SIAM Undergraduate Research Online (SIURO)

Publishing outstanding undergraduate research in applied and computational mathematics

Volume 1, Issue 1

Volume 1, Issue 2

 

 

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]

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 R^m+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]

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]

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 3.1^o 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]

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]

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]

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.