» Vol. 11, Issue 1, 2010 «

Title: An Exploration of the Cantor Set Author: Christopher Shaver, Rockhurst University Author Bio     Abstract: This paper is a summary of some interesting properties of the Cantor ternary set and a few investigations of other general Cantor sets. The ternary set is discussed in detail, followed by an explanation of three ways of creating general Cantor sets developed by the author. The focus is on the dimension of these sets, with a detailed explanation of Hausdorff dimension included, and how they act as interesting examples of fractal sets. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: The Sigma Invariants of the Lamplighter Groups Author: Daniel Allen, University of Maine Farmington Author Bio     Abstract: We compute the Bieri-Neumann-Strebel-Renz geometric invariants, Σn, of the lamplighter groups Lm by using the Diestel-Leader graph DL(m,m) to represent the Cayley graph of Lm. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Early die out events in SIR epidemic models Author: Jonathan Ballone, Montclair State University Author Bio     Abstract: We are interested in the behavior of an SIR epidemic model with respect to low-probability events. Specifically, we want to identify the probability of the early die out of a disease. Ordinary differential equations are commonly used to model SIR systems. However, this approach fails to describe the spontaneous die out event. We develop a Markov SIR model from which the probability of early die out can be captured. Additional simulations reveal that this model agrees closely with the ODE solutions when these low-probability events are ignored. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Cut-sets and Cut-vertices in the Zero-Divisor Graph of ∏Zni Authors: Benjamin Coté, University of Idaho Caroline Ewing, Colorado College Michael Huhn, University of St. Thomas Chelsea Plaut, University of Tennessee-Knoxville Darrin Weber, Millikin Unversity Author Bio     Author Bio     Author Bio     Author Bio     Author Bio     Abstract: We examine minimal sets of vertices which, when removed from a zero-divisor graph, separate the graph into disconnected subgraphs. We classify these sets for all direct products of Γ ∏Zni Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Perfect Distance Stars mod m Authors: Xianrui Meng, Bloomsburg University of Pennsylvania Author Bio     Abstract: A perfect distance tree is a weighted tree with n vertices in which the set of distances between vertices is . We define a tree with n vertices to be a Perfect Distance Tree mod m if the distances (mod m) can be obtained. In this paper, we find that every star with m+1 vertices, where m is odd, labeled with 0, 1, 2, 3, ..., m-1 is a perfect distance tree mod m. The stars obtained from this star by removing the edge labeled 0 or by changing the weight 0 to another weight are also perfect distance trees mod m. By combining stars, we show that every star with km+j vertices can be labeled to be a perfect distance tree mod m, where m is odd, k ≥ 1 and -1 ≤ j ≤ 4. Finally, we show that certain twin-stars (trees of diameter 3) can be labeled as perfect distance trees mod m. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Generalizations of Goursat's Theorem for Groups Author: Kristina Kublik, Mount Allison University Author Bio     Abstract: Petrillo's recent article in the College Mathematics Journal explained a theorem of Goursat on the subgroups of a direct product of two groups. In this note, we extend this theorem to commutative rings, and to modules over commutative rings and fields. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Groups of a Square-Free Order Author: Iordan Ganev, Miami University and Royal Holloway, University of London Author Bio     Abstract: Hölder's formula for the number of groups of a square-free order is an early advance in the enumeration of finite groups. This paper gives a structural proof of Hölder's result that is accessible to undergraduates. We introduce a number of group theoretic concepts such as nilpotency, the Fitting subgroup, and extensions. These topics, which are usually not covered in undergraduate group theory, feature in the proof of Hölder's result and have wide applicability in group theory. Finally, we remark on further results and conjectures in the enumeration of finite groups. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits Author: Tyson Lipscomb, Marshall University Author Bio     Abstract: The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the far-reaching mathematical implications thereof. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: An Exploration of the Application of the Banzhaf Power Index to Weighted Voting Systems Author: Elise Buckley, Juniata College Author Bio     Abstract: In a weighted voting system, each player has a certain number of votes. A coalition of players can pass a measure if the total of their votes meets or exceeds a fixed quota. One example is the United States Electoral College. A player's power is measured by the Banzhaf Power Index, which counts the number of coalitions that need that player's votes to pass a measure. This research looks at how changing the quota affects the players' power in 3, 4, and 5-player weighted voting systems. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP