Undergraduate Speaker Session

The purpose of this research was to study the concepts of Bayes’ Theorem and Total Probability, and how they can be put to use in real-life situations such as finance, medicine, marketing, engineering, and many other. In this talk, we will investigate the change in the stock index with a correlation to the event of covid, a global pandemic. Through the application and analysis of Bayes’ Theorem, it has been found that there is a positive correlation between the S&P 500 stock index price and the federal interest rate. The results were compared to pre-covid and current covid dates.
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In recent years, the study of epidemic analysis has become a more prominent and needed field of study due to the COVID-19 virus that took the world by storm. Epidemic analysis uses a reproductive number, that is the ratio of the mean individuals infected by each infectious person to determine a strategy to guide people through the epidemic. If the number is less than one then the epidemic will die off in time, however when the number is greater than one, a strategy needs to be implemented to lower that number, as seen during the recent COVID-19 pandemic when the CDC sent out guidelines to “flatten the curve”. Epidemic analysis is also responsible for how to spread out limited yet needed resources for infected areas such as personal protective equipment (PPE), medical professionals and disinfectant agents.
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My research investigates the impact of the Covid-19 pandemic on high school students’ ability to progress mathematically in preparation for college math classes. It explores students’ opinions on how the pandemic affected their math learning and analyzes how their actual group math performance at Indiana University of Pennsylvania compares with their collective opinions. Part of my research consists of an online Qualtrics survey that was sent to all Intermediate Algebra Spring 2022 and Fall 2022 students, which is complemented by institutionally provided identity-wiped data through the IUP Department of Mathematical and Computer Sciences. We compare the two cohorts and determine if there was a difference in mathematical knowledge between the two cohorts, and if that difference could be primarily attributed to whether the pandemic lockdown occurred during the sophomore or junior high school year for the students. As seen at IUP, students who have a lower placement in their mathematics course sequence are more likely to switch out of a natural science major and are more likely to not finish their college degree. Studying the negative effects of the pandemic on math knowledge can help to lead to an understanding of how much more remediation might be needed for these students.
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This project will apply basic principles of predictive analytics to provide an explanation of how the Zola algorithm featured in Captain America: The Winter Soldier works. Using data available online, such as financial records, medical histories, and voting patterns, Zola was able to predict who would be least likely to conform to authority and who would therefore pose a threat to the potential HYDRA dictatorship.
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Consider an auction akin to Storage Wars whereby bidders can signal to the auctioneer whether they would like to raise the current bid price or drop out of the auction. Auctions such as these are known as English auctions and can be modeled and analyzed with game theory. In a famous paper, Milgrom and Weber show that under reasonable circumstances English Auctions outperform other standard auctions, but they rely on a particular Nash equilibrium strategy for the English auction. However, Bikhchandani et al show that there is in fact a continuum of equilibrium strategies. In this research, we simulate such auctions whereby bidders make use of different equilibrium strategies and consider the consequences.
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A continuous problem today is wasted food. Due to the strict cosmetic standards that companies are pressing onto foods the rate of waste is continuing to increase. A more common way today to reduce such waste is to sell ugly or imperfect produce. My research will model the rate of sold imperfect and perfect PA peaches compared to the waste produced. Each of the models comprised were based off the Lotka-Volterra model, which is used to determine predator prey relationships.
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Cochlear implants (CIs) are remarkable devices that are able to recreate a sense of hearing for individuals with severe to profound hearing loss. If CIs are in both ears (bilateral CIs), neural responses to stimulation in the both ears can enable users to locate sounds in their surroundings. However, cochlear implants (CIs) users typically do not localize sounds with the same accuracy as unaided listeners, especially in instances of listening in noisy backgrounds. Furthermore, sound localization is further degraded for CI users with early-onset deafness and prolonged sound deprivation. Through this research, we aimed to understand the neural basis for this limitation by implementing a two-part model describing how the auditory nerve (AN) and medial superior olive (MSO) neurons respond to cochlear implant stimuli with varying interaural time differences (ITDs). MSO neurons receive inputs originating from both ears and are particularly sensitive to sub-millisecond timing differences in their inputs . We explored how ITD sensitivity in MSO neurons may change due to early-onset deafness, and extended this model to account for how sound-deprivation may alter spiking dynamics in the MSO. The phenomenological AN model is stochastic and generates auditory nerve spike inputs that drive the responses in the MSO model which performs neural coincidence detection. Our MSO model is a Hodgkin-Huxley-type deterministic model that captures biophysical features of MSO neurons. We used a two-compartment structure for the MSO model to represent the neuron’s input region (dendrite and soma) and spike generating region (axon) and systematically varied synaptic strength and neural excitability to explore coincidence detection in the model. We generated two AN streams with a possible time-shift between them to simulate ITDs that enable real-life sound localization. We found that simulated spiking behavior depended on ITD. Specifically, model MSO neurons responded to inputs with zero ITD with more spikes than in response to inputs with larger ITD. We also observed that under high pulse rate (500pps and 1000pps), fewer spikes occurred than lower pulse rates (100 pps and 200 pps) for sound-deprived simulations. The models created through this research can be applied to further advance understanding of binaural circuitry and may help suggest further improvements in binaural sensitivity for CI users with early-onset deafness.
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This presentation will discuss how to test if a given triangle model is possible, and how to construct triangles, with this criterion: integer triangles with integer distance from each vertex to a point in the center. This will be done generally, with Cevian points, as well as two special cases: symmetric triangles and triangles with the orthocenter as its point. This is applicable for teachers who want to create problems requiring students to find a side length or length of vertex to point.
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An arbitrary collection of lines, together on a flat surface, occasionally seem to describe the shape of a curve. Given an infinite collection of lines in a single plane of the form $a(t)x+b(t)y=c(t)$ for some differentiable functions $a,b,c$ on some open interval, the curve that all the lines describe can be determined and expressed as a function. After describing this curve for a collection of generic lines, the result can be applied in various contexts, such as identifying the shape of light rays reflecting off oddly-shaped mirrors in a 2-dimensional plane.
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A 2020 paper introduced the multicolor forcing process, a generalization of the zero forcing process on graphs. Here, we introduce the notion of a cycle contraction for the multicolor forcing process, generalizing the idea of a color contraction. We then consider the three-cyclic multicolor forcing process on paths, and extend the notion of cycle contraction beyond its original definition in order to derive a recursive sequence of vectors which encodes information about the forcing process. We then use this sequence to prove a tight bound on the propagation time of the three-cyclic multicolor forcing process on paths.
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For a graph G and a set of colors, we denote $P_G(q)$ as the number of ways to properly color the vertices of G using at most $q \ge 1$ colors. This is a well-studied parameter in Graph Theory, and there are many problems involving $P_G(q)$ that have yet to be solved. The general problem we considered is as follows: "Among all graphs with $n$ vertices and $m$ edges, which graph(s) attain the highest value of $P_G(q)$ for a fixed number of $q$ colors?" Linial and Wilf proposed the problem in the mid-1980s. Solving this problem would help to better determine the running time of a well-known algorithm, called the Backtrack Algorithm for the Graph Coloring Problem. We solved the problem for specific values of $n$, $m$, and $q$. Let $T_r(n)$ denote the Turán graph - the complete $r$-partite graph on $n$ vertices with partition sizes as equal as possible. We proved that when the number of colors $q=5$ and the number of vertices $n$ is sufficiently large, the Turán graph $T_3(n)$ has more colorings than any other graph with the same number of vertices and edges as $T_3(n)$. The key to proving this was to solve a difficult quadratically constrained linear optimization problem, which we will present in our talk.
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In Calculus, finding both the derivative and anti-derivative of a function is rudimentary, and much of Calculus focuses on revealing the properties and the applications of these two mathematical concepts. In this presentation I will introduce and define "the cognate" of a function which presents itself as a "sibling" of the traditional derivative (cognate means "of the same parentage"). Preliminary results show that the cognate has properties similar to the derivative, and yet somewhat different as well. The intent of this presentation is to show some of the basic properties and applications of the cognate.
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In this talk we will begin with presenting three crocheted visual representations of the sequences of digits for three famous numbers: pi, e, and phi. The crocheted pieces each use a color assignment to represent the numerical value of each digit. The shapes of the pieces are significant to the number they represent. By studying the similarities and the differences in the three sequences, our visual representations allow us to look for patterns in the randomness from a new point of view. These pieces brought inspiration to create a new visualization of “fake randomness” that we will present as well.
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Group Theory is a very important subject because many physical systems possess symmetries, which can be classified as either continuous or discrete. Examples of continuous symmetries include space translations and rotations in homogeneous systems, Lorentz transformations, internal rotations of quantum mechanical spin, etc. On the other hand, examples of discrete symmetries include parity, charge conjugation, time reversal, etc. We will focus on the connection between group theory and quantum mechanics. In these physical problems, we typically have a system described by a Hamiltonian, which may be very complicated. Symmetry often allows us to make certain simplifications, without knowing the detailed Hamiltonian. To make a connection between group theory and quantum mechanics, we consider the group of symmetry operators PR, which leave the Hamiltonian invariant. The operators PR are said to form the group of Schrodinger’s equation. In conclusion, we will describe some applications of group theory in the field of quantum mechanics.
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In this presentation, we will talk about the Fibonacci Sequence, its background, and some of its applications in our daily lives. This includes the upbringing of the Fibonacci sequence, the various areas in the real world where the sequence can be used, and how the sequence is used to draw significant conclusions that are utilized by scientists today.
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