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A Simple Expression for Multivariate Lagrange Interpolation
Published electronically July 2, 2008
DOI:
10.1137/08S010025
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 R
^{2}
. We also provide examples to show how the formula is used in practice.
Testing for the Benford Property
Published electronically July 2, 2008
DOI:
10.1137/08S010098
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
Published electronically July 11, 2008
DOI:
10.1137/08S010062
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
Published electronically August 22, 2008
DOI:
10.1137/08S010050
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.