Two problems involving high-resolution reconstruction from nonuniformly sampled data in x-ray computed tomography are addressed. A technique based on the theorem for sampling on unions of shifted lattices is introduced which exploits the symmetry property in two-dimensional fan beam computed tomography and permits the reconstruction of images with twice the...
X-ray computed tomography is a noninvasive imaging modality capable of reconstructing exact density values of 3D objects. Computed tomography machines are deployed across the world to provide doctors with an image that reveals more detail than a standard x-ray image. We investigate algorithms based on exact computed tomography reconstruction formulas...
In this thesis, we will study certain generalizations of the classical Shannon Sampling Theorem, which allows for the reconstruction of a pi-band-limited, square-integrable function from its samples on the integers. J. R. Higgins provided a generalization where the integers can be perturbed by less than 1/4, which includes nonuniform and...
Sampling theorems provide exact interpolation formulas for bandlimited
functions. They play a fundamental role in signal processing. A function is called
bandlimited if its Fourier transform vanishes outside a compact set. A generalized
sampling theorem in the framework of locally compact Abelian groups is presented.
Sampling sets are finite unions...
We explore two characteristic features of x-ray computed tomography inversion formulas in two and
three dimensions that are dependent on π-lines. In such formulas the data from a given source
position contribute only to the reconstruction of ƒ(x) for x in a certain region, called the region
of backprojection. The...
In this paper, we provide a thorough proof of most of Bertrand's Theorem. Using Arnold's book "Mathematical methods of classical mechanics" as a backbone and calculus methods demonstrated by Jovanović in his article "A note on the proof of Bertrand's theorem," we show that for masses in a central field...
Facial recognition has become increasingly important in recent years, due to the wide range of applications it has in fields such as security, surveillance, and human-computer interaction. Three popular methods for facial recognition are the Principal Component Analysis (PCA), Karhunen-Loeve Expansions, which is fundamentally a continuous form of PCA but...
In this paper, we discuss two possible modifications to a numerical solution method for a model of microbiologically induced calcite precipitation (MICP). MICP provides a means to seal cracks in the surfaces of geological structures. From a mathematical and computational point of view MICP has very interesting features which make...
In previous papers by Awrejcewicz in 1986 and Narayanan and Jayaraman in 1991, it was claimed that the nonlinear oscillator with dry friction exhibited chaos for several forcing frequencies. The chaos determination was achieved using the characteristic exponent of Lyapunov which requires the right-hand side of the differential equation to...
This thesis explores student understanding of divergence in physics settings. The various uses of divergence in physics are detailed. Textbook approaches to divergence are categorized as geometric, curvilinear coordinate, discussion-based, and algebraic statement. Textbook approaches to the divergence theorem are categorized as geometric, equivalent integrals, mathematical statement with explanation, and...
Time-domain models were developed to predict the response of a tethered buoy
subject to hydrodynamic loadings. A coupled analysis of the interaction of a buoy and
its mooring is included and three-dimensional response is assumed. External loadings
include hydrodynamic forces, tethers tensions, wind loadings and the weight of both
cable...
The growing need for cleaner fuels requires the development of better deep fuel desulfurization methods. The current study presents a reaction model for the mechanism of dibenzothiophene oxidization by dissolved oxygen occurring in a corona discharge microreactor. In the present work, a Finite Volume Method model of the reactor is...
In this work, we consider a convexity splitting scheme for a coupled phase field and energy equation, a modification of Stefan problem. The Stefan problem is a free boundary value problem that models the temperature in a homogeneous multiphase medium. Each phase is modeled using a heat diffusion parabolic partial...
Simulations of combustion and reacting flows often encounter stiffness in the equations governing chemical kinetics. Explicit solvers for these ordinary differential equations offer low computational expense, but typically cannot efficiently handle stiff systems. In contrast, implicit methods demand greater expense but offer unconditional stability—as a result, most reactive-flow solvers rely...
Wachspress rational functions are ratios of polynomials having certain properties which make them good candidates for basis functions in finite element methods. In addition. it had been theorized that Wachspress rational weight functions should Provide a full-resolution discretization of the neutral particle transport equation in thick diffusive regions, but this...
The uptake of hydrogen by lanthanum pentanickel (LaNi₅) to form lanthanum nickel hydride (LaNi₅H₆) is
followed with three-dimensional imaging by neutron tomography. The hydrogen absorption process is
slower than the time needed for acquiring a single radiograph, about 10 s, but fast relative to the time
to acquire a fully-sampled...
Frequency synthesizers are critical components of all communication systems. This thesis considers the issue of undesirable frequency spurs of a relatively recent type of frequency synthesis architecture called digital-to-time conversion (DTC). The DTC-based frequency synthesis architecture has important performance benefits over older frequency synthesizers, such as fast frequency switching, large...
In this dissertation, we study two risk models. First, we consider the dual risk process which models the surplus of a company that incurs expenses at a constant rate and earns random positive gains at random times. When the surplus is invested in a risky asset following a geometric Brownian...
We present a study of the ocean circulation using state of the art numerical and data assimilation techniques. The second chapter of the thesis presents the development and application of generalized inversion to a simple dynamical model of Lake Kinneret. The intent was to develop the necessary tools to implement...
This thesis contains three parts addressing the asymptotic analysis of fluid flow through fully saturated porous medium in the presence of an adjacent thin channel.
In the first part the problem is modeled by Darcy's law in both the porous medium and in the channel. The permeability in the channel...
In this thesis we study mathematical and computational models for phenomena of flow and transport in porous media in the presence of changing pore scale geometries. The differential equations for the flow and transport models at Darcy scale involve the coefficients of permeability, porosity, and tortuosity which depend on the...
Development of efficient methods for the destruction of solid wastes and recovery of valuable resources is needed to support long-duration manned missions in space. In particular, these technologies are required for deployment in hypogravity and microgravity environments, such as at the lunar or Martian surfaces. Gradient Magnetically Assisted Fluidized Bed...
The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow....
In this dissertation, we introduce a family of fully discrete finite difference time-domain (FDTD) methods for Maxwell’s equations in linear and nonlinear materials. Onecategory of methods is constructed using multiscale techniques involving operator splittings. We present the sequential splitting scheme, the Strang Marchuk splitting scheme,the weighted sequential splitting scheme including...
Calcium alginate gels are widely used in the biotechnology, food, and pharmaceutical industries for cell immobilization, food additives, and controlling the release of therapeutic agents. Different gelation conditions can lead to different gel structures which affects the diffusion of solutes in gels, thus mathematical models were developed to describe diffusion...
Magnetically Assisted Fluidized Bed (MAFB) is a novel technology where an external magnetic field with constant gradient interacts with magnetically susceptible particles. The linear magnetic field creates two type of forces: external and interparticle magnetic forces. A theoretical mathematical model based on the interaction of ideal dipoles is proposed to...
Health care providers, including complementary and alternative medical (CAM) practitioners, exert a significant influence on parental pediatric vaccination decisions. Use of CAM therapies is increasing in Oregon. Concomitantly, there has been a decade-long increase in parental vaccine refusal in Oregon, rising from 1 to 5 percent from 2000-2009. For example,...
As the world's power needs grow, the demand for power from renewable resources, such as wind or solar is increasing. One major drawback associated with these renewable resources is that the power output is dependent on environmental factors, such as cloud cover and wind speeds. This allows the possibility of...
Physicists solve problems and communicate their work using many external representations, such as equations, words, diagrams, graphs, sketches, pictures, and more. To learn physics, then, students must learn to use external representations. In this dissertation, I present three manuscripts. Each manuscript discusses how upper-division Paradigms in Physics students use multiple...
In 2013, Lemke Oliver created a list of all eta-quotients which are theta functions. Then in 2016, Folsom, Garthwaite, Kang, Swisher, and Treneer utilized this list of ``eta-theta'' functions along with Zwegers's construction of mock theta functions to create a set of mock modular forms which are also quantum modular...
Mathematical sophistication increases rapidly as students transition from lower- to upper-division physics courses. Complex algebra is one of the mathematical tools that is not introduced or used in lower-division physics courses but is pervasive throughout upper-division courses. In this dissertation, I study middle-division physics students' developing fluency with complex number...
Single-scattering tomography describes a model of photon transfer through a object in which photons are assumed to scatter at most once. The Broken Ray transform arises from this model, and was first investigated by Lucia Florescu, Vadim A. Markel, and John C. Schotland, [2], in 2010, followed by an inversion,...
In this dissertation, a group of vehicle dynamics simulation tools is developed with two primary goals: to accurately represent vehicle behavior and to provide insight that improves the understanding of vehicle performance. Three tools are developed that focus on tire modeling, vehicle modeling and lap time simulation.
Tire modeling is...
This thesis will cover work that I have completed relating to the field of terahertz (THz) science. My work has consisted of generating tunable, narrowband THz pulses in a table-top optical setup and using both narrow- and broadband THz pulses to study various material systems. Broadband THz pulses were used...
The extent to which thermoacoustic data determines the acoustic properties of an object was studied. In the case of one dimensional thermoacoustic imaging it was shown that constant acoustic profiles are uniquely specified from measurements. For a radial thermoacoustic problem we have shown that if the acoustic source is radial...
In source-detector radiation transport simulations, pulse height distributions are a useful metric in assessing the effectiveness of nuclear instrumentation. In the area of spectroscopy, pulse height distributions are used to identify an unknown source. It is widely believed that pulse height distributions cannot be created using deterministic methods. This quantity...
Gallium arsenide shows excellent promise for terahertz generation using mid infrared.
This is for two reasons. First, the indices of refraction for the terahertz (nTHz=3.61 at 1 THz) and mid infrared (nopt=3.431 at 2 μm) are close allowing a long interaction length.
Second, the linear absorption is low at terahertz...
The microscopic, momentum space, optical potential description of spin 1/2 x 1/2 scattering is extended to include the coupling of the singlet-triplet spin channels and
the exact handling of the Coulomb force. Computing performance in constructing
the optical potential and in solving the coupled-channels Lippmann-Schwinger equations
is enhanced by parallelization...
A perfectly matched layer (PML) is widely used to model many different types of wave propagation in different media. It has been found that a PML is often very effective and also easy to set, but still many questions remain.
We introduce a new formulation from regularizing the classical Un-Split...
The effects of alongshore variability in topography (banks and capes) and spatial variability in the wind forcing, including the wind-stress curl, on coastal ocean circulation are studied using a combination of observations and model simulations. Satellite sea surface temperature observations are used to describe the seasonal evolution of temperature fronts...
PURPOSE: We analyze a recently proposed polyenergetic version of the simultaneous algebraic reconstruction technique (SART). This algorithm, denoted pSART, replaces the monoenergetic forward projection operation used by SART with a post-log, polyenergetic forward projection, while leaving the rest of the algorithm unchanged. While the proposed algorithm provides good results empirically,...
The paper reviews percolation and some of its important properties, particularly on the 2-D square lattice. A bilevel lattice is introduced, with a percolation model representing the spread of a forest fire according to characteristics of the forest. It is proven that the value of the laddering probability may determine...
Circular dichroism (CD) spectra were obtained for bacteriophage T4 lysozyme and three of its mutants in the presence and absence of colloidal silica nanoparticles. Mutant lysozymes were produced by substitution of the isoleucine at position 3 with tryptophan, cysteine and leucine. Each substitution resulted in an altered structural stability, quantified...
In this dissertation, we investigate three numerical methods for inverting the Laplace transform. These methods are all based on the trapezoidal-type approximations to the Bromwich integral. The first method is the direct integration method: It is a straightforward application of the trapezoidal rule to the Bromwich integral, followed by convergence...
This is a two-part thesis. The first part is a generalization of vector calculus tools to Minkowski Space, a non-Euclidean 3-dimensional geometry that has a distance function that is not positive definite. We orient a cube in Minkowski Space using the generalized Stokes' Theorem to relate a divergence integral to...
Classical chaotic scattering experiments were performed on ten different four hill potentials, unique under x-axis reflection, given by: V(x,y) = x²y²e⁻⁽ˣ² ⁺ ʸ²⁾ σᵢ where σᵢ was ±1 depending upon quadrant. Our work was based on a paper (9) that only studied the case where σᵢ =1 in all quadrants....
Cellular sets in the Hilbert cube are the intersection of nested sequences of normal
cubes. One way of getting cellular maps on the Hilbert cube is by decomposing the Hilbert
cube into cellular sets and using a quotient map. By using a cellular decomposition of the
Hilbert cube, an example...