## The 2009 Indiana Undergraduate Research Conference

### Parallel session

speakers, titles, and abstracts

Adam Abegg,

Indiana University and St. Louis University

ajabegg@gmail.com

*The Mutational Rate of Emergence of Complex Biological Adaptations *

Within populations, three factors affect evolution: random genetic drift, selection, and mutation. On their own, the three -- and within certain limits the combination thereof -- are well understood. The aim of this project is to expand the limits onto populations with varying mutation rates, while keeping the other two factors well-defined and at play within the system.

Theresa Anderson,

Rose-Hulman Institute of Technology and Univ. of Wisconsin, Madison

tcanderson2@wisc.edu

Brooke Phillips,

Rose-Hulman Institute of Technology and Murray State University

brooke.phillips@murraystate.edu

*Anti-cloaking: The Mathematics of Disguise *

Given a subdomain $D$ of a larger domain, we are able to mathematically describe several transformations to disguise $D$ in the context of impedance imaging, including rotations, enlargements, and translations. A major consequence that follows is that in order to disguise $D$ as another subdomain, both must be enclosed in a suitable ``metamaterial'', which redirects current. Though studied using electricity, these techniques have a wide range of applications such as optics and shock waves.

Myles Baker,

Wabash College and Baylor University

myles_baker@baylor.edu

Sarah Farell,

Wabash College and Bard College

sf457@bard.edu

*An Adaptively Weighted Least-Squares Finite Element Method for Convection Dominated Diffusion PDEs *

Convection-dominated partial differential equations give rise to error as a byproduct of approximation that is difficult to resolve using computational solution methods. We balance this error by developing a new adaptively weighted least-squares finite element method that works in conjunction with adaptive mesh refinement, allowing us to improve solutions in terms of both accuracy and com- putational cost. We extend this method to solving the Navier-Stokes equations.

John Brown,

Indiana University

brownjoh@umail.iu.edu

*Random Sampling of Constrained Phylogenies by a Monte Carlo Markov Chain Method *

To conduct statistical comparison tests in biology, one often needs to generate a set of random phylogenies (evolutionary histories). To improve the tests, we want to be able to draw random trees which fit certain constraints, thus incorporating the available biological information. An algorithm to do this exists, but is not computationally feasible for reasonably large problems. A Monte Carlo Markov Chain method to draw the trees would be an efficient alternative to the existing algorithm.

Mark Burek,

Wabash College and Valparaiso University

Mark.Burek@valpo.edu

*Characterization of Zero-divisor graphs by diameter and girth *

The paper expands on previous work regarding the limitations that diameter and girth impose on zero-divisor graphs by considering all possible realizable diameter-girth combinations of several types of rings and the subrings that contain them. The zero-divisor graph of a ring can be either realizable or not realizable based solely on knowledge of its diameter and girth. Just knowing the girth and diameter of a ring can eliminate all other possibilities of subrings within that ring.

Ben Cote,

Wabash College and University of Idaho

cote@vandals.uidaho.edu

Caroline Ewing,

Wabash College and Smith College

cewing@email.smith.edu

Mike Huhn,

Wabash College and University of St. Thomas

mahuhn@stthomas.edu

Chelsea Plaut,

Wabash College and University of Tennessee - Knoxville

cplaut1@utk.edu

Darrin Weber,

Wabash College and Millikin Universty

dmweber@millikin.edu

*Cut Sets and vertices of zero-divisor graphs *

A cut-set in a graph is a set of vertices which, when removed, separate the graph into two or more connected subgraphs. We seek to classify cut-sets in the zero-divisor graphs of finite commutative rings.

Ben Cote,

Wabash College and University of Idaho

cote@vandals.uidaho.edu

Ben Harvill,

Wabash College

harvillb@wabash.edu

Mike Huhn,

Wabash College and University of St. Thomas

mahuhn@stthomas.edu

Abigail Kirchman,

Wabash College and St. Olaf College

kirchman@stolaf.edu

*On the Bol-Moufang identities*

A loop identity is of the Bol-Moufang type if the same 3 variables appear on both sides of the equation in the same order, one of the variables appears twice on both sides and the remaining two variables appear once on both sides. One way to generalize this definition is to allow different variable orders on one side of the identity, viz. ((xx)y)z=x(y(xz)). We define these to be loops of the Perfect type. We show that the 1215 identities of Perfect type can be classified as the 6 commutative Bol-Moufang varieties and 28 new varieties.

Ashley Crish,

Wabash College and St. Mary's College/University of Notre Dame

acrish@nd.edu

Jessica Keeton,

Wabash College and Simpson College

jessica.keeton@simpson.edu

Simon Rush,

Wabash College and Dallas Baptist University

rushs@mail.dbu.edu

*Finite Element Solution of a Non-Newtonian Blood Flow Model *

We focus on computationally modeling a non-Newtonian fluid using a least squares finite element method. Specifically, the model includes viscoelastic and shear-thinning effects found in blood as well as moderate inertial effects. We compare the numerical solutions of the model we developed with those from the Navier-Stokes equations used to model Newtonian fluids. Our goal is to determine the error in simulating a non-Newtonian fluid with a Newtonian fluid model.

Zachary Doenges,

Indiana University

zaadoeng@indiana.edu

*Confidence intervals of permutations with weighted reversals *

It has been commonplace to use the minimum number of reversals between two genomes as a measure of evolutionary distance. And sorting algorithms for this problem have been successful. However, in the system being studied certain reversals are more likely than others. This project attempts to take this into account and return a measure of the confidence for a particular sequence occurring.

Katrina Glaeser,

Rose-Hulman Institute of Technology and University of California Berkeley

katrinaglaeser@berkeley.edu

*The Digraph of the Square Mapping on Elliptic Curves *

Consider a subgroup of an elliptic curve generated by a point P of order n. It is possible to match any point Q to find an integer k (mod n) such that $Q = kP$ by using a brute force method. By observing patterns in the digraph of the squaring map on the integers modulo n it is possible to perform this matching more efficiently. These techniques can be applied to solving the Elliptic Curve Discrete Log Problem given a complete or partial graph of the square mapping for the elliptic curve points.

Court Hoang,

Rose-Hulman Institute of Technology and Pomona College

cmh02006@mymail.pomona.edu

Katherine Osenbach,

Rose-Hulman Institute of Technology and University of Scranton

osenbachk2@scranton.edu

*Impedance Imaging of Corrosion on a Partially Accessible 2-Dimensional Region: An inverse problem in non-destructive testing *

Given a 2-dimensional region with corrosion on an inaccessible back surface, we develop methods for approximating the corrosion profile using measurements of electrical potential along the accessible portion of the region. These methods have applications in the field of non-destructive testing.

Andrew Hoffman,

Rose-Hulman Institute of Technology and Wabash College

hoffmaan@wabash.edu

*Statistical Investigation of Structure in the Discrete Logarithm *

The absence of an efficient algorithm to solve to the Discrete Logarithm Problem is often exploited in cryptography. While exponentiating with a modulus $(b^x \equiv a \pmod{m})$ is extremely fast with a modern computer, the inverse is decidedly not. At the present time, the best algorithms assume that the inverse mapping is completely random. Yet there is at least some structure such as the fact that 1 maps to itself. To uncover additional structure that may be useful in constructing or refining algorithms, statistical methods are employed to compare mappings, $x \mapsto b^x \pmod{m}$, to random mappings. This talk assumes no background in cryptography or statistics, but some familiarity with modular arithmetic will be useful.

Joseph Kramer-Miller,

Rose-Hulman Institute of Technology and Oberlin College

jkramerm@oberlin.edu

*Symmetries and Automorphisms in Power Digraphs Modulo n *

Given an odd integer n and an integer k, we can form a digraph of the mapping $f(x)=x^k mod n$. We explore the structure of this digraph, focusing on various sorts of symmetries within the graph. The case where n is square-free is of particular interest, where we establish necessary and sufficient conditions for the functional digraph to contain symmetry.

Marc Mace,

Rose-Hulman Institute of Technology and Abilene Christian University

mlm05h@acu.edu

*Mapping the Discrete Logarithm Problem over Composite Moduli *

In an age of digital information, security is of upmost importance. Many encryption schemes, such as the Diffie-Hellman Key Agreement and ElGamal, use a function which maps $x$ to $y$ by $y \equiv gx$ (mod n) for a given $n$ and a generator (or primitive root) $g$. In most cases, this $n$ is a prime number. In some cases, however, $n$ may be a composite number. In particular, we will look at when $n = pk$ and $n = 2*pk$. We will show different techniques of obtaining functional graphs of this mapping and then we look to see does the mapping for the described n values look like a "random map," and, if it does not, what can we observe that would help in solving the Discrete Log Problem?

Jamil Merali,

Indiana University and Northwestern University

jamilmerali2010@u.northwestern.edu

*Growth of Groups, a short introduction *

I will briefly cover the notion of growth of a group, and give some examples.

Ziva Meyer,

Indiana University and New College of Florida

zivakaye@gmail.com

*tba *

tba

Ari Nachison,

Indiana University and UC Santa Barbara

nachison@engr.ucsb.edu

*Psuedo Relativistic Behavior of Fermionic Systems *

Recently it has been shown that scaling arguments imply the pseudo-relativistic behavior of electrons in graphene may be a much more pervasive phenomenon. The pseudo-relativistic behavior manifests itself as one of several universal scaling limit laws that emerge out of a rigorous asymptotic analysis of the Schrodinger equation in the limit when the average nearest neighbor distance to the characteristic length of interaction becomes small. Here we explore the implications of this scaling limit law. In particular, we develop a new computational algorithm that solves the pseudo-relativistic equation via the method of Characteristics. Since the pseudo-relativistic equation it operates on a long timescale. This opens the way to long time rigorous simulation of many fermion systems.

Zachary Norwood,

Indiana University and University of Nebraska-Lincoln

znorwood@huskers.unl.edu

*Counting Involutions in Finite Groups *

Finding groups with the largest possible number of involutions---elements of order 2---was motivated by Allan Edmonds' study of equifacetal simplices. These ``2-maximal" groups have been determined, but we discuss from a purely group theoretic standpoint extension of this classification, basic preliminary results, and possible generalization to elements of prime order for any prime.

Jacek Skryzalin,

Indiana University

jskryzal@umail.iu.edu

*Creating graphs of filled Julia sets for quadratic polynomials *

Let $c$ be in the interior of a component of the Mandelbrot set which is attached to the main cardioid. The filled Julia set of a quadratic polynomial $P_c(z)=z^2+c$ contains those points in the complex plane that do not escape to infinity when acted upon by $P_c$ an infinite number of times. To better understand the dynamics of $P_c$, we can create a graph of the filled Julia set by looking at successive inverse images of the components of the filled Julia set which contain attracting periodic points. After developing an algorithm which describes the graph of the filled Julia set at the $n$th inverse image, we prove that this algorithm works using external rays.

Joseph Thurman,

Indiana University and Vanderbilt University

joseph.d.thurman@vanderbilt.edu

*The Construction of a Complete, Bounded, Negatively-Curved Surface in R^3 *

In 1981 E.R. Rozendorn published a paper describing the construction of a bounded, complete surface in R^3 that was negatively curved except at a countable set of discrete singular saddle points of zero curvature. Building on his work, we are working to find methods to deform Rozendorn's surface in order to remove the singular points and make the surface negatively curved at every point. We are also investigating other methods of modifying Rozendorn's construction through the use of different starting surfaces, and exploring the topological restrictions that dictate what properties such a starting surface must have.

Seth Unruh,

Goshen College

sethku@goshen.edu

*Envy-free divisions*

We consider the division of a single object among several people who are willing to transfer money among themselves. A division is envy free if every person believes they received at least as much value from the object and money as they believe every other person received. We characterize the set of envy-free divisions.

Leah Wolberg,

Indiana University and Bowdoin College

lwolberg@bowdoin.edu

*What is the largest circle you can draw on the front and back of a paper triangle without enclosing any of the three vertices?*

The isoperimetric problem, that of finding the largest area that can be enclosed with a given perimeter, has interested mathematicians since antiquity. The most famous result is the proof that, in the plane, a disk maximizes area for a given perimeter. We'll consider isoperimetric inequalities on polyhedra. While no general solutions are expected, we have a nice result for the doubled triangle.

Chengcheng Yang,

Indiana University and Rice University

cy2@rice.edu

*A Problem of Erdos concerning Lattice Cubes *

The problem is trying to find a largest subset in an N x N x N grid not containing the eight corners of a nontrivial box. Erdos conjectured that for any \epsilon > 0, there is a constant C (depends on \epsilon only) and a subset A in the N x N x N grid with at least CN^ (11/4-\epsilon) elements. However, the best known example today is about CN^ (8/3) elements. The known example is produced using linear algebra over finite fields, but it is not clear whether larger examples can be built in the same algebraic way. The attempt is either to improve the known exponents by algebraic methods or to determine the limits of the method.