# Balanced Sequences and Egyptian Fractions

Summer 2011

**Student: Haoqi Chen**

**Adviser: Professor Teena Carroll**

Haoqi and Professor Carroll received a student/faulty grant from the St. Norbert College Collaborative to write a paper based on the results of their 2010 summer research project. Finding three integers which can be the lengths of sides of a right triangle is an idea that has interested mathematicians from ancient to modern times.

They found a way to generate infinitely many disjoint sequences encoding Pythagorean triples. The recurrence relation which generates these sequences is the same recurrence relation used to find the Egyptian fraction representation of one of a given length using the largest denominator. This connection to Egyptian Fractions is particularly striking because it combines ideas developed separately in Ancient Greece and Ancient Egpyt.

Haoqi Chen investigated this particular subject area since his first semester at St. Norbert. The project was rooted in the Summer Undergraduate Research Program in Mathematics at the college in 2010.

Generalizations of the Koch Curve and Their Dimensions

**Student: Caesar (HanQin) Cai **

**Advisers: John Frohliger and Kevin Murphy**

Unlike traditional geometric shapes, fractals may have non-integer dimensions. The Koch curve is one of the basic fractals, which has dimension log4/log3.

In this research project, Ceasar and his advisers showed how to generate a fractal having any dimension between 1 and 2 by modifying the Koch curve with a focus on a Dimension Equation.

This modification encouraged them to examine three questions: how can the Koch curve be generalized, when can the Dimension Equation be used, and what dimensions are possible with generalizations of the Koch curve. Special attention is paid to space-filling variation. The number of pre-images required to parameterize the generalized Koch curve with dimension 2 was explored.

Caesar (HanQin) Cai is from Shanghai, China, and graduated in 2012 as a double major in mathematics and computer science. During his time at St. Norbert, he was an active math speaker who gave four math talks in the past three years.

Notably, one of these presentations won second place in the Wisconsin Mathematical Modeling Challenge of Fall 2010. Ceasar was also a Mathematics Teaching Assistant, a College tutor, and the President of the St. Norbert Sigma Nu Delta Math Club (elected twice from 2010 through 2012).

Parameterizing the Koch Curve

Student: Brian Pietsch

Advisers: Professors John Frohliger and Kevin Murphy

One of the most famous fractals is the self-similar Koch curve. It is known for having infinite length, and it is generated by infinite iterations of four affine transformations. In this research project, the student/faculty team used these transformations along with a base four addressing scheme to create a well-defined, continuous parameterization of the Koch curve.

Brian Pietsch, a native from Green Bay, Wis., and graduated with a double major in mathematics and physics in 2012. This research project began after Brian's junior year. He gave several talks on this research, and he planned to pursue a graduate degree in mathematics.

At MathFest 2011 in Lexington, Ky., Brian's presentation was judged to be one of the best Pi Mu Epsilon National Honorary Society student talks. In recognition, he was prized a financial award from the American Mathematical Society and the American Statistical Association.

An Extension of the Hiring Program

Student: Nicole Harp

Advisers: Professors John Frohliger and Kevin Murphy

A classic probability problem, the Hiring Problem, also known as the Secretary Problem, looks at the optimal strategy for hiring the best-qualified candidate from a pool of applicants. If the best applicant is hired, it is considered a success. it is dependent on the fact that an applicant cannot be revisited once he or she has been interviewed.

This student/faculty team extended this problem from originally hiring the best applicant, to instead, hiring one of the top (pre-determined) applicants. The probability of success cannot be maximized using basic techniques so they used numerical approximations to explore specific cases.

Further, this team found that by allowing one to hire from the top number of applicants, the maximum probability of a success will increase.

Nicole Harp, a native from Appleton, Wis., is pursuing a double major in mathematics and economics. This summer research project began just prior to her senior year. She plans to complete her degree in three and a half years. Nicole has been an active member of the Sigma Nu Delta Math Club since her first year at St. Norbert. She also teaches tennis during the summers at her former high school.