2013 HRS+ Undergraduate Symposium

36th annual conference, April 19–20, 2013, at Hendrix College

One of the longest-running undergraduate research conferences in mathematics or computer science, the HRS+ Symposium rotates annually between three liberal arts colleges: Rhodes, Hendrix, and Sewanee. Hendrix College is the host in 2013. [History]

Contents:
Directions
Schedule
Selected student abstracts

Directions


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Schedule

Friday, April 19

5:30pm-6:30pm

Dinner in Mills Library (#13 on map)

(best parking is behind Bailey Library (#2))

7:00pm-8:30pm

Welcoming address and invited lecture in M C Acxiom 119 (#15)

Hats, Hamming and Hypercubes
Mike Pinter, Professor, Belmont University, Tennessee
(Hendrix alum, 2012 Tennessee Professor of the Year)

The presentation will begin with a team strategy game using colored hats followed by an error-correcting Hamming code and a corresponding link to hypercubes. We will explore how the game, code and cube are connected to each other as well as being connected to symbolic logic specifically and critical thinking more generally.

8:30pm–

Activities for student guests

Saturday, April 20

9:30am-10:20pm

Student talks in M C Acxiom 119 (#15)

9:00, Michael Todd (Rhodes), “A Non-Euclidean Geometrical Analysis of Music”

9:20, Innocent Bushayija (Hendrix), “On The Gauss-Markov Theorem”

9:40, Matt Cappleman (Sewanee), “The Mathematics of Optical Illusions”

10:00, Edward Fay (BSC), “Practical Folding Solutions to the Fold-Cut Problem”

10:20am-10:40am

Break

10:40am-12:00pm

Student talks in M C Acxiom 119 (#15)

10:40, Christopher Meixell (Rhodes), “An Incentives-Based Analysis of Bidding Rings in Auctions”

11:00, Andrew Conner and Xuejiao Feng (BSC), “Decomposing Polynomials by Composition of Functions”

11:20, Meagan Mansfield (Rhodes), “Modeling the Seasonality of influenza in the United States”

11:40, Lauren Irby (Hendrix), “Paths Through Rectangular Lattice Points”

12:00pm-1:00pm

Lunch in SLTC Bates Room (#18)

Selected student abstracts

Michael Todd (Rhodes), “A Non-Euclidean Geometrical Analysis of Music”

Topology is essentially the study of continuous functions. When analyzing musical properties in various ways, topology is not necessarily the first thing that comes to mind; however, this branch of mathematics is useful in observing the symmetries already present in music. Using the recent studies of Dr. Dmitri Tymoczko, this presentation focuses on the use of manifolds to observe the mathematical and spatial interpretation and characteristics of music. I will display the basic properties and foundations of my topological spaces used to perform analysis on music. Furthermore, I will be presenting the works I am analyzing as well as my observations and data gathered from my examples in this musical application of topological principles.

Innocent Bushayija (Hendrix), “On The Gauss-Markov Theorem”

We consider a population distribution where the dependent variable is a function of several independent variables. Then by random sampling the population, the Sample Regression Function (SRF) slope coefficient estimators are obtained by the method of Ordinary Least Squares (OLS). Then, we prove independently the Gauss-Markov Theorem which states that the SRF estimators, under certain assumptions, are the best linear unbiased estimators of the population slope coefficient estimates.

Edward Fay (BSC), “Practical Folding Solutions to the Fold-Cut Problem”

We present a computer implemented algorithm to align the edges of a polygon drawing onto a common line by a sequence of folds. When cutting along that line, the polygon component is separated from the rest of the paper with a single cut. Our algorithm is a modification and expansion of the simple-fold-and-cut algorithm aimed at computing the minimum necessary fold sequence to fold and cut any polygon containing at least one line of symmetry.

Christopher Meixell (Rhodes), “An Incentives-Based Analysis of Bidding Rings in Auctions”

Bidding rings are thought to be the primary example of collusion in auctions. Since bidding rings act covertly, documentation of ring structure and implementation are virtually nonexistent. Nonetheless, we are able to construct game theoretic auction models that can answer many questions regarding how bidding rings are formed and how they affect auction outcomes. Specifically, I will use these models to describe an incentive-compatible bidding ring mechanism and demonstrate its effectiveness. I will also provide an incentives-based analysis of different bidding ring components using comparative statics. Lastly, I will discuss the collusion-deterrent strategies that an auctioneer could employ and the idea of a collusion-proof auction.

Andrew Conner and Xuejiao Feng (BSC), “Decomposing Polynomials by Composition of Functions”

We explore the notion of decomposing polynomials by composition of functions rather than by the usual factoring by multiplication. We give special attention to the issues of determining whether a polynomial has a nontrivial decomposition and of uniqueness of these decompositions. We link various disconnected papers on the subject into a single narrative. We begin with the seminal work of Ritt from the 1920s, follow with Dorey and Whaples and their attempt to prove Ritt's results through algebraic means, and conclude with recent papers by Rickards and Carter that independently gave much simpler methods and proofs.

Meagan Mansfield (Rhodes), “Modeling the Seasonality of influenza in the United States”

The outbreak of Influenza in the United States fluctuates according to the seasons, and mainly occurs in the winter due to a higher ratio of human-to-human contact rate, drier air, and colder temperatures with relatively low humility. We use the SIR model to show the outbreak according to the different seasons, in order to compare the results and see in which season, the outbreak of Influenza is more likely. Using MATLAB we can generate graphs that map the seasonality of influenza. Using the Latin Hypercube Sampling uncertainty analysis and Partial Ranking Correlation Coefficients sensitivity analysis, we can further see which parameters are vital to the model.

Lauren Irby (Hendrix), “Paths Through Rectangular Lattice Points”

In a rectangular point lattice of dimensions N vertices down and M vertices across, it is possible to define a path through the vertices from the left-hand side to the right-hand side of the lattice. This project explored the properties of these paths, including the number of total paths on a given lattice, the number of paths associated with each node, and the probabilities of these paths.

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