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# 2020

# “Computational modeling of wind effects on a tall building by the finite element method” by Bakdauren Narbayev (Nazarbayev University)

Pedestrian wind comfort plays important role in the urban environment. In our work, we consider a model obtained using the Computational Fluid Dynamics (CFD) around the tall building. Our focus is the Tower of Abu Dhabi Plaza in Nur-Sultan city (Kazakhstan), which will be the tallest building of Central Asia with a height of 382 m. We investigated the effect of the wind velocity for pedestrians solving the incompressible time-dependent Navier-Stokes equations in the deal.II library by the Finite Element Method (FEM). We present numerical simulation results for various scenarios. It has been found that the velocity profile can vary in the domain that creates different pedestrian comfort conditions including the extreme category at places dedicated to the pedestrian walking.

# “Dynamical Characterization & Analysis of the Optimization Algorithms: Linearized Bregman and Iterative Shrinkage Thresholding Algorithm” by Yihua Xu (Georgia Tech)

Sparse recovery optimization algorithms are utilized in machine learning, imaging, and parameter fitting in problems, as well as a multitude of other fields. Compressive sensing, a prominent field in mathematics this past decade, has motivated the revival of sparse recovery algorithms with ?-1 norm minimization. Although small underdetermined problems are substantially well understood, large, inconsistent, nearly sparse systems have not been investigated with as much detail. In this dynamical study, two commonly used sparse recovery optimization algorithms, Linearized Bregman and Iterative Shrinkage Thresholding Algorithm are compared. The dependence of their dynamical behaviors on the threshold hyper-parameter and different entry sizes in the solution suggests complementary advantages and disadvantages. These results prompted the creation of a hybrid method which benefits from favorable characteristics from both optimization algorithms such as less chatter and quick convergence. The Hybrid method is proposed, analyzed, and evaluated as outperforming and superior to both linearized Bregman and Iterative Shrinkage Thresholding Algorithm, principally due to the Hybrid’s versatility when processing diverse entry sizes.

# “Agmon-Type Decay of Eigenfunctions for Schrodinger Operators with a Non-compactly Supported Classical Region” by Hengrui Zhu (Oberlin College)

A classical result by Agmon implies that an eigenfunction of a Schrodinger operator on R^d decays exponentially if the classical region associated with that eigenstate is compactly supported. A Schrodinger operator is the sum of the negative Laplacian and a multiplication operator by a real valued function, known as the potential. The classical region is defined as the subset of R^d for which the potential is less than the eigenvalue.

In this work, we extend the result to Schrodinger operators with an eigenvalue such that the classical region is not compactly supported. We show that assuming integrability of the classical region with respect to an increasing weight function implies L^2-decay of the eigenfunction with respect to the same weight. Here, decay is measured in the Agmon metric which takes into account anisotropies of the potential.

# “Chaotic dynamics in asymmetric lemon billiards” by Hailey Spinella Camryn Colonna (Fairfield University)

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.

# “Dynamics of Cellular Signal Adaptation Mechanisms: A Comparative Study” by Christopher Ryzowicz (New College of Florida)

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.

# “Identifying Winning Strategies for the Game of Cycles Using Computer Programs and Game Trees” by Nicholas Bozentko (Iona College)

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.

# “Factors Impacting Student Teacher Ratios in New York State at the Secondary Level ” by Willow Lopez-Silvers (Iona College)

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.

# “Modeling Park Visitation Using Transformations of the Distance-Type Predictor Variables with LASSO” by Ashley Hall (Western Washington University)

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.

# “Orthogonal polynomials via differential equations” by Nick Juricic (UConn)

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.