Session Index
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Undergraduate Session A
9:35am--11:10am, Zoom Undergrad A
Speakers: Brianna Falby, Maxwell James Billante, Yawen Zhang, Zhengyi Xiao, Jessica Champion, Megan Osborne
Undergraduate Session B
9:35am--11:10am, Zoom Undergrad B
Speakers: Isaac Reiter, Maxwell Norfolk, Yusong Deng, Sabrina Traver, Abhinav Pandey, Isaac Reiter
Undergraduate Session A, Zoom Undergrad A
9:35, Brianna Falby (Elizabethtown College)
Greedy Strategies for 2048
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2048 is an online game that is popular in the mathematical circles. Optimal strategies have been found using a Markov decision process for a reduced version of the game, but to the best of our knowledge, an optimal solution for the full game has not been discovered. The greedy strategy will present a winning percentage of approximately 6%, which is better than most people’s performance percentage.
Close Abstract9:50, Maxwell James Billante (Elizabethtown College)
US Car Accidents Analysis
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In my research project, I analyzed the US Accidents (3.5 million records) A Countrywide Traffic Accidents Dataset on Kaggle to answer the question, “What factors predict the severity of traffic interference?” I will discuss my analysis of this data set, which included data cleaning and processing, missing value imputation, visualization, variable selection, and machine learning model building.
Close Abstract10:05, Yawen Zhang (Elizabethtown College )
Go Big or Go Home
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Many people say that “you should play to win instead of playing not to lose.” However, we show that is not the best strategy in Penney’s game, which is a probability game involving sequences of coin flips where the second player always has an advantage over the first player. However, if both players choose their sequences simultaneously, then game theory paradoxically shows that a conservative player who “plays to win or tie” wins more often than an aggressive player who “plays to win”. Interestingly, that result breaks down for two coin flips with a biased coin with probability 1/φ of being heads, where φ is the golden ratio. In that case, both players win equally often.
Close Abstract10:25, Zhengyi Xiao (Franklin and Marshall College)
A $C_0$-Semigroup Approximation to the Fractional Damped Wave Equations
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The fractional calculus is a branch of mathematical analysis that studies non-integer (real or complex) powers of the derivative and integration. The differential equations involving fractional derivatives are called fractional differential equations (FDE). In this talk, we introduce a class of finite difference method by $C_0$-semigroup operator theory for solving two-sided space-fractional order wave equation of the form:
\begin{equation}
\frac{\partial^2u}{\partial t^2} =c_+(x,t) \frac{\partial ^\alpha u}{\partial_+ x^\alpha} + c_{-}(x,t)\frac{\partial ^\alpha u}{\partial_- x^\alpha} + g(x,t).
\end{equation}
The comparison of numerical results with the exact solutions as well as ordinary explicit and implicit finite difference methods will be presented.
Close Abstract10:40am, Jessica Champion (Shippensburg University)
Mapping the Inequality $a^b > b^a$
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For any real numbers $a$ and $b$, how can the values of $a^b$ and $b^a$ be compared? This seems tedious or impossible to compute by hand, especially for large and non-integer values of $a$ and $b$. Using technology is equally unrewarding as computers and calculators quickly reach their overflow limit due to the extreme size of the computed values. Hence, in order to compare values of $a^b$ and $b^a$, general rules need to be determined. In this presentation, I will summarize my process of compiling the rules on this inequality, using various methods including Python and the Lagrange reversion theorem.
Close Abstract10:55am, Megan Osborne (University of Scranton)
An Unstructured Mesh Approach to Nonlinear Noise Reduction
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In any type of data acquisition, the event of gathering undesirable noise along with desirable data is inevitable. To denoise signals originating from smooth, chaotic attractors, the Air Force Research Laboratory (AFRL) adapted the time-delay embedding theory of Takens’ Theorem (1981) and the causation-detecting method of Convergent Cross Mapping (CCM) to develop a grid-based denoising technique. Given a clean signal from such a dynamical system, AFRL’s technique attempts to denoise a corrupted signal observed from the same system. To improve this grid-based method, we implement an unstructured mesh based on triangulations and Voronoi diagrams that better distributes data over mesh cells and improves the accuracy of the reconstructed signal. Our method achieves statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall Effect Thrusters (HETs) with greater success than the grid-based strategy.
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Undergraduate Session B, Zoom Undergrad B
9:35am, Isaac Reiter (Kutztown University of Pennsylvania)
The VICCard Cipher: Our Contribution to the Field of Playing Card Cryptography
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Spies and soldiers alike have used so-called hand ciphers throughout history to send secret messages using little equipment beyond a pencil and paper. Modern secure communication, by contrast, requires computer software and hardware to execute various cryptosystems. However, a routine calculation shows that the randomness (entropy) of a well-shuffled standard deck of playing cards has the potential to provide security to rival modern cryptosystems. Playing cards have the convenient feature that two suits have 26 cards, so the full deck can encode each uppercase and lowercase letter. For these reasons, Neal Stephenson, author of the novel Cryptonomicon, commissioned security expert Bruce Schneier to create a cryptosystem using playing cards. In this talk, I will discuss Dr. Landquist’s and my contribution to the field of playing card ciphers. After analyzing existing playing card ciphers, we culminated our research by creating our own unique playing card cipher. Entitled VICCard, our cipher takes inspiration from the Nihilist cipher VIC, which was used by Russian spies during the Cold War. I will discuss the workings of the cipher and provide some data regarding the security afforded by our cipher.
Close Abstract9:50am, Maxwell Norfolk (Bloomsburg University)
The Cost of an Integer
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The cost $C_S$ of a positive integer $m$ relative to a set $S$ of binary operations is defined to be the lesser of $m$ and the minimum of $C_S(a)+C_S(b)$ where $a$ and $b$ are positive integers and $m=a \circ b$ for some binary operation $\circ \in S$. The cost of a positive integer measures the complexity of expressing $m$ using the operations in $S$, and is intended to be a simplification of Kolmogrov compelexity. We show that, unlike Kolmolgorov complexity, $C_{S}$ is computable for any finite set $S$ of computable binary operations. Several interestings theorems, sets of operations, and open questions are also discussed.
Close Abstract10:05am, Yusong Deng (Franklin and Marshall College)
On Tiling a Unit Square and The Greatest Sum of Side Lengths
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In 1995 P. Erdos and A. Soifer published several open questions intended to discuss the biggest sum of side lengths when putting $n$ squares into a unit square. In their work, proof of $n=6$ was given. In 2007, W. Staton and B. Tyler gave proof for the case $n=10$. In 2008, I. Praton offered a proof for the case $n=8$. After 2008, little progress was made on this question. The case of $n=7$, and the cases of $n >10$ have not been solved yet. My research gives proofs to the case of $n=7$ and $n=11$ by using the properties of the square. A further target of my research is to give proofs for the case of $12,13,14$. The approaches to this question are diverse. In general, the approach to this question should be not complex. Students with knowledge of college calculus and geometry should be able to understand.
Close Abstract10:25am, Sabrina Traver (King's College)
Monodromy and Chart Representations for Surfaces with Braided Boundaries in the Blowup of $D^2 \times D^2$
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Adapting graphical methods developed by Seiichi Kamada in his paper, “Graphic descriptions of monodromy representations” and in his book "Braid and Knot Theory in Dimension Four", we attempt to classify surfaces with braided boundaries in the blowup of the 4-ball. Our method takes cross sections of the four-dimensional surface and uses their monodromy representations to determine surface equivalence.
Close Abstract10:40am, Abhinav Pandey (Penn State Brandywine)
Geography of Legendrian Knot Mosaics
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Knots mosaics were first introduced by S. J. Lomonaco and L. Kauffman and provide a method for building knots using a set of mosaic tiles. A modified version of these tiles can be used to build Legendrian knots, which are knots that satisfy an extra geometric condition. This talk discusses the effect of stabilization on the mosaic number of a Legendrian knot. We define and classify all “two-cell” stabilizations. Finally, we look specifically at how the mosaic number changes under stabilization for the unknot and both trefoils. This work is joint with Samantha Pezzimenti.
Close Abstract10:55am, Isaac Reiter (Kutztown University of Pennsylvania)
Covering All Our Bases: Exploring Minimum Dominating Sets of Grids
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Dominating sets are a very interesting topic in graph theory. Given a graph, a dominating set D is a subset of the vertices such that every vertex is either in D or adjacent to a vertex in D. Every graph has a domination number, which is the minimum number of vertices required to dominate that graph. Any dominating set of minimum cardinality is called a minimum dominating set. Proving that a graph has a particular domination number can often come down to exhaustively testing almost every possible dominating set. Nonetheless, strategies can be adopted that significantly reduce that number of cases that need to be considered. In this talk, the presenter will discuss some of his methods of proving the domination numbers of particular grids.
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