Physical Sciences and Mathematics Commons^™

Mathematical Modeling For Dental Decay Prevention In Children And Adolescents, Mahdiyeh Soltaninejad

Dissertations

The high prevalence of dental caries among children and adolescents, especially those from lower socio-economic backgrounds, is a significant nationwide health concern. Early prevention, such as dental sealants and fluoride varnish (FV), is essential, but access to this care remains limited and disparate. In this research, a national dataset is utilized to assess sealants' reach and effectiveness in preventing tooth decay, particularly focusing on 2nd molars that emerge during early adolescence, a current gap in the knowledge base. FV is recommended to be delivered during medical well-child visits to children who are not seeing a dentist. Challenges and facilitators in …

Bacterial Motion And Spread In Porous Environments, Yasser Almoteri

Dissertations

Micro-swimmers are ubiquitous in nature from soil and water to mammalian bodies and even many technological processes. Common known examples are microbes such as bacteria, micro-algae and micro-plankton, cells such as spermatozoa and organisms such as nematodes. These swimmers live and have evolved in multiplex environments and complex flows in the presence of other swimmers and types, inert particles and fibers, interfaces and non-trivial confinements and more. Understanding the locomotion and interactions of these individual micro-swimmers in such impure viscous fluids is crucial to understanding the emergent dynamics of such complex systems, and to further enabling us to control and …

Fluid Dynamics Of Interacting Particles: Bouncing Droplets And Colloid-Polymer Mixtures, Lauren Barnes

Dissertations

Interacting particles are a common theme across various physical systems, particularly on the atomic and sub-atomic scales. While these particles cannot be seen with the human eye, insight into such systems can be gained by observing macroscopic systems whose physical behavior is similar. This dissertation consists of three different chapters, each presenting a different problem related to interacting particles, as follows:

Chapter 1 explores chaotic trajectories of a droplet bouncing on the surface of a vertically vibrating fluid bath, with a simple harmonic force acting on the droplet. The bouncing droplet system has attracted recent interest because it exhibits behaviors …

Boundary Integral Equation Methods For Superhydrophobic Flow And Integrated Photonics, Kosuke Sugita

Dissertations

This dissertation presents fast integral equation methods (FIEMs) for solving two important problems encountered in practical engineering applications.

The first problem involves the mixed boundary value problem in two-dimensional Stokes flow, which appears commonly in computational fluid mechanics. This problem is particularly relevant to the design of microfluidic devices, especially those involving superhydrophobic (SH) flows over surfaces made of composite solid materials with alternating solid portions, grooves, or air pockets, leading to enhanced slip.

The second problem addresses waveguide devices in two dimensions, governed by the Helmholtz equation with Dirichlet conditions imposed on the boundary. This problem serves as a …

Continuum Modeling Of Active Nematics Via Data-Driven Equation Discovery, Connor Robertson

Dissertations

Data-driven modeling seeks to extract a parsimonious model for a physical system directly from measurement data. One of the most interpretable of these methods is Sparse Identification of Nonlinear Dynamics (SINDy), which selects a relatively sparse linear combination of model terms from a large set of (possibly nonlinear) candidates via optimization. This technique has shown promise for synthetic data generated by numerical simulations but the application of the techniques to real data is less developed. This dissertation applies SINDy to video data from a bio-inspired system of mictrotubule-motor protein assemblies, an example of nonequilibrium dynamics that has posed a significant …

Deep Hybrid Modeling Of Neuronal Dynamics Using Generative Adversarial Networks, Soheil Saghafi

Dissertations

Mechanistic modeling and machine learning methods are powerful techniques for approximating biological systems and making accurate predictions from data. However, when used in isolation these approaches suffer from distinct shortcomings: model and parameter uncertainty limit mechanistic modeling, whereas machine learning methods disregard the underlying biophysical mechanisms. This dissertation constructs Deep Hybrid Models that address these shortcomings by combining deep learning with mechanistic modeling. In particular, this dissertation uses Generative Adversarial Networks (GANs) to provide an inverse mapping of data to mechanistic models and identifies the distributions of mechanistic model parameters coherent to the data.

Chapter 1 provides background information on …

Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht

Dissertations

Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the …

Stochastic Modeling Of Flows In Membrane Pore Networks, Binan Gu

Dissertations

Membrane filters provide immediate solutions to many urgent problems such as water purification, and effective remedies to pressing environmental concerns such as waste and air treatment. The ubiquity of applications gives rise to a significant amount of research in membrane material selection and structural design to optimize filter efficiency. As physical experiments tend to be costly, numerical simulation and analysis of fluid flow, foulant transport and geometric evolution due to foulant deposition in complex geometries become particularly relevant. In this dissertation, several mathematical modeling and analytical aspects of the industrial membrane filtration process are investigated. A first-principles mathematical model for …

Numerical Methods For Optimal Transport And Optimal Information Transport On The Sphere, Axel G. R. Turnquist

Dissertations

The primary contribution of this dissertation is in developing and analyzing efficient, provably convergent numerical schemes for solving fully nonlinear elliptic partial differential equation arising from Optimal Transport on the sphere, and then applying and adapting the methods to two specific engineering applications: the reflector antenna problem and the moving mesh methods problem. For these types of nonlinear partial differential equations, many numerical studies have been done in recent years, the vast majority in subsets of Euclidean space. In this dissertation, the first major goal is to develop convergent schemes for the sphere. However, another goal of this dissertation is …

Optimization Opportunities In Human In The Loop Computational Paradigm, Dong Wei

Dissertations

An emerging trend is to leverage human capabilities in the computational loop at different capacities, ranging from tapping knowledge from a richly heterogeneous pool of knowledge resident in the general population to soliciting expert opinions. These practices are, in general, termed human-in-the-loop (HITL) computations.

A HITL process requires holistic treatment and optimization from multiple standpoints considering all stakeholders: a. applications, b. platforms, c. humans. In application-centric optimization, the factors of interest usually are latency (how long it takes for a set of tasks to finish), cost (the monetary or computational expenses incurred in the process), and quality of the completed …

Periodic Fast Multipole Method, Ruqi Pei

Dissertations

Applications in electrostatics, magnetostatics, fluid mechanics, and elasticity often involve sources contained in a unit cell C, centered at the origin, on which periodic boundary condition are imposed. The free-space Green’s functions for many classical partial differential equations (PDE), such as the modified Helmholtz equation, are well-known. Among the existing schemes for imposing the periodicity, three common approaches are: direct discretization of the governing PDE including boundary conditions to yield a large sparse linear system of equations, spectral methods which solve the governing PDE using Fourier analysis, and the method of images based on tiling the plane with copies of …

Nystrom Methods For High-Order Cq Solutions Of The Wave Equation In Two Dimensions, Erli Wind-Andersen

Dissertations

An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate …

Type I Error Rate Controlling Procedures For Multiple Hypotheses Testing, Beibei Li

Dissertations

This dissertation addresses several different but related topics arising in the field of multiple testing, including weighted procedures and graphical approaches for controlling the familywise error rate (FWER), and stepwise procedures with control of the false discovery rate (FDR) for discrete data. It consists of three major parts.

The first part investigates weighted procedures for controlling the FWER. In many statistical applications, hypotheses may be differentially weighted according to their different importance. Many weighted multiple testing procedures (wMTPs) have been developed for controlling the FWER. Among these procedures, two weighted Holm procedures are commonly used in practice: one is based …

Model Checks For Two-Sample Location-Scale, Atefeh Javidialsaadi

Dissertations

Two-sample location-scale refers to a model that permits a pair of standardized random variables to have a common distribution. This means that if X₁ and X₂ are two random variables with means µ₁ and µ₂ and standard deviations ?₁ and ?₂, then (X₁ - µ₁)/?₁ and (X₂ - µ₂)/?₂ have some common unspecified standard or base distribution F₀. Function-based hypothesis testing for these models refers to formal tests that would help determine whether or not two samples may have come from some location-scale …

Coherent Control Of Dispersive Waves, Jimmie Adriazola

Dissertations

This dissertation addresses some of the various issues which can arise when posing and solving optimization problems constrained by dispersive physics. Considered here are four technologically relevant experiments, each having their own unique challenges and physical settings including ultra-cold quantum fluids trapped by an external field, paraxial light propagation through a gradient index of refraction, light propagation in periodic photonic crystals, and surface gravity water waves over shallow and variable seabeds. In each of these settings, the physics can be modeled by dispersive wave equations, and the technological objective is to design the external trapping fields or propagation media such …

Advances In Modeling Gas Adsorption In Porous Materials For The Characterization Applications, Max A. Maximov

Dissertations

The dissertation studies methods for mesoporous materials characterization using adsorption at various levels of scale and complexity. It starts with the topic introduction, necessary notations and definitions, recognized standards, and a literature review.

Synthesis of novel materials requires tailoring of the characterization methods and their thorough testing. The second chapter presents a nitrogen adsorption characterization study for silica colloidal crystals (synthetic opals). These materials have cage-like pores in the range of tens of nanometers. The adsorption model can be described within a macroscopic approach, based on the Derjaguin-Broekhoff-de Boer (DBdB) theory of capillary condensation. A kernel of theoretical isotherms is …

Modeling Dewetting, Demixing, And Thermal Effects In Nanoscale Metal Films, Ryan Howard Allaire

Dissertations

Thin film dynamics, particularly on the nanoscale, is a topic of extensive interest. The process by which thin liquids evolve is far from trivial and can lead to dewetting and drop formation. Understanding this process involves not only resolving the fluid mechanical aspects of the problem, but also requires the coupling of other physical processes, including liquid-solid interactions, thermal transport, and dependence of material parameters on temperature and material composition. The focus of this dissertation is on the mathematical modeling and simulation of nanoscale liquid metal films, which are deposited on thermally conductive substrates, liquefied by laser heating, and subsequently …

Modeling And Design Optimization For Membrane Filters, Yixuan Sun

Dissertations

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for: (1) pleated membrane …

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 …

Dominating Functions In Graphs, Maria Talanda-Fisher

Dissertations

Domination in graphs has become one of the most popular areas of graph the- ory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. While this area evidently began with the work by the French mathematician Claude Berge in 1958 and the Norwegian-American mathematician Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of the survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous …

Intermittent Dynamics Of Dense Particulate Matter, Chao Cheng

Dissertations

Granular particle systems are scattered around the universe, and they can behave like solids when there exist strong force-bearing networks, so that the granular system can resist certain stress without deformation. When such a network is not present, particles yield to small stress and behave like a fluid. A wide range of systems exhibit intermittent dynamics as they are slowly loaded, with different dynamical regimes governing many industrial and natural phenomena. While a significant amount of research on exploring intermittent dynamics of granular systems has been carried out, not much is known about the connection between particle-scale response and the …

Eigenvalue Problems For Fully Nonlinear Elliptic Partial Differential Equations With Transport Boundary Conditions, Jacob Lesniewski

Dissertations

Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods …

A Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte

Dissertations

Krylov Subspace Spectral (KSS) Methods have demonstrated to be highly scalable methods for PDEs. However, a current limitation of these methods is the requirement of a rectangular or box-shaped domain. Smooth Extension Embedding Methods (SEEM) use fictitious domain methods to extend a general domain to a simple, rectangular or box-shaped domain. This dissertation describes how these methods can be combined to extend the applicability of KSS methods, while also providing a component-wise approach for solving the systems of equations produced with SEEM.

Efficient Denoising Of High Resolution Color Digital Images Utilizing Krylov Subspace Spectral Methods, Eva Lynn Greenman

Dissertations

The solution to a parabolic nonlinear diffusion equation using a Krylov Subspace Spectral method is applied to high resolution color digital images with parallel processing for efficient denoising. The evolution of digital image technology, processing power, and numerical methods must evolve to increase efficiency in order to meet current usage requirements. Much work has been done to perfect the edge detector in Perona-Malik equation variants, while minimizing the effects of artifacts. It is demonstrated that this implementation of a regularized partial differential equation model controls backward diffusion, achieves strong denoising, and minimizes blurring and other ancillary effects. By adaptively tuning …

Construct Linear Quasi-Interpolants On Infinite Intervals, Johara Farah Albaliwi

Dissertations

In solving the data interpolation problem, which is fundamental in data analysis, we typically deal with the data samples spread in a finite interval [a, b], which results in the operations involving finite-dimensional matrices. There are many interesting results developed under this framework. However, when the data samples are given from an infinite interval [a, ∞) (for certain special types of real-world applications), many existing results would not work anymore due to the special properties of the infinite data samples. A new framework should be established to support the infinite data samples.

In this dissertation, we develop a special tool …

Instantaneous Frequency Estimation And Signal Separation Using Fractional Continuous Wavelet Transform, Abdelbaset R. Zeyani

Dissertations

In the signal processing field, time-frequency representations (TFR's) have intensively been improved to provide effective and powerful tools for reliable signal analysis. One of the most valuable and frequently used tools is Fourier transform (FT) which has been used to study the frequency content of stationary signals in the Fourier domain (FD). However, FT is not sufficient to study the frequency of non-stationary signals. For this particular type of signals to be best analyzed, some transforms such as the short time Fourier transform (STFT) and the continuous wavelet transform (CWT) have been introduced to provide us with a signal representation …

Dances And Escape Of The Vortex Quartet, Brandon Behring

Dissertations

This dissertation considers the linear stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W. Gröbli (1877) and A. E. H. Love (1883) and can be parameterized by a dimensionless parameter related to the geometry of the initial configuration. Simulations by Acheson and numerical Floquet analysis by Tophøj and Aref both indicate, to many digits, that the bifurcation occurs at a value related to the inverse square of the golen ratio. Acheson observed that, after an initial period …

Convergence Of The Boundary Integral Method For Interfacial Stokes Flow, Keyang Zhang

Dissertations

Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. A convergence analysis of the boundary integral method for Stokes flow …

Stochastic Delay Differential Equations With Applications In Ecology And Epidemics, Hebatallah Jamil Alsakaji

Dissertations

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the …

Mathematical Models And Tools To Understand Coupled Circadian Oscillations And Limit Cycling Systems, Guangyuan Liao

Dissertations

The circadian rhythm refers to an internal body process that regulates many body processes including the sleep-wake cycle, digestion and hormone release. The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of the most important properties. There are several scenarios in which circadian oscillators do not directly receive light-dark forcing. Instead they are part of hierarchical systems in which, as \peripheral" oscillators, they are periodically forced by other \central" circadian oscillators that do directly receive light input. Such dynamics are modeled as hierarchical coupled limit cycle systems. Those models usually have a large population, …