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» Vol. 6, Issue 1, 2005 «



Title: The k-Compartment Problem
Author: Scott McEuen, Brigham Young University Author Bio    
Abstract: This article defines a new minimization problem, the k-Compartment Problem, and presents its solution. The k-Compartment Problem is to determine the minimum sum of k+1 line segments that intersect two parallel lines which form k compartments.
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Title: Domains That Do Not Have a Nice Complementary Set of Rays
Author: W. Lauritz Petersen, Brigham Young University Author Bio    
Abstract: We will introduce conditions which are sufficient to guarantee that a closed connected domain in R^2 that has a smooth boundary does not have a nice complementary set of rays.
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Title: Positive Solutions to a Diffusive Logistic Equation with Constant Yield Harvesting
Authors: Tammy Ladner, Millsaps College
Anna Little, Samford University
Ken Marks, Millersville University
Amber Russell, Mississippi State University
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Abstract: We consider a reaction diffusion equation which models the constant yield harvesting of a spatially heterogeneous population which satisfies a logistic growth. In particular, we study the existence of positive solutions subject to a class of nonlinear boundary conditions. We also provide results for the case of Neumann and Robin boundary conditions. We obtain our results via a quadrature method and Mathematica computations.
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Title: A Nonstandard Fourier Inequality
Authors: Jonas Azzam, University of Nebraska-Lincoln
Bobbi Buchholz, University of Nebraska-Lincoln
Ian Grooms, College of William and Mary
Gretchen Hagge, University of Nebraska-Lincoln
Kyle Hays, University of Missouri - Columbia
Greg Norgard, University of Colorado
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Abstract: We consider a class of functions given by a class of generalized Fourier series which arise in the study of sampled-data control. These functions are continuous on the real line, but not differentiable at x=0. We prove that for all sufficiently small x > 0, these functions are larger than a constant times the square root of x.
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Title: Estimation of the Length of a Rod from Thermal Data
Authors: Nathaniel Givens, University of Richmond
Robin Haskins, University of Richmond
Daniel Katz, University of Richmond
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Abstract: Given an initial-boundary value problem as model of a heated rod of unknown length, we consider the inverse problem of determining this length from temperature and heat flux measurements at one end of the rod. This models the situation where one end of the rod is inaccessible. We derive and test two different algorithms to numerically estimate the length of the rod, and demonstrate their performance through numerical examples.
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Title: Fractional Calculus and the Taylor-Riemann Series
Author: Joakim Munkhammar, Uppsala University Author Bio    
Abstract: In this paper we give some background theory on the concept of fractional calculus, in particular the Riemann-Liouville operators. We then investigate the Taylor-Riemann series using Osler's theorem and obtain certain double infinite series expansions of some elementary functions. In the process of this we give a proof of the convergence of an alternative form of Heaviside's series. A Semi-Taylor series is introduced as the special case of the Taylor-Riemann series when \alpha=1/2, and some of its relations to special functions are obtained via certain generating functions arising in complex fractional calculus.
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Title: Non-Destructive Testing of Thermal Resistances for a Single Inclusion in a 2-Dimensional Domain
Authors: Nicholas Christian, University of North Carolina at Asheville
Mathew Johnson, Ball State University
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Abstract: In this paper we examine the inverse problem of determining the amount of corrosion/disbonding which has occurred on the boundary of a single circular (or nearly circular) inclusion D in a two-dimensional domain Omega , using Cauchy data for the steady-state heat equation. We develop an algorithm for reconstructing a function which quantifies the level of corrosion/disbonding at each point on the boundary of D. We also address the issue of ill-posedness and develop a simple regularization scheme, then provide several numerical examples. We also show a simple procedure for recovering the center of D assuming that Omega and D have the same thermal conductivity.
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