Session Index
Click a session title to jump to the abstracts.
Session I-A
2:05pm--3:05pm, Gambet 206
Speakers: Megan McGee, Alvaro Rafael Belmonte, Rey Anaya, Sabrina Traver, William Baar, Chooka Weiss, Sam Mathers
Session I-B
2:05pm--3:05pm, Gambet 214
Speakers: Vanessa Maybruck, Gillian Rose McGuire, Joseph Franks, Peter Liu, Laura Seaberg
Session I-C
2:05pm--3:05pm, Gambet 224
Speakers: Caroline Pathappillil, Malcolm Kenyon, Dhanush Rajesh, Charles Kulick, Beth Anne Castellano, Megan Osborne, Edmond Mbadu
Session I-D
2:05pm--3:05pm, Gambet 232
Speakers: Aubrey Schall, Halee Kemper, Benjamin Skultety, Christopher Blum, Dhruv Dhingani, Chance Mezzaroba, Shabnoor Begum, Kyle Brunner, Maria Bouda, Steven Simpkins, Steven Boothe, Mathew Becker, Teresa Simon , Shea Moore
Session I-A, Gambet 206
2:05pm, Megan McGee (Millersville University)
Study of the recovery of nesting bald eagle population in New Jersey by statistical and mathematical models
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By the 1970’s, New Jersey’s eagle population diminished to just one known nest. Due to the ban of DDT and restoration efforts, New Jersey's eagle population has increased to 185 active pairs by 2018. The NJ Division of Fish and Wildlife’s Endangered and Nongame Species Program (ENSP) have monitored nesting bald eagles since 1982. Data collected was used to quantify the recovery of the nesting population. Mathematical and statistical models were built to analyze the growth of nesting population and productivity rate. The population's asymptotic growth rate was calculated and the carrying capacity was predicted. The results indicate that the bald eagle population in New Jersey grows on average by approximately 18.8% per year. The best model fitting predicts that the active nesting bald eagle population will reach the carrying capacity, which was predicted to be 250, at or after 2036.
Close Abstract2:15pm, Alvaro Rafael Belmonte, Rey Anaya (Morovian College)
2-Super Bondage Number and Network Reliability.
View Abstract
We propose and investigate a generalization of the bondage number of the graph called the $k$-Super Bondage ($k$-SB). The $k$-SB of a graph is the smallest number of edges that, when removed, increases the dominating number by at least $k$. We will show the 2-SB for some graph classes and also show the minimal value of 2-SB over the class of graphs, $G(n,m)$, which consists of graphs with $n$ vertices and $m$ edges.
Close Abstract2:25pm, Sabrina Traver (King's College)
Latin Squares, Transversals, and Determinants: When Sudoku and Linear Algebra Cross Paths
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Latin squares are $n \times n$ dimensional squares in which numbers $1,2,...n$ appear without repetition in each row and column. A Latin square's transversal is any set of $n$ cells containing $\{1, 2, \dots, n\}$ such that no two cells are in the same row or column. Let $D_n$ be the set of all determinants of $n \times n$ Latin squares. In a summer REU project, we analyzed properties of these Latin squares, including properties of $D_n$ and transversals. In the talk, we will discuss the OEIS sequences derived from $D_n$ and the number of transversals an $n \times n$ Latin square has. This research was supported by the National Science Foundation under grants numbered DMS-1852378 and DMS-1560019.
Close Abstract2:35pm, William Baar (Muhlenberg College)
Classifying Configuration Graphs of the Class-0 Petersen Graph
View Abstract
Consider a configuration $S_{G}$ of pebbles on a simple, connected graph $G$. A standard pebbling move removes $2$ pebbles from a vertex in $V(G)$ and adds $1$ pebble to an adjacent vertex. A configuration graph $[S_{G}]$ associated with a configuration $S_G$ is a Hasse diagram whose vertices represent subsequent configurations that can be reached from $S_G$, and whose edges correspond to a single pebbling move in $G$. This talk concentrates on the classification, up to isomorphism, of configuration graphs for the Petersen graph with configurations of ten pebbles.
Close Abstract2:45pm, Chooka Weiss (Muhlenberg College)
Exploration to the Pythagorean Quintuple
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The Pythagorean equation is an equation of the form $x_1^{2}+x_2^{2}+\dots+x_{a−1}^{2}=x_a^2$. The most well known Pythagorean equation is $a^2+b^2=c^2$. All solutions to this equation are Pythagorean triples. This is a 2 dimensional right triangle. If you add another dimension, the equation becomes $a^2+b^2+c^2=d^2$. The solutions to this new equation are called Pythagorean quadruple. When extended again, the equation becomes $a^2+b^2+c^2+d^2=f^2$ and the solutions are called Pythagorean quintuples. It is known all Pythagorean triples and quadruples have a tree-like structure. We show that a similar tree-like structure does not extend to the Pythagorean quintuples.
Close Abstract2:55pm, Sam Mathers (Princeton University)
Transitivity on the Calogero-Moser Variety $C_2$
View Abstract
In this talk, I will discuss some of the key definitions, motivations, and results from an REU project I participated in this past summer. In this project, we considered the Calogero-Moser Varieties $C_n$, defined as orbit spaces of the set $\{(X,Y)\in\mathbb C^{n\times n}\times\mathbb C^{n\times n}: \operatorname{Rank}[XY-YX+I_n]\leq 1\}$ under the group action of $GL_n(\mathbb C)$. These emerge through the study of $n-$particle systems in $\mathbb R$ evolving in time. Since Calogero conjectured such systems are solvable and proved so using matrices of the above form, these matrix sets have been studied in detail, where it was shown that the elements of the space are expressible as polynomials in matrix traces. In our project, we considered the variety of dimension four, which we proved can be expressed as the quotient $C_2:=\mathbb C[x_1,...,x_5]/(x_4^2-x_3x_5-1)$. We then define an action by the group of automorphisms of the free algebra $\mathbb C\langle x,y\rangle$ on $C_2$. It was proven by Berest, et al. that this group action is doubly-transitive, and they conjectured that it is infinitely transitive, which is what our work attempts to prove. We succeed in developing a method that allows us to prove up to seven-transitivity, and which should be generalizable to $n-$transitivity for any positive integer $n$. Specifically, barring the proof of a technical lemma, we indeed show that the action is infinitely transitive.
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Session I-B, Gambet 214
2:05pm, Vanessa Maybruck (Kutztown University of Pennsylvania)
An Analysis and Extension of the Fontan Procedure Model for Cardiovascular Flow
View Abstract
Certain congenital heart defects can lead to the development of only a single pumping chamber, or ventricle, in the heart instead of the usual two ventricles. Individuals with this defect undergo a corrective, three-part surgery, the third step of which is the Fontan procedure, but as the patients age, the Fontan procedure will likely fail. Using computational fluid dynamics and differential equations, Fontan circulation can be modeled to investigate why the procedure fails and how the failure can be maximally prevented. Borrowing from well-established literature on RC circuits, the differential equations models simulate systemic blood flow in a piecewise, switch-like fashion. Here, we generalize the switch system to a 1-D case and investigate the effects of changing parameters on the ODE model. Furthermore, we consider an extension of the ODE model to a PDE model in one spatial dimension and investigate the effects of certain parameters on the overall system.
Close Abstract2:15pm, Gillian Rose McGuire (Temple University)
The Frequency of Agreement Between Advanced Voting Systems
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This talk will cover a few advanced voting systems well recognized by literature with the intent of increasing understanding of how these systems work. The talk will use some constructed examples to illustrate our questions about these systems. We will discuss relevant theory such as Arrow’s Theorem and Sen’s examples. Then, we will explain our experiments and show some of our data related to these questions.
Close Abstract2:25pm, Joseph Franks (Temple University)
Expected Value, Average Trajectory and Minimal Cycles of the 3n+1 Problem
View Abstract
The 3n+1 Problem, also known as the Collatz Conjecture, has been famously referenced by Erdos as “absolutely hopeless.” In this talk, we show several results on average trajectories and minimal cycle length. We will show our data on the difference between expected and measured average trajectories. We will also share some updates on published and unpublished work relating to minimal potential cycle length and least escaping numbers.
Close Abstract2:35pm, Peter Liu (Franklin and Marshall College)
Finding Solutions of $a^{\sqrt b} = b^{\sqrt a}$, $(a,b\in \mathbb Z^+)$ using Calculus
View Abstract
I will show how to solve the equation Finding Solutions of $a^{\sqrt b} = b^{\sqrt a}$, where $a$ and $b$ are positive integers, using calculus. My inspiration comes from the equation $a^b = b^a$, which my professor taught and is often taught in Calculus class.
Close Abstract2:45pm, Laura Seaberg (Haverford College)
An Analogue of K-marked Durfee Symbols for Strongly Unimodal Sequences
View Abstract
We apply methods from Andrews's work on partitions to another combinatorial object: strongly unimodal sequences. Specifically, we define ``$k$-marked unimodal symbols" for unimodal sequences analogously to Andrews's definition of $k$-marked Durfee symbols for partitions. We establish a multivariate rank generating function $U_k(\bf{\zeta_k};q)$ for $k$-marked unimodal symbols, as well as $\mathcal{SCU}_k(q)$ for self-conjugate $k$-marked unimodal symbols, which we also interpret combinatorially in terms of partitions. We then discuss potential quantum modularity properties for $U_k(\bf{\zeta_k};q)$ for certain vectors of roots of unity $\bf{\zeta_k}$, including determining when $U_k(\bf{\zeta_k};q)$ can be defined as a function on a subset of rationals. We conclude with some further observations based on computational data and conjectures.
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Session I-C, Gambet 224
2:05pm, Caroline Pathappillil (Penn State Brandywine)
The Mathematics behind Medical Imaging
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A great deal of abstract mathematics goes into recreating complex structures of the body from images taken with an X ray or a CAT scan. Based on material from Tim Freeman’s book “The Mathematics of Medical Imaging,” I will start by explaining the basic behavior of different machines, such as X rays, as well as some of the math involved in recreating an image. The goal in image reconstruction is to reconstruct the attenuation coefficient function from a set of line integrals that the machine is able to capture. This reconstruction process involves concepts including Beer’s Law, the Radon transform, back projection, the Fourier Transform, and much more. In this talk, I will give a brief overview of this process.
Close Abstract2:15pm, Malcolm Kenyon (Penn State Brandywine)
Expansion of Martin Gardner’s Dollar Bill Trick
View Abstract
In Martin Gardner’s book “Mathematics, Magic and Mystery,” he explains a trick where you can determine the serial number of a dollar bill by having someone add up certain pairs of numbers. In this presentation, I will explain the math behind the trick and how I have modified it to fit other amounts of numbers. I will demonstrate additional applications that can be performed with these different amount of digits.
Close Abstract2:25pm, Dhanush Rajesh (Pennslyvania State University-Brandywine)
Base Number System in Math Magic
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Magicians use mathematics to design tricks that stun and entertain an audience. This presentation will focus on how different base number systems are used in math magic. While researching various math-related tricks, one that particularly intrigued me was Martin Gardner’s “Number 27” card trick. In this trick, a person mentally selects one card and the magician finds it using base-three calculation. In this talk, I will explain the mathematics behind the trick as well as methods of extending it to alternate number bases.
Close Abstract2:35pm, Charles Kulick, Beth Anne Castellano (The University of Scranton)
Exploring Preference Orderings Through Discrete Geometry
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Consider $n + 1$ points in the plane: a set $S$ consisting of $n$ points along with a distinguished vantage point $v$. By measuring the distance from $v$ to each of the points in $S$, we generate a preference ordering of $S$. The maximum number of orderings possible is given by a fourth-degree polynomial (related to Stirling numbers of the first kind), found by Good and Tideman (1977), while the minimum is given by a linear function. We investigate intermediate numbers of orderings achievable by special configurations $S$. This work is motivated by a voting theory application, where an ordering corresponds to a preference list. We also consider this problem for points on the sphere, where our results are similar to what we found for the plane. Other variants of the original problem inspired by voting theory are developed. These include using a weighted distance function and also using two vantage points.
Close Abstract2:45pm, Megan Osborne (University of Scranton)
Probability Modeling of HIV Viral Blips
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Understanding and observing HIV has been an ongoing goal for decades. The hindrance to curing HIV likely stems from the behavior of latent reservoirs in the body. The latent cells which make up these reservoirs are able to carry infection and can become actively infected. At some point, the viral load in the body may fall below the detection level, followed by potential random blips of detectable viral load. Since the virus is able to remain present in latent reservoirs during treatment, it is much more difficult to track and makes viral blips difficult to predict. In light of this information, we plan to carefully model and examine the dynamics of the cell interactions of the virus, especially the latent cells and virions. We plan on using stochastic and probability models to capture the dynamics of HIV given the unexpected behavior of these cell types. This type of modeling will give us an accurate depiction of the behavior of this randomness. In doing this, we propose to provide a practical method to interpret experimental data in the context of the HIV model, focusing on the random behavior of the infection. This research took place at the University of Michigan-Dearborn REU.
Close Abstract2:55pm, Edmond Mbadu (Chestnut Hill College)
The Mathematics behind Mirror Anamorphosis
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Anamorphosis is a distorted projection or perspective. In this talk we will explore the mathematical foundations of mirror anamorphosis and show examples of how both a spherical mirror and cylindrical mirror transform an apparently undistorted image into a flat distorted image.
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Session I-D, Gambet 232
2:05pm, Aubrey Schall, Halee Kemper (University of the Sciences)
Carl Fridrich Gauss’ Contributions to Mathematics
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We will be talking about Carl Friedrich Gauss’s life and some of his major contributions to math. We will go over his fundamental theorem of algebra, fundamental theorem of arithmetic, the first two proofs of quadratic reciprocity, and the theory of constructible polygons including his discovery of how to construct a heptadecagon with just a compass and a ruler.
Close Abstract2:15pm, Benjamin Skultety, Christopher Blum (University of the Sciences)
20 Moves or Less: God's Number
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In this talk, we will be discussing the theory of God's Number. This theory outlines what the minimum number of moves would be to solve any permutation of the cube. All 43 quintillion possible permutations can be solved in 20 moves or less.
Close Abstract2:25pm, Dhruv Dhingani, Chance Mezzaroba, Shabnoor Begum (University of the Sciences)
The Use of Discrete Mathematics in Cryptology
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In this presentation, we will discuss the science of cryptology and the math behind it. Furthermore, we will explain different applications, how they work, and give a short real life demonstration.
Close Abstract2:35pm, Kyle Brunner, Maria Bouda (University of the Sciences)
Applications of Matrices in Chemistry Topics
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Matrices are a subject expressed in discrete mathematics that can be applied to many other sciences such as chemistry. More specifically, matrices can be used to determine the symmetry of coordination complexes by determining their point groups. In addition, matrices can be used to balance chemical equations instead of doing them by hand. Overall, matrices provide insight into the chemistry realm in order to understand complex topics such as these. We plan to provide specific examples regarding how to comprehend chemistry related topics by utilizing matrices as a model to represent them.
Close Abstract2:45pm, Steven Simpkins, Steven Boothe, Mathew Becker (University of the Sciences)
Shor's algorithm
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A very difficult problem in math and computer science is finding the prime factors of a number. There are many algorithms that could be used to achieve this goal. One of which involves using the period of a sequence of the powers of a number in mod N where N is the number that must be decomposed into its primes. Unfortunately, the period must have certain properties and finding a sequence with a period like such is just as computationally difficult as the original problem thus not that useful. However, finding that period on a quantum computer isn’t as difficult with the use of the Inverse Quantum Fourier Transform. The result is that this algorithm on a quantum computer can find the prime factors of a number in polynomial time. Internet security uses very large coprime numbers to keep sensitive information encrypted since normally those coprime numbers are too large to factor so your information should be safe. With quantum computers being an up and coming technology, it could pose a threat to internet security due to their ability to factors these very large numbers in a reasonable amount of time.
Close Abstract2:55pm, Teresa Simon , Shea Moore (University of the Sciences in Philadelphia )
Using Graph Theory to Study Brain Network Organization
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Graph Theory is a discrete mathematical concept in which graphs and structures are used to model and analyze relationships between objects. Recent studies have shown that this theory can be used to understand the brain network organization. Its systems of complex networks can be displayed through different mathematical ways such as topology and modularity. Using different experimental instruments, such as MRI, many recent studies have shown that one can understand the brain network using the graph theory. There is however, challenges and important questions that still need to be addressed in this rapidly moving mathematical connection.
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