Mathematical billiards are important models of dynamical systems from mathematical physics in which point particles collide elastically with fixed boundaries. Chaotic dynamics emerge when the boundary of the billiard table is dispersing or when it contains focusing arcs at sufficient distance to allow a defocusing effect to occur. This project studies a type of billiard known as an asymmetric lemon billiard, comprised of focusing boundaries which seem to violate the usual defocusing condition. Numerical evidence is obtained showing that chaotic dynamics nevertheless occur for a large range of parameter values, extending beyond the range to which analytic proofs apply. This work was completed at Fairfield University during Summer 2020, and was supported by a grant from the National Science Foundation.

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Cells respond to external stimuli and adapt to prolonged exposure to persistent signals to maintain cellular homeostasis. There are a number of regulatory mechanisms to achieve signal adaptation, among them are negative feedback (NFB) and incoherent feed-forward (FFS) mechanisms. We have deterministically and stochastically studied these two mechanisms in terms of their capacity for producing complex dynamics such as oscillation and multiple steady states as well as how they process intrinsic noise and noisy input signals.

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The Game of Cycles (Su, 2020) is an impartial combinatorial game played on a connected planar graph. Each player takes turns marking an edge of an initially undirected graph with a direction subject to certain rules, with the goal of completing a cycle cell. Our research involves finding winning strategies for different classes of game boards. We begin by identifying either first or second player winning strategies for relatively simple boards. To study more complex boards, we implement a playable version of the game that can be run on a computer. We then use this to create a program that evaluates every possible game state of a given board and finally determines which player has a winning strategy. Currently, we have our playable computer game and can generate game trees for smaller boards. Going forward, we plan to implement more boards in our playable game and optimize our program to efficiently identify winning strategies in larger boards.

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Inequality is a growing issue in education. Demographics, location, and the systematic oppression of certain groups appear to have a direct impact on student teacher ratios. These differences are significant and valid. Using mathematical and statistical analysis, there are predictions and analysis of the relationships between student teacher ratios, location, and racial demographics. For example, as a location becomes more urban and grows in size, the student teacher ratio increases. In districts with higher levels of minority populations, student teacher ratios are also higher. This can have an effect on the quality of education for these districts.

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We examine three common transformations (identity, fourth-root, and log) to determine the most suitable transformation for evaluating the importance of certain common features surrounding the Twin Cities Metropolitan Area (TCMA) city parks on park visitation. The distances between these features and city parks are approximately exponentially distributed by noting that their relative locations closely follow the spatial Poisson process. Because a fourth-root transformation improves the normality of exponential random variables, we verify that the fourth-root transformation is considered best by comparing correlation coefficients of the fourth-rooted data to the untransformed and log-transformed data via simulation. Using the TCMA city parks data, we also confirm that the fourth-root transformation improves the bivariate normality. Finally, we show that the fourth-root transformation of distance-type variables improves the probability of selecting the most important features affecting the park visitation using the least absolute shrinkage and selection operator (LASSO) regression.

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We define what it means for polynomials to be orthogonal with respect to an inner product. As an example, we discuss the Legendre polynomials and their construction via the Gram-Schimdt process. Then we turn to a definition of the Legendre polynomials via a differential equation, and use it to prove their orthogonality. This approach generalizes to all classical orthogonal polynomials.

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Among the most important applications of Deep Learning to engineering are those in wireless communication (WC). WC concerns the processing, communication, and transfer of data performed between two or more devices that are not connected by an electrical conductor.
Considering the constantly increasing volume of devices being deployed, the wireless ecosystem is getting saturated. This poses new challenges to WC, since frequency band allocation needs to move from static to dynamic, in order to try to optimize spectrum occupancy.
There have been several recent works in the literature which use Deep Learning to predict radio frequency spectrum occupancy. In this poster presentation, we will concentrate on the analysis of Deep Learning algorithms for spectrum occupancy prediction in interfering wireless systems, using simulated data.
Future work will address the problem of spectrum occupancy with real-time data from the metropolitan Fort Wayne area.

This is a work done by:
L. Le, J. Asher, N. Hinniger, T. Kelly, P. Klopfenstein, M. Masters, S. Owusu, R. Ruble, W.K. Sellers, A. Yano;
Mentor: Prof. A.M. Selvitella, Co-Mentors: Prof. T. Cooklev and Prof. P. Dragnev

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