Session Index
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Applied
9:40am--11:05am
Speakers: Carly Flickinger, Emily Lucy, Annabelle Rogers, Tess Weber, Erik Edwards, Kody Ream, Luke Thatcher, Zian Liu, Qi Miao, Yannan Niu, Xincheng Zhu, Hao Huang, Youyuan Kong, Zixue Wang, Shengxi Zhang, Keeran Ramanathan, Dan Fauni
Combinatorics and Friends
9:40am--11:20am
Speakers: Breille Duncan, Peter Kanjorski, Morgan Zimmerman, Ian Parzyszek, Zhuolin Yao, Lucas Acosta-Morales, Andrew Clickard
Modeling
9:40am--11:20am
Speakers: Jessica Parente, Kiersten Grieco, Brittany Gelb, Lisa Cenek, Gillian Rose McGuire, Madison Kuduk, Evan Malcolm, Jessica McKay
Applied
9:40am, Carly Flickinger, Emily Lucy, Annabelle Rogers, Tess Weber (Arcadia University)
Is there evidence that students cheated when Arcadia’s math placement test went online?
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In the past, incoming first-year Arcadia students took the math placement test in-person under strictly proctored conditions. Due to the pandemic, last summer the test was administered online, unmonitored, with no requirement that students leave their cameras on. We employed several statistical techniques, including ANCOVA, to investigate whether these changes led to a different distribution of scores. We will report our findings and also discuss our experiences using JASP, and Jamovi, two free and open R-based point-and-click alternatives to SPSS.
Close Abstract9:55am, Erik Edwards, Kody Ream, Luke Thatcher (Arcadia University)
A Monte Carlo analysis comparing ANCOVA, gain scores, and post-test only methodologies
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Three common techniques for analyzing pre-test 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.
Close Abstract10:10am, Zian Liu, Qi Miao, Yannan Niu, Xincheng Zhu (Arcadia University)
Health Index-Based Insurance Design for NEW·WORLD
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Our team designed a novel health insurance product by constructing a real-time health surveillance system for insurers in which they can assess a policyholder’s latest health status and make a quick payout if a certain condition has been met, regardless of the actual spending. Through analyzing a CDC survey data set related to behavioral risks, we trained a multinomial logistic regression model. Using the health condition classifications from the model, we further constructed pricing strategies and financial projections for our client NEW·WORLD.
Close Abstract10:35am, Hao Huang, Youyuan Kong, Zixue Wang, Shengxi Zhang (Arcadia University)
Parametric Insurance Product Design
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This report presents a design of a parametric insurance product for individual consumers in two neighboring countries Ambernϊa and Palȍmϊnϊa. Unlike traditional insurance, this product issues a predetermined payout to a policyholder when a pre-agreed event has been triggered. By conducting analyses on given health data in the countries, we first projected individual losses and calculated premiums according to gender, age and risk factor information. Then we defined triggering events and modeled the payout scheme for our product. Comprehensive strategies are also provided for marketing and risk mitigation.
Close Abstract10:50am, Keeran Ramanathan, Dan Fauni (University of the Sciences)
Playing the Changes: A Graph Theory Analysis of Chord Progressions in Hit Songs
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We present an analysis of the music theory of hit songs through the lens of graph theory. It is common in modern music for a song to begin and resolve on the same chord, the root chord of a respective key signature. By assigning vertices to the natural chords in a given key signature, and drawing directed edges for each chord change in a song, we can map the chord progression of songs in a visual format. We hope to be able to categorize the harmonic nature of popular songs from throughout the ages through graphs in this analysis.
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Combinatorics and Friends
9:40am, Breille Duncan (Cedar Crest College)
Exceptional Totient Numbers
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Let us denote the set of positive integers less than $n$ and relatively prime to $n$ as $R(n)$. If $R(n)$ can be partitioned into two sets that have equal sums, $n$ is called super totient. We will discuss exceptional totient numbers, which are defined similarly, and provide a classification for them.
Close Abstract9:55am, Peter Kanjorski (Kutztown University)
Counting the Number of Complete Bipartite Subgraphs in Three-Dimensional Monomial Graphs
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Graph Theory is a mathematical subject that is as intrinsically beautiful as it is applicable in today’s modern, data driven world. Indeed, without its study, much of our understanding of logistical and computer-based networks would not exist, forfeiting the speed and convenience we have come to expect from the services we engage with every single day. One useful insight into graphs is the ability to compare them to each other. If two graphs can be shown to be equivalent, then any knowledge we have of one could be applied to the other. That is where our research comes in. Our investigations centered around algebraically-defined graphs; in particular, we counted how many specific subgraphs each of these graphs can have. Having this knowledge can definitively show that two graphs are not equivalent if they have different counts, or can be a hint that two nonequivalent graphs may have some useful commonality.
Close Abstract10:10am, Morgan Zimmerman, Ian Parzyszek (Messiah University)
Staircase Tableaux
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In this presentation, we will discuss our research on staircase tableaux, mathematical objects that have applications in physics and biochemistry. Staircase tableaux are constructed in a similar way to Sudoku puzzles. Boxes are aligned in a staircase shape and these boxes are filled with alphas and betas depending on a few simple rules. In our research, we determined the probability of staircase tableaux that have an alpha followed by a beta on the main diagonal as well as a beta followed by an alpha. These results are interesting because of the applications of staircase tableaux in other fields.
Close Abstract10:35am, Zhuolin Yao (Muhlenberg College)
Gauss's Reduction Theory and Almost Pythagorean Triple
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Let $f(x,y)$ be a binary quadratic form on $x$ and $y$, that is,
\[f(x,y) = ax^2+bxy+cy^2\]
where $a, b, c \in \mathbb{Z}$. We defined \emph{the discriminant $\Delta(f)$ of} $f$ to be
\[\Delta(f)= b^2-4ac.\]
Recall the group $SL(2,\mathbb{Z})$ consists of 2$\times$2 integral matrix $M$ whose determinant is 1. For $M$ in $SL(2, \mathbb{Z})$, we define another form $Mf$ by setting $Mf(x, y) = f(M [x$ $y])$ where $[x$ $y]$ is a column vector. This defines an equivalent relation on the set of all binary quadratic forms; if $g$ is another form with $g = Mf$ for some $M$ in $SL(2, \mathbb{Z})$, we say that $g$ and $f$ are equivalent, which we denote by $g \sim f$. Then we can show that the forms in the same equivalence share the same discriminant. Gauss proved that, for a fixed $D$, the number
$h(D)$ (called the $\textit{class number}$) of
equivalence classes in the set
\[
\{ f(x, y) \mid \Delta(f) = D \}
\]
always finite.
We defined $(a, b, c)$ is an Almost Pythagorean Triple (APT) if it satisfies $a^2+b^2=c^2+D$ where $a$, $b$, and $c$ are all positive integers without common factors and fixed integer $D$, $D\ne0$. We do the analog of Gauss's reduction theory to APT in this research. If we define two APTs to be equivalent whenever they are connected by the action of certain orthogonal matrices, we can define the $\textit{class number}$ $h(D)$ of APT to be the number of equivalence classes, that is, the number of connected components.
Close Abstract10:50am, Lucas Acosta-Morales (DeSales University)
Properties of Double Choco Puzzles
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We explore a relatively new Japanese puzzle called Double Choco from the perspectives of both Computer Science and Mathematics. On the computing end, we consider various algorithms that aim to solve these puzzles. On the mathematical side of the project, we present properties of solvable as well as unsolvable puzzles.
Close Abstract11:05am, Andrew Clickard (Bloomsburg University of Pennsylvania)
Synthetic Geometry in Simplices with Constant Curvature
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Previous work has been done in metric geometry to find an intrinsic method by which the distance between two points in an $n$-dimensional simplex can be calculated. The natural question, then, arises concerning simplices embedded in spaces with constant positive or negative curvature. We give a simple technique to compute the distance between two points in an $n$-dimensional simplex of constant curvature given only the points' barycentric coordinates and the edge lengths of that simplex, as well as a simple test to verify the legitimacy of a set of positive real numbers as edge lengths of an $n$-dimensional simplex of constant curvature. We also determine a method of finding the barycentric coordinates of the orthogonal projection of a point in a simplex.
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Modeling
9:40am, Jessica Parente (King's College)
Axiom System: Player, Dart, Throw
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Axioms are statements that are accepted to be true, and are used to prove theorems. Euclid created an axiomatic system with five axioms, in which he was able to derive all theorems in Euclidean geometry. I have constructed an axiom system that uses the terms dart, player, and throw. During this presentation, I will provide the consistency model, the independence models, and the proof of one theorem from my axiom system.
Close Abstract9:55am, Kiersten Grieco (King's College)
K5 Axiom System
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An axiomatic system consists of a set of accepted truths, called axioms, used to prove other truths that can be concluded based upon the axioms. These truths are called theorems. This presentation explores the axiom system I developed in a freshman-level logic class. It will include a consistency model, independence models for each axiom, and a full presentation of a proven theorem derived from the axioms.
Close Abstract10:10am, Brittany Gelb, Lisa Cenek (Muhlenberg College & Amherst College)
Interactive Theorem Proving with Lean
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The Lean proof assistant is a tool for verifying and generating proofs using a computer. In the past few years, mathematicians have started using Lean to formalize important results: Gusakov, Mehta, and Miller formalized the proof of Hall’s Marriage Theorem, and Kontorovich and Gomes formalized the statement of the Riemann Hypothesis. In this talk, we will introduce and demonstrate interactive theorem proving. We will share our motivation for learning about proof assistants, and we will discuss our process of formalizing results about primitive Pythagorean triples in Lean.
Close Abstract10:35am, Gillian Rose McGuire (Temple University)
Computational Analysis of the Agreement Between Common and Advanced Voting Methods
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This talk will focus on an analysis of several common and advanced voting systems. The primary source of experimental data is a program written to repeatedly simulate the ranked preferences of members of electorates and then compute the resulting election outcomes using a variety of these systems. The talk will highlight the frequency with which these systems generate the same outcome. The talk will include some basic examples and background theory. We hope to increase awareness of advanced voting systems and to share some insights that could inform the audience's understanding of their utility.
Close Abstract10:50am, Madison Kuduk (Arcadia University)
Modeling gene expression using differential equations
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Gene expression is the process by which the information stored in DNA is converted into a functional gene product, such as protein. In simple cells, this process can be represented as a system of first order differential equations that expresses the rate of change of mRNA and proteins. This model provides insight on the interrelation of transcription and translation, the fundamental functions in gene expression, and acts as a starting point towards a more advanced model for higher order cells.
Close Abstract11:05am, Evan Malcolm, Jessica McKay (University of the Sciences)
Matrixes and the Bond Structure of Molecules
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Graph theory, and specifically the use of matrices, have an incredible application in the chemical sciences. The molecular structure of any compound can be derived from or into a matrix based off of the connections and orientations of the chemical bonds. Such analysis can give way to further understanding and prediction of physicochemical and pharmacological properties of the compound depending on application. Some attributes that can be learnt from this analysis include viscosity, surface tension, and refractive indices. We review the basics of examining molecular topology with this approach and discuss possible applications.
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