Mathematics Commons^™

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

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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

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No abstract provided in this article.

Fibonacci Determinants — 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.

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.

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

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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.

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

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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

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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.

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

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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.

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

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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

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No abstract provided in this article.

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

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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

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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 …

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

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No abstract provided in this article.

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.

A *-Closed Subalgebra Of The Smirnov Class, Stephan Ramon Garcia

Pomona Faculty Publications and Research

We study real Smirnov functions and investigate a certain *-closed subalgebra of the Smirnov class N^+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space H^p ∩ H^p [overscore over the second H^p]. This leads to a natural analog of the Riesz projection on a certain quotient space of L^p for p ϵ (0, 1). We also study a Herglotz-like integral transform for singular measures on the unit circle ∂D.

Microarray Data From A Statistician’S Point Of View, Johanna S. Hardin

Pomona Faculty Publications and Research

No abstract provided.

Inner Matrices And Darlington Synthesis, Stephan Ramon Garcia

Pomona Faculty Publications and Research

We describe and parameterize the solutions of the scalar valued Darlington synthesis problem. In the case of rational data we derive a simple procedure for producing all possible solutions.

Srt Division Algorithms As Dynamical Systems, Mark Mccann, Nicholas Pippenger

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Sweeney--Robertson--Tocher (SRT) division, as it was discovered in the late 1950s, represented an important improvement in the speed of division algorithms for computers at the time. A variant of SRT division is still commonly implemented in computers today. Although some bounds on the performance of the original SRT division method were obtained, a great many questions remained unanswered. In this paper, the original version of SRT division is described as a dynamical system. This enables us to bring modern dynamical systems theory, a relatively new development in mathematics, to bear on an older problem. In doing so, we are able …

The Closed Topological Vertex Via The Cremona Transform, Jim Bryan, Dagan Karp

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We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.

Properties Of One-Point Completions Of A Noncompact Metrizable Space, Melvin Henriksen, Ludvík Janoš, R. G. Woods

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If a metrizable space X is dense in a metrizable space Y, then Y is called a metric extension of X. If T₁ and T₂ are metric extensions of X and there is a continuous map of T₂ into T₁ keeping X pointwise fixed, we write T₁ ≤ T₂. If X is noncompact and metrizable, then (M(X),≤) denotes the set of metric extensions of X, where T₁ and T₂ are identified if T₁ ≤ T₂ and T₂ ≤ T₁, i.e., if there is a homeomorphism of …

C(X) Can Sometimes Determine X Without X Being Realcompact, Melvin Henriksen, Biswajit Mitra

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As usual C(X) will denote the ring of real-valued continuous functions on a Tychonoff space X. It is well-known that if X and Y are realcompact spaces such that C(X) and C(Y ) are isomorphic, then X and Y are homeomorphic; that is C(X) determines X. The restriction to realcompact spaces stems from the fact that C(X) and C(uX) are isomorphic, where uX is the (Hewitt) realcompactifcation of X. In this note, a class of locally compact spaces X that includes properly the class of locally compact realcompact spaces is exhibited such that C(X) determines X. The problem of getting …

Problems From The Cottonwood Room, Matthias Beck, Beifang Chen, Lenny Fukshansky, Christian Haase, Allen Knutson, Bruce Reznick, Sinai Robins, Achill Schürmann

CMC Faculty Publications and Research

This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in polyhedra. Lattice points in homogeneously expanding compact domains. Presented by Lenny Fukshansky (Texas A&M University).