Volume 18

DOI: 10.1137/25S1781917

Authors

Valentin Garcia (Corresponding author – Brown University), Mia Escobar (University of Washington Tacoma)

Project Advisors

Dr. Anastasiia Minenkova (University of Hartford)

Abstract

The existence of infinite families of (N + 1) × (N + 1) Jacobi matrices representing the Hamiltonians of quantum spin chains with and without early state exclusion (ESE) has been shown to exist for any even (N + 1) ≥ 4. However, their existence for odd (N + 1) ≥ 7 has remained an open problem. In Section 3, we consider a chain of qubits experiencing nearest-neighbor interactions with environmental effects and present infinite families of 7 × 7 Jacobi matrices with and without ESE.

DOI: 10.1137/25S1739224

Authors

Vedant Ganesh (Corresponding author – Tanglin Trust School, Singapore)

Project Advisors

Alexander Mushtukov (University of Oxford)

Abstract

The dynamics of accretion in X-ray pulsars (XRPs) are shaped by the surrounding magnetic field. Charged material in the accretion disk impacts the magnetosphere and is constrained to the magnetic field lines due to the strength of the field. The particles’ dynamics are then influenced by extremely large gravitational forces due to high density and proximity, and centrifugal forces due to the corotation of the XRP magnetosphere. At higher mass accretion rates, radiative force appears which heavily influences the magnetic field flow. Accretion flows follow the magnetic field lines to impact the neutron star in a small region near the magnetic poles. We construct a numerical model to help determine the dynamics of accretion flow under these conditions and investigate its effects on the mass accretion rate and pulse profile of XRPs, assuming that the magnetic field is dominated by the dipole geometry. We show that variability in mass accretion rate, and hence pulse profile asymmetry, can be due to the misalignment between the accretion plane normal, rotational, and dipole axes, and we investigate the effects of each angle on pulse profile asymmetry.

DOI: 10.1137/24S1683263

Authors

Joseph Erwin (Corresponding author – University of Southern California)

Project Advisors

John Gunnar Carlsson (University of Southern California)

Abstract

Selecting optimal slide positions is a crucial challenge in trombone performance that impacts both technical execution and musical expression. This paper models trombone slide positioning as a network-based shortest path problem and presents several objective functions to evaluate different positioning strategies. We compare various cost functions, including minimizing distance to standard positions, finding the global shortest path, a greedy approach, minimizing slide reversals, minimizing impulse changes, and maximizing natural slur opportunities. Our findings reveal that the shortest path algorithm consistently performs well across different musical pieces, striking a balance between reducing physical effort and maintaining musical expressiveness. The impulse minimization objective is particularly effective for fast, technical passages, while the ‘Against the Grain’ algorithm excels in slow, melodic passages. The standard positions, though commonly used, are shown to be the least efficient in terms of energy expenditure. Additionally, the greedy algorithm, despite its simplicity, demonstrates robust performance, making it a practical choice for real-time implementation. These insights not only enhance our understanding of trombone performance optimization but also suggest that adopting advanced computational techniques can significantly improve both the efficiency and expressiveness of musical performance. This study underscores the potential of interdisciplinary approaches to enrich musical execution, offering valuable strategies for trombonists and potentially extending to other musical instruments.

DOI: 10.1137/24S1710486

Authors

Alvey Lin (Corresponding author – The University of Hong Kong)

Project Advisors

Zhiwen Zhang (The University of Hong Kong)

Abstract

Time Series Analysis has been given a great amount of study in which many useful tests were developed. The phenomenal work of Engle and Granger [1] in 1987 and Johansen [2] in 1988 has paved the way for the most commonly used cointegration tests so far. Even though cointegrating relationships focus on long-term behaviour and correlation of multiple nonstationary time series, oftentimes we encounter statistical data with limited sample sizes and other information. Thus other tests with empirical advantages may also be of considerable importance. In this paper, we provide an optimisation approach motivated by the Blind Source Separation, or also known as Independent Component Analysis, for cointegration between financial time series. Two methods for cointegration tests are introduced, namely decorrelation for the bivariate case and maximisation of nongaussianity for higher-dimensions. We highlight the empirical preponderances of independent components and also the computational simplicity, compared to common practices of cointegration such as the Johansen’s Cointegration Test [3]. The advantages of our methods, especially the better performances in limited sample size, enable a wider range of application and accessibility for researchers and practitioners to identify cointegrating relationships.

DOI: 10.1137/24S171694X

Authors

Petr Kisselev (Corresponding author – George Mason University)

Project Advisors

Padmanabhan Seshaiyer (George Mason University)

Abstract

Graph convolutional neural networks (GCNs) have shown tremendous promise in addressing data-intensive challenges in recent years. In particular, some attempts have been made to improve predictions of Susceptible-Infected-Recovered (SIR) models by incorporating human mobility between metapopulations and using graph approaches to estimate corresponding hyperparameters. Recently, researchers have found that a hybrid GCN-SIR approach outperformed existing methodologies when used on the data collected on a precinct level in Japan. In our work, we extend this approach to data collected from the continental US, adjusting for the differing mobility patterns and varying policy responses. We also develop the strategy for real-time continuous estimation of the reproduction number and study the accuracy of model predictions for the overall population as well as individual states. Strengths and limitations of the GCN-SIR approach are discussed as a potential candidate for modeling disease dynamics.

DOI: 10.1137/25S1777554

Authors

Raymond Shao (William Mason High School), Aneesh Iyer (William Mason High School), Ramya Rajan (William Mason High School), Vivian Tang (William Mason High School), Mingjia Zhang (William Mason High School)

Project Advisors

Colleen Everett (William Mason High School)

Abstract

In recent years, the United States experienced rising temperatures and increasingly frequent heatwaves, or extended periods of high temperatures, posing a significant risk to the health of residents [1]. Increases in power demand caused by record-breaking temperatures put major stress on city power grids, furthermore increasing the probability of power outages [2] that then leave even more of the population vulnerable to dangerously high temperatures: a cycle that exacerbates the consequences of heatwaves. Thus, it is imperative for the city of Memphis to examine the impact of heat waves and mitigate their risk.

M3 Challenge Introduction

DOI: 10.1137/25S1758295

Authors

Ian Kung (Corresponding author – William & Mary)

Project Advisors

Sarah Day (William & Mary, Williamsburg, VA)

Abstract

As innovation in flight continues to expand and attract commercial and research interest, it is essential to understand unpredictable elements in combustion mechanics closely tied to the longevity of propulsion systems. Past examples and current efforts prioritized by NASA demonstrate the importance of utilizing computational methods to address unpredictability in modern propulsion methods. In particular, research in thermoacoustic instability seeks to mitigate the perturbations which arise from a combination of the combined heat release and acoustic waves in the combustion process. Thermoacoustic instability can lead to functionality concerns in gas turbines, engines and rockets. This topic has increasing relevance to the development of propulsion systems, which frequently employ lean-burning combustion processes. By incorporating greater amounts of air, lean combustion processes have the dual purpose of enhancing performance while also decreasing emissions, but are especially susceptible to instabilities in the form of high-pressure oscillations. This project focuses on the use of Fourier analysis techniques to characterize these oscillations in combustion simulations when changing the exit area opening size, reaction rate, and fuel flow rate. Analysis found relatively straightforward results when variables were manipulated independently, and observed more interesting nonlinear behavior when altered together.

DOI: 10.1137/25S1743971

Authors

Rongxuan Li (Corresponding author – The Chinese University of Hong Kong)

Project Advisors

Professor Yangyang Xu (Rensselaer Polytechnic Institute, Troy, NY)

Abstract

The graph matching problem is a sparse optimization problem, which is also a significant special case of the Quadratic Assignment Problem, with extensive applications in pattern recognition, computer vision, protein alignment, and related fields. As the problem is NP-hard, relaxation and regularization techniques are frequently employed to improve tractability. However, many existing regularizers are nonconvex which poses optimization challenges. In this paper, we propose a linear reweighted regularization framework for solving the relaxed graph matching problem while preserving convexity. By solving a sequence of relaxed problems with the linear reweighted regularization term, one can obtain a sparse solution. Moreover, we establish the local convergence of the proposed method to the solution of the original graph matching problem. Furthermore, we present a practical version of the algorithm by incorporating the projected gradient method. The proposed framework is applied to synthetic instances and demonstrates promising numerical results.

DOI: 10.1137/24S171406X

Authors

Alexandra Whiteside (Corresponding author – University of North Carolina at Chapel Hill), Emily Phan (University of North Carolina at Chapel Hill)

Project Advisors

Karin Leiderman (University of North Carolina at Chapel Hill)

Abstract

Hemostasis is a vital physiological process that prevents excessive bleeding through a cascade of complex biochemical reactions, resulting in blood clot formation. This process involves a coagulation cascade that converts fibrinogen to fibrin by thrombin. Mathematical models, including the Hockin-Mann, Danforth, and Lakshmanan models, have been instrumental in simulating thrombin generation and understanding the coagulation cascade. However, these models often use simplified reaction schemes and neglect the inclusion of intermediate complexes, which can compromise their accuracy. This study demonstrates that such simplifications distort the time scale and peak values of thrombin generation, leading to misleading results. By modifying all three mathematical models to incorporate classical enzyme kinetics and critical intermediate steps, we show that the modifications change the dynamics of thrombin generation and emphasize the need for a more detailed representation of the biochemical interactions involved. This work underscores the importance of employing comprehensive modeling approaches in coagulation studies to achieve a more accurate representation of thrombotic and hemophilic conditions, ultimately aiding in the development of effective therapeutic strategies.

DOI: 10.1137/24S1713685

Authors

Ziling Chen (Corresponding author – Johns Hopkins University)

Project Advisors

Fadil Santosa (Johns Hopkins University)

Abstract

Electromagnetic wave manipulation plays a crucial role in advancing technology across various domains, including photonic device design. This study presents an inverse design approach for a periodic medium that optimizes electromagnetic wave localization at the interface between a layered half-space and a homogeneous half-space. The approach finds a maximally localized mode at a specified frequency and wave number. The mode propagates in the direction of the interface. The design parameters are the permittivity of the layered medium, their relative thicknesses, and the permittivity of the homogeneous half-space. We analyze the problem using the transfer matrix method and apply the particle swarm optimization to find a rapidly decaying mode that satisfies the design constraints. The design process is demonstrated in a numerical example, which serves to illustrate the efficacy of the proposed method.

DOI: 10.1137/23S159857X

Authors

James Kessinger (Corresponding author – Baylor University), Maya Bocanegra (California State University, Northridge)

Project Advisors

Dr. Norma Ortiz-Robinson (Grand Valley State University)

Abstract

In this work, we formulate a data driven optimal control model involving differential equations to capture the population dynamics of the wolf and moose populations in Isle Royale National Park, Michigan. A numerical solver implemented in the Python language is used to obtain solutions, and these are analyzed to identify optimal strategies for augmentation of the declining wolf population.

DOI: 10.1137/24S1710498

Authors

Yutong Bu (Corresponding author – Emory University)

Project Advisors

Julianne Chung (Emory University)

Abstract

Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.

DOI: 10.1137/24S1686000

Authors

Riley Link (Corresponding author – Creighton University) , Ethan Ebbighausen (University of North Carolina Chapel Hill)

Project Advisors

Alen Alexanderian (North Carolina State University), Joseph Hart (Sandia National Laboratories)

Abstract

We consider trajectory optimal control problems in which parameter uncertainty limits the applicability of control trajectories computed prior to travel. Hence, efficient trajectory adjustment is needed to ensure successful travel. However, it is often prohibitive or impossible to recalculate the optimal control in-transit due to strict time constraints or limited onboard computing resources. Thus, we propose a framework for quick and accurate trajectory approximations by using postoptimality sensitivity information. This allows the reduction of uncertain parameter space and an instantaneous approximation of the new optimal controller while using sensitivity data computed and stored pretransit.

DOI: 10.1137/24S1710917

Authors

Warin Watson (Corresponding author – Colorado Mesa University), Cash Cherry (Colorado School of Mines), Rachelle Lang (University of Wisconsin – Madison)

Project Advisors

Lars Ruthotto (Emory University)

Abstract

We demonstrate that automatic differentiation (AD), which has become commonly available in machine learning frameworks, is an efficient way to explore ideas that lead to algorithmic improvement in multi-scale affine image registration and affine super-resolution problems. In our first experiment on multi-scale registration, we implement an ODE predictor-corrector method involving a derivative with respect to the scale parameter and the Hessian of an image registration objective function, both of which would be difficult to compute without AD. Our findings indicate that exact Hessians are necessary for the method to provide any benefits over a traditional multi-scale method; a Gauss-Newton Hessian approximation fails to provide such benefits. In our second experiment, we implement a variable projected Gauss-Newton method for super-resolution and use AD to differentiate through the iteratively computed projection, a method previously unaddressed in the literature. We show that Jacobians obtained without differentiating through the projection are poor approximations to the true Jacobians of the variable projected forward map and explore the performance of other approximations in the problem of super-resolution. By addressing these problems, this work contributes to the application of AD in image registration and sets a precedent for further use of machine learning tools in this field.

DOI: 10.1137/24S1703392

Authors

Jimmy Vineyard (Corresponding author – University of Wisconsin - Madison), Peizhe Li (University of Wisconsin – Madison), Seungyeon Oh (University of Wisconsin – Madison), Jack Maloney (University of Wisconsin – Madison)

Project Advisors

Haley Colgate Kottler (University of Wisconsin - Madison), Amy Cochran (University of Wisconsin - Madison)

Abstract

The two-stage Markov task, widely-used for measuring model-based relative to model-free learning in humans, has faced skepticism regarding its effectiveness. We suggest a modification to better distinguish the two learning approaches. Our revised task incorporates an additional phase for learning local contingencies, mirroring a strategy from machine learning for separating model-free from model-based algorithms. We evaluated the effectiveness of our revised task through simulations, employing model-free and model-based strategies.

DOI: 10.1137/24S1717993

Authors

Justin Le (Corresponding author – Arizona State University)

Project Advisors

Sebastien Motsch (Arizona State University), Johannes Brust (Arizona State University)

Abstract

We provide an overview of the diffusion model as a method to generate new samples. Generative models have been recently adopted for tasks such as art generation (Stable Diffusion, Dall-E) and text generation (ChatGPT). Diffusion models in particular apply noise to sample data and then “reverse” this noising process to generate new samples. We will formally define these noising and denoising processes, then present algorithms to train and generate with a diffusion model. Afterward, we will explore a potential application of diffusion models in improving classifier performance on imbalanced

DOI: 10.1137/25S172954X

Authors

Aiden Rohrbach (Corresponding author – University of Rochester)

Project Advisors

Xuwen Chen (University of Rochester)

Abstract

We address a Landau temperature fluctuation problem by applying Bernstein polynomials and the Weierstrass approximation theorem. We consider the problem detailed in [X. Wang and Q.H. Liu, Temperature fluctuations for a finite system of classical spin-1/2 particles, Annals of Physics 322 (2007), 2168-2178] in which temperature fluctuations diverge as the temperature of a particle system tends to 0. An application of Bernstein polynomials allows us to solve this issue by proving that the m’th moment of the configurational temperature of a finite particle system tends to the temperature of the surroundings, implying the fluctuations in the temperature must tend to 0.

DOI: 10.1137/24S1692241

Authors

Xiaoxing Yu (Corresponding author – Pomona College)

Project Advisors

Christina J. Edholm (Scripps College)

Abstract

Mathematical models of language change promise to provide insight on how community-wide changes emerge from individual speaker interactions. We develop an agent-based model to test the validity of the gravity theory of linguistic diffusion. Our model uses an exemplar model and the functional-teleological explanation of chain shifts to build up a picture of how individual agents change their vowel systems, and studies the emergent patterns across language communities. We find that the predictions of the gravity model hold true in a subset of cases determined at least in part by patterns of interaction and the particularities of how vowels are produced and perceived. To be specific, the gravity model’s predictions will hold true in situations where two interacting communities have categories within each other’s tolerance thresholds.

Supplementary Materials #S169224R

DOI: 10.1137/24S1675503

Authors

Murphy John (Corresponding author – University of New Mexico), Samantha Mackley (University of Missouri)

Project Advisors

Dr. Mihhail Berezovski (Emory-Riddle Aeronautical University)

Abstract

Accurately predicting the demand for aviation is a complex problem that is essential for the success of private aviation providers. Factors such as seasonality and location affect the demand for private flights, but high-demand events and holidays introduce additional and often unexpected influences on these services. In European destinations, travel is heavily characterized by high-demand events and holidays. This research utilizes detailed characterization data centered in Europe containing over 1.1 million private flights between 2,016 locations from 2018 and 2019. Leveraging advanced data analysis techniques, this project constructs a spatio-temporal forecasting model to accurately predict the demand for private jet travel during high-demand events and holidays in European destinations. This research delivers valuable insights to providers of private aviation, enabling them to proactively respond to market fluctuations and optimize their operational strategies.

DOI: 10.1137/24S168324X

Authors

Alexander Le (Corresponding author – United States Air Force Academy)

Project Advisors

Dr. Maila Hallare (United States Air Force Academy), Maj. A.J. Wallerstein (United States Air Force Academy)

Abstract

Knowing baseplate kinematics during mortar fire allows for more accurate shelling during military operations. During mortar fire, ignition causes a rapid increase in barrel pressure, exerting a recoil force onto the baseplate and causing it to sink into the ground. This sinkage causes the barrel to move and fire shells slightly off trajectory. Accordingly, shells may miss their targets or hit unintentional ones. Using second-order linear differential equations, mortar kinematics was analyzed as an externally-forced spring-mass-damper system. Soil properties accounted for springiness and damping in this model, with recoil as the external forcing function. Simulations of the differential equations show that mortar movement was significantly minimized with larger baseplate area, and that launch angle did not significantly affect vertical sinkage. This project recommends that mortar baseplate area should be maximized to allow for more accurate artillery strikes.

DOI: 10.1137/24S1681069

Authors

Aidan Tillman (Corresponding author – Northeastern University), Yuzhi Liu (University of Michigan Ann Arbor)

Project Advisors

Stuart Brorson (Northeastern University)

Abstract

We define algorithms for computing the eigenvalue counting functions (ECFs) of the Laplacian with Dirichlet and Neumann boundary conditions for a variety of convex 2D and 3D domains using floating point precision in Python. These domains include the square, disk, ball, cube, and cylinder. We use these computed ECFs to validate Polya’s conjecture on these domains. The conjecture has not been proven in completeness on the ball, so this work presents numerical evidence to the positive. We also use these calculations to study the potential geometric information in the lower-order terms of the asymptotics by comparing ECFs between different shapes. We do this by subtracting the like terms in the asymptotics of the ECFs to draw out the lower-order information. In the 3D case, a difference can be seen in the lower-order asymptotics of a cube and cylinder with the same volume and surface area. In particular, the cube’s o(k2) term has a larger variance than the cylinder’s term, and in both cases, the term is mostly positive.

Supplementary Material

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