The study of differentiation of integrals has led to the study of maximal functions. In the development of harmonic analysis, the most powerful result connected with Lebesgue's theorem was that of the Hardy-Littlewood Maximal Theorem. This maximal theorem implies Lebesgue's theorem, and the maximal function and its variants have played...
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...
The micropolar equations of motion are developed, in a very general form, for a
mixture of fluid and solid particles. The fluid is allowed to move differently than the solids,
and the solids are modeled as a micropolar continuum. That is, the solid particles have an
additional degree of freedom...
Understanding mathematics and teaching mathematics involve
numerous factors, one of which may be an individual's spatial ability.
This paper examines research conducted on the relationship between
spatial abilities and mathematics, gender differences in the area of
spatial ability, the types of experiences that may affect one's spatial
ability, and issues...
The generalized variational principle of Herglotz defines the functional whose extrema are sought by a differential equation rather than an integral. It reduces to the classical variational principle under classical conditions. The Noether theorems are not applicable to functionals defined by differential equations. For a system of differential equations derivable...
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...
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...
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...
An insurance company, having an initial capital u, receives premiums continuously and pays claims of random sizes at random times. A classical result states that if the rate of premium, c, exceeds the average of the claims paid per unit time, ⋋μ, then the ruin probability decays exponentially fast as...
Arising from an investigation in Hydrodynamics, the Korteweg-de Vries equation demonstrates existence of nonlinear waves that resume their profile after interaction. In this thesis, the classical equations governing wave motion are the starting point for the development of an analogue of the KdV that describes the evolution of a wave...
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...
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 recursive and stochastic representation of solutions to the Fourier transformed Navier-Stokes equations, as introduced by [34], is extended in several ways. First, associated families of functions known as majorizing kernels are analyzed, in light of their apparently essential role in the representation. Second, the theory is put on a...
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...
The focus of this grounded theory research was to investigate the problems that those groups closest to students placed in mathematics classes by mathematics ability have and how those parties work to resolve the problems. The main problem found was a conflict between educators and parents over which students deserve...
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 LIGO interferometers have reached their designed sensitivity level which is greater than any other interferometer ever built. At this point, it is important to have a proper understanding of small perturbations that might still exist (until now these issues did not come to the forefront because other more critical...
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...
This thesis considers one of the classical problems in the actuarial mathematics literature, the decay of the probability of ruin in the collective risk model. The
claim number process N(t) is assumed to be a renewal process, the resulting model
being referred as the Sparre Andersen risk model. The inter-claim...
Water is one of the most biologically and economically important substances on Earth. A significant portion of Earth's water subsists in the subsurface. Our ability to monitor the flow and transport of water and other fluids through this unseen environment is crucial for a myriad of reasons.
One difficulty we...
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...
Consider a simple Markov process on [0,T]. The occupation times were extensively studied in the case of T increasing to infinity. Here we develop a method of computing the distribution for occupation times when T is small via integral equations and integral transforms.
This thesis contains three manuscripts addressing the application of stochastic processes to the analysis and solution of partial differential equations (PDEs) in mathematical physics.
In the first manuscript, one dimensional diffusion and Burgers equation are considered. The Fourier transform of the solution to each PDE is represented as the expected...
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...
This thesis examines the mixing times for one-dimensional interacting particle systems. We use the coupling method to study the mixing rates for particle systems on the circle which move according to specific permutations e.g., transpositions and 3-cycles.
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 purpose of this study was to document the development of pre-service teachers' Technology Specific Pedagogy as they learned to teach mathematics with technology during their initial licensure program. The study investigated the pre-service teachers' learning using both a social and a psychological perspective of teacher learning. Two research questions...
Encryption is essential to the security of transactions and communications, but
the algorithms on which they rely might not be as secure as we all assume. In this
paper, we investigate the randomness of the discrete exponentiation function used
frequently in encryption. We show how we used exponential generating functions...
The purpose of this study was to address the implementation fidelity of one part of a professional development model developed by the Northwest Regional Educational Laboratory (NWREL). Specifically, this research investigates middle school teachers’ use of a formative feedback guide developed by NWREL staff, examining the reliability with which teachers...
Temperature data from above and below the Cougar Dam collected by the U.S. Geological Survey prior to the construction of the temperature control structure was analyzed to determine how the di®ering temperature regimes a®ect the growth and survival of threatened spring- run Chinook salmon. An ARIMA time-series model was used...
Factorization of integers is an important aspect of cryptography since it can be used as an
attack against some of the common cryptographic methods being used. There are
numerous methods in existence for factoring integers. Some of these are faster than
others for general numbers, while others work best on...
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...
Future satellite missions like NASA’s upcoming Magnetospheric Multiscale (MMS) mission are targeting reconnection diffusion regions at the Earth’s magnetopause. These diffusion regions are small compared to the total surface area of the magnetopause. Furthermore, the location of the diffusion region depends on external parameters such as the current state of...
It is necessary to encode data when transmitting over a noisy channel in order for
errors to be detected and corrected. List decoding algorithms provide all code words
within a specified distance of a received word in order to be sufficiently robust for
cases when two or more code words...
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...
In this thesis, we investigate the problem of simulating Maxwell's equations in dispersive dielectric media. We begin by explaining the relevance of Maxwell's equations to
21st century problems. We also discuss the previous work on the numerical simulations of
Maxwell's equations. Introductions to Maxwell's equations and the Yee finite difference...
For a certain class of Z²-actions, we provide a proof of a conjecture that the ratio of the Perron eigenvalues of the transfer matrices of the free boundary restrictions converge to the entropy of that action. Also, a novel method for computing the entropy of Z²-actions is conjectured.
In contemporary art, some artists enter into working relationships with other
groups or institutions to create a final piece. For example, an artist may be invited by a
museum to take part in an exhibition, or an artist may work with a city council to create a
public artwork. However,...
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.
A striking feature in the study of Riemannian manifolds of positive sectional curvature
is the narrowness of the collection of known examples. In this thesis, we examine the
structure of the cohomology rings of three families of compact simply connected seven dimensional
Riemannian manifolds that may contain new examples of...
In this paper we examine some of the developments concerning the Gauss class number problems and build a solid understanding of the class number. First we will develop some background knowledge necessary to understand the problem, specifically the theory of quadratic forms and quadratic fields and how the class number...
If P is an integer polynomial denote the degree of P by ∂(P) and let H(P) be the maximum of the absolute value of the coefficients of P. Define Λ(P)=2[superscript ∂(P)]H(P) and for a fixed prime p let C[subscript p] denote the completion of the algebraic closure of the p-adic...
Geometric Problems become increasingly intractable and difficult to visualize as the number of dimensions increases beyond three. Inductions from lower dimensional spaces are possible yet often awkward. This thesis shows how elementary linear algebra, vector calculus, and combinatorics offer improved methods for calculating the dihedral angles of n-simplicies and proving...
The existence of generalized symmetries of Maxwell's equations in Gödel's Universe is investigated. It is shown that their existence is in turn tied to the existence of certain spinorial objects called Killing spinors.
The conformal algebra, corresponding to valence (1; 1) Killing spinors, for Gödel's Universe is discussed in detail....
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...