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...
Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to the modular group, and Bannai, Kojima, and Miezaki's partial proof for the case of the Fricke group of level 2, we show that the zeros of certain modular functions for some low-level genus zero groups associated to...
In this dissertation, we begin by presenting the result of F. K. C. Rankin and Swinnerton-Dyer on the location of the zeros of the Eisenstein series for the full modular group in the standard fundamental domain. Their simple but beautiful argument shows that all zeros are located on the lower...
This thesis is an exploration of writing a music album from a singer-song writer perspective. It contains the process of writing lyrics and music, inspiration from other artists, as well as performance. It goes through the emotional ebbs and flows of creative writing, ground work for recording an album and...
String theory, one of the more popular approaches to quantizing gravity, is a highly complex
theory, involving high level mathematics and physics. But the basic ideas of string theory are not inaccessible, even to the undergraduate. This document acts as report to my thesis project, the writing of A Detailed...
The goal of this research project is to determine the fractal nature, if any, which
surface water waves exhibit when viewed on a microscopic scale. Due to the
relatively recent development of this area of mathematics, a brief introduction to the
study of fractal geometry, as well as several examples...
Microbial ecology has been transformed by metagenomics, the study of the genetic in-formation in entire communities of organisms. In the following we develop metagenomic tools arising from the classic Wasserstein metric as applied to questions regarding the diversity between microbial communities. We provide a novel proof of the characteriza-tion of...
A concept image is that collection of all images, pictures, symbols, definitions
and properties associated with any given mathematical concept. One of the
most important components in the mental representation of concepts in the concept
image of advanced mathematical thinkers is visualization. This component, in
turn, is indispensable in the...
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....
The author shows that a necessary and sufficient condition
for a convex polyhedron to be representable as a finite vector sum
of line segments is that each of its faces possesses central symmetry.
College Algebra is a prerequisite for calculus and is thus an important stepping stone in the careers of STEM-intending undergraduates. However, College Algebra has low pass rates across the United States, interrupting students’ pathways to success. To address this concern, a research-oriented university in the Northwest United States restructured its...
In this work, we provide a detailed analysis of a discrete time regime switching financial market model with jumps. We consider the model under two different scenarios: known and unknown initial regime. For each scenario we investigated conditions that guarantee the model's completeness. We find that the model under consideration...
The standard way of representing numbers on computers gives rise to errors
which increase as computations progress. Using p-adic valuations can reduce
error accumulation. Valuation theory tells us that p-adic and standard valuations
cannot be directly compared. The p-adic valuation can, however, be used in
an indirect way. This gives...
Machine learning models are powerful tools which may aid in the prediction of survival outcomes of cancer patients. This study evaluated eight classification models and eight regression models on their ability to predict survival outcomes on breast cancer and prostate cancer data sets from the SEER database. The most accurate...
Machine learning models are powerful tools which may aid in the prediction of survival outcomes of cancer patients. This study evaluated eight classification models and eight regression models on their ability to predict survival outcomes on breast cancer and prostate cancer data sets from the SEER database. The most accurate...
Microcomputers are being used in teaching and learning
mathematics. This paper examines ways that computers can be used
to enhance teaching in precalculus mathematics. The advantages of
computers discussed are: 1) instant access to graphs, 2) examples
that are not oversimplified to make computations manageable, 3)
encouragement of hypothesizing and...
In this paper we develop an upscaling technique for non-Darcy flow in porous media. Non-Darcy model of flow applies to flow in porous media when large velocities occur. The well-posedness results for theory of quasilinear elliptic partial differential equations. To discretize the model we used lowest order Raviart-Thomas mixed finite...
In this paper a direct, constructive proof of the equivalence of the Normal Algorithm and Turing machine using the Turing machines NAS (Normal Algorithm Simulator) and NAC (Normal Algorithm Converter) is presented. The Turing machine NAS can simulate any particular Normal Algorithm, and NAG can convert the quintuples of a...
In this paper, we are concerned with the very general notion of
a universal algebra. A universal algebra essentially consists of a set
A together with a possibly infinite set of finitary operations on. A.
Generally, these operations are related by means of equations, yielding
different algebraic structures such as...
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...
Sequential machines uniquely determine directed graphs. A path in a sequential machine may be specified by a starting state and an input sequence. A uniform Hamiltonian touring sequence (UHTS) is an input sequence that specifies a Hamiltonian path regardless of the starting state. We present a polynomial time algorithm that...
In the study of uniform convergence, one is led naturally to
the question of how uniform convergence on subsets relates to uniform convergence on the whole space. This paper develops theorems on how pointwise convergence relates to uniform convergence
on finite sets, how uniform convergence on finite subsets relates to...
Generalization is a fundamental mathematical practice across all disciplines and content areas (Amit & Klass, 2005; Lannin, 2005; Pierce, 1902; Vygotsky, 1986; Ellis, Lockwood, Tillema & Moore, 2017). While a considerable amount of research has been conducted on students' generalizing activity in algebraic contexts (Amit & Klass 2005; Becker &...
In this study I viewed video and corresponding transcripts of two students solving introductory combinatorial problems. Using an adapted version of Harel’s (2008) concepts of Result Pattern Generalization and Process Pattern Generalization, I analyzed the work done by the two students. Both students primarily worked through the problem How many...
We develop several a-priori and a-posteriori error estimates for two-grid finite element discretization of coupled elliptic and parabolic systems with a set of parameters P. We present numerical results that verify the convergence order of the numerical schemes predicted by the a-priori estimates. We present numerical results that verify the...
A simplification of the proof of the maximum principle of
Pontryagin is obtained for constrained and unconstrained optimal control
problems. Two numerical methods for solving optimal control problems
with guaranteed error bounds using the maximum principle of Pontryagin
and interval analysis are derived.
The ring disclination is a topological defect that may be suitable for light polarization inside of a nematic liquid crystal. Due to its stability and chirality, the ring disclination could also allow for theoretical applications to quantum and classical field theories as a model for fundamental particles. In order to...
We identify all translation covers among triangular billiards surfaces. Our main tools are the J-invariant of Kenyon and Smillie and a property of triangular billiards surfaces, which we call fingerprint type, that is invariant under balanced translation covers.
This dissertation arose out of an awareness of difficulties undergraduate linear algebra students encounter when solving linear algebra problems from novel, non-isomorphic settings, even when the problems could be solved with matrix representations and similar procedures as problems from a more familiar setting. This mixed-methods study utilized both traditional and...
Let G be a finite group, G₂ be a Sylow 2-subgroup of G, and L/K be a G-Galois extension. We study the trace form qL/K of L/K and the question of existence of a self-dual normal basis. Our main results are as follows: (1) If G₂ is not abelian and...
An analytical model is developed to address the question of how different disturbance
regimes affect the mean and variance of landscape carbon storage in forest ecosystems. Total landscape carbon is divided into five pools based on the processes from which they are derived and based on their temporal dynamics. Formulae...
First, topological vector spaces are examined from a partial
order structure derived from neighborhood bases of the origin. This
structure is used to produce a minimal vector norm for every
Hausdorff locally convex space.
Then, topological vector spaces are examined to find translation
invariant measures with respect to which functions...
This dissertation investigates the structure and topological properties of cyclicallypresented groups. First, a family of groups called groups of type Z is considered. Withfew exceptions, the finiteness, asphericity, fixed point, and 3-manifold spine problemsare solved. Most groups of type Z have a central element of infinite order fixed by theshift....
The classical two-dimensional wavemaker problem is formulated for
linear waves. Two conformal mappings are applied to the mathematical
formulation to transform the wavemaker problem into a unit disk. It is then
shown that this technique cannot produce in practice a numerical
representation of the fluid motion throughout time for any...
ThermoSolver is an educational thermodynamics software program designed to be both
easy to use and useful in that it permits the user to make nontrivial chemical engineering
thermodynamic calculations. The software program accompanies the textbook
Engineering and Chemical Thermodynamics by Milo Koretsky, and is available for free
download from the...
The Van der Pauw method for transport measurements of conductivity and carrier concentration were tested on indium-tin-oxide (ITO) films and silicon wafers, and then implemented on Cu10xAgxZn2Sb4S13 thin films. ITO was used as a test case for high temperature transport measurements because it a well characterized semi-metal. The mobility of...
Using thermodynamic principles, the general relationship describing the equilibrium vapor content in the gas phase above a saline liquid and across a curved liquid-gas interface is developed. Since high salt concentration also affects the intensive and extensive liquid properties, it is also necessary to account for these effects in liquid...
The classical theory of elasticity and plasticity does
not recognize explicitly the existence of a "transition
zone" between elastic and plastic states, which instead,
makes extensive use of ad-hoc, semi-empirical laws, such as
yield conditions, at the "yield surface" to match both the
extreme states. In the present investigation, it...
A general theory of micromorphic materials was developed by
Eringen for the prediction of continuum behavior of materials with
inner structure, such as, granular solids, composite materials,
anisotropic and polymeric fluids. Recently Eringen derived balance
laws of micromorphic mechanics from a different point of view. His
derivation of the master...
This thesis explores the vibrational behavior of the main components of sound production in the violin using a continuum mechanics approach. The author provides a mathematical description of the regions in the vibrating continuum, and begins to develop a system of equations governing their behavior, focusing on the air in...
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...
Under certain conditions, a neural network may be trained to perform a
specific task by altering the weights of only a portion of the synapses.
Specifically, it has been noted that certain three layer feed-forward networks may
be trained to certain tasks by adjusting only the synapses to the output...
Let X be a set. Given any preorder < on the set X, there corresponds a family of subsets of X, namely, WIx E x} where L = {y y E X y <x} such that, for all elements x and y of X, x <y iff L ( L....
A fundamental question related to any Lie algebra is to know its subalgebras. This is
particularly true in the case of E6, an algebra which seems just large enough to contain the algebras which describe the fundamental forces in the Standard Model of particle physics. In this situation, the question...
The flow of incompressible, viscous fluids in R³ is governed by the non-linear Navier-Stokes equations. Two common linearizations of the Navier-Stokes equations, the Stokes equations and the Oseen equations, are studied in this thesis using probabilistic methods.
The incompressibility condition presents new challenges for the well known theory relating partial...
As part of an HHMI funded research project, I showed that overexpression (OE) of actin (ACT2) in Arabidopsis thaliana alters the expression of genes involved in plant immunity. Others have previously shown that knockout (KO) mutants of actin depolymerizing factor 4 are affected in the same gene as ACT2-OE. For...
This study undertakes to determine the existence or nonexistence
of an implication in either direction between any two out of
nine different modes of convergence, with the use of any subset of a
set of ten auxiliary hypotheses. The functions are real finite-valued
measurable functions defined on an arbitrary abstract...
This paper is the record of an exploration of two quadratic
number fields. The first section is devoted to the field with elements
of the form a +b√3 where a and b are rational numbers.
This field contains an integral domain in which unique factorization
holds. The second section is...
This paper records a study of two quadratic number fields. In the first field, denoted by Ra[[square root] 11], the unique factorization theorem holds. In the second field, denoted by Ra[[square root] 10], it is demonstrated that the unique factorization theorem does not hold and therefore ideals are introduced to...
The electromagnetic field in a cone of arbitrary slant
height with a symmetrically placed time harmonic ring source is
studied. Through the use of the modified Helmholtz equation as
an intermediate, we obtain the solution of the semi-infinite
cone directly from the finite cone. To demonstrate the need
for the...
The study into specific properties of the partition function has been a rich topic for number theorists for many years. Much of the current work involving the arithmetic properties of the partition function and their seed in some keen observations of Ramanujan.In particular he discovered what are referred to as...
Loop quantum gravity is a framework for quantizing general relativity which imposes a discrete structure on space. In exploring the structure of loop quantum gravity, researchers have investigated loop quantum cosmology, a toy model with reduced degrees of freedom while maintaining a discrete space. Polymer quantum mechanics is a framework...
In this thesis, problems arising in pumping fluids
from the ground are treated. The mathematical theory is
derived for both the time dependent and independent cases.
A numerical method is developed to solve these problems.
Some particular examples are presented and computer plots
are drawn to illustrate the movement of...
The two dimensional wavemaker problem on a finite domain is derived
for nonlinear waves. A numerical method based on the method of lines is
developed and applied to two test problems, the nonlinear surface pressure
distribution problem and the nonlinear full-flap wavemaker problem. The
solutions yield information about the fluid...
A long running problem in mathematical biology is the prediction of extinction events, a specialized case of the larger global stability problem found in differential equations and dynamical systems theory. A central technical question is how to introduce the randomness observed in real ecological systems not accounted for in deterministic...
The author studies the class of rectangular arrangements in
terms of two binary relations on the objects of the arrangement.
He shows how a univalent matrix determines a unique rectangular
arrangement, and how each rectangular arrangement is associated
with one, two, or four distinct matrices, according to the number
of...
A teacher's mathematical knowledge for teaching has been shown to have a positive correlation with their students' success (Monk, 1994). So, when half of the students that start out in a STEM major switch out to a non-STEM major before graduation to in large part to instructors pedagogical methods and...
The nonlinear Schrödinger equation is a well-known partial differential equation that provides a successful model in nonlinear optic theory, as well as other applications. In this dissertation, following a survey of mathematical literature, the geometric theory of differential equations is applied to the nonlinear Schrödinger equation. The main result of...
We present an analysis of the relationship between spectral lag and luminosity in time-resolved segments of long gamma-ray bursts detected by BATSE, an experiment aboard the Compton Gamma Ray Observatory satellite. For full bursts, there is a well-established correlation between the lag, which is easily computed, and the total burst...
The LA Book is a free-to-students Linear Algebra textbook written at the introductory
level and suitable for teaching MTH341 at Oregon State University. It functions both as
a traditional, printed book (rendered via LaTeX) and as an online textbook (rendered via
HTML), viewable via any modern web-browser. It is built...
The first published notion that the j-function was in any way related to the Monster came in 1979, when Conway and Norton noted in [CN79] that each coefficient in the q- expansion of the j-function could be written as a (nontrivial) integral linear combination of the dimensions of irreducible representations...
A technique of differentiation with respect to the distance to
the boundary of an outer parallel-body is applied to known measures of
sets of p-dimensional linear spaces which intersect a general convex
body in n-dimensional euclidean space in order to obtain an appropriate
definition of the measures of sets of...
We construct an implicit derivative matching (IDM) technique for restoring the accuracy of the Yee scheme for Maxwell's equations in dispersive media with material interfaces in one dimension. We consider media exhibiting orientational polarization, which are represented using a Debye dispersive model, examples of which are water and living tissue....
An important problem in computer graphics is to determine where contour lines and ridges appear in a surface constructed from a triangle mesh. In this presentation we will investigate a new answer to this problem – the horizon measure. The horizon measure determines the likelihood of contour lines to appear...
This paper has two objectives. Firstly, we present the homological properties as defined by de Rham using a notion of current. We show this with the aid of the Eilenberg and Steenrod Axioms. The cohomology is defined in the usual fashion from algebraic topology. The second goal is the relationship...
Methamphetamine has flooded the media for the past two decades however, this
drug has impacted the nation for many decades prior. Since its synthesis in 1893,
methamphetamine has appealed to various aspects of society including soldiers,
housewives, college students, businessmen, truck drivers, drugged crazed hippies, and
athletes. The extensive effects,...
The purpose of this descriptive case study analysis was to provide portraits of the heuristics students used and difficulties they encountered solving conditional probability problems prior to and after two-week instruction on sample space, probability, and conditional probability. Further analysis consisted of evaluating the data in relation to a previously...
This thesis contains a collection of properties of the greatest integer part function which were obtained by an extensive literature search. A few original properties are stated and proved and some of the properties which were found unproved in the literature are proved.
We analyze some symmetries of the octonionic multiplication table, expressed in terms of the Fano plane. In particular, we count how many ways the Fano plane can be labeled as the octonionic multiplication table, all corresponding to a specified octonion algebra. We show that only 28 of these labelings of...
The goal of this research project is to determine the fractal nature, if any, which
certain surface water waves exhibit when viewed on a microscopic scale. We make
use of the mathematical formulation of non-viscous fluids describing their physical
properties. Using these expressions and including boundary conditions for free
surfaces...
This work contains a brief history of the four color problem
from 1840 to 1890. This includes Kempe's attempted proof of the
problem as well as maps which illustrate Heawood's discussion of
Kempe's error. The remaining part is a discussion of Kempe's and
Story's work on patching out maps. Story...
Consider a transformation group G operating on a space X
and a G- invariant function f defined on a G- invariant subset of
X. By imposing suitable conditions on X, G, f and A, the
author derives sufficient conditions for extending f invariantly to
the whole space, and thus generalizing...