Volume 1, Issue 1

DOI: 10.1137/08S010050

Authors

Stuart Boersma (Central Washington University)

Project Advisors

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.

DOI: 10.1137/08S010062

Authors

Justine M. McMillan (Department of Mathematics & Statistics, Acadia University, Wolfville, NS, Canada), Megan J. Lickley (Department of Mathematics & Statistics, Acadia University, Wolfville, NS, Canada)

Project Advisors

Richard Karsten, 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° 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.

DOI: 10.1137/08S010025

Authors

Kamron Saniee (New Providence High School, New Providence, NJ)

Project Advisors

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.

DOI: 10.1137/08S010098

Authors

Daniel P. Pike (School of Mathematical Sciences, Rochester Institute of Technology, Rochester, New York)

Project Advisors

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.