As Flat As Possible

How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students build geometric and analytic understanding of solutions of certain elliptic and parabolic partial differential equations

A Liouville-Gelfand Equation for k-Hessian Operators

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand problem from combustible gas dynamics and prescribed curvature problems

The Liouville-Bratu-Gelfand Problem for Radial Operators

We determine precise existence and multiplicity results for radial solutions of the Liouville–Bratu–Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators

Turing Patterns on Growing Spheres: The Exponential Case

We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations

Monotone Solutions of a Nonautonomous Differential Equation for a Sedimenting Sphere

We study a class of integrodifferential equations and related ordinary differential equations for the initial value problem of a rigid sphere falling through an infinite fluid medium. We prove that for creeping Newtonian flow, the motion of the sphere is monotone in its approach to the steady state solution given by the Stokes drag. We discuss this property in terms of a general nonautonomous second order differential equation, focusing on a decaying nonautonomous term motivated by the sedimenting sphere proble

Traveling Waves and Shocks in a Viscoelastic Generalization of Burgers\u27 Equation

We consider traveling wave phenomena for a viscoelastic generalization of Burgers\u27 equation. For asymptotically constant velocity profiles we find three classes of solutions corresponding to smooth traveling waves, piecewise smooth waves, and piecewise constant (shock) solutions. Each solution type is possible for a given pair of asymptotic limits, and we characterize the dynamics in terms of the relaxation time and viscosity

Approximations of Continuous Newton\u27s Method: An Extension of Cayley\u27s Problem

Continuous Newton\u27s Method refers to a certain dynamical system whose associated flow generically tends to the roots of a given polynomial. An Euler approximation of this system, with step size h=1, yields the discrete Newton\u27s method algorithm for finding roots. In this note we contrast Euler approximations with several different approximations of the continuous ODE system and, using computer experiments, consider their impact on the associated fractal basin boundaries of the root

R₀ Analysis of a Spatiotemporal Model for a Stream Population

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R0 for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays the groundwork for connecting R0 to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data

The Art of Teaching Mathematics

On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity

Rheumatoid arthritis, item response theory, Blom transformation, and mixed models

We studied rheumatoid arthritis (RA) in the North American Rheumatoid Arthritis Consortium (NARAC) data (1499 subjects; 757 families). Identical methods were applied for studying RA in the Genetic Analysis Workshop 15 (GAW15) simulated data (with a prior knowledge of the simulation answers). Fifty replications of GAW15 simulated data had 3497 ± 20 subjects in 1500 nuclear families. Two new statistical methods were applied to transform the original phenotypes on these data, the item response theory (IRT) to create a latent variable from nine classifying predictors and a Blom transformation of the anti-CCP (anti-cyclic citrinullated protein) variable. We performed linear mixed-effects (LME) models to study the additive associations of 404 Illumina-genotyped single-nucleotide polymorphisms (SNPs) on the NARAC data, and of 17,820 SNPs of the GAW15 simulated data. In the GAW15 simulated data, the association with anti-CCP Blom transformation showed a 100% sensitivity for SNP1 located in the major histocompatibility complex gene. In contrast, the association of SNP1 with the IRT latent variable showed only 24% sensitivity. From the simulated data, we conclude that the Blom transformation of the anti-CCP variable produced more reliable results than the latent variable from the qualitative combination of a group of RA risk factors. In the NARAC data, the significant RA-SNPs associations found with both phenotype-transformation methods provided a trend that may point toward dynein and energy control genes. Finer genotyping in the NARAC data would grant more exact evidence for the contributions of chromosome 6 to RA