Undergraduate Speaker Session

Session Index

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Session A
2:05pm--3:00pm, McGowan 201
Speakers: Carlie Banchi, Shurong Huang, Chenrui Xu, Nirali Patel, John Gabel, Phillip Pierfy, JunXuan Jiang, Yuqi Zheng, Taylor Byers, Dongjian Hu, Emily Rusack, John Zinzer, William Buhring, Daniel Raskay, Ziqin Xu

Session B
2:05pm--3:00pm, McGowan 204
Speakers: Angela Wesneski, John Kampmeyer, Yike Gong, Nart Shalqini, Zhengyi Xiao

Session C
2:05pm--3:00pm, McGowan 207
Speakers: Courtney Rikoskie, Madison Miller, Nick Kubishin, Ryan Althoff, Daniel Diethrich, Emily Wichert, Lindsay Kane

Session D
2:05pm--3:00pm, McGowan 221
Speakers: James Silva, Kirsten Replogle, Giovanni Di Cicco, Aditya Kommoju, Alexander Vetter, Zbynek Gold, Gillian Evers, Tim White

Session A, McGowan 201

2:05pm, Carlie Banchi, Shurong Huang, Chenrui Xu (Arcadia University)
A Monte Carlo analysis comparing ANCOVA, gain scores, and post-test only methodologies
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Three common techniques for analyzing pretest-post-test designs involving one control and one treatment group are: t-test comparing post-test scores only, t-test comparing gain scores, and analysis of covariance. In this talk we give the results of our R-based simulations, which indicate that the relative power of the techniques is a function of the pretest/posttest correlation coefficient.
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2:16pm, Nirali Patel, John Gabel (Arcadia University)
Using propensity scores in quasi-experimental analyses
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Propensity scores represent the conditional probability of being in the treatment group given the covariate(s). They are used to match the control and treatment groups in the statistical analysis of observational data. We will review several techniques related to propensity scores and will compare them to other techniques including ANCOVA.
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2:27pm, Phillip Pierfy, JunXuan Jiang, Yuqi Zheng (Arcadia University)
Actual Versus Effective Sample Size
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Data are often clustered, such as students in classrooms. In this case, the effective sample size, which is a function of the intraclass correlation coefficient, is less than the actual sample size. Our R-based simulations demonstrate that researchers should use the effective sample size in order to preserve accurate Type-I error rates.
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2:38pm, Taylor Byers, Dongjian Hu, Emily Rusack, John Zinzer (Arcadia University)
An analysis of Women with Type 2 Diabetes Mellitus (T2DM) on Self-Care, Time Management, and Distress
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In reducing potential complications related to T2DM and maintaining health, self-care plays a vital role. In this analysis, we are using SPSS and R to determine the relationships between and among diabetes self-care, time management, and distress in women with T2DM.
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2:49pm, William Buhring, Daniel Raskay, Ziqin Xu (Arcadia University)
Severe Accident Risk Factors
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Using PennDot’s public records, we are investigating risk factors that are most likely to cause major injuries or fatalities. Analyzing severity by looking at the most extreme accidents is an important component of pricing insurance.
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Session B, McGowan 204

2:05pm, Angela Wesneski (Elizabethtown College)
Partial Conjugation of Pure Braids
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Classification of links is a long-standing problem in knot theory. We know from work of Habegger and Lin that classifying links up to link homotopy can be achieved by considering partial conjugations of corresponding pure braids. This presentation will briefly introduce links, braids, link homotopy, and some simple computations in the relevant pure braid group structure.
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2:16pm, John Kampmeyer (Elizabethtown College)
Discrete Approximations of Continuous Markov Chains
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Markov chains are a powerful tool for modeling stochastic processes that evolve over time. These objects are described by transition matrices, which represent the probabilities of transitioning between states at each moment. When the time parameter is continuous, this matrix is the matrix exponential, which is a rather difficult quantity to compute. We can, however, approximate continuous-time Markov chains with discrete-time chains whose transition matrices are easier to work with. In this presentation, we measure how close these approximations are to the matrix exponential in their short-term behavior. We also study the long-term behavior phenomena of stationary distributions and limiting distributions for both of these matrices and how they relate to one another.
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2:27pm, Yike Gong (Franklin & Marshall College)
Two-slit camera
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We analyze the properties of images generated by two-slit cameras by using both projective geometry and coordinate systems. The two-slit camera is a projection model that maps $\mathbb{R}^3$ to $\mathbb{R}^2$ (or $\mathbb{P}^3$ to $\mathbb{P}^2$) through two slits that are skew. As a specialization of the camera models, the two-slit camera is of interest to areas such as multi-view geometry, computer vision, and photogrammetry. We start with a simple model where the picture plane is parallel to both slits that are orthogonal to each other. Then we consider the case where the slits are not parallel to the picture plane.
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2:38pm, Nart Shalqini (Franklin and Marshall College)
Non-unique factorizations
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The fundamental theorem of Arithmetic (FTA) states that every positive integer can be written as a product of primes in an “essentially” unique way. The quest to solving Fermat’s last theorem led to other integer-like systems where unique factorization fails to hold. In this talk, we introduce three number systems that illustrate where the FTA fails. We see that even though FTA fails to hold, the factorization into “prime” elements is much nicer in one of those three than in the other two. In particular, we develop an invariant which measures how far a number system is from having unique factorization.
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2:49pm, Zhengyi Xiao (Franklin and Marshall College)
Gauss-Bonnet theorem applied to Flatland
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In Flatland: A Romance of Many Dimensions written by Edwin Abbott, A Square, the main character, lived in a 2-dimensional world called Flatland where he always wondered about the existence of a 3-dimensional world. Then one day A Sphere, a 3-dimensional creature, brought him to Spaceland, where A Square confirmed what he suspected and saw the whole picture of Flatland. But if no one had helped him, how could he have known the shape of Flatland? Carl Friedrich Gauss came up with the idea of Gaussian curvature in the 19th century and realized that the integral of the Gaussian curvature over a region always equals 2$\pi$ times its Euler characteristic. This relationship is the Gauss-Bonnet theorem published by Pierre Ossian Bonnet in 1848. This presentation is going to show how a two-dimensional creature can apply this theorem to reveal the shape of their universe without stepping into the three-dimensional world.
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Session C, McGowan 207

2:05pm, Courtney Rikoskie (King's College)
Axiom System
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During my first semester at King’s College, I took a course called Logic and Axiomatics. In this course, I developed an axiom system from which I derived three theorems. After gaining more experience in the field of mathematics, I have returned to the system and revised it. I will be presenting the system, its consistency models, and its independence models. Further, I will prove one of the theorems.
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2:16pm, Madison Miller (King's College)
Axiom System Project
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When I was a freshman at King’s College, I took Logic and Axiomatics. This class required us to create our own axiom systems as a part of the final project. Three years later, I had the opportunity to revisit the project. Over the past few weeks, I worked on revising my axiom system. It was rewarding to improve the earlier versions and I can see the system has developed over the years into the final version.
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2:27pm, Nick Kubishin (King's College)
Axiom System Project
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As a freshman student at King’s College, I had the pleasure of taking a class called Logic and Axiomatics. In that class, I created an axiom system as the final project for that class. Now, as a junior mathematics major, I have revisited my axiom system and made refinements and corrections from knowledge I have gained from three years of mathematics courses at King’s. I have constructed and will be presenting my axioms, a consistency model, independence models, and a theorem that I have proven based on these axioms.
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2:38pm, Ryan Althoff, Daniel Diethrich, Emily Wichert (Messiah College)
Statistics on Type-B Permutation Tableaux
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Type-B permutation tableaux are combinatorial objects consisting of rows and columns of cells that are filled according to certain rules, much like a Sudoku puzzle. These objects, introduced by Lam and Williams, have interesting applications in particle physics and biochemistry. In this talk, we will present multiple statistics on type-B tableaux, which highlight the distinctive properties of these objects. We will make connections between our results on type-B tableaux and those previously determined for regular permutation tableaux and will conclude with a brief discussion of the aforementioned applications.
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2:49pm, Lindsay Kane (Misericordia University)
Fibonacci Graphs - An Investigation
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The sequence of Fibonacci numbers is a well-known, easily defined sequence of integers $(f_{0}=1,f_{1}=0,f_{n}=f_{(n-1)}+f_{(n-2)} \mbox{for } n>1)$. Despite its simple look, this sequence has connections to art, nature, the stock market, and many areas of mathematics. One way that people consider properties of the Fibonacci numbers is by determining a quantity that these numbers count. For example, one popular quantity is that $f_{n}$ counts the number of tilings on a strip of n spaces by squares and dominoes. For example $f_{3}=3$ and there are three possible tilings of a strip of 3 spaces. A Fibonacci graph $F_{n}$ is a graph with a vertex for each such tiling of n spaces and has an edge between two such tilings if one can be obtained from another by switching a domino with a pair of squares (or vice versa) or by switching the order of an adjacent pair of a square and a domino. Fibonacci graphs are a new construct (as far as is currently known) and therefore they needed to be investigated.
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Session D, McGowan 221

2:05pm, James Silva (Muhlenberg College)
The Topological Structure of a Smooth Function
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A smooth (infinitely differentiable) function $f$ of $n$ variables is called a Morse function, if every critical point is non-degenerate (the Hessian) is different from zero at this point. For a Morse function of two variables, all critical points are local maxima, local minima, or saddle points. It is not difficult to show that a Morse function on a compact domain has finitely many critical points. The topological structure of a two-variable Morse function on a compact domain can be associated with a tree, called an $A$-tree, whose vertices are the connected components of the level surfaces of the function. Each vertex is of degree 1 (local maximum or minimum) or 3 (saddle point). Counting the number of Morse functions associated with each tree is equivalent to playing a game of ``plates and olives'' in which a plate or olive is added, combined, or removed depending on whether the associated vertex is a local maximum, local minimum, or saddle point. In this talk we consider the minimum number of ways in which a game of plates and olives can be resolved to an empty plate in $m$ moves, if the starting position contains $k$ plates and $l$ olives for a specific family of $A$-trees.
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2:16pm, Kirsten Replogle (Cedar Crest College)
Using Network Theory to Model Disease Spread in Animal Populations
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Network theory is a part of graph theory used to study relationships in different fields. It is used in epidemiology to study the spread of diseases. Network theory can be used to predict the spread based upon the centralities of different nodes. This project is focusing on disease patterns in wildlife populations. Specifically, looking at the spread of chronic wasting disease in white tail deer populations. The graph chosen to start with was created though a random graph generator and has 17 nodes. The approach is to use different types of centralities to determine the impact of certain nodes on the spread of a disease. The centralities investigated were degree, betweeness, and closeness centrality.
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2:27pm, Giovanni Di Cicco, Aditya Kommoju (Penn State Brandywine)
Exploring and Extending the Impossible Card Location Trick
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Typically, executing a magic trick involves some sort of sleight-of-hand manipulation, or perhaps, a prop with a secret compartment. However, the truly magical tricks involve a third technique - mathematics.* In this talk, we explore a variation of the “Impossible Card Location Trick.” In this classic card trick, a spectator selects three cards at random while the magician deals three piles with the remaining cards. The spectator puts their chosen cards on the piles and then hides them away by making a cut in each deck. The magician stacks the cards all together and successively splits the deck until the chosen cards are miraculously the last to appear. In this talk, we demonstrate our modified version of this trick and provide a detailed proof of the mechanics behind it. Based on these techniques, we also give an extension of the trick to allow the spectator to choose the number of cards in the initial pile dealt by the magician. *Based on Doug Ensley’s classification of magic as described in his invited talk at the Spring 2014 EPaDel meeting.
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2:38pm, Alexander Vetter, Zbynek Gold, Gillian Evers (Villanova University)
Coefficients of the Peak Algebra
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We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in S_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. We can partition the set of permutation in $S_n$ according to the positions in which the permutations have peaks. If we then formally sum the permutations that have peaks at the same positions, the elements we get form a linear basis for a subalgebra (called the peak algebra) of the group algebra of $S_n$ over $\mathbb{Q}$. In this talk, we explore the coefficients that appear as you expand the product of two such basis elements as a linear combination of all basis elements.
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2:49pm, Tim White (Elizabethtown College)
An Alternative Way to Price Exotic Financial Options
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One of the most important aspects of financial options is how they are priced. Although there are a variety of methods for pricing basic financial options, the two of the most utilized are the Binomial Option Pricing method and the Black-Scholes Formula. The Binomial Option Pricing method requires the assumption that asset prices only increase or decrease by a certain amount in a time-period. This method also requires the creation of binomial trees to track the asset and option prices. In contrast, the Black-Scholes Formula is a general formula and does not hold the assumption that stocks only go up or down by a certain amount. When looking at more complex exotic options, they are almost always priced via the Black-Scholes Formula. This is partly because the Binomial Option Pricing method is too calculation-heavy; however, through programming languages, such as R, some computations of the Binomial Option Pricing method become more feasible. Due to this, comparisons between these two pricing methods can be made for exotic options.
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