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» Vol. 8, Issue 2, 2007 «



Title: The Circuit Partition Polynomial with Applications and Relation to the Tutte and Interlace Polynomials
Author: Andrea Austin, Saint Michael's College Author Bio    
Abstract: This paper examines several polynomials related to the field of graph theory including the circuit partition polynomial, Tutte polynomial, and the interlace polynomial. We begin by explaining terminology and concepts that will be needed to understand the major results of the paper. Next, we focus on the circuit partition polynomial and its equivalent, the Martin polynomial. We examine the results of these polynomials and their application to the reconstruction of DNA sequences. Then we introduce the Tutte polynomial and its relation to the circuit partition polynomial. Finally, we discuss the interlace polynomial and its relationship to the Tutte and circuit partition polynomials.
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Title: Intrinsically S1 3-Linked Graphs and Other Aspects of S1 Embeddings
   
Author: Andrew Brouwer, SUNY College at Potsdam
Rachel Davis, Le Moyne College
Abel Larkin, SUNY College at Potsdam
Daniel Studenmund, Haverford College
Cherith Tucker,Southern Nazarene University
Author Bio    
Author Bio    
Author Bio    
Author Bio    
Author Bio    
Abstract: A graph can be embedded in various spaces. This paper examines S1 embeddings of graphs. Just as links can be defined in spatial embeddings of graphs, links can be defined in S1 embeddings. Because linking properties are preserved under vertex expansion, there exists a finite complete set of minor minimal graphs such that every S1 embedding contains a non-split 3-link. This paper presents a list of minor minimal intrinsically S1 3-linked graphs, along with methods used to find and verify the list, in hopes of obtaining the complete minor minimal set. Other aspects of S1 embeddings are also examined. 1
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Title: Orientability of Phylogenetic Network Graphs
Authors: Ethan Cecchetti, Lexington High School in Massachusetts Author Bio    
Abstract: Traditionally, genetic history of species has been modeled using phylogenetic trees. Recently, scientists have begun using phyolgenetic networks to model more complex occurrences, such as hybridization, which cannot be displayed by trees. Phylogenetic networks are represented by network graphs which are trivalent, directed graphs without directed circuits. In this paper we discuss the mathematics of network graphs. Given an unoriented trivalent graph, we determine a necessary and sufficient condition for orienting the graph as a network graph.
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Title: Course Function Value Theorems
Authors: Jared Duke, Brigham Young University
Chul-Woo Lee, Brigham Young University
Author Bio    
Author Bio    
Abstract: Coarse functions are functions whose graphs appear to be continuous at a distance, but in fact may not be continuous. In this paper we explore examples and properties of coarse functions. We then generalize several basic theorems of continuous functions which apply to coarse functions.
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Title: Zero-Divisor Graphs of Zn and Polynomial Quotient Rings over Zn                                                
Authors: Daniel Endean, St. Olaf College
Erin Manlove, St. Olaf College
Kristin Henry, St. Olaf College
Author Bio    
Author Bio    
Author Bio    
Abstract: Critical to the understanding of a graph are its chromatic number and whether or not it is perfect. Here we prove when G (Zn), the zero-divisor graph of Zn, is perfect and show an alternative method to Duane for determining the chromatic number in those cases. We go on to determine the chromatic number for G(Zp[x]/< xn>) where p is prime and show that an isomorphism exists between this graph and G(Zpn).
   
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Title: The Burnside Group B(3,2) as a Two-Relator Quotient of C3*C3                                                
Author: Matthew Farrelly, Siena College Author Bio    
Abstract: We prove that the free Burnside Group B(3,2) has order 27 and is isomorphic to < a,b | a3, b3 (ab)3, (b-1a)3 > . The technique of our proof is also used to show that < a,b | a3, b3, a2 (ba)nb2 > is a semidirect product Cn2+n+1 x C3 .
   
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Title: Improving the Mathematical Model of the Tacoma Narrows Bridge
Authors: Brian Fillenwarth, University of Evansville Author Bio    
Abstract: In this paper, we investigate the mathematical model for the torsional rotation of the Tacoma Narrows Bridge derived by P.J. McKenna. Through modifying this model and programming various cases of these modifications using Matlab, we explore how the Tacoma Narrows Bridge would respond to different loading conditions that may have occurred the day the bridge collapsed. From this we are able to gain a better understanding of how the bridge actually behaved prior to its collapse, and can see possible reasons for the ultimate collapse
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Title: The Discrete Logarithm Problem and Ternary Functional Graphs
Author: Christina Frederick, University of Maryland
Max Brugger , Oregon State University
Author Bio    
Author Bio    
Abstract: Encryption is essential to the security of transactions and communications, but the algorithms on which they rely might not be as secure as we all assume. In this paper, we investigate the randomness of the discrete exponentiation function used frequently in encryption. We show how we used exponential generating functions to gain theoretical data for mapping statistics in ternary functional graphs. Then, we compare mapping statistics of discrete exponentiation functional graphs, for a range of primes, with mapping statistics of the respective ternary functional graphs.
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Title: Reducibility of Second Order Differential Operators with Rational Coefficients
Author: Joe Geisbauer, University of Arkansas- Fort Smith Author Bio    
Abstract: This paper will provide several results for the reducibility of second order differential operators. More specifically, we will discuss second order operators that factor into two first order operators with either; one constant coefficient and one rational function coefficient, two polynomial coefficients, or two rational function coefficients with regular singularities. Furthermore, we will only consider operators with one or two regular singularities when considering rational function coefficients. The discussion will provide tests for reducibility of the operators as well.
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Title: Eigenvalues of Non-Backtracking Walks in a Cycle with Random Loops
Author: Ana Pop, University of British Columbia Author Bio    
Abstract: In this paper we take a very special model of a random non-regular graph and study its non-backtracking spectrum. We study graphs consisting of a cycle with some random loops added; the graphs are not regular and their non-backtracking spectrum does not seem to be confined to some one-dimensional set in the complex plane. The non-backtracking spectrum is required in some applications, and has no straightforward connection to the usual adjacency matrix spectrum for general graphs, unlike the situation for regular graphs. Experimentally, the random graphs' spectrum appears similar in shape to its deterministic counterpart, but differs because the eigenvalues are visibly clustered, especially with a mysterious gap around Re(l)=1.
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Title: Least-Perimeter Partitions of the Sphere
Author: Conor Quinn, Williams College Author Bio    
Abstract: We consider generalizations of the honeycomb problem to the sphere S2 and seek the perimeter-minimizing partition into n regions of equal area. We provide a new proof of Masters' result that three great semicircles meeting at the poles at 120 degrees minimize perimeter among partitions into three equal areas. We also treat the case of four equal areas, and we prove under various hypotheses that the tetrahedral arrangement of four equilateral triangles meeting at 120 degrees minimizes perimeter among partitions into four equal areas.
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Title: An Exploration of the Approximation of Derivative Functions via Finite Differences
Author: Brian Jain, Baylor University
Andrew Sheng, Westwood HS, Austin
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Author Bio    
Abstract: Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent approximations to various derivative functions, including those used in modeling important physical processes on uniform grids. However, our research reveals that difference approximations on uniform grids cannot be applied blindly on nonuniform grids, nor can difference formulas to form consistent approximations to second derivatives. At best, they may lose accuracy; at worst they are inconsistent. Detailed consistency and error analysis, together with simulated examples, will be given.
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Title: On Fourier Series Using Functions Other than Sine and Cosine
Author: Henry Scher, University of Maryland Author Bio    
Abstract: An important aspect of Fourier series is that sin(x), cos(x) and all of their dilations sin(jx) and cos(jx) for all j create an orthogonal basis of the Hilbert space of periodic square-integrable functions with period 2 p . In this paper, we define the notion of dilation basis and prove that only a pair of orthogonal sinusoidal functions can generate an orthogonal dilation basis of this space.
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