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» Vol. 7, Issue 2, 2006 «



Title: Population Dynamics with Nonlinear Diffusion
Author: David Perry, New York University
Jessica Schaefer, Northern Arizona University
Brian Schilling, Mississippi State University
Matthew Williams, Clarkson University
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Abstract: We consider reaction diffusion models in population dynamics where the per capita growth rate is a logistic type or a weak Allee type. In particular, we study the effects of nonlinear diffusion (arising due to aggregative population movements) on the steady states. We obtain our results via the quadrature method.
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Title: Formulas for Computable and Noncomputable Functions
Author: Samuel Alexander, University of Arizona Author Bio    
Abstract: We explore the problem of writing explicit formulas for integer functions. We demonstrate that this can be done using elementary machinery for a wide class of functions. Constructive methods are given for obtaining formulas for computable functions and for functions in the arithmetical hierarchy. We include a short background on computability theory.
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Title: Two Questions on Continuous Mappings
Authors: Xun Ge, Suzhou University Author Bio    
Abstract: In this paper, it is shown that a mapping from a sequential space is continuous iff it is sequentially continuous, which improves a result by relaxing first-countability of domains to sequentiality. An example is also given to show that open mappings do not imply Darboux-mappings, which answers a question posed by Wang and Yang.
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Title: Subgroup Lattices That Are Chains
Authors: Amanda Jez, King's College Author Bio    
Abstract: A group G has a subgroup lattice that is a chain if for all subgroups H and K of G, we have that H is a subset of K or K is a subset of H. In this article, we first provide elementary proofs of results describing groups whose subgroup lattices are chains, and then generalize this concept to look at groups in which the subgroup lattice can be constructed by pasting together chains.
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Title: Demystifying Functions: The Historical and Pedagogical Difficulties of the Concept of the Function
Authors: Melanie Jones, Trinity University Author Bio    
Abstract: In this study, the author discusses the concept of function from a historical and pedagogical perspective. The historical roots, ranging from ancient civilizations all the way to the twentieth century, are summarized. The author then details several different function representations that have emerged over the course of the concept's history. Special attention is paid to the idea of abstraction and how students understand functions at different levels of abstraction. Several middle school, high school, and college textbooks are then analyzed and evaluated based on their portrayal of the function concept. The author describes several common misconceptions that students have about functions and finally proposes a short educational module designed to help older high school students grow to a deeper level of understanding of this complex and often misunderstood concept.
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Title: On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell
Author: Jarrod Anthony Cunningham, University of South Alabama
Nancy Ho, Mills College
Karen Lostritto, Brown University
Jon Anthony Middleton, SUNY Buffalo
Nikia Tenille Thomas, Morgan State University
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Abstract: In 1659, John Pell and Johann Rahn wrote a text which explained how to find all integer solutions to the quadratic equation u2 - d v2 = 1. In 1909, Axel Thue showed that the cubic equation u3 - d v3 = 1 has finitely many integer solutions, so it remains to examine their rational solutions. We explain how to find "large" rational solutions i.e., a sequence of rational points (un, vn) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y2 = x3 - D. The rational points on an elliptic curve form an abelian group, so a "large" rational point (u,v) maps to a rational point (x,y) of "approximate" order 3. Following an idea of Zagier, we explain how to compute such rational points using continued fractions of elliptic logarithms.

We divide our discussion into two parts. The first concerns Pell's quadratic equation. We give an informal discussion of the history of the equation, illuminate the relation with the theory of groups, and review known results on properties of integer solutions through the use of continued fractions. The second concerns the more general equation uN - d vN = 1. We explain why N = 3 is the most interesting exponent, present the relation with elliptic curves, and investigate properties of rational solutions through the use of elliptic integrals.

This project was completed at Miami University, in Oxford, OH as part of the Summer Undergraduate Mathematical Sciences Institute (SUMSRI).

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Title: Linear Feedback Shift Registers and Cyclic Codes in SAGE
Authors: Timothy Brock, United States Naval Academy Author Bio    
Abstract: This talk will discuss the history of linear feedback shift registers (LFSR) in cryptographic applications and will attempt to implement an algorithm in SAGE and Python to create a linear feedback shift register sequence (LFSR sequence) in cryptography. Also, this talk will describe an implementation of the Berlekamp-Massey Iterative Algorithm in SAGE and Python. This algorithm will be able to use the Linear Feedback Shift Register sequence generated by the first algorithm to find the sequence's connection polynomial. I will attempt to show that the connection polynomial of a given LFSR sequence is the reverse of a generator polynomial of the cyclic code of length p , where p is also the period of the LFSR sequence. This will provide a connection between cyclic error-correcting codes and LFSR sequences.
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Title: Upper Bound for Ropelength of Pretzel Knots
Author: Safiya Moran, Columbia College, South Carolina Author Bio    
Abstract: A model of the pretzel knot is described. A method for predicting the ropelength of pretzel knots is given. An upper bound for the minimum ropelength of a pretzel knot is determined, and shown to improve on existing upper bounds.
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Title: On Polya's Orchard Problem
Author: Alexandru Hening, International University Bremen, Germany
Michael Kelly, Oklahoma State University
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Abstract: In 1918 Polya formulated the following problem: ``How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?" (Polya and Szego [2]). We study a more general orchard model, namely any domain that is compact and convex, and find an expression for the minimal radius of the trees. As examples, solutions for rhombus-shaped and circular orchards are given. Finally, we give some estimates for the minimal radius of the trees if we see the orchard as being 3-dimensional.
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Title: Moments of the Distribution of Okazaki Fragments
Author: Krzysztof Bartoszek, Gdansk Univeristy of Technology
Justyna Singerska, Gdansk Univeristy of Technology
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Abstract: This paper is a continuation of [1] which provides formulae for the probability distributions of the number of Okazaki fragments at time t during the process of DNA replication. Given the expressions for the moments of the probability distribution of the number of Okazaki fragments at time t in the recursive form, we evaluated formulae for the third and fourth moments, using Mathematica, and obtained results in explicit form. Having done this, we calculated the distribution's skewness and kurtosis.
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Title: Optimizing a Volleyball Serve
Author: Dan Lithio, Hope College
Eric Webb, Case Western Reserve University
Author Bio
Abstract: An effective service in volleyball is crucial to a winning strategy. A good serve either will not be returned, resulting in the point, or it will be returned weakly, giving the serving team the advantage. One objective of an effective serve is to give the receivers as little time as possible to react. In this paper we construct a model of a served volleyball and use it to determine how to serve so that, after crossing the net, the ball hits the desired location in the minimal amount of time.
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Title: Randomly Generated Triangles whose Vertices are Vertices of Regular Polygons
Author: Anna Madras, Drury University in Springfield, Missouri
Shova KC, Hope College
Author Bio
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Abstract: We generate triangles randomly by uniformly choosing a subset of three vertices from the vertices of a regular polygon. We determine the expected area and perimeter in terms of the number of sides of the polygon. We use combinatorial methods combined with trigonometric summation formulas arising from complex analysis. We also determine the limit of these equations to compare with a classical result on triangles whose vertices are on a circle.
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Title: Stabilizing a Subcritical Bifurcation in a Mapping Model of Cardiac-Membrane Dynamics
Author: Matthew Fischer, Duke University
Colin Middleton, Duke University
Author Bio
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Abstract: In this paper we study an iterated map that describes action potential durations (acronym: APD) in a single cardiac cell. In particular, we are interested in alternans, a term which refers to phase locked period-two APDs. Under certain parameter values, alternans are theoretically possible but are unstable and therefore not seen under normal pacing conditions. We would like to stabilize alternans under these conditions using feedback. In essence, a feedback scheme uses information about previous iterates of an iterated map function to perturb future iterates in order to force stability. This paper builds on previous work on feeback control, but in the somewhat different context here, a new feedback scheme must be constructed.
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Title: Self-Quasi-Regularity in Certain Rings
Author: Allen Hoffmeyer, Georgia College and State University Author Bio
Abstract: Let R be an associative ring, not necessarily commutative and not necessarily having unity. Recall an element x in R is called quasi-regular if and only if solutions y and z exist for the equations x+ y - x*y = 0 and x + z - z*x = 0. In this case y=z, and the unique element y is called the quasi-inverse for x. It is well known that J(R), the Jacobson radical of R, is the unique largest ideal in R consisting entirely of quasi-regular elements. In this paper, we explore the implications of the case x is its own quasi-inverse. We call such elements self-quasi-regular. We determine some properties of sq(R), the set of all self-quasi-regular elements, for a general ring, and also compare this set to J(R). Then, we completely characterize the set sq(R) for all homomorphic images of Z, the integers, including the cardinality and membership of the set sq(Zn) for each choice of n.
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Title: Isoperimetric Regions in Spaces
Author: Michelle Lee, Williams College Author Bio
Abstract: We study the isoperimetric problem, the least-perimeter way to enclose given area, in various surfaces. For example, in two-dimensional Twisted Chimney space, a two-dimensional analog of one of the ten flat, orientable models for the universe, we prove that isoperimetric regions are round discs or strips. In the Gauss plane, defined as the Euclidean plane with Gaussian density, we prove that in halfspaces ya vertical rays minimize perimeter. In Rn with radial density and in certain products we provide partial results and conjectures.
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Title: Proper Colorings and p-Partite Structures of the Zero Divisor Graph
Author: Anna Duane, Carlton College Author Bio
Abstract: Let Γ(Zm) be the zero divisor graph of the ring Zm. In this paper we explore the p-partite structures of Γ(Zm), as well as determine a complete classification of the chromatic number of Γ(Zm). In particular, we explore how these concepts are related to the prime factorization of m.
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Title: Combinatorics of the Figure Equation on Directed Graphs
Author: Taylor Coon, University of Rochester Author Bio
Abstract: There are many ways of calculating a graph’s characteristic polynomial; a lesser known method is a formula called the figure equation. The figure equation provides a direct link between a graph’s structure and the coefficients of its characteristic polynomial. This method does not use determinants, but calculates the characteristic polynomial of any graph by counting cycles. We give a complete combinatorial analysis of four increasingly complex graph families, which yields closed formulae for their characteristic polynomial. In this paper, we introduce the figure equation, prove formulae for the line, cyclic, ladder, and dihedral graphs, and examine connections among these graph families including isospectrality and graph covering maps.
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