Fall 2016 and Spring 2017 Capstone Research
Carryless Arithmetic Mod 10
We all started our mathematics’ journey with 1 + 1 = 2. Then we moved on to 6 + 6 = 12 and learned the importance of the “carry” digit. But what if 6 + 6 = 2 instead of 12? How does that change our understanding of mathematics, and specifically, the concepts of squares and prime numbers? This idea is the basis of Carryless arithmetic, where we don’t have to worry about those pesky carry digits. During this session, we will explore the fundamental concepts of operations and prime numbers. Additionally, we will look at the rules for determining when two numbers squared equal one another, as well as formulating the number of distinct squares based on the digit length of the underlying number.
Investigating Rho, Two-Fold Sumsets
Given a finite abelian group $G$, and a positive integer $m$, how small can a sumset of an $m$-subset of $G$ be? This value is represented by the function in additive combinatorics, $\rho$, studied by Cauchy in the early 19th century and by several mathematicians over the years. After introducing $\rho$ with some examples, we will investigate the closely related function $\rho\hat$, for which less results are known. We will mention several results without proof and discuss bounds given by famous mathematicians. Finally, we will take a look at the maximum values that $\rho\hat$ can have; we call this new function $\rho\hat$ max. We will begin our talk by introducing some basics of Group theory, Combinatorics, and Number Theory to set us up properly for the three functions: $\rho$, $\rho\hat$, and $\rho\hat$ max.
Traffic Jams on a Circular Road
Traffic congestion has become a vital problem in recent years. Frequent causes include bad weather conditions, accidents, and the ever-increasing number of cars. Unexpectedly, though, some researchers argue that traffic tie-ups can happen even in off-peak traffic times with no adverse conditions. They attribute this to bad behavior on the part of the drivers themselves. I will present a car-following model to simulate how driver behavior leads to traffic jams, along with theoretical and numerical results related to traffic congestion.
Holy Inpainting, Batman! Wavelets in Digital Image Inpainting
Image inpainting refers to repairing a damaged area of an image, and originally described the process artists used to restore paintings. Digital inpainting can be accomplished with algorithms based on partial differential equations (PDE) and radial basis functions (RBF). A Wavelet transformation decomposes an image into two parts, the high pass portion and low pass portion, which can be addressed individually using traditional inpainting methods. In this talk, we will explore an algorithm that uses the Haar wavelet transform in combination with PDE or RBF methods to substantially improve inpainting results.
Ensuring a Secure Transmission
Since the dawn of the information era, people have turned to encryption strategies to protect their data and send secured messages, but are they truly secure from attacks? We will explore the effectiveness of current encryption methods, then propose a new method in which traditional encryption techniques are combined with modern, wavelet-based steganography. In executing this procedure, we will ensure a secure and secret transmission of data within the confines of an image without being perceptible to the eye. We will then discuss the results of implementing this improved method, and finally analyze an example of a transformed image with an encoded message.
The Classification of Division Algebras
A division algebra structure over R is a bilinear product such that the multiplication of two nonzero elements yields a nonzero element. One may wonder when Rn admits the structure of a division algebra. This question is equivalent to several other interesting questions, such as the existence of n-1 linearly independent tangent vector fields on the sphere. It is a classical theorem by Kervaire, Milnor, and Bott that Rn admits the structure of a division algebra exactly when n = 1, 2, 4, or 8; we are already familiar with two of these cases, R and R2 = C, While this is a strictly algebraic result, every proof of this fact seems to require some topology. Time permitting, we will explore this interesting connection in Hopf’s proof that n must be a power of two.
An Explanation of the Simplex Method
Linear programming is the mathematical technique of determining the best value obtainable of a function whose domain is restricted by various equality and inequality constraints. The simplex method is a powerful iterative procedure for solving linear programming problems by repeatedly solving a system of linear equations, stopping if the solution proves to be optimal, and otherwise proceeding to check the next linear system. In this paper, we provide a comprehensive explanation of how the simplex method works, and proofs of the properties that are the backbone of the method.
The Centennial Conference Traveling Salesman Problem
What would be the shortest way to start in Gettysburg and visit all ten Centennial Conference Schools before returning home? What if we could send multiple people out to visit the various schools? If we could, how many should we send? This is the goal of the Traveling Salesman Problem (TSP) and Multiple Traveling Salesman Problem (mTSP). We introduce common constraints for the mTSP, and discuss how changing these constraints can impact solutions. Finally, we analyze variations of an mTSP for ten Centennial Conference colleges/universities and make conclusions as to the best location for home base, or “depot” school.
Frobenius Problem for Polynomials
Given relatively prime positive integers a1, a2, · · · , an, the Frobenius Problem aims at finding the largest positive integer that cannot be expressed as a non-negative integer combination of a1, a2, · · · , an. Also, in abstract algebra, we will have a chance to learn about polynomial rings over a field, we will see that there are a lot of similarities between integers and polynomial rings over a field. We will discuss the development of Frobenius Problems. We will also translate some results of the classical Frobenius Problem into a polynomial ring. We call such a problem the Frobenius Problem for Polynomials.
The Twin Prime Conjecture and Sieve Theory
The twin prime conjecture, made by Euclid more than two thousand years ago, is a famous conjecture that has not been proven. Though little progress was made on it until the nineteenth century, recently several mathematicians have shown promising results. This paper focuses on analyzing sieve theory, the root of this breakthrough research. An overview of recent developments is also presented.
Presidential Primaries: Using Optimization to Measure Democracy
Democracy is one of the fundamental values of American government, but how democratic are our elections? This research explores the mathematics behind the American primary election system. Through Operations Research techniques, we study the unique methods of delegate apportionment by the national political parties and delegate assignment by the individual states. Specifically, we build a series of models that calculate the minimum proportion of the popular vote a candidate must win in order to win the primary election, based on each combination of party delegate apportionment and state delegate assignment. We determine that the method with the highest minimum percentage is the most democratic and then explore the mathematic reasoning behind the results, answering some common questions voters might ask.
Investigating the NBA's Three-Point Revolution
Since the National Basketball Association's (NBA) introduction of the three-point line in 1979, the three-point shot has steadily grown in importance and influence in the NBA, to the point where some teams are now taking more than a third of their shot attempts from behind the three-point line. Some people argue that this trend is "bad" for the game of basketball, as it is making the sport more and more dependent upon the single skill of three-point shooting. Leaving aside this argument, this talk will examine the current trend of three-point shooting in the NBA from a statistical point of view. Additionally, using a combination of traditional basketball statistics and more modern player tracking data, we will analyze and attempt to predict the results of increasing the distance of the three-point line -- a proposal that is gaining popularity among critics of the growing reliance on the three-point shot in the NBA.
What is Persistent Homology?
With a decent number of points extracted from an image of an object, our brain would be able to recognize the object without further information. Our brain or eyes does a marvelous job on taking the sense of data of individual points and assembling them into a coherent image of a continuum. However how can we be certain about the structure of the data? Furthermore, how do we learn the structure of higher dimensional data? Persistent homology is an algebraic tool for measuring topological features of shapes. It infers high-dimensional structure from low-dimensional representations. We will learn the fundamental concepts of persistent homology such as simplicial homology and Rip complexes. We will also explore a method of computing and analyzing persistent homology using barcodes in time series data.
P-adic Numbers and Dividing a Square into Equal Area Triangles
Could you imagine a world where 2199 and 2 are closer than 3 and 2? Could you imagine a world where every triangle is an isosceles triangle? That world exists if we extend the field of rational numbers using a matric induced by a p-adic absolute value. We work on some examples to get familiar with this bizarre world and investigate some interesting properties it has. In the end, we use the idea of p-adic numbers to prove Monsky’s theorem, which says that we cannot divide a square into an odd number of equal area triangles.
The Mystery of NBA Schedules
The National Basketball Association (NBA) is a major professional basketball league composed of 30 teams located in the United States and Canada. With 1,230 games played each season, the NBA needs to create a schedule that not only has teams playing each other a certain number of times, but also makes sure that no team gains an advantage from their schedule. For example, how would you feel if your favorite team had to travel a significantly longer distance during the season than the team you hate most? The goal of this project is to find a schedule that minimizes travel distance for the league as a whole.