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# Mathematics

## Periodic Solutions to a Piecewise Linear Second Order Differential Equation

In this work we construct nontrivial solutions to the boundary value problem (BVP) -u''(x) = μu+(x) - νu-(x) for 0 < x < 2π, u(0) = u(2π), u'(0) = u'(2π), where u denotes twice-differentiable real valued function, u+ its positive part and u- its negative part. The real parameters μ and ν are determined so that the BVP has nontrivial solutions. Every nontrivial solution to the BVP has a corresponding ordered pair of the form (μ, ν); the set of all such ordered pairs is called the Fucik Spectrum. We derive the Fucik Spectrum and use the results to compute the inner products of nontrivial solutions to the BVP with eigenfunctions for the corresponding linear problem, -u''(x) = λu (x) for 0 < x < 2π, u(0) = u(2π), u'(0) = u'(2π).
Funding provided by: Pomona College Mathematics Dept.

## Solving Two-Point Boundary Value Problems via the Hilbert Space Approach

Our study is of the two-point boundary value problem: -u''(x) = g(x,u) for 0 < x < π, (1) u(0) = 0 = u(π), where g(x,s) is some function of x ε (0, π) and s ε R. We want to determine conditions on g that will guarantee the existence of a least one solution, u: [0, π] → R, to problem (1) in some appropriate function space Starting with the space of continuous functions, we develop the idea of Hilbert spaces and examine the concept of weak solutions. The Riesz Representation Theorem (RRT) for Hilbert spaces is used to show that, in the case in which g(x,s)=f(x) is independent of s, problem (1) has a weak solution for a general class of functions f. We then explore how ideas behind the proof of RRT lead to an approach that can be used in more general problems.
Funding provided by: Pomona College SURP

## Modeling the Degree Distribution of a Fractal Transportation Network

Rodriguez, Juan Diego ('09);  Hubler, Alfred*;  Ketisch, Pia*
*University of Illinois at Urbana-Champaign, Urbana IL

There are many examples in nature of evolving tree-like structures. One non-biological example of a system which produces ramified pattern formations is a system of metal ball bearings in castor oil under an electric field. We give a graph-theoretical description of this electromechanical system, and compare it to various models for tree formation: the propagation front model (PFM), minimum spanning tree graph (MST), and diffusion limited aggregation (DLA). We find that the degree distributions of DLA and MST are independent of the number of nodes and are close to the experiment. In addition, the Strahler order distribution of PFM correlates well with the experiment and the external path length of DLA correlates well with the experiment.
Funding provided by: National Science Foundation Grant No. NSF PHY 01-40179 and NSF DMS 03-25939 ITR

## The Search for an Origin: The Ancestral Periodic Orbit and Sacker Bifurcation

Zhu, Tammy ('10);  Elderkin, Richard

This is a project about the search for an origin. The project began with a doughnut-shaped surface called a torus. This torus, defined by three differential equations and discovered by Robby Gerrity (‘09), describes relationships between predators and prey. My goal was to understand where this torus came from. My hypothesis was that it came from an ancestral periodic orbit through a Sacker bifurcation, which occurs when a small change in parameters generates a big change in the dynamic system, causing the ancestral periodic orbit to spawn a torus by a swap in stability. By modifying the parameters of the differential equations, I found a narrowing nest of tori converging to an ostensible ancestral periodic orbit. My findings support but fall short of proving my initial hypothesis. Further confirmation that the torus indeed results from a Sacker bifurcation would proffer this torus as a real-life example of this uncommon bifurcation.
Funding provided by: The Norris Foundation

## The Two-Locus, Two-Allele Model With Selection

Zimmerman, Scott ('09);  Radunskaya, Ami;  Cavalcanti, Andre

The two-locus, two-allele model of population genetics describes intergenerational changes in haplotype frequencies under selection and recombination. The model discrete dynamical system consists of three polynomial equations in three variables and ten parameters. We have investigated the number of possible equilibria for the model, conjectured to be 15 (Copeland, 2007). We are attempting to find a Groebner basis with Lexicographic monomial ordering for the system from which we would determine the maximum number of equilibria. It has been shown that Buchberger's Algorithm will find such a basis in finite time (Copeland, 2007), but memory constraints have prevented its calculation. We have found a Groebner basis with Degree Reverse Lexicographic ordering and are attempting to convert this to a Lexicographic-ordered basis by various methods. We are also developing a set of documents and programs that can be used as tools in teaching and research using the model in biologically reasonable situations.
Funding provided by: Howard Hughes Medical Institute