Applied Mathematics Commons^™

The Computational Study Of Fly Swarms & Complexity, Austin Bebee

Senior Theses

A system is considered complex if it is composed of individual parts that abide by their own set of rules, while the system, as a whole, will produce non-deterministic properties. This prevents the behavior of such systems from being accurately predicted. The motivation for studying complexity spurs from the fact that it is a fundamental aspect of innumerable systems. Among complex systems, fly swarms are relatively simple, but even so they are still not well understood. In this research, several computational models were developed to assist with the understanding of fly swarms. These models were primarily analyzed by using the …

Classifying Textual Fast Food Restaurant Reviews Quantitatively Using Text Mining And Supervised Machine Learning Algorithms, Lindsey Wright

Undergraduate Honors Theses

Companies continually seek to improve their business model through feedback and customer satisfaction surveys. Social media provides additional opportunities for this advanced exploration into the mind of the customer. By extracting customer feedback from social media platforms, companies may increase the sample size of their feedback and remove bias often found in questionnaires, resulting in better informed decision making. However, simply using personnel to analyze the thousands of relative social media content is financially expensive and time consuming. Thus, our study aims to establish a method to extract business intelligence from social media content by structuralizing opinionated textual data using …

A Practical Guide To Big Data, Ekaterina Smirnova, Andrada Ivanescu, Jiawei Bai, Ciprian M. Crainiceanu

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Big Data is increasingly prevalent in science and data analysis. We provide a short tutorial for adapting to these changes and making the necessary adjustments to the academic culture to keep Biostatistics truly impactful in scientific research.

Analysis Of Daily Precipitation Data From Selected Sites In The United States, Sahar Ahmed

Theses, Dissertations and Culminating Projects

Global warming is a contentious topic since modern climate records only exist for the last 100 years in contrast to ice-core analysis that establishes ice ages tens of thousands of years ago. Nevertheless, patterns associated with events such as El Niño Southern Oscillation (ENSO), precipitation, tornadoes, and snowfall amounts over the last century can provide a useful and objective indicator of climate “change”. This project focuses on daily precipitation totals for the state of New Jersey over the last 100 to 150 years from nineteen meteorological recording stations and involves large data sets with a million observations. This research utilizes …

Inertial Particle Transport By Lagrangian Coherent Structures In Geophysical Flows, Alexa Aucoin

Theses, Dissertations and Culminating Projects

Lagrangian Coherent Structures (LCS) provide a skeleton for the underlying structures in geophysical flows. It is known that LCS govern the movement of fluid particles within a flow, but it is not well understood how these same LCS influence the movement of inertial particles within a fluid flow. In this thesis, we consider two geophysical flows, the double-gyre model, and a single-layer quasi-geostrophic PDE model. In particular, we use finite-time Lyapunov exponents (FTLE) to characterize the attracting and repelling LCS for these models and show how inertial particles aggregate with respect to LCS. We numerically investigate the dynamics of inertial …

Seasonal Switching Affects Bacterial-Fungal Dominance In An Ecological System, Kristin Carfora

Theses, Dissertations and Culminating Projects

We consider a model inspired by producer-herbivore-decomposer soil food webs and determine the effect of ecological parameters on the decomposer pool. In particular, we observe how seasonal changes in the stoichiometric quality of the producer coupled with the efficiency of herbivory over the calendar year can induce a shift in the composition of the decomposer pool. Decomposers have a significant effect on the movement of essential nutrients throughout an ecosystem; we further determine how this shift between a bacterially dominated decomposer pool and a fungally dominated pool affects primary production and relative distribution of biomass of the other compartments.

Rapid Generation Of Jacobi Matrices For Measures Modified By Rational Factors, Amber Sumner

Dissertations

Orthogonal polynomials are important throughout the fields of numerical analysis and numerical linear algebra. The Jacobi matrix J for a family of n orthogonal polynomials is an n x n tridiagonal symmetric matrix constructed from the recursion coefficients for the three-term recurrence satisfied by the family. Every family of polynomials orthogonal with respect to a measure on a real interval [a,b] satisfies such a recurrence. Given a measure that is modified by multiplying by a rational weight function r(t), an important problem is to compute the modified Jacobi matrix Jmod corresponding to the new measure from knowledge of J. There …

Automatic Construction Of Scalable Time-Stepping Methods For Stiff Pdes, Vivian Ashley Montiforte

Master's Theses

Krylov Subspace Spectral (KSS) Methods have been demonstrated to be highly scalable time-stepping methods for stiff nonlinear PDEs. However, ensuring this scalability requires analytic computation of frequency-dependent quadrature nodes from the coefficients of the spatial differential operator. This thesis describes how this process can be automated for various classes of differential operators to facilitate public-domain software implementation.

Properties And Convergence Of State-Based Laplacians, Kelsey Wells

Department of Mathematics: Dissertations, Theses, and Student Research

The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peridynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different …

Power Corrections To Tmd Factorization For Z-Boson Production, I. Balitsky, A. Tatasov

Physics Faculty Publications

A typical factorization formula for production of a particle with a small transverse momentum in hadron-hadron collisions is given by a convolution of two TMD parton densities with cross section of production of the final particle by the two partons. For practical applications at a given transverse momentum, though, one should estimate at what momenta the power corrections to the TMD factorization formula become essential. In this paper we calculate the first power corrections to TMD factorization formula for Z-boson production and Drell-Yan process in high-energy hadron-hadron collisions. At the leading order in N_c power corrections are expressed in …

Harmonic Functions And Harmonic Measure, David Mcdonald

Honors Scholar Theses

The purpose of this thesis is to give a brief introduction to the field of harmonic measure. In order to do this we first introduce a few important properties of harmonic functions and show how to find a Green’s function for a given domain. Following this we calculate the harmonic measure for some easy cases and end by examining the connection between harmonic measure and Brownian motion.

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Energy Calculations And Wave Equations, Ellen R. Hunter

Graduate Theses/Dissertations

The focus of this thesis is to show how methods of Fourier analysis, in particular Parseval’s equality, can be used to provide explicit energy calculations for solutions of wave equations in one dimension. These calculations are discussed for simple examples and then extended to fit the general wave equation with Robin boundary conditions. Ideas from Sobolev space theory are used to provide justification of the method.

Godunov-Type Upwind Flux Schemes Of The Two-Dimensional Finite Volume Discrete Boltzmann Method, Leitao Chen, Laura Schaefer

Publications

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major …

On The Qualitative Theory Of The Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes, Habeeb Abed Kadhim Aal-Rkhais

Theses and Dissertations

We consider nonlinear second order degenerate or singular parabolic equation ut − a(um)xx + buβ + c(up)x = 0, a, m, β, p > 0, b, c ∈ R describing reaction-diffusion-convection processes arising in many areas of science and engineering, such as filtration of oil or gas in porous media, transport of thermal energy in plasma physics, flow of chemically reacting fluid, evolution of populations in mathematical biology etc. We apply the methods developed in U.G. Abdulla, Journal of Differential Equations, 164, 2(2000), 321-354 for the reaction-diffusion equation (c = 0) and prove the existence, uniqueness, boundary regularity and comparison theorems …

Hierarchical Bayesian Regression With Application In Spatial Modeling And Outlier Detection, Ghadeer Mahdi

Graduate Theses and Dissertations

This dissertation makes two important contributions to the development of Bayesian hierarchical models. The first contribution is focused on spatial modeling. Spatial data observed on a group of areal units is common in scientific applications. The usual hierarchical approach for modeling this kind of dataset is to introduce a spatial random effect with an autoregressive prior. However, the usual Markov chain Monte Carlo scheme for this hierarchical framework requires the spatial effects to be sampled from their full conditional posteriors one-by-one resulting in poor mixing. More importantly, it makes the model computationally inefficient for datasets with large number of units. …

Boundary Value Problems In A Multidimensional Box For Higher Order Linear And Quasi-Linear Hyperbolic Equations, Noha Aljaber

Theses and Dissertations

Boundary value problems in a multidimensional box for higher order linear hyperbolic equations are considered. The concept of associated problems are introduced. For general boundary value problems there are established: (i) Necessary and sufficient conditions for a linear problem to have the Fredholm property in two–dimensional case; (ii) Necessary and sufficient conditions of well–posedness in two–dimensional case; (iii) Unimprovable sufficient conditions for a linear problem to have the Fredholm property; (iv) Unimprovable sufficient conditions of well–posedness and α–well–posedness; (v) Effective sufficient conditions of unqie solvability of two–point, periodic and Dirichlet type problems. (iv) Unimprovable conditions of unique solvability of two …

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices …

The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski

Portland Institute for Computational Science Publications

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of …

The Pope's Rhinoceros And Quantum Mechanics, Michael Gulas

Honors Projects

In this project, I unravel various mathematical milestones and relate them to the human experience. I show and explain the solution to the Tautochrone, a popular variation on the Brachistochrone, which details a major battle between Leibniz and Newton for the title of inventor of Calculus. One way to solve the Tautochrone is using Laplace Transforms; in this project I expound on common functions that get transformed and how those can be used to solve the Tautochrone. I then connect the solution of the Tautochrone to clock making. From this understanding of clocks, I examine mankind’s understanding of time and …