Published electronically January 26, 2010
DOI: 10.1137/09S010496

Author: S. Minerva Venuti (George Mason University)
Sponsor: Padmanabhan Seshaiyer (George Mason University)

Abstract: A mathematical modeling for the interaction of blood flow with the arterial wall surrounded by cerebral spinal fluid is developed. The blood pressure acting on the inner arterial wall is modeled using a Fourier Series, the arterial wall is modeled using a spring-mass system, and the surrounding cerebral spinal fluid is modeled via a simplified Navier-Stokes equation. The resulting coupled system of partial differential equations for this fluid structure interaction with appropriate boundary conditions are solved first analytically using Laplace Transform and then numerically using an implicit finite difference scheme. The solutions are also investigated using computational tools. An application of the model studied to intracranial saccular aneurysms is also presented.

Published electronically March 2, 2010
DOI: 10.1137/09S010538

Author: Thomas Stephens (George Mason University)
Sponsor: Thomas Wanner (George Mason University)

Abstract: The classical phase-field model represents a coupling of an Allen-Cahn type nonlinear equation with a standard diffusion equation and has been proposed to describe uniform phase separation in a pure substance. This research considers an extension of that model which incorporates a more accurate approximation to the diffuse interface between states of matter through the use of a nonlocal operator. Results of simulations under this model are compared with those of the classical model in order to understand the effects of the nonlocal contribution. Attention has been given to the behavior of the underlying temperature field during early phase separation. We use the tools of computational homology to quantitatively compare patterns in the phase field under both models. Simulations show that the complicated patterns in the phase field persist longer during the solidification process in the nonlocal extension of the classical model.

Published electronically April 28, 2010
DOI: 10.1137/10S010648

Author: Jeremiah Jones (Arizona State University)
Sponsor: Carl Gardner (Arizona State University)

Abstract: Ionic channels play an important role in regulating the cell's membrane potential and internal charge. This paper will focus on a continuum model of the KcsA potassium channel. We will derive the Poisson-Nernst-Planck (PNP) equations in general and then provide computational solutions for a 1D KcsA channel using experimentally determined parameters. The solution to the PNP equations consists of the time-dependent charge densities of each ion, coupled with the electric potential. The results will be used to solve for the time-dependent current in the channel in response to different time-dependent voltage signals, which can be compared with experimental data.

Published electronically May 13, 2010
DOI: 10.1137/09S010435

Authors: Mary E. Broadbent (Amherst College), Martin Brown (University of California, Berkeley), and Kevin Penner (University of Pittsburgh)
Sponsors: Ilse Ipsen (North Carolina State University), Rizwana Rehman (North Carolina State University)

Abstract: Subset selection is a method for selecting a subset of columns from a real matrix, so that the subset represents the entire matrix well and is far from being rank deficient. We begin by extending a deterministic subset selection algorithm to matrices that have more columns than rows. Then we investigate a two-stage subset selection algorithm that utilizes a randomized stage to pick a smaller number of candidate columns, which are forwarded for to the deterministic stage for subset selection. We perform extensive numerical experiments to compare the accuracy of this algorithm with the best known deterministic algorithm. We also introduce an iterative algorithm that systematically determines the number of candidate columns picked in the randomized stage, and we provide a recommendation for a specific value. Motivated by our experimental results, we propose a new two stage deterministic algorithm for subset selection. In our numerical experiments, this new algorithm appears to be as accurate as the best deterministic algorithm, but it is faster, and it is also easier to implement than the randomized algorithm.

Published electronically May 26, 2010
DOI: 10.1137/09S010459

Authors: Mike Egesdal (University of California, Los Angeles), Chris Fathauer (Harvey Mudd College), Kym Louie (Harvey Mudd College), and Jeremy Neuman (University of California, Los Angeles)
Sponsors: George Mohler (University of California, Los Angeles) and Erik Lewis (University of California, Los Angeles)

Abstract: Gang violence has plagued the Los Angeles policing district of Hollenbeck for over half a century. With sophisticated models, police may better understand and predict the region's frequent gang crimes. The purpose of this paper is to model Hollenbeck's gang rivalries. A self-exciting point process called a Hawkes process is used to model rivalries over time. While this is shown to fit the data well, an agent based model is presented which is able to accurately simulate gang rivalry crimes not only temporally but also spatially. Finally, we compare random graphs generated by the agent model to existing models developed to incorporate geography into random graphs.

Published electronically July 6, 2010
DOI: 10.1137/10S010594

Author: Jeremy Semko (Duke University)
Sponsor: Thomas Witelski (Duke University)

Abstract: Thin layers of slow-moving, viscous fluids coating hydrophobic surfaces are shaped by the competing forces of disjoining pressure and surface tension. These forces form the fluid layer into an array of discrete droplets connected by an ultra thin layer. However, the droplet array is unstable, and the droplets will interact with one another. To determine the structure and properties of steady droplets, we use the Reynolds' PDE in one dimension and phase-plane methods. We can then analyze the unstable droplet system by utilizing paired ODEs. Numerical solutions show how the droplets interact to produce movement and mass exchange, giving rise to coarsening events which reduce the number of droplets in the system. These events occur when a droplet collapses into the ultra thin layer or when two droplets collide, and thus, merge. Using numerical simulations and statistical analysis of their results, we aim to gain a better understanding of the dynamics of this system including the factors that influence coarsening events such as parameters and initial conditions.

Published electronically July 15, 2010
DOI: 10.1137/10S010582

Author: Mark Spanier (North Dakota State University)
Sponsor: Marian Bocea (North Dakota State University)

Abstract: The Euler-Lagrange equations associated to the problem of minimizing a power-law functional acting on symmetrized gradients are identified. A formal derivation of the limiting system of partial differential equations stemming from these equations as p tends to infinity is provided. Our computations are reminiscent of the derivation of the infinity Laplace equation starting from the p-Dirchlet integral.

Published electronically July 27, 2010
DOI: 10.1137/09S010563

Author: Dennis S. Fillebrown (Bucknell University)
Sponsor: Linda B. Smolka (Bucknell University)

Abstract: We numerically solve a fourth order nonlinear partial differential equation derived by Craster and Matar [Craster and Matar, J. Fluid Mech. 553, 85 (2006)] that models a viscous fluid flowing down the outside of a vertical fiber in order to investigate the initial formation of perturbations along the fluid free surface. We compare numerical results of their model to existing experimental data [Smolka et al., Phys. Rev. E 77, 036301 (2008)]. In the simulations, perturbations consistently coalesced with neighboring perturbations during their initial formation, whereas in the experiments no coalescence was observed during this time period. We find that the amplitude growth follows two distinct exponential functions (referred to as phases I and II) in the initial formation of perturbations; in the experiments the data follows only one exponential function. The switch in growth rates from phase I to II is influenced by the coalescence of neighboring perturbations. The wavelength varies throughout the time that a perturbation forms so that the flow is unstable to a range of wavenumber. We compare the growth rates of several perturbations during phase II from the simulations to experimental data. Of the six data sets, the range of growth rate and wavenumber were in excellent agreement in two of the data sets and in fair to good agreement with two other data sets. For the last two data sets, there was no overlap in growth rate and little overlap in wavenumber. We find that linear stability results developed from Craster and Matar's Stokes flow model are in excellent qualitative agreement with the growth of the perturbations measured during phase II in simulations for all six data sets. Finally, we find that data from simulations consistently overpredict the final perturbation amplitude measured in experiments.

Published electronically July 29, 2010
DOI: 10.1137/09S010526

Author: Taylor Hines (Arizona State University)
Sponsor: Eric Kostelich (Arizona State University)

Abstract: When developing an accurate model of the development of glioblastomas multiforme, it is important to account not only for the invasion and diffusion of tumor cells into healthy tissue but also the resulting mass effect and brain tissue deformation. This motivates the model presented here, which implements the finite element method to solve a boundary value problem defined through classical continuum mechanics. Intended to improve existing models of tumor invasion, this model predicts the mass-effect of an invading tumor in heterogeneous brain tissue. Several parameters, taken from existing literature, dictate the behavior of differing types of brain matter. The model operates on a two-dimensional (2D) domain and outputs the displacement of brain tissue as a result of the pressure surrounding and within the tumor (peri-tumor pressure).

Published electronically August 4, 2010
DOI: 10.1137/10S010697
M3 Challenge Introduction

Authors: Andrew Das Sarma, Jacob Hurwitz, David Tolnay, and Scott Yu (Montgomery Blair High School, Silver Spring, MD)
Sponsor: David Stein (Montgomery Blair High School, Silver Spring, MD)

Executive Summary: With so much political and economic interest behind accurate results for the United States Census, the Census Bureau has implemented several strategies for dealing with the particularly pesky problem of undercounting, or the exclusion of certain individuals from the Census: These include sampling the population after the Census to gauge how many people were excluded, guessing the values for missing data, and examining public records to estimate the breakdown of the population. Of these, we found that only the last two are sufficiently helpful to merit use, whereas the first strategy of post-Census sampling can lead to error greater than what it was intended to remedy.

Of course, even with perfectly reliable Census results, proper political representation cannot be attained without a system that distributes seats in the House of Representatives in a manner that addresses the particularities of the population. Evaluating six methods (Hill, Dean, Webster, Adams, Jefferson, and Hamilton-Vinton) that Congress has historically considered for dividing seats in the House, we found that the Hamilton-Vinton method surpasses others in the arena of fair apportionment.

After Congress, the next bearers of responsibility are the fifty states of the Union, which are constitutionally charged with drawing district lines that demarcate regions for their representatives. In regard to this process, we suggest that states commit to a system that impartially divides the state according to population density. Such a system, we hold, will serve the common good of the state by achieving the democratic goal that our representative democracy should reflect the sentiments of the American people.

Published electronically September 17, 2010
DOI: 10.1137/10S010636

Author: Michael R. Atkins (George Mason University)
Sponsors: Daniel Anderson (George Mason University), Maria Emelianenko (George Mason University), Yuri Mishin (George Mason University)

Abstract: Microstructural evolution is a phenomenon of paramount importance in various areas of industry; its understanding is critical for designing materials with superior properties. The nonlinear and metastable nature of this mesoscale phenomenon has given rise to various models that attempt to describe it. One type of model that is often used is a phase-field model. Here we present a method to determine a constant related to the solid-liquid interface thickness found in the model via atomistic simulation.

Published electronically September 29, 2010
DOI: 10.1137/10S010612

Authors: Anil Damle (University of Colorado at Boulder) and Geoffrey Colin Peterson (University of Colorado at Boulder)
Sponsors: James Currry (University of Colorado at Boulder) and Anne Dougherty (University of Colorado at Boulder)

Abstract: We examine the eigenstructure of generalized isosceles triangles and explore the possibilities of analytic solutions to the general eigenvalue problem in other triangles. Starting with work based off of Brian McCartin's paper on equilateral triangles, we first explore the existence of analytic solutions within the space of all isosceles triangles. We find that this method only leads to consistent solutions in the equilateral case. Next, we develop criteria for the existence of complete solutions in other triangles. We find that complete solutions are guaranteed in the equilateral, right isosceles and 30-60-90 triangles. We then use a method developed by Milan Prager to formulate solutions in the right isosceles triangle through folding transformations of solutions in the square.