Volume 18

DOI: 10.1137/24S1694586

Authors

Jacob Vogelpohl (Corresponding author – Morehead State University)

Project Advisors

Rus May (Morehead State University)

Abstract

We make connections between the well-known gambler’s ruin problem and the enumeration of bounded Motzkin and Dyck paths. We start with a basic recurrence relation for a variation of the gambler’s ruin that permits ties, derive explicit formulas for the corresponding probability generating functions, explain the correspondence between this ruin variation and Motzkin paths, and obtain algebraic and rational expressions for the generating functions that enumerate height-bounded Motzkin and Dyck paths.

DOI: 10.1137/24S1637283

Authors

Lauren Eriksen (Corresponding author – Purdue University), Julian Bennett (North Carolina State University)

Project Advisors

Xingjie Li (University of North Carolina at Charlotte)

Abstract

In this paper, we develop a series of modified Susceptible-Infected- Removed (SIR) models to study the COVID-19 pandemic, assess the effectiveness of government protocols, and investigate the rationale behind them in Florida from March to June 2020. Our research involves the application of multiple regressions to pinpoint and identify Covid-19 lockdown periods, followed by a series of modifications to the SIR model and parameters fitting from the data. This involved creating new model approximations, such as the time-delayed SIR model and the reinfected SIR model, in order to take into account factors such as incubation and reinfection, and get the lowest error discrepancy possible for our infection rate. We were able to conclude that as we included more factors, our error rate became lower and our simulation results became more accurate compared to the actual data. We could also identify outlier metropolitan areas and draw certain conclusions on the level of government performance. We then moved on to find the correlations, if any, between infection rates and external factors. We looked at demographic and weather data to demonstrate whether correlations existed. We found that there are a few factors with high correlations, including graduate education and low temperatures.

DOI: 10.1137/24S1636472

Authors

Shuaicheng Tong (Corresponding author – University of California, Los Angeles), Linghai Liu (Brown University), Lisa Zhao (University of California, Berkeley)

Project Advisors

Samy Wu Fung (Colorado School of Mines)

Abstract

Recent efforts in applying implicit networks to solve inverse problems in imaging have achieved competitive or even superior results when compared to feedforward networks. These implicit networks require only constant memory during backpropagation, regardless of the number of layers. However, they are not necessarily easy to train because they require backpropagating through a fixed point, which leads to computationally expensive gradient calculations. In particular, this process requires solving a large linear system whose size is determined by the number of features in the fixed point iteration. To circumvent such calculations, Jacobian-free Backpropagation (JFB) was recently proposed. This paper explores its application in image deblurring problems. Our results show that JFB is comparable to the classical TV method, Plug-n-Play, and Deep Equilibrium at a reduced computational cost.

DOI: 10.1137/24S1666331

Authors

Haozhi Hong (Corresponding author – Queen’s University), Zoey Drassinower (Queen’s University), Ari Fialkov (Queen’s University), Daniel Forestell (Queen’s University), Emily Hunter (Queen’s University)

Project Advisors

Catherine Pfaff (Queen’s University)

Abstract

This paper focuses on how a soccer team can progress the ball up the field from the defensive third to the attacking third. We define a “safe configuration” of soccer players as one in which the ball possessor is part of a collection of players on the team that are pairwise connected by open passing lanes. We then study how teams can shift between these configurations in progressing the ball up the field. We provide some evidence of the benefits of “safe configurations.”

DOI: 10.1137/24S1691600

Authors

Elias Hohl (Corresponding author – ETH Zürich)

Project Advisors

Giovanni Felder (ETH Zürich)

Abstract

We find analytical solutions of the Laplace equation on a rectangular domain with Dirichlet-Neumann boundary condition type transition occurring on a side of the rectangle. The problem can be tackled either by cutting the domain in two parts and solving the obtained coupled Laplace equations using a Fourier series approach, or by creating a conformal mapping based on Christoffel-Schwarz and Mobius transformations between the original domain and another rectangular domain where the solution of the problem is less challenging. Apart from solving for the temperature field, we also compute a coefficient, dependent only on the geometry of the domain, from which the thermal resistance of a system with the given cross-section can be computed. We conduct an experiment to confirm the validity of our model. Using our results from the Christoffel-Schwarz method, we derive regularity results for the Fourier coefficients of the coupling method. We not only determine the order of decay, but also find formulas for explicit upper bounds of the absolute value of the coefficients. These results can be used to obtain maximum error bounds for a numerical computation considering only a finite number of Fourier coefficients. Furthermore, we prove that the solutions obtained via the coupling method converge to the actual solution.

DOI: 10.1137/23S1553479

Authors

Noah Dahle (Corresponding author – University of Tennessee, Knoxville), Kristina Wilson (University of Tennessee, Knoxville)

Project Advisors

Dr. Olivia Prosper (University of Tennessee, Knoxville)

Abstract

It is common for a mapping platform to relay information to the user about the traffic in the area. Traffic conditions have different degrees of traffic intensity that are represented by colors. We have studied Google Maps traffic data through a simulation to model these conditions. Our methodology explores the prospect of how different traffic condition colors will imply different travel times. This modeling was a means to develop an user interface to determine the fastest path between two points in a particular area of Knoxville, Tennessee. We accomplished this by using Dijkstra’s Algorithm. The final product is a program that accepts and corrects user input for day of week, time, and two points, and the output is the path the user should take, along with the travel time according to the Google Maps traffic conditions at that day and time. This model can be applied to different cities to study their traffic patterns or in city planning.

Supplementary Materials