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» Vol. 9, Issue 2, 2008 «



Title: Duursma Zeta Functions of Type IV Virtual Codes
Author: Sarah Catalano, U S Naval Academy Author Bio    
Abstract: This research project involves an investigation of error correcting codes. The safe and reliable transfer of information depends on coding theory. The problem of transferring dependable information is important and this research project attempts to continue incremental progress in the field. Specifically, the project will emphasize formally self dual codes and will expand upon Duursma's ideas in Extremal Weight Enumerators and Ultraspherical Polynomials.

Considerable work has been devoted to the study of self dual codes. Iwan M. Duursma has written numerous papers on the matter and the greater part of this project is centered on his work. In 1999, Iwan M. Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. The modest goal of Midn Catalano's project is to go through Duursma's papers and evaluate its relevance for formally self dual codes. Therefore, the research projects aims to extend Duursma's work in Extremal Weight Enumerators and Ultraspherical Polynomials to formally self dual codes. More specifically, the project expands Duursma's work to zeta functions of formally self dual codes of Type IV.

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Title: L(d,2,1)-Labeling of Simple Graphs
Author: Jean Clipperton, Simpson College Author Bio    
Abstract: Radio signal interference can be modeled using distance labeling where the labels assigned to each vertex depencd on the distance between vertices and the strength of the radio signal. This paper considers three levels of signal intereference within a graph, G, and the L(d,2,1)-labeling number for paths, cycles, complete graphs, and complete bipartite grpahs is determined.
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Title: Zero-divisor Graphs of Localizations and Modular Rings
Authors: Thomas Cuchta, Marshall University
Kathryn A. Lokken, University of Wisconsin, Madison
William Young, Purdue University
Author Bio    
Author Bio    
Author Bio    
Abstract: In this paper, we examine the algebraic properties of localizations of commutative rings and how localizations affect the zero-divisor graph’s structure of modular rings. We also classify the zero-divisor graphs of modular rings with respect to both the diameter and girth of their resultant zero-divisor graphs.
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Title: A Reverse Sierpinski Number Problem
Authors: Daniel Krywaruczenko, University of Tennessee at Martin
Author Bio    
Abstract: A generalized Sierpinski number base b is an integer k>1 for which gcd(k+1,b-1)=1, k is not a rational power of b, and kbn+1 is composite for all n>0. Given an integer k>0, we will seek a base b for which k is a generalized Sierpinski number base b. We will show that this is not possible if k is a Mersenne number. We will give an algorithm which will work for all other k provided that there exists a composite in the sequence {(k2m+1)/gcd(k+1,2)} for m ≥ 0.
   
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Title: Counting Containment Partitions
Authors: Nathan Langholz, St. Olaf College
Joe Usset, St. Olaf College
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Author Bio    
Abstract: The study of integer partitions has wide applications to mathematics, mathematical physics, and statistical mechanics. We consider the problem of ?nding a generalized ap- proach to counting the partitions of an integer n that contain a partition of a ?xed integer k. We use generating function techniques to count containment partitions and verify exper- imental results using a self-made in program Mathematica. We have found explicit solutions to the problem for general n with k=1, 2, 3, 4, 5, 6. We also discuss open questions and ideas for future work.
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Title: Extreme Rays of AND-Measures in Circuit Complexity
Author: Edward Lui, University of British Columbia Author Bio    
Abstract: This paper is motivated by the problem of proving lower bounds on the formula size of boolean functions, which leads to lower bounds on circuit depth. We know that formula size is bounded from below by all formal complexity measures. Thus, we study formula size by investigating AND-measures, which are weakened forms of formal complexity measures. The collection of all AND-measures is a pointed polyhedral cone; we study the extreme rays of this cone in order to better understand AND-measures. From the extreme rays, we attempt to discover useful properties of AND-measures that may help in proving new lower bounds on formula size and circuit depth. This paper focuses on describing some of the properties of AND-measures, especially those that are extreme rays. Furthermore, it describes some algorithhms for finding the extreme rays.
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Title: Order Dimension of Subgroups
Authors: Iordan Ganev, Miami University, Oxford, Ohio Author Bio    
Abstract: The number of different orders of nonidentity elements in a group is limited by the number divisors of the order of the group. This upper bound can be made more specific for proper subgroups, and can be calculated from the prime power factorization of the group's order. Some groups have subgroups with the highest possible number of different orders for nonidentity elements. This property can be characterized and general results exist for several families of groups.
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Title: Beyond Burnside's Lemma
Author: Lucas Wagner, Concordia College, Moorhead, MN
Author Bio    
Abstract: An extension of Burnside's lemma is presented along with its suggested implementation in computer code. The extension is along the lines of de Bruijn's work, which itself is a generalization of Polya's theory of counting. As an example, in addition to counting the number of distinguishable colorings of a checkerboard if rotations and reflections are allowed, our extension allows the colors themselves to be permuted. The historical context is briefly discussed. Examples are given along the way to illuminate the discussion.
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Title: A Combinatorial Proof of an Identity of Andrews
Author: Katie Evans, St. Olaf College
Trygve Wastvedt, St. Olaf College
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Author Bio    
Abstract: We give a combinatorial proof of an identity originally proved by G. E. Andrews. The identity simplifies a mock theta function first discovered by Rogers.
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Title: An Alternating Sum of Alternating Sign Matrices
Author: Nathan Williams, Carlton College
Author Bio    
Abstract: An alternating-sign matrix (ASM) is a square matrix with entries from {-1, 0,1}, row and column sums of 1, and in which the nonzero entries in each row and column alternate in sign. ASMs have many non-trivial parameters and symmetries that reveal their significant combinatorial structure. In this note, we will prove an identity that relates one parameter and one symmetry.
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Title: Formulas for Fibonacci-Like Sequences Produced by Pascal-Like Triangles
Author: Hiroshi Matsui, Kwansei Gakuin High School, Japan
Toshiyuki Yamauchi, Kwansei Gakuin High School, Japan
Author Bio    
Author Bio    
Abstract: In this paper we are going to present three formulas to express Fibonacci-like sequences with the Fibonacci sequence. We constructed Pascal-like triangles using probabilities of a game, and these Pascal-like triangles can be considered generalizations of the well known Pascal's triangle. Using these triangles, we can make Fibonacci-like sequences. We discovered an interesting formula for these Fibonacci-like sequences. We could generalize this formula, and got some interesting formulas that combine these Fibonacci-like sequences and the Fibonacci-sequence. These formulas can reveal very interesting relationships between Fibonacci-like sequences and the Fibonacci sequence, and we can expect a rich possibility of the research from these Fibonacci-like sequences.
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Title: Graphs with Four Independent Crossings Are Five Colorable
Author: Nathan Harman, University of Massachusetts Amherst Author Bio    
Abstract: Albertson conjectured that if a graph can be drawn in the plane in such a way that any two crossings are independent, then the graph can be 5-colored. He proved it for up to three independent crossings. We prove this for four crossings by showing that any such graph has an independent set of size 4 with one vertex in each crossing, and give an example to show that this method fails for five independent crossings.
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