There are many opportunities for Kenyon's undergraduates to do research in mathematics. Some students take independent studies to pursue independent projects and collaborative work with faculty.

Successful Juniors and Seniors sometimes enroll in the Honors Program in Mathematics to do in-depth work in an area of interest. Kenyon's Summer Science Scholar (SSS) Program provides financial support to undergraduates who want to work on research projects in the summer. Students participating in the SSS program work in collaboration with a faculty mentor for 8 to 10 weeks. Learn more about summer science...

Here are some examples of recent undergraduate research projects:

**Nick Connolly (Class of 2015)**

**Abstract: **Coding theory is the branch of mathematics interested in the reliable transfer of information. Error correcting codes are designed to detect and correct errors that occur during the transmission of a message due to noise. Linear codes have the mathematical structure of a vector space over a finite field. Every linear code has three fundamental parameters which determine its quality: length, dimension, and minimum distance. A code's minimum distance determines its error correcting capacity. For a given length and dimension, there exists an upper bound on the value of the minimum distance of a code; the best known linear codes are those with a minimum distance as close as possible to this bound. In this project, we attempt to construct new linear codes with larger minimum distances than the previously best known codes by exploiting the algebraic structures of constacyclic and quasi-twisted codes. For a given length and finite field, we exhaustively construct all constacyclic codes and record those codes with the highest minimum distance for a given length and dimension. We then use those best constacyclic codes to construct 1-generator quasi-twisted codes. Finally, we compare the minimum distance of these quasi-twisted codes against the best known linear codes with the goal of discovering new linear codes with better parameters. We have been able to find 96 new codes with this method which have been added to the online database of best known linear codes.

**Lila Greco (Class of 2015)**

**Abstract:** This project explores Brownian motion, a model of random motion, in the plane. Given a domain in the complex plane and a basepoint in the domain, start a Brownian traveler at that basepoint. The ℎ-function of the domain gives information about where the Brownian traveler is likely to first hit the boundary of the domain. I computed the ℎ-functions for several families of domains analytically using conformal mapping. I found instances in which non-smoothness in the boundary of a domain can be detected in the h-function. I also proved the pointwise convergence of two different sequences of ℎ-functions. Finally, using simulations of Brownian motion, I approximated the ℎ-functions for another family of domains. By improving the speed of the simulations, I was able to gather more data and obtain more accurate approximations of the ℎ-functions.

**Sam Justice (Class of 2015)**

**Abstract: **We generalize the so-called “Four Numbers Game” to planar graphs. In our generalization, the steps of the game alternate between the graph and its planar dual. To illustrate our generalized game, we present a case study of the game involving the self-dual 5-Wheel graph. We also discuss results for a couple of other self-dual planar graphs, and conclude with some suggestions for avenues of further exploration.

**Robin Belton (Class of 2016) Faculty Mentor: Assistant Professor of Mathematics, Marie Snipes**

**Abstract: **In Calculus one learns how to approximate the definite integral with left, right, midpoint, and trapezoid Riemann Sums. The error of these sums can be described in terms of the first and second derivatives of the integrand,* f*. The definite integral describes the area under the graph of *f*. Our project explores numerical techniques for approximating the volume obtained when *f* is rotated about the -axis. We define left, right, midpoint, trapezoid, and Simpson approximations for this setting. Then we examine the error for these methods and create bounds.

**Zach Weiner (Class of 2016)**

**Abstract:** For a positive integer* n*, the abundancy index *I(n)* is defined to be the sum of its divisors divided by the number itself, or *σ*(*n*)/*n*. The function *I*:N→Q⋂(1,∞) is not onto; rationals not in the range of *I* are called “abundancy outlaws." Identifying and characterizing abundancy outlaws could prove helpful to better understand the existence of odd perfect numbers, a question over 2000 years old. In our research, we consider rationals of the form (*σ*(*n*)+*t*)/*n* where *t* is a positive integer, to produce and characterize as-yet undiscovered outlaws.

**Tsvetan Asamov (Class of 2008)****Faculty Mentor: Assistant Professor of Mathematics, Nuh Aydin**

This work has been accepted for publication in Designs, Codes and Cryptography **Abstract:** This work introduces an algorithm, called progressive dimension growth (PDG), for the construction of linear codes with a pre-specified length and a minimum distance. A number of new linear codes over GF(5) have been discovered via this algorithm.

**William Stanton (Class of 2008) ****Faculty Mentor: Professor of Mathematics, Judy Holdener**

**Abstract:** The abundancy index of a positive integer/ /$n$ is defined to be the rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of divisors function $\sigma(n)=\sum_{d|n}d$. An abundancy outlaw is a rational number greater than 1 that fails to be in the image of of the map $I$. In this paper, we consider rational numbers of the form $(\sigma(N)+t)/N$ and prove that under certain conditions such rationals are abundancy outlaws. (Will presented this research at the annual summer meeting for the Mathematical Association of America and won a Pi Mu Epsilon student presentation award for giving one of the best talks at the meeting. Will also won a Goldwater Scholarship based on this work.)

**Agnese Melbarde (Class of 2008)**

**Abstract:** Kenyon College Biology department has been investigating the relationship

MR = a(BW)^b

for Manduca sexta caterpillars. Empirical evidence suggests some differences in the estimates of 0.67 or 0.75 for b proposed by previous researchers. Apart from modeling the data set, we examine sample size determination problems and measurement spacing issues by using resampling techniques and comparing the variability of the slope coefficient.

**Lee Kennard (Class of 2007) and Matthew Zaremsky (Class of 2007)****Faculty Mentor: Professor of Mathematics, Judy Holdener **

**Abstract:** In a recent paper, Ma and Holdener used turtle geometry and polygon maps to show that the Thue-Morse sequence encodes the von Koch curve. In the final paragraph of this same paper, they ask whether or not there exist certain generalized Thue-Morse sequences that also encode the curve. Here we answer this question in the affirmative, providing an infinite family of words that

generate generalized Thue-Morse sequences encoding the von Koch curve. (Matt and Lee received Franklin Miller Awards for this research.)

**Student: Jun Ma (Class of 2005) ****Faculty Mentor: Professor of Mathematics, Judy Holdener**

**Abstract:** We reveal an unexpected connection between the Thue-Morse sequence and the Koch snowflake. Using turtle geometry and polygon maps, we realize the Thue-Morse sequence as the limit of polygonal curves in the plane. We then prove that a sequence of such curves converges to the Koch snowflake in the Hausdorff metric. In the final section we consider generalized Thue-Morse sequences and provide a characterization of those that encode curves converging to the Koch snowflake. (Jun won Kenyon's /Tomsich Science Award for this research, which appeared in the journal / Fractals / .)

**Student: Stillian Ghaidarov (Class of 2004)****Faculty Mentor: Assistant Professor of Mathematics, Keith Howard**

**Abstract:** Wavelet Transforms provide powerful techniques of converting continuous analog data sets to a digital framework. A particularly important application is the ability to compress data to allow for more compact and efficient storage. With this project we study the application of Image compression and recovery via the Haar Wavelet transform giving particular emphasis to the storage of recovery of medical images.

**Student: Eric Kahn (Class of 2004)****Faculty Mentor: Assistant Professor of Mathematics, Keith Howard**

**Abstract:** A semigroup of operators is a single-parameter set of operators, {T(t) | t= 0} possessing the following two properties when applied to an element x:

1. T(t+s)x = T(t)T(s)x |
(Semigroup Property) |

2. T(0)x = x |
(Identity Property) |

Semigroups of operators are important, as they are an abstract representation of the exponential function. Reflecting the importance of exponentials in solving differential equations, semigroups are particularly useful in analyzing abstract differential equations, in particular the Abstract Cauchy Problem. Semigroups also provide an important framework for studying chaotic behavior exhibited by a linear system. With this project we investigated the general properties of semigroups with emphasis on the study of chaotic systems.

**Student: Amy Wagaman (Class of 2003)****Faculty Mentor: Associate Professor Mathematics, Judy Holdener**

**Abstract:** A turtle sequence is a word constructed from an alphabet of two letters: F, which represents the forward motion of a turtle in the plane, and L, which represents a counterclockwise turn. In this work, we investigate such sequences and establish links between the combinatoric properties of words and the geometric properties of the curves they generate. In particular, we classify periodic turtle sequences in terms of their closure (or lack thereof). This work was published in the International Journal of Mathematics and Mathematical Sciences in June of 2003.

**Student: Tsvetan Asamov (Class of 2008)****Faculty Mentor: Assistant Professor of Mathematics, Nuh Aydin **

This work is published Asamov, T. and Aydin, N. 2007. LDPC codes of arbitrary girth. In Proceedings of the 10th Canadian Workshop on Information Theory (CWIT 2007), June 6-8 2007, Edmonton, Alberta, Canada, 69-72**Abstract:** For regular, degree two LDPC codes, there is a strong relationship between high girth and performance. This article presents a greedy algorithm, called successive level growth (SLG), for the construction of LDPC codes with arbitrarily specified girth. The simulation results show that our codes exhibit significant coding gains over randomly constructed LDPC codes and in some cases outperform PEG codes in the additive white Gaussian noise channel.

**Student: Ben Johnson (Class of 2008)****Faculty Mentor: Assistant Professor of Mathematics, Nuh Aydin**

**Abstract:** One of the most important problems in coding theory is to construct codes with the best possible parameters. Recently, many new codes with best-known parameters have been discovered among the class of quasi-cyclic (QC) and quasi-twisted (QT) codes by computer search. A commonly used search algorithm in this effort is improved and optimized in this work. Also, multi-twisted (MT) codes, a larger class of codes that contains QT codes as a subclass and to which the algorithm applies, is introduced. Moreover, the algorithm also enumerates all 1-generator MT codes with given parameters.

As part of this work, a general theory of ideal and coset representatives is developed in a module of the form V = R/ x R/x..., R/ where m_i is i n R and R is a Euclidean domain. This leads to efficient algorithms which under suitable conditions decide when < heta_1> = < theta_2 and when alpha_1 + <\theta> = alpha_2 + <\theta> by reducing such ideals and cosets to a canonical form, and enumerate all such canonical forms and thus all ideals of a fixed V and all cosets of a fixed ideal.

Algebra Algebraic Geometry Analysis Image Processing Logic Mathematical Biology |
Mathematical Modeling Number Theory Pattern Analysis Probability Statistics Theoretical Computer Science |

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