Applied Mathematics Commons^™

Optimal Information Design In Two-Sided Trade, Pradhi Aggarwal

The Yale Undergraduate Research Journal

In a two-sided market with a broker, the broker can influence the buyer’s and seller’s optimal trading behaviour through strategic information design. We study the impact of information about waiting times on riders and drivers in a rideshare market. We consider three information regimes: the first in which no information about time is revealed, the second in which true waiting times are communicated, and finally an intermediate regime in which agents are only told whether their waiting time falls within a high or low category. We evaluate the optimality of each information regime by maximizing welfare and revenue for each …

Applications Of Bayesian Inference For Modelling Dynamic Instability In Neuronal Dendrite Morphogenesis, Daniel Fridman

The Yale Undergraduate Research Journal

Neurons are complex biological systems which develop intricate morphologies and whose dendrites are essential in receiving and integrating input signals from neighboring neurons. While much research has been done on the role of dendrites in neuronal development, a further understanding of dendrite dynamics can provide insight into neural development and the cellular basis of neurological diseases such as schizophrenia, Down’s syndrome, and autism. The Jonathon Howard lab hypothesizes that microtubules are a primary driving force in dendrite dynamics. Since it is known that microtubules display dynamic instability, rapidly transitioning between growth, paused, and shrinking states, the Howard lab proposes a …

Bernstein-Sato Polynomials In Commutative Algebra, Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez-Betancourt

Department of Mathematics: Faculty Publications

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.

Euler's Three-Body Problem, Sylvio R. Bistafa

Euleriana

In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain …

Client Access Feature Engineering For The Homeless Community Of The City Of Portland, Oswaldo Ceballos Jr

altREU Projects

Given the severity of homeless in many cities across the country, the project at hand attempts to assist a service provider organization called Central City Concern (CCC) with their mission of providing services to the community of Portland. These services include housing, recovery, health care, and jobs. With many different types of services available through the works of CCC, there exists an abundance of information and data pertaining to the individuals that interact with the CCC service system. The goal of this project is to perform an exploratory analysis and feature engineer the existing datasets CCC has collected over the …

On The Application Of Principal Component Analysis To Classification Problems, Jianwei Zheng, Cyril Rakovski

Mathematics, Physics, and Computer Science Faculty Articles and Research

Principal Component Analysis (PCA) is a commonly used technique that uses the correlation structure of the original variables to reduce the dimensionality of the data. This reduction is achieved by considering only the first few principal components for a subsequent analysis. The usual inclusion criterion is defined by the proportion of the total variance of the principal components exceeding a predetermined threshold. We show that in certain classification problems, even extremely high inclusion threshold can negatively impact the classification accuracy. The omission of small variance principal components can severely diminish the performance of the models. We noticed this phenomenon in …

Algebraic, Computational, And Data-Driven Methods For Control-Theoretic Analysis And Learning Of Ensemble Systems, Wei Miao

McKelvey School of Engineering Graduate Student Theses & Dissertations

In this thesis, we study a class of problems involving a population of dynamical systems under a common control signal, namely, ensemble systems, through both control-theoretic and data-driven perspectives. These problems are stemmed from the growing need to understand and manipulate large collections of dynamical systems in emerging scientific areas such as quantum control, neuroscience, and magnetic resonance imaging. We examine fundamental control-theoretic properties such as ensemble controllability of ensemble systems and ensemble reachability of ensemble states, and propose ensemble control design approaches to devise control signals that steer ensemble systems to desired profiles. We show that these control-theoretic properties …

Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo

Department of Mathematics: Faculty Publications

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

Empirical Fitting Of Periodically Repeating Environmental Data, Pavel Bělík, Andrew Hotchkiss, Brandon Perez, John Zobitz

Spora: A Journal of Biomathematics

We extend and generalize an approach to conduct fitting models of periodically repeating data. Our method first detrends the data from a baseline function and then fits the data to a periodic (trigonometric, polynomial, or piecewise linear) function. The polynomial and piecewise linear functions are developed from assumptions of continuity and differentiability across each time period. We apply this approach to different datasets in the environmental sciences in addition to a synthetic dataset. Overall the polynomial and piecewise linear approaches developed here performed as good (or better) compared to the trigonometric approach when evaluated using statistical measures (R^{2 …}

Sparse Domination Of The Martingale Transform, Michael Scott Kutzler

Mathematics & Statistics ETDs

Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators. The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator. Sparse operators are based on a sparse family which is simply a subset of a dyadic grid. We also show the A2 conjecture for the Martingale Transform which follows from the sparse domination of the Martingale Transform and the A2 conjecture for sparse operators.

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Growth-Profile Configuration For Specific Deformations Of Tubular Organs: A Study Of Growth-Induced Thinning And Dilation Of The Human Cervix, Kun Gou, Seungik Baek, Marvin M.F. Lutnesky, Hai-Chao Han

Mathematics Faculty Publications (Archived)

Growth is a significant factor that results in deformations of tubular organs, and particular deformations associated with growth enable tubular organs to perform certain physiological functions. Configuring growth profiles that achieve particular deformation patterns is critical for analyzing potential pathological conditions and for developing corresponding clinical treatments for tubular organ dysfunctions. However, deformation-targeted growth is rarely studied. In this article, the human cervix during pregnancy is studied as an example to show how cervical thinning and dilation are generated by growth. An advanced hyperelasticity theory called morphoelasticity is employed to model the deformations, and a growth tensor is used to …

Fast Multipole Methods For Wave And Charge Source Interactions In Layered Media And Deep Neural Network Algorithms For High-Dimensional Pdes, Wenzhong Zhang

Mathematics Theses and Dissertations

In this dissertation, we develop fast algorithms for large scale numerical computations, including the fast multipole method (FMM) in layered media, and the forward-backward stochastic differential equation (FBSDE) based deep neural network (DNN) algorithms for high-dimensional parabolic partial differential equations (PDEs), addressing the issues of real-world challenging computational problems in various computation scenarios.

We develop the FMM in layered media, by first studying analytical and numerical properties of the Green's functions in layered media for the 2-D and 3-D Helmholtz equation, the linearized Poisson--Boltzmann equation, the Laplace's equation, and the tensor Green's functions for the time-harmonic Maxwell's equations and the …

A Fast Method For Computing Volume Potentials In The Galerkin Boundary Element Method In 3d Geometries, Sasan Mohyaddin

Mathematics Theses and Dissertations

We discuss how the Fast Multipole Method (FMM) applied to a boundary concentrated mesh can be used to evaluate volume potentials that arise in the boundary element method. If $h$ is the meshwidth near the boundary, then the algorithm can compute the potential in nearly $\Ord(h^{-2})$ operations while maintaining an $\Ord(h^p)$ convergence of the error. The effectiveness of the algorithms are demonstrated by solving boundary integral equations of the Poisson equation.

Finite Element Approximation Of Solutions Of The Equations Of Electroporoelasticity, Yu Hu

Mathematics Theses and Dissertations

In this thesis we consider the solution of the equations of electroporoelasticity, which are a combination of Maxwell's equations and the poroelasticity equations. Included is a description of suitable initial and boundary conditions, weak formulation of the equations, and the error estimate for a general numerical method.

Ensemble Data Fitting For Bathymetric Models Informed By Nominal Data, Samantha Zambo

Dissertations

Due to the difficulty and expense of collecting bathymetric data, modeling is the primary tool to produce detailed maps of the ocean floor. Current modeling practices typically utilize only one interpolator; the industry standard is splines-in-tension.

In this dissertation we introduce a new nominal-informed ensemble interpolator designed to improve modeling accuracy in regions of sparse data. The method is guided by a priori domain knowledge provided by artificially intelligent classifiers. We recast such geomorphological classifications, such as ‘seamount’ or ‘ridge’, as nominal data which we utilize as foundational shapes in an expanded ordinary least squares regression-based algorithm. To our knowledge …

Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo

Department of Mathematics: Faculty Publications

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

On A Stochastic Model Of Epidemics, Rachel Prather

Master's Theses

This thesis examines a stochastic model of epidemics initially proposed and studied by Norman T.J. Bailey [1]. We discuss some issues with Bailey's stochastic model and argue that it may not be a viable theoretical platform for a more general epidemic model. A possible alternative approach to the solution of Bailey's stochastic model and stochastic modeling is proposed as well. Regrettably, any further study on those proposals will have to be discussed elsewhere due to a time constraint.

Computational Analysis To Study The Efficiency Of Shear Activated Nano-Therapeutics In The Treatment Of Atherosclerosis, Nicholas Jefopoulos

Theses, Dissertations and Culminating Projects

Strokes are the fifth leading cause of death in the United States and can cause long-term disabilities in patients who survive a stroke. The vast majority of these strokes are ischemic, primarily caused by intracranial atherosclerosis. Most therapies to combat intracranial atherosclerosis simply manage it and do not remove the buildup of plaque. Targeted shear-activated nanotherapeutics are currently being developed to remove these plaques. We discuss the roles that aggregate particle density, aggregate particle diameter, vessel geometry, stenosis shape and breakup threshold play in the efficiency of this new technology. Computational studies were performed to test these parameters in three …

A Modified Preconditioned Conjugate Gradient Method For Approximating The Scattering Amplitude, Samson Ayo

Master's Theses

In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system Ax=b and an adjoint system A^Ty=g. Once these two systems are solved, the scattering amplitude, defined by g^Tx=y^Tb is easily obtained.

We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare …

Report: Spatial Facilitation-Inhibition Effects On Vegetation Distribution And Their Associated Patterns, Daniel D'Alessio

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Changes in the spatial distribution of vegetation respond to variations in the production and transportation mechanisms of seeds at different locations subject to heterogeneities, often because of soil characteristics. In semi-arid environments, the competition for water and nutrients pushes the superficial plant’s roots to obtain scarce resources at long ranges. In this report, we assume that vegetation biomass interacts with itself in two different ways, facilitation and inhibition, depending on the relative distances. We present a 1-dimensional Integro-difference model to represent and study the emergence of patterns in the distribution of vegetation.