Mathematics Commons^™

Teaching Time Savers: Some Advice On Giving Advice, Michael E. Orrison Jr.

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There are always a lot of questions that need to be answered at the beginning of a course. When are office hours? What are the grading policies? How many exams will there be? Will late homework be accepted? We have all seen the answers to these sorts of questions form the bulk of a standard course syllabus, and most of us feel an obligation (and rightly so) to provide such information.

Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz

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No abstract provided in this article.

Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin

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By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8F_n − 4 when n is odd and is 8F_n − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice.

Teaching Time Savers: Style Points, Michael E. Orrison Jr.

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When I began as an assistant professor, I had a pretty good sense of how much time it would take for me to prepare for each class. After a few conversations with my new colleagues, I even had a good sense of how much time I should devote to tasks like office hours and committee work. Somewhere in the middle of grading my first exam, though, it became painfully clear that I had underestimated the amount of time I would need to grade exams!

Hole Dynamics In Polymer Langmuir Films, James C. Alexander, Andrew J. Bernoff, Elizabeth K. Mann, J. Adin Mann Jr., Lu Zou

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This article develops a model for the closing of a gaseous hole in a liquid domain within a two-dimensional fluid layer coupled to a Stokesian subfluid substrate, and compares this model to experiments following hole dynamics in a polymer Langmuir monolayer. Closure of such a hole in a fluid layer is driven by the line tension at the hole boundary and the difference in surface pressure within the hole and far outside it. The observed rate of hole closing is close to that predicted by our model using estimates of the line tension obtained by other means, assuming that the …

Teaching Time Savers: A Recommendation For Recommendations, Michael E. Orrison Jr.

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I admit it — I enjoy writing recommendation letters for my students. I like
learning about their hopes and dreams, where they have been and where they want to go. A recommendation letter is an opportunity to remind myself how much my students can grow while they are in college, and how much I have grown as an instructor, advisor, and mentor.

The Local Gromov–Witten Invariants Of Configurations Of Rational Curves, Dagan Karp, Chiu-Chu Melissa Liu, Marcos Mariño

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We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of ℙ¹’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of ℙ³ at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the …

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.

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.

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 …

Double Birthday Magic Square, Arthur T. Benjamin

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No abstract provided.

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.

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.

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.

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

• …

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 …

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, …

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).