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» Vol. 17, Issue 1, 2016 «



Title: Unique minimal forcing sets and forced representation of integers by quadratic forms
Author: Tahseen Rabbani, University of Virginia Author Bio    
Abstract: For subsets of natural numbers S and W, we say S forces W if any integer-matrix positive definite form which represents every element of S over the integers also represents every element of W over the integers. In the context of a superset S*, S is referred to as a unique minimal forcing set of W if for any subset S0 of S*, we have that S0 forces W if and only if S is a subset of S0. In 2000, Manjul Bhargava used his own novel method of “escalators" to prove the unique minimal forcing set of the natural numbers is T={1, 2, 3, 5, 6, 7, 10, 14, 15}, which was a refinement of the celebrated Conway-Schneeberger Fifteen Theorem. We use Bhargava's theory of escalators to develop an algorithm which determines whether a positive integer, interpreted as a singleton in the natural numbers, has a unique minimal forcing set within T and to establish several infinite families of positive integers without unique minimal forcing sets in T.
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Title: Subrings of C Generated by Angles
Author: Jackson Bahr, Carnegie Mellon University
Arielle Roth, Elizabethtown College
Author Bio    
Author Bio    
Abstract: Consider the following inductively defined set. Given a collection U of unit magnitude complex numbers, and a set initially containing just 0 and 1, through each point in the set, draw lines whose angles with the real axis are in U. Add every intersection of such lines to the set. Upon taking the closure, we obtain R(U). We investigate for which U the set R(U) is a ring. Our main result holds when 1 is in U and the cardinality of U is at least 4. If P is the set of real numbers in R(U) generated in the second step of the construction, then R(U) equals the module over Z[P] generated by the set of points made in the first step of the construction. This lets us show that whenever the pairwise products of points made in the first step remain inside R(U), it is closed under multiplication, and is thus a ring.
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Title: Constructing Probability Distributions Having a Unit Index of Dispersion
Author: Omar Talib, Modern College of Business and Science, Muscat, Oman Author Bio    
Abstract: The index of dispersion of a probability distribution is defined to be the variance-to-mean ratio of the distribution. In this paper we formulate initial value problems involving second order nonlinear differential equations, the solutions of which are moment generating functions or cumulant generating functions of random variables having specified indices of dispersion. By solving these initial value problems we will derive relations between moment and cumulant generating functions of probability distributions and the indices of dispersion of these distributions. These relations are useful in constructing probability distributions having a given index of dispersion. We use these relations to construct several probability distributions having a unit index of dispersion. In particular, we demonstrate that the Poisson distribution arises very naturally as a solution to a differential equation.
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Title: A Permutation Model with Finite Partitions of the Set of Atoms as Supports
Author: Benjamin Baker Bruce, St. Olaf College Author Bio    
Abstract: The method of permutation models was introduced by Fraenkel in 1922 to prove the independence of the axiom of choice in set theory with atoms. We present a variant of the basic Fraenkel model in which supports are finite partitions of the set of atoms, rather than finite sets of atoms. Among our results are that, in this model, every well-ordered family of well-orderable sets has a choice function and that the union of such a family is well-orderable.
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Title: Reversing A Doodle
Author: Bryan A. Curtis, Metropolitan State University of Denver Author Bio    
Abstract: The radius r neighborhood of a set X, denoted Nr(X), is the collection of points within a distance r of X. We discuss some of the properties preserved by the radius r neighborhood in Rn. In particular, we find a collection of sets which have a unique pre-image when mapped under Nr. This problem has interesting ties to convex geometry.
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Title: Prime Labeling of Small Trees with Gaussian Integers
Author: Hunter Lehmann, Seattle University
Andrew Park, Seattle University
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Author Bio    
Abstract: A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural numbers such that any two adjacent vertices have relatively prime labels. Here we extend the idea of prime labeling to the Gaussian integers, which are the complex numbers whose real and imaginary parts are both integers. We begin by defining an order on the Gaussian integers that lie in the first quadrant. Using this ordering, we show that all trees of order at most 72 admit a prime labeling with the Gaussian integers.
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Title: Randomness Extractors -- An Exposition
Author: Wei Dai, College of Creative Studies, University of California, Santa Barbara Author Bio    
Abstract: Randomness is crucial to computer science, both in theory and applications. In complexity theory, randomness augments computers to offer more powerful models. In cryptography, randomness is essential for seed generation, where the computational model used is generally probabilistic. However, ideal randomness, which is usually assumed to be available in computer science theory and applications, might not be available to real systems. Randomness extractors are objects that turn “weak” randomness into almost “ideal” randomness (pseudorandomness). In this paper, we will build the framework to work with such objects and present explicit constructions. We will discuss a well-known construction of seeded extractors via universal hashing and present a simple argument to extend such results to two-source extractors.
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Title: Cantor’s Proof of the Nondenumerability of Perfect Sets
Author: Laila Awadalla, University of Missouri — Kansas City Author Bio    
Abstract: This paper provides an explication of mathematician Georg Cantor’s 1883 proof of the nondenumerability of perfect sets of real numbers. A set of real numbers is denumerable if it has the same (infinite) cardinality as the set of natural numbers {1, 2, 3, …}, and it is perfect if it consists only of so-called limit points (none of its points are isolated from the rest of the set). Directly from this proof, Cantor deduced that every infinite closed set of real numbers has only two choices for cardinality: the cardinality of the set of natural numbers, or the cardinality of the set of real numbers. This result strengthened his belief in his famous continuum hypothesis that every infinite subset of real numbers had one of those two cardinalities and no other. This paper also traces Cantor’s realization that understanding perfect sets was key to understanding the structure of the continuum (the set of real numbers) back through some of his results from the 1874–1883 period: his 1874 proof that the set of real numbers is nondenumerable, which confirmed Cantor’s intuitive belief in the richness of the continuum compared to discrete subsets (such as the set of natural numbers) and proved that there was more than one “size” of infinite cardinal; his 1878 proof that continuous domains of different dimensions (such as a one-dimensional line and a two-dimensional surface) surprisingly have the same cardinality; and his 1883 definition of a continuum as a set that is connected (all of one piece) and perfect.
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Title: An Examination of Richard Dedekind’s “Continuity and Irrational Numbers”
Author: Chase Crosby, University of Missouri — Kansas City Author Bio    
Abstract: This paper explicates each of the seven sections of mathematician Richard Dedekind’s 1858 essay “Continuity and Irrational Numbers”, which he eventually published in 1872. In this essay, he provides a simple, completely arithmetic proof of the continuity of the set of real numbers, a property on which the validity of many mathematical theorems, especially those in calculus, depend. The intent of this paper is to familiarize the reader with the details of Dedekind’s argument, which is exceptionally easy to follow and self-contained. Although the real numbers were often imagined as points lying on an infinite line, as a calculus instructor in Zürich, Switzerland, Dedekind became deeply troubled by the need to reference geometry when teaching his students concepts such as functions and limits. This inspired him to develop a rigorous arithmetic foundation for the set of real numbers, in which, through the use of what are now called “Dedekind cuts,” he cleverly defines both rational and irrational numbers, and demonstrates how they fit together to form the continuum of real numbers. Alternative viewpoints and criticisms of his work exist, and one is briefly discussed at the conclusion of the paper, though it is noted that Dedekind’s essay accomplishes the goal he set for himself in its preface.
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Title: Graphs with characteristic-dependent well-covered dimension
Author: Joseph Burdick, Humboldt State University Author Bio    
Abstract: The dimension of the well-covered space of certain graphs depends upon characteristic of the field of scalars of the vector space. We investigate graphs that have this characteristic-dependent well-covered dimension and show how more of these graphs can be constructed.
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Title: Kidney Paired Donation: Optimal and Equitable Matchings in Bipartite Graphs
Author: Robert John Montgomery, St. Lawrence University Author Bio    
Abstract: If a donor is not a good match for a kidney transplant recipient, the donor/recipient pair can be combined with other pairs to find a sequence of pairings that is more effective. The group of donor/recipient pairs, with information on the potential effectiveness of each match, forms a weighted bipartite graph. The Hungarian Algorithm allows us to find an optimal matching for such a graph, but the optimal outcome for the group might not be the most equitable for the individual patients involved. We examine several modifications to the Hungarian method which consider a balance between the optimal score for the group and the most uniformly equitable score for the individuals.
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Title: SI Dynamics of Disease Spread
Author: Catherine Northrup, Carthage College
Elisabeth Rutter, Carthage College
Kerry Stapf, Carthage College
Author Bio    
Author Bio    
Author Bio    
Abstract: Imagine you are walking down a crowded hallway. You aren't in contact with everyone all at once. You talk to or simply pass by different people at different times as you walk down the hall. These connections would best be represented using a temporal network. In this work, we examine temporal networks to determine the behavior of disease spread across these networks and how it differs from the behavior of static networks. We use differential equations for mean field approximations to theoretically model how infection spreads throughout a temporal network. We extend our model to incorporate network structure by deriving a degree-based mean field theory. We then validate our theories with simulations in Mathematica. We also look into including multiple rounds of infections to see how it affects the spreading behavior. From our results we are able to determine how the temporal aspect affects the rate of spread of the disease and the overall size of the infected population.
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Title: On the existence of normal subgroups of prime index
Author: Brooklynn Szymoniak, Saginaw Valley State University Author Bio    
Abstract: In this article, we characterize finite groups having normal subgroups of a given prime index. Precisely, we prove that if p is a prime divisor of a finite group G, then G has no normal subgroup of index p if and only if G=G’Gp, where Gp is the subgroup of G generated by all elements of the form gp for any g in G and G’ is the derived subgroup of G. We also extend a characterization of finite groups with no subgroups of index 2 by J.B. Nganou to infinite groups. We display an example to show that for a prime index greater than 2 the characterization does not hold.
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Title: Cops and Robbers on Oriented Graphs
Author: Elise Darlington, Morningside College
Claire Gibbons, Morningside College
Kelly Guy, Morningside College
Jon Hauswald, Morningside College
Author Bio    
Author Bio    
Author Bio    
Author Bio    
Abstract: Cops and Robbers is a turn-based game traditionally played on graphs. Similar to the classic children's game, one cop and one robber move along edges in a graph with the cop trying to catch the robber, and the robber trying to evade capture. In this article, we extend this game to oriented graphs. Although a complete characterization of 1-cop-win graphs is known, there is not yet a corresponding characterization for oriented graphs. Necessary conditions are described for an oriented graph to be 1-cop-win, and several results are provided toward finding sufficient conditions.
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