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.

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.

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

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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.

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.

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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.

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.

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

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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 …

Estimating Winning Probabilities In Backgammon Races, Andrew M. Ross, Arthur T. Benjamin, Michael Munson '94

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In modern backgammon, it is advantageous to know the chances each player has of winning, and to be able to compute the chances without the aid of calculators or pencil and paper. A simple model of backgammon is used to approximate those chances, and a readily computable and sufficiently accurate approximation of that is developed. From there, the model is compared to simulated backgammon games, and the previous approximation is modified to fit the real data.

Seeking Bang-Bang Solutions Of Mixed Immuno-Chemotherapy Of Tumors, Lisette G. De Pillis, K Renee Fister, Weiqing Gu, Craig Collins, Michael Daub, David Gross '08, James Moore '07, Benjamin Preskill '09

All HMC Faculty Publications and Research

It is known that a beneficial cancer treatment approach for a single patient often involves the administration of more than one type of therapy. The question of how best to combine multiple cancer therapies, however, is still open. In this study, we investigate the theoretical interaction of three treatment types (two biological therapies and one chemotherapy) with a growing cancer, and present an analysis of an optimal control strategy for administering all three therapies in combination. In the situations with controls introduced linearly, we find that there are conditions on which the controls exist singularly. Although bang-bang controls (on-off) reflect …

Teaching Time Savers: Is Homework Grading On Your Nerves?, Lisette G. De Pillis, Michael E. Orrison Jr.

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You have probably heard it said that we learn mathematics best when we do mathematics, or that mathematics is not a spectator sport. For most of our students, this means that their mathematics courses will involve a fair amount of homework. This homework is often used to evaluate individual student progress, but it can also be used, for example, as a catalyst for discussion, to emphasize a point made in class, and to identify common misunderstandings throughout the class as a whole. There is, however, the matter of grading homework.

Domain Relaxation In Langmuir Films, James C. Alexander, Andrew J. Bernoff, Elizabeth K. Mann, J. Adin Mann Jr., Jacob R. Wintersmith '06, Lu Zou

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We report on theoretical studies of molecularly thin Langmuir films on the surface of a quiescent subfluid and qualitatively compare the results to both new and previous experiments. The film covers the entire fluid surface, but domains of different phases are observed. In the absence of external forcing, the compact domains tend to relax to circles, driven by a line tension at the phase boundaries. When stretched (by a transient applied stagnation-point flow or by stirring), a compact domain elongates, creating a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape will then relax slowly …

Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen

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We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.

Sp-Scattered Spaces: A New Generalization Of Scattered Spaces, Melvin Henriksen, Robert M. Raphael, R. G. Woods

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The set of isolated points (resp. P-points) of a Tychonoff space X is denoted by Is(X) (resp. P(X)). Recall that X is said to be scattered if Is(A) ≠ ∅ whenever ∅ ≠ A ⊂ X. If instead we require only that P(A) has nonempty interior whenever ∅ ≠ A ⊂ X, we say that X is SP-scattered. Many theorems about scattered spaces hold or have analogs for SP-scattered spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces …

Removing Sets From Connected Spaces While Preserving Connectedness, Melvin Henriksen, Amir Nikou

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As per the title, the nature of sets that can be removed from a product of more than one connected, arcwise connected, or point arcwise connected spaces while preserving the appropriate kind of connectedness is studied. This can depend on the cardinality of the set being removed or sometimes just on the cardinality of what is removed from one or two factor spaces. Sometimes it can depend on topological properties of the set being removed or its trace on various factor spaces. Some of the results are complicated to prove while being easy to state. Sometimes proofs for different kinds …

Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins

CMC Faculty Publications and Research

Abstract. Let N ≥ 2 and let 1 < a(1) < ... < a(N) be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as Sigma(N)(i=1) a(i) x(i) where x(1),..., x(N) are non-negative integers. The condition that gcd(a(1),..., a(N)) = 1 implies that such a number exists. The general problem of determining the Frobenius number given N and a(1),..., a(N) is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.

Review: An Intertwining Property For Positive Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: On A Class Of Reflexive Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.