Mathematics Commons^™

The Linear Complexity Of A Graph, David L. Neel, Michael E. Orrison Jr.

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The linear complexity of a matrix is a measure of the number of additions, subtractions, and scalar multiplications required to multiply that matrix and an arbitrary vector. In this paper, we define the linear complexity of a graph to be the linear complexity of any one of its associated adjacency matrices. We then compute or give upper bounds for the linear complexity of several classes of graphs.

Combinatorial Interpretations Of Spanning Tree Identities, Arthur T. Benjamin, Carl R. Yerger

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We present a combinatorial proof that the wheel graph Wn has L2n − 2 spanning trees, where Ln is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci number. Our proofs avoid the use of induction, determinants, or the matrix tree theorem.

A Framework For Inclusive Teaching In Stem Disciplines, Lois Reddick, Wayne Jacobson, Angela Linse, Darryl Yong

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A wide body of literature exists recounting the ways in which inclusive teaching practices and principles benefit students and positively impact learning, student retention, and professional development across disciplines. However, STEM faculty do not readily accept the traditional approach of examining course content from multiple perspectives as relevant to their course content or useful in their teaching. In this chapter, we propose a Framework for Inclusive Teaching in STEM Disciplines that reflects the contexts of teaching in these disciplines, and extends James Banks’ Five Dimensions of Multicultural Education to the distinct needs of STEM faculty in their classes. We also …

The Maximal Regular Ideal Of Some Commutative Rings, Emad Abu Osba, Melvin Henriksen, Osama Alkam, Frank A. Smith

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In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring R has an ideal M (R) consisting of elements a for which there is an x such that axa=a, and maximal with respect to this property. Considering only the case when R is commutative and has an identity element, it is often not easy to determine when M(R) is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of a or 1-a has a von Neumann inverse, …

Spatial Tumor-Immune Modeling, Lisette G. De Pillis, D G. Mallet, Ami E. Radunskaya

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In this paper, we carry out an examination of four mechanisms that can potentially lead to changing morphologies in a growing tumor: variations in nutrient consumption rates, cellular adhesion, excessive consumption of nutrients by tumor cells and immune cell interactions with the tumor. We present numerical simulations using a hybrid PDE-cellular automata (CA) model demonstrating the effects of each mechanism before discussing hypotheses about the contribution of each mechanism to morphology change.

Some Promising Approaches To Tumor-Immune Modeling, Lisette G. De Pillis, Ami E. Radunskaya

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Mathematical models of tumor-immune interactions provide an analytical framework in which to address specific questions regarding tumor-immune dynamics. We present a brief summary of several approaches we are currently exploring to model tumor growth, tumor-immune interactions, and treatments. Results to date have shown that simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge, are able to reproduce data from published studies. We additionally present some of our current efforts in the investigation of optimal control to aid in determining improved treatment strategies.

Spectral Analysis Of The Supreme Court, Brian L. Lawson, Michael E. Orrison, David T. Uminsky

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The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small voting bodies like the United States Supreme Court. Our goal is to show how simple ideas from linear algebra can come together to say something interesting about voting. And what could be more simple than where our story begins— with counting.

The Linking Probability Of Deep Spider-Web Networks, Nicholas Pippenger

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We consider crossbar switching networks with base b (that is, constructed from b x b crossbar switches), scale k (that is, with b^k inputs, b^k outputs, and b^k links between each consecutive pair of stages), and depth l (that is, with l stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least k stages. We assume that each vertex is independently idle with probability q, the vacancy probability. We assume that b ≥ 2 and the vacancy probability q are fixed, and that k …

Communicating Applied Mathematics: Four Examples, Daniel E. Finkel, Christopher Kuster, Matthew Lasater, Rachel Levy, Jill P. Reese, Ilse C. F. Ipsen

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Communicating Applied Mathematics is a writing- and speaking-intensive graduate course at North Carolina State University. The purpose of this article is to provide a brief description of the course objectives and the assignments. Parts A–D of of this article represent the class projects and illustrate the outcome of the course:

• The Evolution of an Optimization Test Problem: From Motivation to Implementation, by Daniel E. Finkel and Jill P. Reese

• Finding the Volume of a Powder from a Single Surface Height Measurement, by Christopher Kuster

• Finding Oscillations in Resonant Tunneling Diodes, by Matthew Lasater

• …

Residue Class Rings Of Real-Analytic And Entire Functions, Marek Golasiński, Melvin Henriksen

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Let A(ℝ) and E(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if m is a maximal ideal of A(ℝ), then A(ℝ)/m is isomorphic either to the reals or a real closed field that is an η1-set, while if m is a maximal ideal of E(ℝ), then E(ℝ)/m is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of a classical characterization of algebraically closed …

Reflections Acting Efficiently On A Building, Michael E. Orrison

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We show how Radon transforms may be used to apply efficiently the class sum of reflections in the finite general linear group GLn(Fq) to vectorsin permutation modules arising from the action of GLn(Fq) on the building oftype An−1(Fq).

Optimal Therapy Regimens For Treatment-Resistant Mutations Of Hiv, Weiqing Gu, Helen Moore

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In this paper, we use control theory to determine optimal treatment regimens for HIV patients, taking into account treatment-resistant mutations of the virus. We perform optimal control analysis on a model developed previously for the dynamics of HIV with strains of various resistance to treatment (Moore and Gu, 2005). This model incorporates three types of resistance to treatments: strains that are not responsive to protease inhibitors, strains not responsive to reverse transcriptase inhibitors, and strains not responsive to either of these treatments. We solve for the optimal treatment regimens analytically and numerically. We find parameter regimes for which optimal dosing …

Double Birthday Magic Square, Arthur T. Benjamin

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No abstract provided.

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.

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.

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

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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.

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

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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.

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.

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.

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.

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

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No abstract provided in this article.

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.

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.

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 …

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 …

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 …