Hudson River Undergraduate Conference Spring 2013
16 students and 6 faculty made the 6 hour drive to Williams College to attend the 20th annual Hudson River Undergraduate Mathematics conference. Images from the weekend.
Of the attendees, 9 students and 1 faculty member presented on the following topics:
Probability Of Making Street Networks One-Way
Consider the task of making all the streets of a network one-way so that a driver may still travel from any given intersection to any other. In terms of graph theory, we seek to orient each edge of a connected graph so that the resulting digraph is strongly connected. In this talk we will investigate the probability that randomly orienting each edge of an undirected graph will yield a strongly connected digraph and its implications for making street networks one-way. We use a computer program to simulate random orientations of the edges and test each resulting digraph for strong connectedness. This analysis enables us to answer the following question: given a fixed set of vertices and a limited number of edges, where should we place the edges to maximize the probability of obtaining a strongly connected orientation?
Using Non-Linear Regression to Model Autocorrelation in Plutonium Management Models
Non‐linear regression is a style of regression function that uses a non‐linear function of the parameters to model data. Each Y is modeled as a random variable and given a mean by a non‐linear function commonly based on a known general form with unknown (and estimated) parameters. One use of non‐linear regression is for modeling the autocorrelation seen in the residual heat, carry‐over effect, from using calorimeters to monitor plutonium at the Los Alamos National Laboratory (LANL). This presentation will introduce the method of non‐linear regression and discuss how it can be applied to help LANL better manage nuclear materials.
Using Next Generation matrices to investigate disease spread in a system with an inhomogeneous population distribution
Standard SIR Models for disease spread assume that all members of the population have an equal chance to be infected by, or pass a disease on to, any other member of the population. However, the real world is not that simple. We are using compartmental epidemic models to investigate how the spread of a disease changes when you separate the population into several groups that have separate intra- and inter-compartmental contact rates. This is done by using what is called next generation matrices and it allows us to monitor exactly how many new infections occur in each different compartment as the results of a single infection in any of the compartments. One application of this project is to model how a disease will spread if it starts in the center of a city as opposed to somewhere outside the city.
A Paradoxical Decomposition of the Real Number Line
In 1924, Banach and Tarski demonstrated that two arbitrary point sets with nonempty interior in Euclidean n-space for n ≥ 1 are equivalent to each other using a countable decomposition. It is also well known that it is impossible to replicate the Banach-Tarski paradox (i.e. using finitely many subsets) for R using rigid motions. In this poster, we propose a free group similar to the one used in proving the Banach-Tarski paradox whose generators are not rigid motions, but piecewise isometires based on infinite permutation cycles. Using the free group on these generators, we are able to partition the real number line into a disjoint union of subsets that can be transformed via these maps and then reassembled into two identical copies of the original line. The result is a paradoxical finite decomposition of the real number line.
Burning Bridges: A Graph Theory Game
Burning Bridges is a two-player game on a connected graph. The players alternate turns by moving along edges from vertex to vertex. A move must be made to a vertex adjacent to the one previously selected, and each edge can only be used once. The object of the game is to reach a position in which the other player is not able to move, resulting in their loss. For certain graphs, the outcome is clear just by inspection. For example, the first player always wins on a cycle graph with an odd number of edges. There are strategies for other graphs as well, such as complete graphs, which we will discuss so that you are able to defeat your neighbor in Burning Bridges.
Fractals, Chaos, and Newton's Method
Often introduced during a first semester Calculus course, Newton's Method (also known as the Newton-Raphson method) is an iterative technique used to approximate solutions to f(x) = 0. The method is simple: start with an initial guess x0, run it through N(x) = x - f(x)=f0(x) obtaining N(x0) = x1, then rinse and repeat to create the sequence (xn) converging to a solution of f(x) = 0. We will briefly discuss why this method works and then move on to the more interesting question: are there initial guesses that do not work? To fully understand Newton's Method, we will move to the more natural setting of the complex plane where (often) beautiful fractal images result.
A Chromatic Duel on Simple Graphs
We will provide a brief survey of the historical and mathematical background of graph edge-coloring as well as several known results pertaining to the edge-coloring of simple graphs. We will then introduce the following edge coloring game. Given a particular simple graph G, two players alternate coloring an edge of G using one of k available colors in such a way that the coloring remains proper; that is to say, no two adjacent edges receive the same color. The game is won by the player who makes the last valid move that maintains the proper edge-coloring of G. We will analyze the winning strategies for several classes of graphs, including both paths and cycles. Our talk will conclude with a discussion of further avenues of exploration regarding the edge-coloring game.
Compartmental Models: An Epidemiological Application of Ordinary Differential Equations
Ordinary Differential Equations can be used to create epidemiological models to study the spread of infections. The traditional SIR model splits a population into three separate categories: Susceptible, Infected, and Removed. This model provides a basic understanding of the progress of an infection, however, it overlooks important details. By splitting these categories into smaller subgroups, it becomes possible to understand the various stages of the infection and study it within the contexts of populations with varying interaction and transmission rates.
Divisor Graphs for Egyptian Fractions
Egyptian fractions have the form 1/n, where n is a positive integer. Given a set of Egyptian fractions that sum to one we can create a divisor graph by labeling each vertex with a denominator, then connecting any pair of vertices that share a common factor. We have decided to allow repeated denominators, as in 1/2 + 1/4 + 1/4. We are investigating which connected graphs can be generated using this method and whether there is an inherent property of a graph that can be used to determine whether it arises from a corresponding Egyptian fraction set. We have found that it is possible to generate complete graphs, all star graphs, and probably paths of any length. Additionally, we have found that cycle graphs are seemingly impossible to construct according to our method, due to the requirements of which vertices must be relatively prime.