Physical Sciences and Mathematics Commons^™

Understanding Waveguides In Resonance, Pieter Johannes Daniel Vandenberge

Dissertations and Theses

Several important classes of modern optical waveguides, including anti-resonant reflecting and photonic bandgap fibers, make use of geometries that guide energy in low refractive index material, a property that makes them of significant interest in numerous applications, notably including high-power delivery and guidance. These waveguides frequently exhibit resonance phenomena, in which their ability to propagate an input signal is sharply curtailed at particular operating frequencies. In this work we detail new advances in understanding these resonance effects and their implications for numerical modeling of these structures.

Part 1 focuses on the fields of slab waveguides, relatively simple structures for which …

Forest Generating Functions Of Directed Graphs, Ewan Joaquin Kummel

Dissertations and Theses

A spanning forest polynomial is a multivariate generating function whose variables are indexed over both the vertex and edge sets of a given directed graph. In this thesis, we establish a general framework to study spanning forest polynomials, associating them with a generalized Laplacian matrix and studying its properties. We introduce a novel proof of the famous matrix-tree theorem and show how this extends to a parametric generalization of the all-minors matrix-forest theorem. As an application, we derive explicit formulas for the recently introduced class of directed threshold graphs.

We prove that multivariate forest polynomials are, in general, irreducible and …

Equidistant Sets In Spaces Of Bounded Curvature, Logan Scott Fox

Dissertations and Theses

Given a metric space (X,d), and two nonempty subsets A,B ⊆ X, we study the properties of the set of points of equal distance to A and B, which we call the equidistant set E(A,B). In general, the structure of the equidistant set is quite unpredictable, so we look for conditions on the ambient space, as well as the given subsets, which lead to some regularity of the properties of the equidistant set. At a minimum, we will always require that X is path connected (so that E( …

Linear Nearest Neighbor Flocks With All Distinct Agents, Robert G. Lyons

Dissertations and Theses

This dissertation analyzes the global dynamics of 1-dimensional agent arrays with nearest neighbor linear couplings. The equations of motion are second order linear ODE's with constant coefficients. The novel part of this research is that the couplings are different for each agent. We allow the forces to depend on the relative position and relative velocity (damping terms) of the agents, and the coupling magnitudes differ for each agent. Further, we do not assume that the forces are "Newtonian'" (i.e., the force due to A on B equals minus the force of B on A) as this assumption does not apply …

An Analysis Of Capillary Flow In Finite Length Interior Corners, Samuel Shaw Mohler

Dissertations and Theses

We analyze the mathematical robustness of slow massively parallel interior corner flows in low gravity environments. An interior corner provides a preferential orientation in low gravity environments. This is a luxury usually only found on earth. It also provides a passive pumping mechanism due to geometry of a conduit. The driving force for this flow is a pressure difference due to local surface curvature gradients. An alternative reasoning is that due to the geometrical constraints the interior corner surface energy is unbounded below. This results in the liquid wicking into corners indefinitely. Interior corner flow's main quantity of interest is …

Numerical Techniques And Simulations For Studying Various High Power Optical Fiber Amplifiers, Particularly For Ytterbium (Yb+3), And Thulium (Tm+3) Doped Fibers, Tathagata Goswami

Dissertations and Theses

In this dissertation we present a simplified scalar numerical model, derived from Maxwell's field equations, for the fiber laser amplifier simulations. Maxwell's equations are reduced using a technique called Coupled Mode Theory (CMT).

The reduced model is made more efficient through a new scale model, referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber. The equivalent short fiber can be viewed as a fiber made using artificial (nonphysical) material properties that in some sense compensates for its reduced length. The computations can be accelerated by a factor approximately equal to the …

A Posteriori Error Estimates For Maxwell's Equations Using Auxiliary Subspace Techniques, Ahmed El Sakori

Dissertations and Theses

The aim of our work is to construct provably efficient and reliable error estimates of discretization error for Nédélec (edge) element discretizations of Maxwell's equations on tetrahedral meshes. Our general approach for estimating the discretization error is to compute an approximate error function by solving an associated problem in an auxiliary space that is chosen so that:

-Efficiency and reliability results for the computed error estimates can be established under reasonable and verifiable assumptions.

-The linear system used to compute the approximate error function has condition number bounded independently of the discretization parameter.

In many applications, it is some functional …

Guided Reinvention As A Context For Investigating Students' Thinking About Mathematical Language And For Supporting Students In Gaining Fluency, Kristen Vroom

Dissertations and Theses

Fluency with mathematical language is important for students' engagement in many disciplinary practices such as defining, conjecturing, and proving; yet, there is growing evidence that mathematical language is challenging for undergraduate students. This dissertation study draws on two design experiments with pairs of students who were supported to encode their mathematical meanings with more formal language. I aimed to investigate the teaching and learning of mathematical language, and particularly the language in statements with multiple quantifiers, by engaging students in this type of activity. In the first paper, I investigated the complex ways in which the students in my study …

Convex And Nonconvex Optimization Techniques For Multifacility Location And Clustering, Tuyen Dang Thanh Tran

Dissertations and Theses

This thesis contains contributions in two main areas: calculus rules for generalized differentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Banach spaces. These calculus rules are useful for further applications to nonsmooth optimization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class …

Leveraging Model Flexibility And Deep Structure: Non-Parametric And Deep Models For Computer Vision Processes With Applications To Deep Model Compression, Anthony D. Rhodes

Dissertations and Theses

My dissertation presents several new algorithms incorporating non-parametric and deep learning approaches for computer vision and related tasks, including object localization, object tracking and model compression. With respect to object localization, I introduce a method to perform active localization by modeling spatial and other relationships between objects in a coherent "visual situation" using a set of probability distributions. I further refine this approach with the Multipole Density Estimation with Importance Clustering (MIC-Situate) algorithm. Next, I formulate active, "situation" object search as a Bayesian optimization problem using Gaussian Processes. Using my Gaussian Process Context Situation Learning (GP-CL) algorithm, I demonstrate improved …

Necessary Conditions For Stability Of Vehicle Formations, Pablo Enrique Baldivieso Blanco

Dissertations and Theses

Necessary conditions for stability of coupled autonomous vehicles in R are established in this thesis. The focus is on linear arrays with decentralized vehicles, where each vehicle interacts with only a few of its neighbors. Decentralized means that there is no central authority governing the motion. Instead, each vehicle registers only velocity and position relative to itself and bases its acceleration only on those data. Explicit expressions are obtained for necessary conditions for asymptotic stability in the cases that a system consists of a periodic arrangement of two or three different types of vehicles, i.e. configurations as follows: ...2-1-2-1 or …

A New Finite Difference Time Domain Method To Solve Maxwell's Equations, Timothy P. Meagher

Dissertations and Theses

We have constructed a new finite-difference time-domain (FDTD) method in this project. Our new algorithm focuses on the most important and more challenging transverse electric (TE) case. In this case, the electric field is discontinuous across the interface between different dielectric media. We use an electric permittivity that stays as a constant in each medium, and magnetic permittivity that is constant in the whole domain. To handle the interface between different media, we introduce new effective permittivities that incorporates electromagnetic fields boundary conditions. That is, across the interface between two different media, the tangential component, E_r(x,y), …

Generalized Differential Calculus And Applications To Optimization, R. Blake Rector

Dissertations and Theses

This thesis contains contributions in three areas: the theory of generalized calculus, numerical algorithms for operations research, and applications of optimization to problems in modern electric power systems. A geometric approach is used to advance the theory and tools used for studying generalized notions of derivatives for nonsmooth functions. These advances specifically pertain to methods for calculating subdifferentials and to expanding our understanding of a certain notion of derivative of set-valued maps, called the coderivative, in infinite dimensions. A strong understanding of the subdifferential is essential for numerical optimization algorithms, which are developed and applied to nonsmooth problems in operations …

Computational Algorithms For Improved Representation Of The Model Error Covariance In Weak-Constraint 4d-Var, Jeremy A. Shaw

Dissertations and Theses

Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and …

Orthomorphisms Of Boolean Groups, Nichole Louise Schimanski

Dissertations and Theses

An orthomorphism, π, of a group, (G, +), is a permutation of G with the property that the map x → -x + π(x) is also a permutation. In this paper, we consider orthomorphisms of the additive group of binary n-tuples, Zⁿ₂. We use known orthomorphism preserving functions to prove a uniformity in the cycle types of orthomorphisms that extend certain partial orthomorphisms, and prove that extensions of particular sizes of partial orthomorphisms exist. Further, in studying the action of conjugating orthomorphisms by automorphisms, we find several symmetries within the orbits and stabilizers of this action, and …

Tree Graphs And Orthogonal Spanning Tree Decompositions, James Raymond Mahoney

Dissertations and Theses

Given a graph G, we construct T(G), called the tree graph of G. The vertices of T(G) are the spanning trees of G, with edges between vertices when their respective spanning trees differ only by a single edge. In this paper we detail many new results concerning tree graphs, involving topics such as clique decomposition, planarity, and automorphism groups. We also investigate and present a number of new results on orthogonal tree decompositions of complete graphs.

The Design And Validation Of A Group Theory Concept Inventory, Kathleen Mary Melhuish

Dissertations and Theses

Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated …

An Investigation Of The Role Of Alternate Numeration Systems In Preservice Teacher Mathematics Content Courses, Jodi I. Fasteen

Dissertations and Theses

Alternate numeration systems are common in preservice teacher (PST) mathematics curricula, but there is limited research on how to leverage alternate systems to promote the development of mathematical knowledge for teaching. I analyzed the role of alternate numeration systems in three ways. I conducted a thematic analysis of current PST textbooks to consider the role of alternate numeration systems in written curricula. I conducted a teaching experiment to analyze PSTs' mathematical activity as they engaged with a base five task sequence to reinvent an algorithm for multiplication. And I introduced problematizing mathematical contexts as a design heuristic, situating this within …

Global Resource Management Of Response Surface Methodology, Michael Chad Miller

Dissertations and Theses

Statistical research can be more difficult to plan than other kinds of projects, since the research must adapt as knowledge is gained. This dissertation establishes a formal language and methodology for designing experimental research strategies with limited resources. It is a mathematically rigorous extension of a sequential and adaptive form of statistical research called response surface methodology. It uses sponsor-given information, conditions, and resource constraints to decompose an overall project into individual stages. At each stage, a "parent" decision-maker determines what design of experimentation to do for its stage of research, and adapts to the feedback from that research's potential …

Elementary Teacher Candidates' Images Of Mathematics, Diverse Students, And Teaching: An Exploratory Study With Implications For Culturally Responsive Mathematics Education, Bernd Richard Ferner

Dissertations and Theses

Children from many culturally diverse backgrounds do not achieve in mathematics at the same rates as their counterparts from the dominant White, European-American culture (Gay, 2010). This so-called achievement gap is an artifact of an educational system that continues to fail to provide equal learning opportunities to culturally diverse children (Ladson-Billings, 2006; Nieto & Bode, 2011). Teachers who employ culturally responsive teaching (Gay, 2010) may help to close this opportunity gap and hence, the achievement gap. This study investigated, "How do elementary teacher candidates perceive teaching mathematics in a multicultural environment"; Using a critical constructivism research paradigm, this qualitative instrumental …

A Parabolic Equation Analysis Of The Underwater Noise Radiated By Impact Pile Driving, Nathan Laws

Dissertations and Theses

Impact pile driving can produce extremely high underwater sound levels, which are of increasing environmental concern due to their deleterious effects on marine wildlife. Prediction of underwater sound levels is important to the assessment and mitigation of the environmental impacts caused by pile driving. Current prediction methods are limited and do not account for the dynamic pile driving source, inhomogeneities in bathymetry and sediment, or physics-based sound wave propagation.

In this thesis, a computational model is presented that analyzes and predicts the underwater noise radiated by pile driving and is suitable for shallow, inhomogeneous environments and long propagation ranges. The …

Establishing Foundations For Investigating Inquiry-Oriented Teaching, Estrella Maria Salas Johnson

Dissertations and Theses

The Teaching Abstract Algebra for Understanding (TAAFU) project was centered on an innovative abstract algebra curriculum and was designed to accomplish three main objectives: to produce a set of multi-media support materials for instructors, to understand the challenges faced by mathematicians as they implemented this curriculum, and to study how this curriculum supports student learning of abstract algebra. Throughout the course of the project I took the lead investigating the teaching and learning in classrooms using the TAAFU curriculum. My dissertation is composed of three components of this research. First, I will report on a study that aimed to describe …

Short-Term Plasticity At The Schaffer Collateral: A New Model With Implications For Hippocampal Processing, Andrew Hamilton Toland

Dissertations and Theses

A new mathematical model of short-term synaptic plasticity (STP) at the Schaffer collateral is introduced. Like other models of STP, the new model relates short-term synaptic plasticity to an interaction between facilitative and depressive dynamic influences. Unlike previous models, the new model successfully simulates facilitative and depressive dynamics within the framework of the synaptic vesicle cycle. The novelty of the model lies in the description of a competitive interaction between calcium-sensitive proteins for binding sites on the vesicle release machinery. By attributing specific molecular causes to observable presynaptic effects, the new model of STP can predict the effects of specific …

Stochastic Orders In Heterogeneous Samples With Applications, Maochao Xu

Dissertations and Theses

The statistics literature has mostly focused on the case when the data available is in the form of a random sample. In many cases, the observations are not identically distributed. Such samples are called heterogeneous samples. The study of heterogeneous samples is of great interest in many areas, such as statistics, econometrics, reliability engineering, operation research and risk analysis.

Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc.

In this dissertation, most of the work is …

Model Reduction For Simulation, Optimization And Control, Oleg Edward Roderick

Dissertations and Theses

Many tasks of simulation, optimization and control can be performed more efficiently if the intermediate complexity of the numerical model is reduced. In our work, we investigate model reduction, as applied to reaction-transport systems of atmospheric chemistry. We use a Proper Orthogonal Decomposition-based approach to extract information from a set of model observations, and to project the model equations onto a reduced order space chosen in such a way that the essential model behavior is preserved in the solution of the reduced version. We examine and improve many features of the method. In particular, we show how to measure sensitivities …

Quantum Multiplexers, Parrondo Games, And Proper Quantization, Faisal Shah Khan

Dissertations and Theses

A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property.

The main result in this dissertation is the development of quantizations of …

Quaternions, Octonions, And The Quantization Of Games, Aden Omar Ahmed

Dissertations and Theses

We present an effect on classical games that is obtained by replacing the notion of probability distribution with the notions of quantum superposition and measurement. Our particular focus will be on two and three player games where each player has precisely two pure strategic choices. Games in normal form are represented as "payoff" functions.

Game quantization requires the extension of these functions to much larger domains. The main result of this work is the co-ordinatization of these extended functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then …

Data Assimilation, Adaptive Observations And Applications, Humberto Carlos Godinez Vasquez

Dissertations and Theses

Sensitivity analysis, data assimilation and targeting observation strategies are methods that are applied to various complex mathematical models of fluid dynamics. In this research we investigate new directions to improve on the current strategies used to deploy additional observational resources (targeting strategies) for data assimilation in dynamical systems of fluid mechanics.

Targeting strategies aim to determine optimal locations where additional observations will improve the solution of the data assimilation process by identifying regions where state errors in the model have a high potential to grow.

Properly accounting for nonlinear error growth is an unresolved issue in targeted observations for numerical …

Students' Reasoning About The Concept Of Limit In The Context Of Reinventing The Formal Definition, Craig Alan Swinyard

Dissertations and Theses

Many researchers (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit plays as a foundational concept in analysis. The vast majority of topics encountered in calculus and undergraduate analysis are built upon understanding the concept of limit and being able to work flexibly with its formal definition (Bezuidenhout, 2001). The purpose of this study was to: (1) Develop insight into students' reasoning about limit in relation to their engagement in instruction designed to support their reinventing the formal definition of limit, and; (2) Inform the design of principled instruction that might support students' attempts to …

Graduate Teaching Assistants' Statistical Knowledge For Teaching, Jennifer Ann Noll

Dissertations and Theses

This dissertation explores graduate teaching assistants’ (TAs’) statistical knowledge for teaching. Data collection methods that enabled the exploration of TA s’ statistical knowledge for teaching include: (a) a task-based web survey administered to 68 TAs from 18 universities across the United States; and, (b) a series of three taskbased interviews with a subset of five TAs from the larger survey population. Through and Strauss, 1967), I investigated the ways in which TAs reason about sampling tasks, and how they think about teaching and student learning in relation to sampling ideas. Building on past research in statistics education on K-12 and …