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...
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...
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.
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...
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...
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...
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...
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...
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...
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.
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.
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....
This paper is a continuation of William Zell's thesis, A Model of Non-Euclidean Geometry in Three Dimensions. The purpose of that thesis was to show that the axioms of non-Euclidèan geometry are consistent if Euclidean geometry an& hence arithrnetic is consistent. Mr. Zell. discussed the axioms of connection and order...
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...
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...
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....
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...
We study the stability properties of, and the phase error present in, several higher order (in space) staggered finite difference schemes for Maxwell's equations coupled with a Debye or Lorentz polarization model. We present a novel expansion of the symbol of finite difference approximations, of arbitrary (even) order, of 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...
This research seeks to answer the question, "What does it mean for a student to
understand the concept of derivative?" A structured way to describe an individual student's
understanding of derivative is developed and applied to analyzing the evolution of that
understanding for each of nine high school seniors during...
We use a combinatorial approach to study the trajectory of a light ray constrained to Euclidian plane R^2 with random reflecting obstacles placed throughout R^2. For the 2D Lorentz lattice gas (LLG) model we derive an analogue of Russo's formula of increasing events in percolation.
In recent years quantum random walks have garnered much interest among quantum information researchers. Part of the reason is the prospect that many hard problems can be solved efficiently by employing algorithms based on quantum random walks, in the same way that classical random walks have played a central role...
We construct a quantum interchange walk, related to classical walks with memory. This gives us a coinless discrete walk, while the origin in classical walks with memory offers the promise of use of existing tools from classical memoried walks. This approach readily reproduces all standard approaches. We briefly discuss its...
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...
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...
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...
We consider some mathematical problems involving the asymptotic analysis
of rooted tree structures. River channel networks, patterns of electric discharge,
eletrochemical deposition and botanical trees themselves are examples of such naturally
occuring structures. In this thesis we will study the width function aymptotics
of some random trees as well as...
We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell’s
equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order
formulation for the electric field with an ordinary differential equation for the electric polarization added...
Advective skew dispersion is a natural Markov process defined ned
by a di ffusion with drift across an interface of jump discontinuity in
a piecewise constant diff usion coeffcient. In the absence of drift this
process may be represented as a function of -skew Brownian motion
for a uniquely determined...
For cell-like upper semicontinuous(usc) decompositions G of finite dimensional manifolds M, the decomposition space M/G turns out to be an ANR provided M/G is finite dimensional ([Dav07], page 129 ). Furthermore, if M/G is finite dimensional and has the
Disjoint Disks Property (DDP), then M/G is homeomorphic to M ([Dav07],...
Product densities have been widely used in the literature to give a
concrete description of the distribution of a point process. A rigorous
description of properties of product densities is presented with examples to
show that in some sense these results are the best possible. Product
densities are then used...
We discuss a mathematical model arising in the melting
of a fluid in two spatial dimensions and in time. The
model leads to a free boundary value problem for
determining the location of the interface as well as the
temperature distribution. The movement of the interface
depends on the temperatures,...
Large deviation theory has experienced much development and interest in
the last two decades. A large deviation principle is the exponential decay of the
probability of increasingly rare events and the computation of a rate or entropy
function which measures the rate of decay. Within the probability literature there
has...
This paper examines the probability that a random polynomial of specific degree over a field has a specific number of distinct roots in that field. Probabilities are found for random quadratic polynomials with respect to various probability measures on the real numbers and p-adic numbers. In the process, some properties...
Quantum field theory has enjoyed much success, indeed for ordinary flat spacetime applications it has been experimentally verified to a great deal of accuracy. However on general cursed spacetime backgrounds there is no canonical method for constructing a quantum field theory. In 1975, Abhay Ashtekar and Anne Magnon made progress...
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...