Digital Commons Network^™

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.

Strings, Chains, And Ropes, Darryl H. Yong

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Following Antman [Amer. Math. Mon., 87 (1980), pp. 359–370], we advocate a more physically realistic and systematic derivation of the wave equation suitable for a typical undergraduate course in partial differential equations. To demonstrate the utility of this derivation, three applications that follow naturally are described: strings, hanging chains, and jump ropes.

Emergence Of The Fuzzy Horizon Through Gravitational Collapse, Anand Murugan '07, Vatche Sahakian

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For a large enough Schwarzschild black hole, the horizon is a region of space where gravitational forces are weak; yet it is also a region leading to numerous puzzles connected to stringy physics. In this work, we analyze the process of gravitational collapse and black hole formation in the context of light-cone M-theory. We find that, as a shell of matter contracts and is about to reveal a black hole horizon, it undergoes a thermodynamic phase transition. This involves the binding of D0 branes into D2’s, and the new phase leads to large membranes of the size of the horizon. …

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.

How Effective Is Security Screening Of Airline Passengers?, Susan E. Martonosi, Arnold Barnett

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With a simple mathematical model, we explored the antiterrorist effectiveness of airport passenger prescreening systems. Supporters of these systems often emphasize the need to identify the most suspicious passengers, but they ignore the point that such identification does little good unless dangerous items can actually be detected. Critics often focus on terrorists' ability to probe the system and thereby thwart it, but ignore the possibility that the very act of probing can deter attempts at sabotage that would have succeeded. Using the model to make some preliminary assessments about security policy, we find that an improved baseline level of screening …

Certifications Offered By Cost Estimating Organizations, Donald S. Remer, Karen M. Ahle, Kevin J. Alley, John Silny, Karen Hsin

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There has been an increasing trend in professional and engineering circles to place greater emphasis on official credentials. These credentials commonly come in the form of certificates – documented recognition by a professional body that an engineer or other professional has the qualifications and technical knowledge to be a practitioner in that field. These certificates are somewhat analogous to merit badges in scouting – the certification is evidence that the holder has a certain minimum level of competence in the subject area. This review will help you decide what certifications are applicable to you and the requirements to obtain a …

In Situ Muscle Power Differs Without Varying In Vitro Mechanical Properties In Two Insect Leg Muscles Innervated By The Same Motor Neuron, Anna N. Ahn, Kenneth Meijer, Robert J. Full

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The mechanical behavior of muscle during locomotion is often predicted by its anatomy, kinematics, activation pattern and contractile properties. The neuromuscular design of the cockroach leg provides a model system to examine these assumptions, because a single motor neuron innervates two extensor muscles operating at a single joint. Comparisons of the in situ measurements under in vivo running conditions of muscle 178 to a previously examined muscle (179) demonstrate that the same inputs (e.g. neural signal and kinematics) can result in different mechanical outputs. The same neural signal and kinematics, as determined during running, can result in different mechanical functions, …

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!

Mean Field Effects For Counterpropagating Traveling Wave Solutions Of Reaction-Diffusion Systems, Andrew J. Bernoff, R. Kuske, B. J. Matkowsky, V. Volpert

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In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a …

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 …

Role Of Beat Noise In Limiting The Sensitivity Of Optical Coherence Tomography, Richard C. Haskell, David Liao, Adam E. Pivonka, Tera L. Bell, Brendan R. Haberle, Barbara M. Hoeling, Daniel C. Petersen

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The sensitivity and dynamic range of optical coherence tomography (OCT) are calculated for instruments utilizing two common interferometer configurations and detection schemes. Previous researchers recognized that the performance of dual-balanced OCT instruments is severely limited by beat noise, which is generated by incoherent light backscattered from the sample. However, beat noise has been ignored in previous calculations of Michelson OCT performance. Our measurements of instrument noise confirm the presence of beat noise even in a simple Michelson interferometer configuration with a single photodetector. Including this noise, we calculate the dynamic range as a function of OCT light source power, and …

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 …

Erratum: The Structure Of Alkali Halide Dimers: A Critical Test Of Ionic Models And New Ab Initio Results, T. Törring, S. Biermann, J. Hoeft, Richard J. Mawhorter, Robert J. Cave, C. Szemenyei

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It has come to our attention that some of the ab initio results presented are incorrect due to errors in the Cs and C1 basis sets, and a small error in the binding energy of Rb2F2. The corrected results are presented below for the species that were affected, modifying the results in Table III of the original paper. Only those values which are different from the results of the original Table III are included. Note that some of these results are used for comparison with the ionic models in later tables. In addition, some HF data quoted in Tables V …

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.

Transverse Priority Phase Sensitive Optical Coherence Tomography, Jeff Fingler, Jon Williams, Zahid Yaqoob, Changhuei Yang, Richard C. Haskell, Scott E. Fraser

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A variation on the standard time domain optical coherence tomography (TDOCT) system is presented. Using an inexpensive piezoelectric stack to modulate the reference mirror position, the amplitude and phase of the sample reflection is determined without scanning. With the primary scan in the transverse direction, en face and B-scan OCT images can be readily produced with phase information. This project plans to use the dynamic phase information to add an extra level of contrast to the images, based on the motion of the scatterers.

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 …

An Optimal Brain Can Be Composed Of Conflicting Agents, Adi Livnat, Nicholas Pippenger

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Many behaviors have been attributed to internal conflict within the animal and human mind. However, internal conflict has not been reconciled with evolutionary principles, in that it appears maladaptive relative to a seamless decision-making process. We study this problem through a mathematical analysis of decision-making structures. We find that, under natural physiological limitations, an optimal decision-making system can involve “selfish” agents that are in conflict with one another, even though the system is designed for a single purpose. It follows that conflict can emerge within a collective even when natural selection acts on the level of the collective only.

The Motion Of A Thin Liquid Film Driven By Surfactant And Gravity, Michael Shearer, Rachel Levy

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We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock …

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

• …

Office-Based Optical Coherence Tomographic Imaging Of Human Vocal Cords, Elizabeth J.; Haskell, Richard C. Orwin, Shuguang Guo, River Hutchison, Ryan P. Jackson, Anu Kohli, Tristan Sharp, Zhongping Chen, Brian J. F. Wong

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Optical coherence tomography (OCT) is an evolving noninvasive imaging modality and has been used to image the larynx during surgical endoscopy. The design of an OCT sampling device capable of capturing images of the human larynx during a typical office based laryngoscopy examination is discussed. Both patient’s and physician's movements were addressed. In vivo OCT imaging of the human larynx is demonstrated. Though the long focal length limits the lateral resolution of the image, the basement membrane can still be readily distinguished. Office-based OCT has the potential to guide surgical biopsies, direct therapy, and monitor disease. This is a promising …

Hairy Strings, Vatche Sahakian

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Zero modes of the world-sheet spinors of a closed string can source higher order moments of the bulk supergravity fields. In this work, we analyze various configurations of closed strings focusing on the imprints of the quantized spinor vacuum expectation values onto the tails of bulk fields. We identify supersymmetric arrangements for which all multipole charges vanish; while for others, we find that one is left with Neveu-Schwarz–Neveu-Schwarz, and Ramond-Ramond dipole and quadrupole moments. Our analysis is exhaustive with respect to all the bosonic fields of the bulk and to all higher order moments. We comment on the relevance of …

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 …