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» Vol. 15, Issue 1, 2014 «



Title: Points of Ninth Order on Cubic Curves
Author: Leah Balay-Wilson, Smith College
Taylor Brysiewicz, Northern Illinois University
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Abstract: In this paper we geometrically provide a necessary and sufficient condition for points on a cubic to be associated with an infinite family of other cubics who have nine-pointic contact at that point. We then provide a parameterization of the family of cubics with nine-pointic contact at that point, based on the osculating quadratic.
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Title: Squaring The Circle In The Hyperbolic Disk
Author: Noah Davis, Aquinas College Author Bio    
Abstract: Bolyai ended his 1832 introduction to non-Euclidean geometry with a strategy for constructing regular quadrilaterals (squares) and circles of the same area. In this article, we provide the steps for these constructions in the Poincare disk. We come to the surprising conclusion that Bolyai's strategy of building the circle and square separately is the only way to perform the constructions. That is, we cannot in general construct the square from the circle, nor vice versa.
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Title: Simple Surface Singularities, their Resolutions, and Construction of K3 Surfaces
Author: Graham Hawkes, University of North Carolina at Chapel Hill Author Bio    
Abstract: This paper describes, in detail, a process for constructing Kummer K3 surfaces, and other "generalized" Kummer K3 surfaces. In particular, we look at how some well-known geometrical objects such as the platonic solids and regular polygons can inspire the creation of singular surfaces, and we investigate the resolution of those surfaces. Furthermore, we will extend this methodology to examine the singularities of some complex two-dimensional quotient spaces and resolve these singularities to construct a Kummer K3 and other generalized Kummer K3 surfaces.
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Title: A Classification of Quadratic Rook Polynomials
Author: Alicia Velek, York College of Pennsylvania Author Bio    
Abstract: Rook theory is the study of permutations described using terminology from the game of chess. In rook theory, a generalized board B is any subset of the squares of an n x n square chessboard for some positive integer n. Rook numbers count the number of ways to place non-attacking rooks on a generalized board, and the rook polynomial is a polynomial that organizes the rook numbers for a board B. In our research, we classified all quadratic polynomials that are the rook polynomial for some generalized board B.
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Title: Total Linking Numbers of Torus Links and Klein Links
Author: Michael A. Bush, The College of Wooster
Katelyn R. French, The College of Wooster
Joseph R. H. Smith, The College of Wooster
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Author Bio    
Author Bio    
Abstract: We investigate characteristics of two classes of links in knot theory: torus links and Klein links. Formulas are developed and confirmed to determine the total linking numbers of links in these classes. We find these relations by examining the general braid representations of torus links and Klein links.
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Title: Improving Conservation Techniques Through Graph Theoretic Models
Author: Rob Hammond, Saint Michael's College Author Bio    
Abstract: While concerns for local and global ecological issues increase, there is a growing need for the conservation of functioning ecosystems; as land management evolves in scope and emphasis, land managers are in need of new tools. Graph theory, a relatively efficient mathematical modeling approach, has been used for modeling and analyzing an array of networks and as a result proving itself as a potential framework for landscape modeling with its adaptable measures. Through an in-depth review of two unique examples of graph theoretic habitat modeling, we will see how these models compare to the more complex and biologically accurate spatially explicit population models, and what they can tell us about habitats that have had no previous analysis. It will then be seen that graph theoretic modeling approaches are important to habitat conservation research in various ways.
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Title: Deflection of an elliptically loaded vortex sheet by a flat plate
Author: Jason O. Archer, University of New Mexico Author Bio    
Abstract: We investigate the behavior of vortex flows in the presence of obstacles using numerical simulations. Specifically, we simulate the evolution of an elliptically loaded vortex sheet in the presence of a stationary flat plate in its path. The plate is represented by a number of point vortices whose strength is such that they cancel the normal fluid velocity on the plate. The sheet is approximated by a number of smoothed point vortices called vortex blobs. The resulting system of ordinary differential equations is solved using the 4th order Runge-Kutta method. In our simulations, we vary the initial distance d from the vortex sheet to the plate, the angle φ of the plate relative to the sheet, and the numerical smoothing parameter δ. We study the effects these parameters have on the vortex sheet evolution, including the positions of the vortex centers and the vortex sheet midpoint. We also compare with results derived from a simpler model using only two point vortices instead of a whole sheet. Our main conclusions regard the effect of the distance d, which reduces the total distance traveled as it is increased, and the angle φ, which significantly affects the vortex trajectory after it encounters the plate.
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Title: The KMO Method for Solving Non-homogenous, mth Order Differential Equations
Author: David Krohn, University of Notre Dame
Daniel Mariño-Johnson, University of Maryland, College Park
John Paul Ouyang, University of Maryland, College Park
Author Bio    
Author Bio    
Author Bio    
Abstract: This paper shows a simple tabular procedure \added{derived from the method of undetermined coefficients} for finding a particular solution to differential equations of the form \sum_{j=0}^m a_j\frac{d^j y}{dx^j} = P(x)e^{\alpha{}x}. This procedure reduces the derivatives of the product of an arbitrary polynomial and an exponential to rows of constants representing the coefficients of the terms. The rows are each multiplied by aj and summed to produce a mth order differential equation such that its solution is the polynomial part of the particular solution of the above equation. Solving this corresponding differential equation determines the coefficients of the polynomial. The underlying algebra of this conversion and its formulaic implication are then discussed. Using the formula derived, the particular solution is found. This procedure is based on but different than the method of undetermined coefficients because while the method of undetermined coefficients requires substitution of a product of a polynomial, Q, and an exponential into the differential equation immediately, this procedure is derived from the examination of the substitution of the product of any function and an exponential. This allows for a richer understanding of the relationship between the differential equation for y and the differential equation for Q. Ultimately this method is better than the method of undetermined coefficients because it is more straightforward. In any case, both methods solve the same problem but KMO is faster.
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Title: A Hormone Therapy Model for Breast Cancer Using Linear Cancer Networks
Author: Michelle McDuffie, Western Carolina University Author Bio    
Abstract: Hormone therapy is a viable technique used to treat endocrine receptor positive cancers. Using the recently developed theory of cancer networks, we create a mathematical model describing the growth of an estrogen-receptive cancer governed by a linear cancer network. We then present hormone therapy as a drug that blocks the estrogen receptors of different cells in the network. Depending on the effectiveness of the drug, the model predicts coexistence of healthy and cancerous cells as well as a cure state. In the case of coexistence, the carrying capacities of all cancerous cells are reduced by hormone therapy, increasing effectiveness of other treatments such as surgery.
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Title: Graceful Labelings of Pendant Graphs
Author: Alessandra Graf, Northern Arizona University Author Bio    
Abstract: In 1967, Alexander Rosa introduced a new type of graph labeling called a "graceful labeling." This paper will provide some background on graceful labelings and their relation to certain types of graphs called "pendant graphs." We will also present new results concerning a specific type of graceful labeling of pendant graphs as well as further areas of research.
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Title: Non-parametric Statistics for Quantifying Differences in Discrete Spectra
Author: Alexander M. DiBenedetto, University of Evansville Author Bio    
Abstract: This paper introduces three statistics for comparing discrete spectra. Abstractly, a discrete spectrum (histogram with n bins) can be thought of as an ordered n-tuple. These three statistics are defined as comparisons of two n-tuples, representing pair-wise, ordered comparisons of bin heights. This paper defines all three statistics and formally proves the first one is a metric, while providing compelling evidence the other two are metrics. It shows that these statistics are gamma distributed, and for n ≥ 10, approximately normally distributed. It also discusses a few other properties of all three associated metric spaces.
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Title: Discretizing the SI Epidemic Model
Author: Kacie M. Sutton, Marshall University Author Bio    
Abstract: Epidemic models are used as a tool to analyze the behaviors of biological diseases and how they spread. In the SI epidemic model, where S represents the group of susceptibles and I represents the group of infectives, the numerical outputs can be nonintegers, which creates obstacles in applying these results to our biological reality. Here, we discretize the model output values by applying various combinations of integerizing (Round, Ceiling, Floor). These discretized values allow the results of the SI model to be applied to reality in terms of whole person outputs. Nine potential discretized SI models are formed with the combinations of integerizing. We eliminate several of these potential models because they do not meet the fundamental property of the SI model -- fixed population size. We compare the properties of the three models that meet the fundamental property with the properties of the original (nondiscretized) SI model. Several unexpected results appear, such as a basic reproduction number, R_0, for two of the three discretized models; the original SI model has no such R0 value.
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Title: Surface-area-minimizing n-hedral Tiles
Author: Paul Gallagher, University of Pennsylvania
Whan Ghang, Massachusetts Institute of Technology
David Hu, Georgetown University
Zane Martin, Williams College
Maggie Miller, University of Texas at Austin
Byron Perpetua, Williams College
Steven Waruhiu, University of Chicago
Author Bio    
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Abstract: We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal "orientation-preserving" tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism).
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Title: The Volume of n-balls
Author: Jake Gipple, University of Missouri Author Bio    
Abstract: In this short paper, we compute the volume of n-dimensional balls in ℝn. The computations rely on techniques from multivariable calculus and a few properties of the gamma function.
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