Digital Commons Network^™

Regular Solutions To Elliptic Equations, Alfonso Castro, Jon T. Jacobsen

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A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is …

An Undergraduate Consortium For Addressing The Leaky Pipeline To Computing Research, James C. Boerkoel Jr., Mehmet Ergezer

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Despite an increasing number of successful interventions designed to broaden participation in computing research, there is still significant attrition among historically marginalized groups in the computing research pipeline. This experience report describes a first-of-its-kind Undergraduate Consortium (UC; https://aaai-uc.github.io/about) that addresses this challenge by empowering students with a culmination of their undergraduate research in a conference setting. The UC, conducted at the AAAI Conference on Artificial Intelligence (AAAI), aims to broaden participation in the AI research community by recruiting students, particularly those from historically marginalized groups, supporting them with mentorship, advising, and networking as an accelerator toward graduate school, AI research, …

Stable Trace Ideals And Applications, Haydee Lindo, Hai Long Dao

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We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications.

Infinitely Many Radial Solutions For A P-Laplacian Problem With Indefinite Weight, Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez

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We prove the existence of infinitely many sign changing radial solutions for a p-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem …

Quantifying Controllability In Temporal Networks With Uncertainty, James C. Boerkoel Jr., Lindsay Popowski, Michael Gao, Hemeng Li, Savana Ammons, Shyan Akmal

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Controllability for Simple Temporal Networks with Uncertainty (STNUs) has thus far been limited to three levels: strong, dynamic, and weak. Because of this, there is currently no systematic way for an agent to assess just how far from being controllable an uncontrollable STNU is. We provide new insights inspired by a geometric interpretation of STNUs to introduce the degrees of strong and dynamic controllability - continuous metrics that measure how far a network is from being controllable. We utilize these metrics to approximate the probabilities that an STNU can be dispatched successfully offline and online respectively. We introduce new methods …

Dynamic Control Of Probabilistic Simple Temporal Networks, James C. Boerkoel Jr., Michael Gao, Lindsay Popowski

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The controllability of a temporal network is defined as an agent’s ability to navigate around the uncertainty in its schedule and is well-studied for certain networks of temporal constraints. However, many interesting real-world problems can be better represented as Probabilistic Simple Temporal Networks (PSTNs) in which the uncertain durations are represented using potentially-unbounded probability density functions. This can make it inherently impossible to control for all eventualities. In this paper, we propose two new dynamic controllability algorithms that attempt to maximize the likelihood of successfully executing a schedule within a PSTN. The first approach, which we call MIN-LOSS DC, finds …

Boundary Homogenization And Capture Time Distributions Of Semipermeable Membranes With Periodic Patterns Of Reactive Sites, Andrew J. Bernoff, Daniel Schmidt, Alan E. Lindsay

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We consider the capture dynamics of a particle undergoing a random walk in a half- space bounded by a plane with a periodic pattern of absorbing pores. In particular, we numerically measure and asymptotically characterize the distribution of capture times. Numerically we develop a kinetic Monte Carlo (KMC) method that exploits exact solutions to create an efficient particle- based simulation of the capture time that deals with the infinite half-space exactly and has a run time that is independent of how far from the pores one begins. Past researchers have proposed homogenizing the surface boundary conditions, replacing the reflecting (Neumann) …

Numerical Approximation Of Diffusive Capture Rates By Planar And Spherical Surfaces With Absorbing Pores, Andrew J. Bernoff, Alan E. Lindsay

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In 1977 Berg and Purcell published a landmark paper entitled Physics of Chemore- ception, which examined how a bacterium can sense a chemical attractant in the fluid surrounding it [H. C. Berg and E. M. Purcell, Biophys J, 20 (1977), pp. 193–219]. At small scales the attrac- tant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg and Purcell modeled the target as a sphere with a set of small circular targets (pores) that …

Infinitely Many Stability Switches In A Problem With Sublinear Oscillatory Boundary Conditions, Alfonso Castro, Rosa Pardo

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We consider the elliptic equation −u+u = 0 with nonlinear boundary condition ∂u ∂n = λu + g(λ, x, u), where g(λ,x,s) s → 0, as |s|→∞ and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.

Existence Of Solutions To A Semilinear Elliptic Boundary Value Problem With Augmented Morse Index Bigger Than Two, Alfonso Castro, Ivan Ventura

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Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in [A. Castro, J. Cossio and J.M. Neuberger, Sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041--1053], we prove the existence of a solution with augmented Morse index at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.

Mathematics For Human Flourishing, Francis Su

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Why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

First Passage Statistics For The Capture Of A Brownian Particle By A Structured Spherical Target With Multiple Surface Traps, Alan E. Lindsay, Andrew Bernoff, Michael J. Ward

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We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small …

Race, Space, And The Conflict Inside Us, Francis Su

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Talking about race is hard. Our nation is wrestling with some open wounds about race. These sores have been around a while, but they have been brought to light recently by technology, politics, and an increasingly diverse population. And regardless of the outcome of the U.S. presidential election, we will all need to work at healing these sores, not just in our personal lives, but in our classrooms and in our profession.

Freedom Through Inquiry, Francis Su

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I delivered this speech at the Inquiry‐Based Learning Forum & 19th Annual Legacy of R.L. Moore Conference on August 4, 2016. It is partly an homage to an influential teacher, partly an excuse to articulate what makes some styles of teaching so effective, and partly an excuse to talk about difficult issues facing our nation and our classrooms today.

Multi-Year Optimization Of Malaria Intervention: A Mathematical Model, Harry J. Dudley, Abhishek Goenka '15, Cesar J. Orellana '17, Susan E. Martonosi

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Malaria is a mosquito-borne, lethal disease that affects millions and kills hundreds of thousands of people each year, mostly children. There is an increasing need for models of malaria control. In this paper, a model is developed for allocating malaria interventions across geographic regions and time, subject to budget constraints, with the aim of minimizing the number of person-days of malaria infection.

Simulating Surfactant Spreading: Impact Of A Physically Motivated Equation Of State, Dina Sinclair '17, Rachel Levy, Karen E. Daniels

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For more than two decades, a single model for the spreading of a surfactant-driven thin liquid film has dominated the applied mathematics literature on the subject. Recently, through the use of fluorescently-tagged lipids, it has become possible to make direct, quantitative comparisons between experiments and models. These comparisons have revealed two important discrepancies between simulations and experiments: the spatial distribution of the surfactant layer, and the timescale over which spreading occurs. In this paper, we present numerical simulations that demonstrate the impact of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension. …

Existence Of Positive Solutions For A Semipositone P-Laplacian Problem, Alfonso Castro, Djairo G. De Figueredo, Emer Lopera

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We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

Guidelines For Good Mathematical Writing, Francis Su

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Communicating mathematics well is an important part of doing mathematics. Many of us know from writing papers or giving talks that communicating effectively not only serves our audience but also clarifies and structures our own thinking. There is an art and elegance to good writing that every writer should strive for. And writing, as a work of art, can bring a person great personal satisfaction.

Within the MAA, we value exposition and mathematical communication. In this column, I’m sharing the advice I give my students to help them write well. There are more extensive treatments (e.g., see Paul Halmos’s How …

To The Mathematical Beach, Francis Su

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What context am I missing that hinders my connection with my students? How often do I take the time to get to know their backgrounds? What are the primary experiences that shaped them, and do those present obstacles or opportunities for learning? And in what ways does the mathematical beach say “open to all” but still feel restricted?

These questions appear unrelated to mathematics, but if we ignore their effects, some of our students will not flourish.

Direct, Biomimetic Synthesis Of (+)-Artemone Via A Stereoselective, Organocatalytic Cyclization, Eric D. Nacsa, Brian C. Fielder, Shannon P. Wetzler, Veerasak Srisuknimit, Jonathan P. Litz, Mary J. Van Vleet, Kim Quach, David A. Vosburg

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We present a four-step synthesis of (+)-artemone from (–)- linalool, featuring iminium organocatalysis of a doubly diastereoselective conjugate addition reaction. The strategy follows a proposed biosynthetic pathway, rapidly generates stereochemical complexity, uses no protecting groups, and minimizes redox manipulations.

Counting On R-Fibonacci Numbers, Arthur Benjamin, Curtis Heberle

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We prove the r-Fibonacci identities of Howard and Cooper using a combinatorial tiling approach.

Probing The Inverted Classroom: A Study Of Teaching And Learning Outcomes In Engineering And Mathematics, Nancy K. Lape, Rachel Levy, Darryl Yong

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Flipped classrooms have started to become commonplace on university campuses. Despite the growing number of flipped courses, however, quantitative information on their effectiveness remains sparse. Active learning is a mode of instruction that focuses the responsibility of learning on learners. Multiple studies have shown that active learning leads to better student outcomes. Given that instructors in flipped classrooms are generally able to create more opportunities for students to apply or practice course material, we hypothesized that students in a flipped classroom would exhibit more learning compared to students in an unflipped class. This case study describes our research comparing …

Combinatorial Proofs Of Fibonomial Identities, Arthur Benjamin, Elizabeth Reiland

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We provide a list of simple looking identities that are still in need of combinatorial proof.

Op-Ed: Solve This Math Problem: The Gender Gap, Francis Su

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Women may not face such blatant impediments to doing math and science today. But Mirzakhani's achievement aside, we are still a long way from adequately recognizing the outstanding work of women.

Energy Driven Pattern Formation In Planar Dipole-Dipole Systems In The Presence Of Weak Noise, Jaron P. Kent-Dobias '14, Andrew Bernoff

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We study pattern formation in planar fluid systems driven by intermolecular cohesion (which manifests as a line tension) and dipole-dipole repulsion which are observed in physical systems including ferrofluids in Hele-Shaw cells and Langmuir layers. When the dipolar repulsion is sufficiently strong, domains undergo forked branching reminiscent of viscous fingering. A known difficulty with these models is that the energy associated with dipole-dipole interactions is singular at small distances. Following previous work, we demonstrate how to ameliorate this singularity and show that in the macroscopic limit, only the relative scale of the microscopic details of a system are relevant, and …

The Scientist–Reporter Collaboration: A Guide To Working With The Press, Rachel Levy, Flora Lichtman, David L. Hu

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Science, technology, engineering, and mathematics (STEM) to the public can be challenging. Often, the language that researchers use among themselves is technical and difficult for non-experts to decipher. But as you probably know, communicating your research to non-experts is becoming mandatory. In a direct sense, funding agencies often require outreach for grant fulfillment. There are indirect benefits as well: Conveying the joy of discovery and the relevance of scientific results builds scientific literacy among the public---which of course includes both students who will eventually do research of their own and people who elect the policy makers who allocate funding. How …

Aftermath: Every Math Major Should Take A Public-Speaking Course, Rachel Levy

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Rachel Levy argues that all mathematics majors should learn the art of public speaking.

Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño

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We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem

Entropy Driven Crystal Formation On Highly Strained Substrates, John R. Savage, Stefan F. Hopp, Rajesh Ganapathy, Sharon J. Gerbode, Andreas Heuer, Itai Cohen

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In heteroepitaxy, lattice mismatch between the deposited material and the underlying surface strongly affects nucleation and growth processes. The effect of mismatch is well studied in atoms with growth kinetics typically dominated by bond formation with interaction lengths on the order of one lattice spacing. In contrast, less is understood about how mismatch affects crystallization of larger particles, such as globular proteins and nanoparticles, where interparticle interaction energies are often comparable to thermal fluctuations and are short ranged, extending only a fraction of the particle size. Here, using colloidal experiments and simulations, we find particles with short-range attractive interactions form …

Mathematical Modeling Of The Regulatory T Cell Effects On Renal Cell Carcinoma Treatment, Lisette G. De Pillis, Trevor Caldwell '12, Elizabeth Sarapata '13, Heather Williams '12

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We present a mathematical model to study the effects of the regulatory T cells (Treg) on Renal Cell Carcinoma (RCC) treatment with sunitinib. The drug sunitinib inhibits the natural self-regulation of the immune system, allowing the effector components of the immune system to function for longer periods of time. This mathematical model builds upon our non-linear ODE model by de Pillis et al. (2009) [13] to incorporate sunitinib treatment, regulatory T cell dynamics, and RCC-specific parameters. The model also elucidates the roles of certain RCC-specific parameters in determining key differences between in silico patients whose immune profiles allowed them to …