Summer 2014 Past Projects
Pattern Avoidance in Double Lists
Dr. Lara Pudwell
The motivation for this paper is to continue combinatorics research in the area of pattern avoidance. We have constructed a subset of words to study called “double lists” which are based on the standard permutations used in counting pattern avoiding lists. We will begin this paper with an introduction to pattern avoidance for those unfamiliar. However, the majority of this paper will focus on our own research of pattern avoidance within double lists.
The Effect of Risk Taking Behavior Induced by Recovery in Epidemic Models
Dr. Daniel Maxin
Individuals facing an infectious disease, consciously or not, evaluate the contamination risk and engage or avoid risky situations or behaviors. The risk taking behavior of any individual is probably a very complex functional response that depends of many factors such as: morbidity of the disease, infectiousness, mode of transmission, existence and eﬃcacy of treatment, etc. There are multiple studies that show individuals have dynamic responses to transmission risks that change with varying circumstances. For example, the advent of antibiotics makes all of us less concerned with exposure risks to common pathogens. Nobody is worried with getting tick bites in the woods since a course of penicillin would cure a possible exposure to Lyme disease. To the contrary, knowledge about the severity of disease (measured by number of infectious people, virrulence, lack of eﬀective treatment) may cause susceptible individuals to be more cautious and reduce their exposure risk. From these remarks we can talk about two general ways in which behavioral attitudes inﬂuence the spread of a disease:
Estimating Option Prices with Heston's Stochastic Volatility Model
Dr. Hugh Gong
Options are a type of financial derivative. This means that their price is not based directly on an asset’s price. Instead, the value of an option is based on the likelihood of change in an underlying asset’s price. More specifically, an option is a contract between a buyer and a seller. This contract gives the holder the right but not the obligation to buy or sell an underlying asset for a specific price (strike price) within a specific amount of time. The date at which the option expires is called the date of expiration.
Summer 2013 Past Projects
Mathematical Modeling in Ecology: What Killed the Mammoth?
Dr. Alex Capaldi
During the paleolithic, mammoths, as well as other mega mammals, went extinct. The exact reasons for this have been debated for some time, and current hypotheses point to causes such as disease, climate change, over hunting by humans, or some combination thereof. However, recent developments in mathematical ecology may be able to elucidate the matter. A recent study has given strong evidence in support of the hypothesis that the Neanderthals' extinction was due to strong competition from modern humans. The goal of this project is to follow in the footsteps of the Neanderthal study and consider the similar question as to whether the mammoth's extinction was anthropogenic.
Prerequisites: At least one of the following: differential equations, statistics, programming experience.
Estimating the Volatility in the Black-Scholes Formula
Dr. Hui Gong
Black-Scholes formula has been a huge success since it's introduced. However, the formula has a component of the volatility, which is not deterministic and available immediately for application of this formula. We can estimate this volatility via multiple models or methods, such as parametric approaches with different probabilistic models, nonparametric approaches and time series models approaches, such as smoothing techniques. We will derive these estimates and incorporate into the Black-Scholes formula.
Prerequisites: an introduction of statistics, understanding the basic probability distributions (uniform, normal, Chi-square and etc.), the concepts of mean, variance, and correlation.
Dr. Zsuzsanna Szaniszlo
Paper: 4-equitable Tree Labelings
In this project we will look at edge-vertex graphs and investigate a certain assignment of numbers to the edges and vertices. These so-called labeling problems were introduced in the 1960s as possible tools for solving graph decomposition problems. Since then different labelings were studied for their own sake and for other applications as well. In the summer project we will investigate when we can distribute labels (numbers) equally in a graph. Answering this question for trees would settle the famous graceful tree conjecture.
Prerequisite: a proof based course
Summer 2012 Projects
Mathematical Models of Infectious Disease
Dr. Alex Capaldi
Paper: Mathematical Models of Infectious Disease
In a world where infectious diseases, such as the H1N1 flu, bird flu or SARS, pose a tangible public health risk, it is vital to understand the dynamics of outbreaks. We will see how to gain information about epidemics by using an array of mathematical tools including differential equation models, stochastic models and network models. Different control strategies for containing diseases such as vaccination and quarantine and how they can be optimized using mathematics will be introduced. An infectious disease topic from current events (such as the swine flu pandemic, the whooping cough outbreaks in California, or even the mythology of the spread of vampires in a human population) will chosen to be studied in depth. Prerequisites: At least one of the following: differential equations, statistics, programming experience.
Pattern Avoidance in Trees
Dr. Lara Pudwell
Paper: Pattern Avoidance in Trees
A current hot topic in combinatorics is enumerating structures that avoid certain patterns. In 2008, Rowland defined contiguous pattern avoidance in binary trees. In the 2010 VERUM program, Gabriel, Peske, and Tay extended Rowland's results to ternary trees. For all this previous work, the trees were rooted and ordered, and the tree patterns were contiguous. Any of these three conditions can be relaxed to produce a related, but new and as-yet-unstudied counting problem. We will begin by considering trees that contain non-contiguous tree patterns, and work to generalize our results from there. Prerequisites: a course in combinatorics, discrete math, or elementary graph theory would be helpful.
Computer Monitored Problem Solving Dialogues
Dr. Michael Glass
Paper: Computer Monitored Problem Solving Dialogues
Is it possible to monitor small groups of students solving a problem online? A common instructional method in mathematics classrooms is to have students work together exploring some phenomenon while the teacher walks around the room monitoring progress and assisting as needed. As instruction moves online, two issues are how to adequately replace the manipulatives-and-worksheet physical problem- solving environment and how to get the computer to "see" what the teacher sees by looking over shoulders, using clickstream data and the students' conversation. Research questions include sensing student affective states (e.g. bored, engaged, confused), student cognitive states (e.g. what aspects of the math do they understand), and degree and quality of collaboration. Prerequisites: strong programming abilities in a high level language.
Summer 2010 Projects
The logistic two-sex model without pair-formation.
Dr. Daniel Maxin
Paper: The Impact of Sexually Abstaining Groups on Persistence of Sexually Transmitted Infections in Populations with ephemeral pair bonds"
It has been shown that the isolation from reproduction may induce a locally asymptotically stable disease free equilibrium in an endemic situation. For human populations, a gender-structured model includes single individuals (females and males) and couples. In animal populations however, various species do not form stable pairs and the mating is a consequence of direct encounter among individuals of different genders. The main objective of this project is to develop and analyze a two-sex logistic model without pairs and to extend this model to an epidemic model and verify if and under what conditions previous results are still valid. A second objective is to analyze in more detail the mathematical and biological properties of various fertility functions both theoretical and against real data.
Prequisites: Differential Equations
Generalized primes on the Mosaic of an Integer
Dr. Rick Gillman
Paper: Mosiac Arithmetic
The mosaic of an integer is the configuration of primes obtained by repeated applications of the Fundamental Theorem of Arithmetic to a positive integer n and any composite exponents. The concept of the mosaic was introduced by Mullin, in a series of papers in the early 1960s. Various arithmetic functions dependent only on the primes in the mosaic were investigated by Girse and Gillman. During the summer of 2007, a team of undergraduate research students investigated this same structure and developed several potential concepts of divisors, allowing the set of arithmetic functions defined on mosaics to be expanded greatly. This team will continue to develop these notions of divisibility, eventually leading to more fully developed theory of mosaics.
Prerequisites: an elementary number theory course would be helpful.
Pattern Avoidance in Ternary Trees
Dr. Lara Pudwell
Paper: Pattern Avoidance in Ternary Trees
For this project we will consider rooted
ordered trees avoiding other trees. In
2008, Rowland defined pattern avoidance in binary trees (i.e. trees where each
vertex has either 0 or 2 children). The
standard combinatorial sequences appear when enumerating such trees, and there
exist bijections between these trees and other common combinatorial
objects. There is also a method to
determine equivalence classes of binary trees based on the number of trees that
avoid them. This team will extend these
known results about pattern avoidance to ternary trees (i.e. trees in which
each vertex has 0 or 3 children).
Prerequisites: linear algebra, or another proof based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.
Functions on the Mosaic of n
Director: Rick Gillman
Paper: Mosaics: A Prime-al Art
Any integer n can be uniquely factored into a product of prime powers. Each of the resulting exponents greater than 1 can, in turn, be factored into a product of prime powers. Iterating this process until there are no composite exponents results in an array of numbers called the mosaic of n. In this project we will define new arithmetic functions on the mosaic of n, and investigate their arithmetic and algebraic properties.Prerequisites: linear algebra or another proof-oriented course. A course in elementary number theory would be helpful.
Understanding Mathematics Tutoring Dialogue (Computer Science project)
Director: Michael Glass
Paper: Computerized Tutoring
As part of building dialogue-based computer tutors for mathematics, this project works on developing the techniques for computer understanding of student utterances. During the course of a tutoring conversation students can perform discourse actions such as checking whether some idea is true, asking a question, expressing confusion, and so on. Examining transcripts of tutoring sessions, this project will work on software methods for guessing the intentions behind a student's utterance. This summer's experiments may involve both numerical and symbolic methods, such as statistical models, latent semantic analysis, and finite-state machines.Prerequisites: one mathematics class above calculus. Students should have strong programming abilities including experience in a high level language, for example Python or List.
Distributions of Interest Disfor Quantifying Reasonable Doubt and their Applications
Director: Dr. James Caristi
Paper: Distributions of Interest for Quantifying Reasonable Doubt and Their Applications
The concept of reasonable doubt is a standard of our legal system; however, it is a standard that is not well defined. Differences in the way reasonable doubt is applied in different courts and states, as well as ambiguities in its different definitions, suggest that the standard puts pressure on due process and equal protection concerns. This paper explores probability distributions that will aid in the understanding of the American legal system as it is today, what reasonable doubt means under this system, and how reasonable doubt should be defined.
Systems of Matrix Equations
Director: Dr. Patrick Sullivan
Paper: On the Properties A(m,n) for Subspaces of C(kk)
This project will study properties of complex matrices. The students will explore what subsets of matrices have certain properties related to rank one matrices. We will be studying when arrays of matrices can be solved simultaneously as rank one matrices up to a certain equivalence.
Director: Dr. Zsuzsanna Szaniszlo
Paper: L(3,2,1)-Labeling of Simple Graphs
An L(3,2,1)-labeling is a simplified model for the channel assignment problem. It is a natural generalization of the widely studied L(2,1)-labeling.
An L(3,2,1)-labeling of a graph G is a function f from the vertex set of the graph to the set of positive integers such that for any two vertices x,y, if d(x,y)=1 then | f(x)-f(y) | ≥3; if d(x,y)=2, then | f(x)-f(y) | ≥2; and if d(x,y)=3, then | f(x)-f(y) | ≥1. The L(3,2,1)-labeling number k(G) of G is the smallest positive integer k such that G has an L(3,2,1)-labeling number for paths, cycles, caterpillars, n-ary trees, complete graphs and complete bipartite graphs. We also present an upper bound for k(G) in terms of the maximum degree of G.
Mathematical Models in Traffic Assignment and Congestion Pricing
Director: Dr. Lihui Bai
Paper: A Genetic Algorithm for the Minimum Tollbooth Problem
This project uses rigorous mathematical models to study traffic assignment in urban transportation networks. Traffic assignment distributes vehicles in a transportation network so that certain criterion are satisfied. The study of traffic assignment models can be used in traffic congestion management, where traffic planners want to minimize the total travel delay for a given transportation network.
Director: Dr. Kimberly Pearson
Paper: Virtually Cyclic Subgroups of Three-Dimensional Cyrstallographic Groups
An enumeration of the virtually cyclic subgroups of the three-dimensional crystallographic groups is given. Additionally, we offer explanations of the underlying group theory and develop several exclusion theorems which simplify our calculations.
Power Distributions in Weighted Voting Systems
Director: Dr. Rick Gillman
Paper: Using Sets of Winning Coalitions to Generate Feasible Banzhaf Power Distributions
Given a weighted voting system, the Banzhaf Power Index can be used to determine the power distribution of the individual voters. We are interested in the converse of this problem: given a collection of voters, can a weighted voting system be constructed which has a prescribed power distribution? This problem has been solved for a system with four voters, but still open in more general settings.