2013 Mathematics and Statistics Capstone Presentations

All talks are in 421 Neville Hall.

Date Time Speaker Title
9:00-9:20am Travis Goodwin The Concepts of Communication Theory
9:25-9:45am Matthew Brenc Dirichlet's Theorem on Primes in Arithmetic Progressions
9:00-9:20am Jodi Harnden A Perspective on Polyominoes
9:25-9:45am Hannah Woodruff Primes of the Form x2+y2
9:00-9:20am Seth Albert Fractal Dimension and the Wavelet Transform Modulus Maxima method
9:25-9:45am Dean Pelletier A Little to Say About RSA
9:00-9:20am Paige Gallagher Introduction to Fibonacci Numbers
9:25-9:45am Matthew Breton Vibrations of a Drumhead
9:00-9:20am Cormick Frizzell An Introduction to Card Counting
9:25-9:45am Timothy Buchak Mathematics of Sudoku
9:00-9:20am Timothy Michaud Fractal Coastlines
9:25-9:45am Adam Smith Quadratic Reciprocity
9:00-9:20am Jean Stevens Gaussian Integers
9:25-9:45am Jennifer Wood Google's PageRanking System
9:00-9:20am Stuart Lawson The Analytic Continuation of The Riemann Zeta Function
9:25-9:45am Nicole Curtis-Bray Applications of Bessel Functions in Electrical Engineering
9:00-9:20am Anne Witick Fractals: A Basic Overview
9:25-9:45am Mahadi Osman Composites in Diferent Bases that Remain Composite After Changing Digits.
Capstone papers due

Abstracts (by last name)

Seth Albert: "Fractal Dimension and the Wavelet Transform Modulus Maxima method"
An introduction to fractal dimension and a look at characterizing multifractal surfaces by its singularities through the Wavelet Transform Modulus Maxima method.

Matthew Brenc: "Dirichlet's Theorem on Primes in Arithmetic Progressions"
Johann Dirichlet proved in 1837 that for coprime integers a and k, there exist infinitely many prime numbers congruent to a modulo k. In this talk I will provide an outline of Dirichlet's proof, focusing in particular on his key insight that the characteristic function of the integers congruent to a mod k can be written as a linear combination of multiplicative functions on the group (Z/kZ)*.

Matthew Breton: "Vibrations of a Drumhead"
This talk will focus on how to describe the vibrations of a drumhead using polar coordinates.

Timothy Buchak: "Mathematics of Sudoku"
My talk will be on the mathematics behind the popular puzzle game Sudoku. I will summarize what work has been done related to Sudoku, and look at some of the more advanced solution algorithms. I will also briefly talk about the relation between Sudoku and graph theory.

Nicole Curtis-Bray: "Applications of Bessel Functions in Electrical Engineering"
An introduction to the origins and applications of Bessel functions and waveguides. In addition, Maxwell's equations for electromagnetic waves will be used to derive Bessel's equation.

Cormick Frizzell: "An Introduction to Card Counting"
I will discuss a few methods of card counting and will prove parts of the Fundamental Theorem of Card Counting.

Paige Gallagher: "Introduction to Fibonacci Numbers"
This presentation will be an introduction to Fibonacci numbers, their properties, applications and other interesting mathematical connections these numbers share.

Travis Goodwin: "The Concepts of Communication Theory"
An introduction to the basic concepts that were developed by Claude E. Shannon, and consequently mathematically proved, in his famous paper, "The Mathematical Theory of Communication".

Jodi Harnden: "A Perspective on Polyominoes"
An introduction to Solomon W. Golomb's polyominoes and their tiling capabilities. There will also be a discussion on how generating functions play a role in computing the possible tilings of specific rectangles with trominoes.

Stuart Lawson: "The Analytic Continuation of The Riemann Zeta Function"
This presentation will provide a detailed walkthrough of one of the many proofs of the analytic continuation of the Riemann Zeta Function; that is, it will show how the zeta function can be extended in a way that its domain is (nearly) all of the complex plane, and not just numbers whose real part is greater than 1. This particular proof utilizes the theta function and the Mellin transform, as well as Poisson summation and Fourier transforms.

Tim Michaud: "Fractal Coastlines"
A look at how coastlines are self-similar under different magnifications and thus are fractal in nature. We will also look at the fractal dimension of the coast of Maine and how it compares to that of other coastlines.

Mahadi Osman: "Composites in Diferent Bases that Remain Composite After Changing Digits."
Filaseta et. al. proved that there are infinitely many composite numbers that remain composite after changing any digit in the decimal expansion by constructing an infinite arithmetic progression of such composite numbers. We show that there are infinitely many composite numbers in base b that have this property for b=2,...,9. We then attempt to show that there are composite numbers in base b that remain composite after replacing any two adjacent digits in the base b expansion.

Dean Pelletier: "A Little to Say About RSA"
RSA cryptography is a modern and secure form of secret communication. In this talk I will cover the basics of RSA cryptography and how it differs from private key cryptography. I will also discuss enciphering keys and algorithms as well as how to develop the deciphering key, and how having one does not give a person the ability to find the other.

Anne Witick: "Fractals: A Basic Overview"
An exploration of the evolution of fractals, with emphasis on the history and development. Examines some famous fractals and the Iterated Function System.

Jennifer Wood: "Google's PageRanking System"
An introduction to Google's search engine and the PageRank equation that calculates the popularity score of webpages by applying the power method to the Google Matrix.

Hannah Woodruff: "Primes of the Form x2+y2"
In this talk, I will be proving Fermat's observation that an odd prime p can be written as x2+y2 if and only if p≡1 mod 4. This proof will be based on Euler's proof which he completed in 1749.