SIAM Undergraduate Research Online (SIURO)
Publishing outstanding undergraduate research in applied and computational mathematics
Volume 1, Issue 1
Volume 1, Issue 2
Volume 2, Issue 1
Volume 2, Issue 2
Volume 3
Volume 1, Issue 1
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
A Simple Expression for Multivariate Lagrange Interpolation [PDF, 414KB]
Published electronically July 2, 2008.
Author: Kamron Saniee (New Providence High School, New
Providence, NJ)
Sponsor: Richard Glahn (New Providence High School,
New Providence, NJ)
Abstract: We derive a simple formula for constructing
the degree n multinomial function which interpolates
a set of {n+m \choose n} points in Rm+1,
when the function is unique. The formula coincides with the standard Lagrange
interpolation formula if the points are given in R2.
We also provide examples to show how the formula is used in practice.
Testing for the Benford Property [PDF, 341KB]
Published electronically July 2, 2008.
Author: Daniel P. Pike (School of Mathematical Sciences,
Rochester Institute of Technology, Rochester, New York)
Sponsor:
David L. Farnsworth (School of Mathematical
Sciences, Rochester Institute of Technology, Rochester, New York)
Abstract: Benford’s Law says that many naturally occurring
sets of observations follow a certain
logarithmic law. Relative frequencies of the first significant digits k are
log(1 + 1/k) for k = 1, 2,
..., 9, where the base of the logarithm is ten. Financial and other auditors
routinely check data
sets against this law in order to investigate for fraud. We present the principal
underlying
mechanism that produces sets of numbers with the Benford property. Examples
in which each
observation consists of a product of variables are given. Two standard statistical
tests that are
useful for testing compliance with Benford’s Law are outlined. A new
Minitab macro, which
implements both statistical tests and produces a graphical output, is presented.
The Potential of Tidal Power from the Bay of Fundy [PDF, 18MB]
Published electronically July 11, 2008.
Author: Justine M. McMillan and Megan J. Lickley (Department
of Mathematics & Statistics, Acadia University, Wolfville, NS, Canada)
Sponsor: Richard Karsten and Ronald Haynes
Abstract: Large tidal currents exist in the Minas Passage, which
connects the Minas Basin
to the Bay of Fundy off the north-western coast of Nova Scotia. The strong currents
through this deep, narrow channel make it a promising location for the generation
of electrical power using in-stream turbines. Using a finite-volume numerical
model,
the high tidal amplitudes throughout the Bay of Fundy are simulated within a
root mean square difference of 8 cm in amplitude and 3.1o in phase.
The bottom friction in the Minas Passage is then increased to simulate the presence
of turbines and an estimate of the extractable power is made. The simulations
suggest that up to
6.9 GW of power can be extracted; however, as a result, the system is pushed
closer
to resonance which causes an increase in tidal amplitude of over 15% along the
coast
of Maine and Massachusetts. The tides in the Minas Basin will also experience
a
decrease of 30% in amplitude if the maximum power is extracted. Such large changes
can have harmful environmental impacts; however, the simulations also indicate
that
up to 2.5 GW of power can be extracted with less than a 6% change in the tides
throughout the region. According to Nova Scotia Energy, 2.5 GW can power over
800,000 homes.
Expository
Moving Forward by Traveling in Circles [PDF, 735KB]
Published electronically August 22, 2008.
Author: Stuart Boersma (Central Washington University)
Abstract: The purpose of this paper is to introduce the reader to the mathematical construct known as holonomy. Holonomy is a measurement of the change in a certain angle as one travels along a curve. For this paper, we will consider two physical situations which involve "traveling in a circle" and comparing an initial and final measurement of an angle. In the first case we will see how this angular displacement can be used to prove that the Earth rotates! In the second example we explore the workings of a nineteenth century cartographic instrument. In both cases, traveling in circles yields interesting mathematical information.
Volume 1, Issue 2
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
Basins of Attraction and Perturbed Numerical Solutions using Euler's Method [PDF, 1MB]
Published electronically September 2, 2008.
Author: Hendrik Orem (Harvey Mudd College)
Sponsor: Professor Rachel Levy (Harvey Mudd College)
Abstract: Small uncertainties in a dynamical system due to imperfect measurements or variations in the environment can dramatically impact the long term behavior of a trajectory. This phenomenon is studied in a population competition model by introducing a random error term into a numerical solver and investigating the effect on the behavior of solutions. Two methods for analyzing the impact of a random term are demonstrated.
Modeling the Fluid Flow around Airfoils Using Conformal Mapping [PDF, 1MB]
Published electronically October 6, 2008.
Authors: Nitin R. Kapania, Katherine Terracciano, Shannon Taylor (Franklin W. Olin College of Engineering)
Sponsor: Burt S. Tilley (Franklin W. Olin College of Engineering)
Abstract: The modeling of fluid interactions around airfoils
is diffcult given the complicated,
often non-symmetric geometries involved. The complex variable technique of
conformal
mapping is a useful intermediate step that allows for complicated airfoil flow
problems to be solved as problems with simpler geometry. In this paper, we
use the conformal mapping technique to model the fluid flow around the NACA
0012, 2215, and 4412 airfoils by using the Joukowsky transformation to link
the flow solution for a cylinder to
that of an airfoil. The flow around a cylinder was derived with the superposition
of elementary potential flows
using an inviscid, incompressible fluid model. Lift calculations
as a function of angle of attack for each airfoil were obtained using the transformed flow
solutions and fundamental theories of aerodynamics. These calculations are
compared
against lift calculations provided by the thin airfoil method. Lift calculations
for the
NACA 0012 airfoil match well with expected results, while there is a discrepancy
at
low angles of attack for the 2215 and 4412 airfoils.
Numerical Wave Scattering
Taking Account of Energy Dissipation and Media Stiffness as Modeled by
the Telegraph Equation
[PDF,
1.9MB]
Published electronically December 9, 2008.
Authors: Sebastian Acosta and Pedro Acosta (Brigham Young University)
Sponsor: Dr. Vianey Villamizar (Brigham Young University)
Abstract: The telegraph equation is employed to model wave fields taking into account energy dissipation and media stiffness. The timeharmonic scattered waves generated by a line source incident upon cylindrical obstacles of arbitrary cross-section are studied. Solutions are found to depend strongly on the relative values of the frequency, damping, and stiffness coefficients. These coefficients are also found to have a significant effect on the far-field pattern. The analytical solution for a circular cylinder is reviewed. An approximate finite-difference solution is also obtained for the case of a two-dimensional scatterer with an arbitrary cross-section. Details are given for both soft and hard boundary conditions. The main feature of the numerical scheme is its computational efficiency based on the coupling between boundary conforming grids and a curvilinear coordinates version of the Dirichlet-to-Neumann non-reflecting boundary condition.
Volume 2, Issue 1
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
Tubuloglomerular Feedback-Mediated Dynamics
in Three Coupled Nephrons
[PDF,
1.2MB]
Published electronically March 31, 2009.
Author: Tracy L. Stepien (State University of New York)
Sponsor: E. Bruce Pitman (State University of New York)
Abstract: A model of three coupled nephrons branching from a common cortical radial artery is developed to further understand the effects of equal and unequal coupling on tubuloglomerular feedback. The integral model of Pitman et al. (2002), which describes the fluid flow up the thick ascending limb of a single, short-looped nephron of the mammalian kidney, is extended to a system of three nephrons through a model of coupling proposed by Pitman et al. (2004). Analysis of the system, verified by numerical results, indicates that stable limit-cycle oscillations emerge for sufficiently large feedback gain magnitude and time delay through a Hopf bifurcation, similar to the single nephron model, yet generally at lower values. Previous work has demonstrated that coupling induces oscillations at lower values of gain, relative to uncoupled nephrons. The current analysis extends this earlier finding by showing that asymmetric coupling among nephrons further increases the likelihood of the model nephron system being in an oscillatory state.
Volume 2, Issue 2
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
A Mathematical Model of the Immune System's Role in Obesity-Related
Chronic Inflammation
[PDF, 361KB]
Published electronically August 10, 2009.
Authors: Pablo Díaz, Michael Gillespie, Justin Krueger, José Pérez, Alex Radebaugh, Toby Shearman, Garret Vo, and Christine Wheatley (University of North Carolina at Greensboro, St. Augustine's College, Miami University, University of Puerto Rico, Mayagüez Campus, Bucknell University, Virginia Tech, Montana State University, Alma College)
Sponsors: J. Bassaganya-Riera, J. Borggaard , J. Burns , E. Cliff , A. Guri, S. Faulkner, R. Hontecillas-Magarzo, A. Jarrah, C. Koelling, R. Laubenbacher, H. Mortveit, L. Zietsman (Virginia Bioinformatics Institute at Virginia Tech and Interdisciplinary Center for Applied Mathematics at Virginia Tech)
Abstract: Obesity is quickly becoming a pandemic. The low-grade chronic inflammation associated with obesity leads to health risks such as cancer, heart disease, and type 2 diabetes mellitus. To better understand the progression of obesity-related chronic inflammation, mice were fed either a high-fat or low-fat diet over 140 days. At Days 0, 35, 70, and 140, the percentages of macrophage subsets, CD4+ T cells, and regulatory T cells infiltrating the intra-abdominal white adipose tissue (WAT) were examined. Monocyte chemoattractant protein-1 (MCP-1) mRNA expression in WAT was also quantified. Additionally, glucose-normalizing ability was examined by administering peritoneal glucose tolerance tests. A system of ordinary differential equations models this system. The model consists of 8 differential equations, has 25 parameters, and has 1 forcing function. Tools used to characterize the model include parameter estimation, sensitivity analysis, and stability analysis. Based on the data provided, the system describes the growth of adipocyte size and chronic inflammation over 105 days beginning at Day 35, which is approximately when the adipose cells become hypertrophic, or too large to function normally. The model shows that without intervention, chronic inflammation escalates and the related health problems persist.
Misclassification Rates in Hypertension Diagnosis
due to Measurement Errors
[PDF, 229KB]
Published electronically August 10, 2009.
Authors: Camila Friedman-Gerlicz and Isaiah Lilly (Claremont McKenna College, California State University at Sacramento)
Sponsor: Xianggui Qu (Oakland University)
Abstract: Using a mixture of two normal distributions, we estimate the false positive and false negative errors in the diagnosis of hypertension. Parameters in the mixture are estimated by the expectation-maximization (EM) algorithm. It is shown that both errors depend on cutoff points. Repeated measurements reduce both errors dramatically. The number of repeated measurements is recommended through a simulation study.
A Numerical Study of Generalized Multiquadric
Radial Basis Function Interpolation
[PDF, 500KB]
Published electronically October 16, 2009.
Author: Maggie E. Chenoweth (Marshall University)
Sponsor: Scott A. Sarra (Marshall University)
Abstract: This work focuses on the generalized multiquadric (GMQ) radial basis function. The GMQ is derived from the multiquadric (MQ), which is used in radial basis function (RBF) interpolation. This is a relatively new field of research, and many properties of the GMQ are still unknown. Numerical experiments will be performed involving the GMQ, and results will be analyzed to gain further understanding into this type of function.
Finding the Maximum Modulus of a Polynomial on the Polydisk Using a Generalization
of Stečkin's Lemma
[PDF, 424KB]
Published electronically October 28, 2009.
Author: Gabriel De La Chevrotière (McGill University)
Sponsor: Stephen Drury (McGill University)
Abstract: This paper is a generalization of the work of J.J. Green in Calculating the maximum modulus of a polynomial using Stečkin's Lemma. This lemma is generalized to higher dimensions and is used in an algorithm to locate the absolute global max of a polynomial on the polydisk. How to apply this algorithm to the real sphere and the complex ball is also explained.
Volume 3
Click on title to view PDF of paper or right click to download paper.
Undergraduate Student Research
Modeling, Analysis and Computation of
Fluid Structure Interaction Models
for Biological Systems
[PDF, 445KB]
Published electronically January 26, 2010.
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.
Influences On Pattern Formation During Non-Isothermal Phase Separation in Local and Nonlocal Phase-Field Models
[PDF, 613KB]
Published electronically March 2, 2010.
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.