Physical Sciences and Mathematics Commons^™

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

All HMC Faculty Publications and Research

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 …

Quadratic Forms And Height Functions, Lenny Fukshansky

CMC Faculty Publications and Research

The effective study of quadratic forms originated with a paper of Cassels in 1955, in which he proved that if an integral quadratic form is isotropic, then it has non-trivial zeros of bounded height. Here height stands for a certain measure of arithmetic complexity, which we will make precise. This theorem has since been generalized and extended in a number of different ways. We will discuss some of such generalizations for quadratic spaces over a fixed number field as well as over the field of algebraic numbers. Specifically, let K be either a number field or its algebraic closure, and …

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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.

All HMC Faculty Publications and Research

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.

Investigation Of Carbon Nanotube Growth Using A Nozzle Cvd Method, James Mcfarland

Pomona Senior Theses

This work uses a modification of the chemical vapor deposition (CVD) technique to study the effects of source gas flow geometry (and the corresponding parameters) on carbon nanotube growth. Our approach is to flow the carbon-containing source gas through a nozzle, projecting the gas stream onto targeted regions of the substrate. This technique not only allows the potential for localized nanotube growth, but also offers an interesting opportunity to provide an experimental test of theoretical nanotube growth models.

The Viscous Catenary, John Koulakis

Pomona Senior Theses

Variational techniques are used to develop a theory for the time evolution of a thin strand of viscous fluid suspended from two points. The shape of the strand is approximated to be a parabola and energy conservation is used to derive a differential equation modeling the change in height over time. Data is collected with a high resolution camera and a strobe light to obtain the position and shape of the strand over multiple intervals of time. Three very different and unexpected types of behaviors are observed depending on the initial thickness and shape of the filament. The approximation fits …

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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.

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

No abstract provided.

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

• …

Hairy Strings, Vatche Sahakian

All HMC Faculty Publications and Research

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 …

Tunneling Through Weak Interactions:  Comparison Of Through-Space-, H-Bond-, And Through-Bond-Mediated Tunneling, Westin Kurlancheek '03, Robert J. Cave

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Results from ab initio electronic structure theory calculations on model systems allow for the detailed comparison of tunneling through covalently bonded contacts, hydrogen bonds, and van der Waals contacts. Considerable geometrical sensitivity as well as an exponential distance dependence of the tunneling is observed for tunneling through various nonbonded contacts. However, the fundamental result from the present study is that at most a modest difference is observed between tunneling mediated by H-bonds and tunneling mediated by van der Waals contacts at typical distances for each type of interaction. These results are considered in relation to the pathways model of Beratan …

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

All HMC Faculty Publications and Research

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).

Siegel’S Lemma With Additional Conditions, Lenny Fukshansky

CMC Faculty Publications and Research

Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …

Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky

CMC Faculty Publications and Research

Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.

Data Mining Techniques To Study Therapy Success With Autistic Children, Gondy A. Leroy, Annika Irmscher, Marjorie H. Charlop

CGU Faculty Publications and Research

Autism spectrum disorder has become one of the most prevalent developmental disorders, characterized by a wide variety of symptoms. Many children need extensive therapy for years to improve their behavior and facilitate integration in society. However, few systematic evaluations are done on a large scale that can provide insights into how, where, and how therapy has an impact. We describe how data mining techniques can be used to provide insights into behavioral therapy as well as its effect on participants. To this end, we are developing a digital library of coded video segments that contains data on appropriate and inappropriate …