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» Vol. 7, Issue 1, 2006 «



Title: Thermal Imaging of Circular Inclusions within a Two-Dimensinoal Region
Author: Shannon Talbott, University of Akron
Hilary Spring,Mount Holyoke College
Author Bio    
Author Bio    
Abstract: The ability to study the interior of an object without destroying it is an important industrial tool. One method of recent interest is thermal imaging . The idea is to use heat energy as a kind of ''x-ray'', to form an image of the interior of an object without causing damage to the object. More precisely, one applies a controlled source of heat energy to the exterior boundary of the object, then monitors the temperature of the object's boundary over time. This measured boundary temperature is influenced by the internal structure of the object. For example, an internal crack or void may block the flow of heat energy, and the heat is forced to flow around the defect. The goal is to determine the internal structure---e.g., locate cracks---from this exterior temperature data.
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Title: Differential Geometry of Manifolds with Density
Author: Ivan Corwin, Harvard University
Neil Hoffman, University of Texas
Stephanie Hurder, Harvard University
Vojislav Sesum, Williams College
Ya Xu, Williams College
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Abstract: We describe extensions of several key concepts of differential geometry to manifolds with density, including curvature, the Gauss-Bonnet theorem and formula, geodesics, and constant curvature surfaces.
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Title: Zeta Functions on Kronecker Products of Graphs
Authors: Rachel Reeds, Wellesley College Author Bio    
Abstract: Ihara introduced the zeta function of a p-adic matrix group in 1966 and the idea was generalized to finite graphs by Hashimoto in 1989. In her dissertation, Debra Czarneski explores the properties of graphs that are or are not determined by the zeta function. This paper defines a Kronecker product of finite graphs and explores the question: given a pair of graphs with equal zeta functions, if we take their Kronecker product with a third graph, is the equality of the zeta function preserved?
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Title: Defining a Zeta Function for Cell Products of Graphs
Authors: Zuhair Khandker, Princeton University Author Bio    
Abstract: The Riemann Zeta Function has been successfully and promisingly generalized in various ways so that the concept of zeta functions has become important in many different areas of research. In particular, work done by Y. Ihara in the 1960s led to the definition of an Ihara Zeta Function for finite graphs. The Ihara Zeta Function has the nice property of having three equivalent expressions: an Euler product form over ``primes" of the graph, an expression in terms of vertex operators on the graph, and an expression in terms of arc operators on the graph. In this paper we present two possibilities for generalizing the Ihara Zeta Function to cell products of graphs. We start with a background discussion of the Ihara Zeta Function and cell products. Then we present our generalized zeta functions and prove some properties about them. Our hope is that the ideas presented in this paper will stimulate further ideas about using the nice properties of the Ihara Zeta Function as a model for defining zeta functions more generally on higher dimensional geometric objects.
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Title: The Secret Santa Problem
Authors: Matthew White, Tandem Friends School Author Bio    
Abstract: In this paper, we will investigate the Secret Santa problem, a combinatorics problem involving derangements with at least one two-cycle. We will first consider the probability that a permutation in a set of derangements has at least one two-cycle, and then generalize the result for derangements with at least one cycle of size q or smaller and derangements with at least one q-cycle. We will first solve for the probabilities by using recurrence relations, and will then provide them in non-recursive form. Next, we will reexamine the eight-year-old solution to the Secret Santa problem, demonstrating an error in the original authors’ approach. We will solve for the error term, and generalize the results. Finally, we will provide secondary results, including an enumeration of the properties of a class of recurrence relations to which derangements and n! belong.
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Title: On the Second Twist Number
Author: Gabriel Murillo, UC Riverside Author Bio    
Abstract: It has been shown that the twist number of a reduced alternating knot can be determined by summing certain coefficients in the Jones Polynomial.  In the discovery of this twist number, it became evident that there exist higher order twist numbers which are the sums of other coefficients.  Some relations between the second twist number and the first are explored while noting special characteristics of the second twist number.
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Title: Finding Minimal Length Representatives in Thompson's group F
Authors: Micah Miller, Bowdoin College Author Bio    
Abstract: Cleary and Taback devised a method called the nested traversal method to construct minimal length representatives for positive and negative elements in Thompson's group.  We show how to use the nested traversal method to construct minimal length representatives for a larger class of elements of this group.
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Title: Detecting Forged Handwriting with Wavelets and Statistics
Author: Beverly Lytle, Allegheny College
Caroline Yang, Duke University
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Abstract: In this paper, we present a method of analyzing handwriting samples (distinguishing between authentic samples and forgeries) and address the question of whether or not the handwriting of different individuals contains a mathematical signature. The method discussed is a modified version of a similar approach towards art presented in ``A Digital Technique for Art Authentication'' by Lyu et al. We give a short history of art and handwriting forgery and characteristics examined in order to determine authenticity. We also examine the rationale behind the Lyu et al.'s method and the changes that we have made in our application to handwriting analysis.
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Title: On A Sequence of Cantor Fractals
Author: Mohsen Soltanifar, K.N. Toosi University of Technology Author Bio    
Abstract: In this paper we discuss some topological and geometrical properties of terms in a sequence of Cantor fractals and the limit of the sequence in order to obtain an exact relation between positive real numbers and Hausdorff dimensions of fractals of Euclidean spaces.
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Title: Level Sets of Arbitrary Dimension Polynomials with Positive Coefficients and Real Exponents
Author: Spencer Greenberg, Columbia University Author Bio    
Abstract: In this paper we consider the set of positive points at which a polynomial with positive coefficients, arbitrary dimension n, and real powers is equal to a fixed positive constant c. We find that when this set is non-empty and is bestowed with the relative Euclidean topology coming from R^n, it is homeomorphic to a codimension one piecewise linear set that depends only on the polynomial's powers. This piecewise linear set can, in a certain sense, be interpreted as a bijectively mapped version of the original set as the constant c approaches infinity. In addition to this result, we provide a condition on the polynomial powers for testing if the solution space is homeomorphic to the n-1 dimensional sphere S^(n-1), and derive piecewise linear inner and outer bounds for our solution set. These provide bounds in the sense that each point in our solution set that lies on any particular ray originating at the origin is trapped between a unique inner and outer bound point also on that ray. While this paper provides insight into only a specific type of polynomial slice, an appropriate generalization of these observations might one day lead to improved techniques for analyzing slices of high dimension polynomials in general, objects which appear frequently throughout mathematics. 
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Title: Human Face Recognition Technology Using the Karhunen-Loeve Expansion Technique
Author: Anthony Giordano, Regis University
Michael Uhrig, Regis University
Author Bio
Author Bio
Abstract: We will explore the area of face recognition using the partial singular value decomposition of a matrix and test some of its successes and limitations. We constructed a database consisting of 130 pictures of 65 individuals, and then used the Karhunen-Loéve (KL) Expansion method to relate pictures from outside the database to those in the database. While this method was generally very successful, we were able to test and define several of its limitations.
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Title: Comparison of Numerical Techniques for Euclidean Curvature
Author: Derek Dalle, University of Minnesota Author Bio
Abstract: This paper begins with a comparison of second-order numerical approx­imations to Euclidean curvature, and verifies that some of the approx­imations are invariant to Euclidean transformations.  Also, higher-order Euclidean invariant numerical techniques are developed and tested.  Consideration is given to strengths and weaknesses of each algorithm.
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Title: Modeling DNA Using Knot Theory:  An Introduction
Author: Jenny Tompkins, University of Texas at Tyler Author Bio
Abstract: An application of mathematics to the field of molecular biology is introduced. More specifically, how knot theory can be used to model DNA recombination is explained. DNA is a long, thin molecule found tightly packed inside the nucleus of a cell. As a result of the tight packing in such a confined space the DNA becomes tangled and knotted inside the cell. Thus, the molecules must be topologically manipulated in order for vital life processes to take place. Nature’s answer to the tangling problem is enzymes. Enzymes play an important roll in affecting the topology of DNA. One way an enzyme may act on DNA is by a process called site-specific recombination. This paper will discuss the tangle model for site-specific recombination.
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Title: Generalization of Vieta's Formula
Author: Al-Jalila Al-Abri, Sultan Qaboos University in Oman Author Bio
Abstract: In his beautiful monograph, Mark Kac gave a proof of Vieta's formula using the Rademacher functions and their independence property. In the first chapter of his book, he gave as an exercise to state and prove a generalization of Vieta's formula. In this paper we give a proof based on Kac's proof for Vieta's formula, and to the best of our knowledge, this generalization has only been achieved by Kent E. Morrison using the Fourier transform and delta distributions.
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