Inflow Rates on Horizontal Wells
Dr. Ken Luther

Horizontal wells are often used to collect drinking water, as their ability to withdraw large amounts of water can be larger than similarly sized vertical wells. Typically, horizontal wells are drilled near surface water bodies, such as lakes or rivers, and are often placed in networks of multiple wells - also known as a radial collector well.  In modeling horizontal wells, we usually assume the wells are truly horizontal.  However, these wells can also be installed at an angle from the horizontal.  One possible modeling project is to determine, given a well of certain length and radius near a given water body, what influence the angle from horizontal has on the total amount of water than can be removed by the well.
Prequisites: Differential Equations

The logistic two-sex model without pair-formation.
Dr. Daniel Maxin
Paper: The Impact of Non-reproductive Groups in Two-sex Demographis and Epidemic Logistic Models without Pair Formation

It has been shown that the isolation from reproduction may induce a locally asymptotically stable disease free equilibrium in an endemic situation. For human populations, a gender-structured model includes single individuals (females and males) and couples. In animal populations however, various species do not form stable pairs and the mating is a consequence of direct encounter among individuals of different genders.  The main objective of this project is to develop and analyze a two-sex logistic model without pairs and to extend this model to an epidemic model and verify if and under what conditions previous results are still valid. A second objective is to analyze in more detail the mathematical and biological properties of various fertility functions both theoretical and against real data.
Prequisites: Differential Equations

Generalized primes on the Mosaic of an Integer
Dr. Rick Gillman
Paper: Mosiac Arithmetic

The mosaic of an integer is the configuration of primes obtained by repeated applications of the Fundamental Theorem of Arithmetic to a positive integer n and any composite exponents. The concept of the mosaic was introduced by Mullin, in a series of papers in the early 1960s. Various arithmetic functions dependent only on the primes in the mosaic were investigated by Girse and Gillman. During the summer of 2007, a team of undergraduate research students investigated this same structure and developed several potential concepts of divisors, allowing the set of arithmetic functions defined on mosaics to be expanded greatly. This team will continue to develop these notions of divisibility, eventually leading to more fully developed theory of mosaics.
Prerequisites: an elementary number theory course would be helpful.

Pattern Avoidance in Ternary Trees
Dr. Lara Pudwell
Paper: Pattern Avoidance in Ternary Trees 

For this project we will consider rooted ordered trees avoiding other trees.  In 2008, Rowland defined pattern avoidance in binary trees (i.e. trees where each vertex has either 0 or 2 children).  The standard combinatorial sequences appear when enumerating such trees, and there exist bijections between these trees and other common combinatorial objects.  There is also a method to determine equivalence classes of binary trees based on the number of trees that avoid them.  This team will extend these known results about pattern avoidance to ternary trees (i.e. trees in which each vertex has 0 or 3 children).
Prerequisites: linear algebra, or another proof based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.