- Past Projects
**Summer 2014 Past Projects****Pattern Avoidance in Double Lists**

Dr. Lara PudwellA permutation of length n is an arrangement of the numbers {1,...,n}. There are n! permutations of length n (for example, the 3! permutations of {1,2,3} are 123, 132, 213, 231, 312, and 321), but it is more interesting to answer questions of the form "How many permutations have (a given property)?"

Pattern-avoiding permutations have been a popular and increasingly useful area of study over the past few decades. Permutation p (of length n) contains permutation q (of length m) if there exist m digits of p that are in the same relative order as the digits of q. For example, p=15243 contains two copies of q=312 because the digits 524 (resp. 523), like 312, are in the order large, small, medium. If p doesn't contain q then p is said to avoid q.

The number of permutations of length n avoiding 12 (resp. 21) is 1. The number of permutations of length n avoiding 123 (or any other pattern of length 3) is given by binomial(2n,n)/(n+1). Many other variations and generalizations have been studied that produce nice enumerative (counting) results.

In this project we will look at special lists of the numbers {1,...,n} that contain 2 copies of each digit, and determine the number of such lists that avoid a given pattern.

**The Influence of Risk-taking Behavior on the Evolution of Infectious Diseases****Dr. Daniel Maxin**We study several epidemic models where the infection transmission term includes a risk-taking behavior component. By this component we mean the totality of measures that an individual will undertake in order to either avoid or reduce his/her chances of infection if the disease is perceived to be dangerous or to increase the exposure risk if the disease is perceived as mild. The main objective is to study the influence of risk-taking behavior in a variety of models (both one-sex or gender-structured). Of particular importance is investigating whether this behavioral component may change the course of the epidemic from disease clearance to persistence.

**Estimation of Option Prices with Stochastic Volatility Models****Dr. Hui Gong**An option gives you a right to buy or sell an asset in a future time at a contract price. However, you need to purchase this right. So how much should you pay to buy this option in that you still can profit in the future? For any asset, its prices are time series data and can be modeled by different time series models. Recently there has been a growing interest in using stochastic volatility models in pricing options. This project is to study several stochastic volatility models. The research will include: deriving option pricing formulas for these stochastic volatility models; estimating the parameters in these models with the assistance of software MATLAB and/or R; and comparing the pricing outcomes. The project will also use simulation-based technique to investigate the correlations between the option pricing and the parameters of the stochastic volatility models.

**Summer 2013 Past Projects****Mathematical Modeling in Ecology: What Killed the Mammoth?**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.

Dr. Alex CapaldiPrerequisites: At least one of the following: differential equations, statistics, programming experience.

**Estimating the Volatility in the Black-Scholes Formula**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.

Dr. Hui GongPrerequisites: an introduction of statistics, understanding the basic probability distributions (uniform, normal, Chi-square and etc.), the concepts of mean, variance, and correlation.

**Graph Labelings**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****Quandles and Generalized Colorings of Knots**

Dr. Paul Drube**Paper: A Partial Ordering of Knots**Knot theory is a branch of topology that studies how one may embed a circle within three-dimensional space. Since knots may become quite complicated, much of modern knot theory involves the development of invariants that can distinguish between different equivalence classes of knots. One basic class of knot invariants are the Fox n-colorings of knots, which count the ways that one may appropriately “decorate” the strands of a knot diagram with elements of a cyclic abelian group. In recent years, the mathematical structure underlying these Fox n-colorings has been greatly generalized utilizing abstract binary operations known as quandles. After a brief introduction to knot theory, this project will explore basic properties about quandles and their associated knot invariants, hopefully developing interesting results about the generalized colorability invariants of specific classes of knots.Prerequisites: linear algebra; an introductory proof course or basic abstract algebra (group theory) may also be helpful.**Vertical transmission in two-sex epidemic models with isolation from reproduction**

Dr. Daniel Maxin

Paper: Is more better? Higher sterilization of infected hosts need not result in reduced pest population sizeVertical transmission is the transmission of an infection from mother to child at birth. During the academic year 2008-2009, Tim Olson and Adam Shull (students at Valparaiso University) worked on a project with me to study the influence of isolation from reproduction on sexually transmitted infections with vertical transmission. One of the questions addressed in their project was the behavior of the disease in the case of 100% vertical transmission (i.e. all newborns from infected mothers acquire the disease). A surprising result showed that, under several conditions on the vital parameters, the isolation from reproduction may prevent a Susceptible Extinction situation (a case when the disease eliminates the entire healthy population) even when the infection rate is large. For simplicity the model did not separate the individuals by gender (to keep the number of equations low). Our project consists in establishing a two-sex version of this model that follows the dynamics of females and males. The main objective is to investigate if the result described above holds in the more realistic frame-work of gender structured models and whether the result is independent from various types of mating functions commonly used in the literature. A secondary objective is to improve the original model with features that were not included so far such as: diseases with recovery or temporary isolation from reproduction.Prerequisites: Differential Equations**Generalized Pattern Avoidance in Trees**

Dr. Lara PudwellA current hot topic in combinatorics is enumerating structures that avoid certain patterns. In the 2010 VERUM program, Gabriel, Peske, and Tay studied contiguous pattern avoidance in ternary trees. In the 2011 VERUM program, Dairyko, Tyner, and Wynn studied non-contiguous pattern avoidance in binary trees. There are a number of variations on tree-pattern avoidance that remain to be explored by modifying what types of trees or what types of patterns are considered. We will consider a new type of tree pattern that is a hybrid of contiguous patterns and non-contiguous patterns. In particular, counting problems in the 2010 project always resulted in algebraic generating functions while counting problems in the 2011 project always resulted in rational generating functions. We will try to characterize whether hybrid patterns fall under the algebraic or rational paradigm and extend our work from there.Prerequisites: linear algebra, or another proof based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.**Summer 2011 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**Computer Monitored Problem Solving Dialogues

Paper:

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**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.**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"Prequisites: Differential Equations

The mosaic of an integer is the configuration of primes obtained by repeated applications of the Fundamental Theorem of Arithmetic to a positive integer**Generalized primes on the Mosaic of an Integer**

Dr. Rick Gillman

Paper: Mosiac Arithmetic*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.In this project we will investigate analytic (closed form) solutions to three dimensional steady state groundwater flow problems. While numerical solutions to groundwater flow problems are more commonly used in practice, especially in the three-dimensional realm where analytic solutions are cumbersome or even nonexistent, the construction of some analytic solutions is essential so that numerical models can be validated. The specific problem at hand will involve flow to a well in a stratified aquifer, and/or interaction between a horizontal well and a surface water body.**Summer 2007**r

Groundwater Flow

Director: Ken Luthe**Paper:**Self-Influencing Interpolation in Groundwater Flow*Prerequisites*: multivariable and vector calculus, linear algebra, and differential equations.Exposure to partial differential equations is a plus, and experience with MATLAB is a double plus.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.**Functions on the Mosaic of n**

**Director: Rick Gillman**

**Paper:**Mosaics: A Prime-al Art*Prerequisites*: linear algebra or another proof-oriented course. A course in elementary number theory would be helpful.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.**Understanding Mathematics Tutoring Dialogue**(Computer Science project)

**Director: Michael Glass**

**Paper**: Computerized Tutoring

*Prerequisites*: one mathematics class above calculus. Students should have strong programming abilities including experience in a high level language, for example Python or List.**Summer 2006**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.**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

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.

**Systems of Matrix Equations**

**Director: Dr. Patrick Sullivan**

**Paper**: On the Properties A(m,n) for Subspaces of C(kk)

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.

**Graph Labelings**

**Director: Dr. Zsuzsanna Szaniszlo**

**Paper:**L(3,2,1)-Labeling of Simple Graphs

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.**Summer 2005**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.**Mathematical Models in Traffic Assignment and Congestion Pricing**

**Director: Dr. Lihui Bai**

**Paper:**A Genetic Algorithm for the Minimum Tollbooth Problem

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.**Crystallographic Groups**

**Director: Dr. Kimberly Pearson**

**Paper:**Virtually Cyclic Subgroups of Three-Dimensional Cyrstallographic Groups

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.**Power Distributions in Weighted Voting Systems**

**Director: Dr. Rick Gillman**

**Paper:**Using Sets of Winning Coalitions to Generate Feasible Banzhaf Power Distributions