
Title:

Population Dynamics with Nonlinear Diffusion



Author:

David Perry, New York University
Jessica Schaefer, Northern Arizona University
Brian Schilling, Mississippi State University
Mathew Williams, Clarkson State University

Author Bio
Author Bio
Author Bio
Author Bio



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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

Two Questions on Continuous Mappings



Authors:

Xun Ge, Suzhou University

Author Bio



Abstract:

Ihara introduced the zeta function of a padic matrix group in 1966 and the idea was generalized to finite graphs by Hashimoto in 1989. In her dissertation, Debra Czarneski explores the properties of graphs that are or are not determined by the zeta function. This paper defines a Kronecker product of finite graphs and explores the question: given a pair of graphs with equal zeta functions, if we take their Kronecker product with a third graph, is the equality of the zeta function preserved?



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:

Word.doc


»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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

Author Bio
Author Bio
Author Bio
Author Bio
Author Bio



Abstract:

In 1659, John Pell and Johann Rahn wrote a text which explained how to find all integer solutions to the quadratic equation u^{2}  d v^{2} = 1. In 1909, Axel Thue showed that the cubic equation u^{3}  d v^{3} = 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 (u_{n}, v_{n}) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y^{2} = x^{3}  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 u^{N}  d v^{N} = 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).



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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 BerlekampMassey 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 errorcorrecting codes and LFSR sequences.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

On Polya's Orchard Problem



Author:

Alexandru Hening, International University Bremen, Germany
Michael Kelly, Oklahoma State University

Author Bio
Author Bio



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 rhombusshaped and circular orchards are given.
Finally, we give some estimates for the minimal radius of the trees
if we see the orchard as being 3dimensional.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

Moments of the Distribution of Okazaki Fragments



Author:

Krzysztof Bartoszek, Gdansk Univeristy of Technology
Justyna Singerska, Gdansk Univeristy of Technology

Author Bio
Author Bio



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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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
Author Bio



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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

Stabilizing a Subcritical Bifurcation in a Mapping Model of CardiacMembrane Dynamics



Author:

Matthew Fischer, Duke University
Colin Middleton, Duke University

Author Bio
Author Bio



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 periodtwo 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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

SelfQuasiRegularity 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 quasiregular 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 quasiinverse for x.
It is well known that J(R), the Jacobson radical of R, is the unique largest ideal in R
consisting entirely of quasiregular elements.
In this paper, we explore the implications of the case x is
its own quasiinverse. We call such elements
selfquasiregular. We determine some properties of sq(R), the set of all selfquasiregular
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(Z_{n}) for each choice of n.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

Isoperimetric Regions in Spaces



Author:

Michelle Lee, Williams College

Author Bio



Abstract:

We study the isoperimetric problem, the leastperimeter way to enclose given area,
in various surfaces. For example, in twodimensional Twisted Chimney space,
a twodimensional 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 y ≥ a vertical rays minimize perimeter. In R^{n} with radial
density and in certain products we provide partial results and conjectures.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

Title:

Proper Colorings and pPartite Structures of the Zero Divisor Graph



Author:

Anna Duane, Carlton College

Author Bio



Abstract:

Let Γ(Z_{m}) be the zero divisor graph of the ring Z_{m}. In this paper we explore the ppartite structures of Γ(Z_{m}), as well as determine a complete classification of the chromatic number of Γ(Z_{m}). In particular, we explore how these concepts are related to the prime factorization of m.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP

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.



Article:

Downloadable PDF



Additional Downloads:



»
BACK TO TOP
