaved solutions of general relativity have given physicists one of the few laboratories to test candidate quantum theories of gravity. Though little is known about exactly what lies beyond the event horizon, and what the ultimate fate of matter that falls in to a black hole is, we know a few intriguing and elegant semi-classical results that have kept physicists occupied. Among these are the known black hole entropy …

On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Illinois Number Theory Fest, May 2007.

The Effects Of Spin-Orbit Coupling On Gravitational Wave Uncertainties, C. L. Wainwright

Pomona Senior Theses

Paper discusses the expected uncertainty of orbital parameters of binary stars as measured by the space-based gravitational wave observatory LISA (Laser Interferometer Space Antenna) and how the inclusion of spin in the model of the binary stars affects the uncertainty. The uncertainties are found by calculating the received gravitational wave from a binary pair and then performing a linear least-squares parameter estimation. The case of a 1500 solar mass black hole that is 20 years from coalescing with a 1000 solar mass black hole--both of which are 50 x 10^6 light years away--is analyzed, and the results show that the …

A Simple Organic Solar Cell, Gordon Patrick Whyburn

Pomona Senior Theses

Finding renewable sources of energy is becoming an increasingly important component of scientific research. Greater competition for existing sources of energy has strained the world’s supply and demand balance and has increased the prices of traditional sources of energy such as oil, coal, and natural gas. The experiment discussed in this paper is designed to identify and build an inexpensive and simple method for creating an effective organic solar cell.

Solar Energy Research And Development In California, Brett T. Close

Pomona Senior Theses

The energy crisis of 2001, high prices for gas and electricity and worries of climate change have caused a growing awareness about energy issues in California. The problems are clear. This paper looks at the next step of finding and implementing solutions. In this case the contribution that solar photovoltaic and solar thermal generation could make toward solving the problem. This paper looks at technological change, the current state of solar energy research, current government policies on solar energy, and finally makes policy recommendations to meet the stated problem.

A Combinatorial Proof Of Vandermonde's Determinant, Arthur T. Benjamin, Gregory P. Dresden

All HMC Faculty Publications and Research

No abstract provided in this article.

Approximations Of Continuous Newton's Method: An Extension Of Cayley's Problem, Jon T. Jacobsen, Owen Lewis '05, Bradley Tennis '06

All HMC Faculty Publications and Research

Continuous Newton's Method refers to a certain dynamical system whose associated flow generically tends to the roots of a given polynomial. An Euler approximation of this system, with step size h=1, yields the discrete Newton's method algorithm for finding roots. In this note we contrast Euler approximations with several different approximations of the continuous ODE system and, using computer experiments, consider their impact on the associated fractal basin boundaries of the roots

A Combinatorial Solution To Intertwined Recurrences, Arthur T. Benjamin, Michael D. Hirschhorn

All HMC Faculty Publications and Research

We provide combinatorial derivations of solutions to intertwined second order linear recurrences (such as a_n = pb_n-1 + qa_n-2, b_n = ra_n-1 + sb_n-2) by counting tilings of length n strips with squares and dominoes of various colors and shades. A similar approach can be applied to intertwined third order recurrences with coefficients equal to one. Here we find that all solutions can be expressed in terms of tribonacci numbers. The method can also be easily extended to solve and combinatorially comprehend kth order Fibonacci recurrences.

Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn

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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.

The Lsb Theorem Implies The Kkm Lemma, Gwen Spencer '05, Francis E. Su

All HMC Faculty Publications and Research

No abstract provided in this article.

Complex Symmetric Operators And Applications Ii, Stephan Ramon Garcia, Mihai Putinar

Pomona Faculty Publications and Research

A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ|T|, where J is an auxiliary conjugation commuting with |T| = √{T*T). We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ|T| also extends to the class of unbounded C-self adjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms …

Spatial And Temporal Expression Of Vegetation And Atmospheric Variability From Stable Carbon And Nitrogen Isotope Analysis Of Bat Guano In The Southern United States, Christopher M. Wurster, Donald A. Mcfarlane, Michael I. Bird

WM Keck Science Faculty Papers

Stable isotopes of faeces contain information related to the animals feeding ecology. The use of stable isotope values from subfossil faeces as a palaeoenvironmental indicator depends on how faithfully the animal records their local environment. Here we present insectivorous bat guano δ¹³C and δ¹⁵N values from a precipitation gradient across the southern United States and northern Mexico to compare with local vegetation and climate. We find δ¹³C values to be an excellent predictor of expected C₄/CAM vegetation, indicating that the bats are non-selective in their diet. Moreover, we find bat guano δ …