Digital Commons Network^™

A Survey Of The Kinetic Monte Carlo Algorithm As Applied To A Multicellular System, Michael Richard Laughlin

Theses and Dissertations

We explore the origins and implementation of the Kinetic Monte Carlo method on a system of cells suspended in a liquid media. The situation presented herein has applications in the emerging field of biofabrication, which may have lasting impacts in medical science. The theory behind the method is explained in detail, starting with its emergence in the 1960s, and two major improvements to the scaling of the method are presented, along with a restriction to a special case. Finally, we give the results of several simulations.

Fast Methods For Variable-Coefficient Peridynamic And Non-Local Diffusion Models, Che Wang

Theses and Dissertations

In previous studies, scientists developed the classical solid mechanic theory. The model has been widely used in scientific research and practical production. The main assumption of the classical theory of solid mechanics is that all internal forces act through zero distance. Because of this assumption, the mathematical model always leads to partial differential equations, which meet with problems when describing the spontaneous formation of discontinuities and other singularities. A peridynamic model was proposed as a reformation of solid mechanics [40, 41, 43, 44, 45], which leads to a non-local framework that does not explicitly involve the notion of deformation gradients, …

Modeling, Simulation, And Applications Of Fractional Partial Differential Equations, Wilson Cheung

Theses and Dissertations

The Black-Scholes model is commonly used to track the price of European options with respect to maturity in many financial markets. This model degenerates into a partial differential equation that relates the European-style option price to the underlying price and time of expiry. Black-Scholes assumes that underlying prices satisfy a geometric Brownian motion.

After the U.S. stock market crash of 1987, this assumption becomes inaccurate as it fails to represent the behavior of S&P 500 European vanilla option prices. Specifically, under the measure of moneyness, the volatility smirk does not flatten out and the resulting conditional return distribution does not …

Relative Divergence, Subgroup Distortion, And Geodesic Divergence, Hung Cong Tran

Theses and Dissertations

In the first part of this dissertation, we generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of divergence, and lower relative divergence generalizes a definition of Cooper-Mihalik. While the lower divergence of Cooper-Mihalik can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this dissertation, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We …

Making Mathematics Memorable, Meaningful, And Fun: Activities To Enhance Precalculus, Nat White

Theses and Dissertations

To master material, students need to make it their own. As teachers, we should structure their interactions with mathematics in ways that are memorable, meaningful, and fun. One way to do this is to provide activities that stretch beyond the textbook and lead students to think and talk to one another about mathematics. This thesis contains a set of activities designed to enhance a precalculus course, along with solutions and feedback on each activity.

Multi-Level Optimal Design Using Game Theory With Model Updating By Low Discrepancy Sampling, Yanchen Xu

Theses and Dissertations

The Design of Experiment (DOE) based response surface methodology (RSM) is a commonly used technique for solving optimization problems. The traditional DOE method has some shortcomings when used to update the RSM model. This thesis aims to develop a new DOE technique to solve the model updating problems in design optimization. Toward this end, a new DOE based RSM method is proposed to solve this problem by using low-discrepancy sequence method to generate the additional data points needed to update the model to replace the traditional factor and level based DOE method.

Tested on a couple of numerical example problems, …

The Solenoid And Warsawanoid Are Sharkovskii Spaces, Tyler Willes Hills

Theses and Dissertations

We extend Sharkovskii's theorem concerning orbit lengths of endomorphisms of the real line to endomorphisms of a path component of the solenoid and certain subspaces of the Warsawanoid. In particular, Sharkovskii showed that if there exists an orbit of length 3 then there exist orbits of all lengths. The solenoid is the inverse limit of double covers over the circle, and the Warsawanoid is the inverse limit of double covers over the Warsaw circle. We show Sharkovskii's result is true for path components of the solenoid and certain subspaces of the Warsawanoid.

Compression Bodies And Their Boundary Hyperbolic Structures, Vinh Xuan Dang

Theses and Dissertations

We study hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. We consider individual hyperbolic structures as well as special regions in the space of all such hyperbolic structures. We use some properties of the boundary hyperbolic structures on C to establish an interesting property of cusp shapes of tunnel number one manifolds. This extends a result of Nimershiem in [26] to the class of tunnel number one manifolds. We also establish convergence results on the geometry of compression bodies. This extends the work of Ito in [13] from the punctured-torus case …

The Bourbaki-Jacobson Correspondence, Jose R. Vera

Theses and Dissertations

A general ring theoretic correspondence between subrings of the endomorphism ring of the additive group of a commutative field will be established. This correspondence (called Bourbaki-Jacobson Correspondence) provides the ordinary Galois correspondence when applied to specific group rings. Throughout this thesis, we will work with finite dimensional field extensions.

Alternating Direction Implicit Method With Adaptive Grids For Modeling Chemotaxis In Dictyostelium Discoideum, Christopher F. Loomis

Theses and Dissertations

Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons …

Heun Polynomials In The Construction Of Vector Valued Slepian Functions On A Spherical Cap, Thomas Anthony Ventimiglia

Theses and Dissertations

I summarize the existing work on the problem of finding vector valued Slepian functions on the unit sphere: separable vector fields whose energy is concentrated within a compact region; in this case, a spherical cap. The radial and tangential components are independent for an appropriate choice of basis, and for each component the problem is recast as that of finding real eigenfunctions of an integral operator. There exist Sturm-Liouville differential operators that commute with these integral operators and hence share their eigenfunctions. Therefore, the radial and tangential eigenfunctions are solutions to second order linear ODEs. After introducing the Heun differential …

A Weak Simpson Method For A Class Of Stochastic Differential Equations And Numerical Stability Results, Ram Sharan Adhikari

Theses and Dissertations

This work proposes a novel weak Simpson method for numerical solution for a class of stochastic differential equations. We show that such a method has weak convergence of order one in general and weak convergence of order three under certain additional assumptions. This work also aims to determine the mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and stepsizes for the numerical method where errors propagation are under control in well-defined sense are given. The main results are …

Random Iteration Of Rational Maps, Jesse Dean Feller

Theses and Dissertations

Random and non-autonomous iteration has been a subject of interest in Mathematics that has received some attention in the last few decades. The earliest paper on random iteration in the complex setting was written by Fornaess and Sibony. They have shown that given a family of functions $\{f_c\}_{c \in \W}$ where $\W$ is a small open set, for almost every z the random iteration is stable on a subset of $\W^\N$ of full probability measure. Later, Hiroki Sumi further extended these results to a more general situation using rational semigroups. We will show that the results of Fornaess and Sibony …

Multivariate Hilbert Series Of Lattice Cones And Homogeneous Varieties, Wayne Andrew Johnson

Theses and Dissertations

We consider the dimensions of irreducible representations whose highest weights

lie on a given finitely generated lattice cone. We present a rational function representing

the multivariate formal power series whose coefficients encode these dimensions.

This result generalizes the formula for the Hilbert series of an equivariant

embedding of an homogeneous projective variety. We use the multivariate generating

function to compute Hilbert series for the Kostant cones and other affine and

projective varieties of interest in representation theory. As a special case, we show

how the multivariate series can be used to compute the Hilbert series of the three

classical families …

A Nonabelian Landau-Ginzburg B-Model Construction, Ryan Thor Sandberg

Theses and Dissertations

The Landau-Ginzburg (LG) B-Model is a significant feature of singularity theory and mirror symmetry. Krawitz in 2010, guided by work of Kaufmann, provided an explicit construction for the LG B-model when using diagonal symmetries of a quasihomogeneous, nondegenerate polynomial. In this thesis we discuss aspects of how to generalize the LG B-model construction to allow for nondiagonal symmetries of a polynomial, and hence nonabelian symmetry groups. The construction is generalized to the level of graded vector space and the multiplication developed up to an unknown factor. We present complete examples of nonabelian LG B-models for the polynomials x^2y + y^3, …

Nonlocally Maximal Hyperbolic Sets For Flows, Taylor Michael Petty

Theses and Dissertations

In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

Evaluation And Refinement Of Generalized B-Splines, Ian Daniel Henriksen

Theses and Dissertations

In this thesis a method for direct evaluation of Generalized B-splines (GB-splines) via the representation of these curves as piecewise functions is presented. A local structure is introduced that makes the GB-spline curves more amenable to the integration used in constructing bases of higher degree. This basis is used to perform direct computation of piecewise representation of GB-spline bases and curves. Algorithms for refinement using these local structures are also developed.

Octahedral Extensions And Proofs Of Two Conjectures Of Wong, Kevin Ronald Childers

Theses and Dissertations

Consider a non-Galois cubic extension K/Q ramified at a single prime p > 3. We show that if K is a subfield of an S_4-extension L/Q ramified only at p, we can determine the Artin conductor of the projective representation associated to L/Q, which is based on whether or not K/Q is totally real. We also show that the number of S_4-extensions of this type with K as a subfield is of the form 2^n - 1 for some n >= 0. If K/Q is totally real, n > 1. This proves two conjectures of Siman Wong.

Topics Pertaining To The Group Matrix: K-Characters And Random Walks, Randall Dean Reese

Theses and Dissertations

Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject.

Analysis Of Multiple Collision-Based Periodic Orbits In Dimension Higher Than One, Skyler C. Simmons

Theses and Dissertations

We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

Investigations Into Non-Degenerate Quasihomogeneous Polynomials As Related To Fjrw Theory, Scott C. Mancuso

Theses and Dissertations

The motivation for this paper is a better understanding of the basic building blocks of FJRW theory. The basics of FJRW theory will be briefly outlined, but the majority of the paper will deal with certain multivariate polynomials which are the most fundamental building blocks in FJRW theory. We will first describe what is already known about these polynomials and then discuss several properties we proved as well as conjectures we disproved. We also introduce a new conjecture suggested by computer calculations performed as part of our investigation.

Semiparametric Estimation Of The Survival Function In The Presence Of Covariates, Madlen Gebauer

Theses and Dissertations

The main interest of survival analysis is to estimate the distribution function of the survival time based on observations of a random sample. In this thesis, a semiparametric estimator is used not only to estimate the survival probability, but also to consider the influence of explanatory variables within the estimation. Therefore, the weighted maximum likelihood estimator of the conditional survival function is derived and a corresponding pointwise likelihood ratio confidence band is developed. Subsequently, the established estimator is compared to a similar estimator which was proposed by Iglesias-Pérez and de Ũna-Álvarez (2008). Since the idea of this paper arose in …

On The Dimension Of Group Boundaries, Molly Ann Moran

Theses and Dissertations

The goal of this dissertation is to find connections between the small-scale dimension (i.e. covering dimension and linearly controlled dimension) of group boundaries and the large scale dimension (i.e. asymptotic dimension and macroscopic dimension) of the group. We first show that generalized group boundaries must have finite covering dimension by using finite large-scale dimension of the space. We then restrict our attention to CAT(0) group boundaries and develop metrics on the boundary that allow us to study the linearly controlled dimension. We then obtain results relating the linearly controlled dimension of CAT(0) boundaries to the large scale dimension of the …

Using A Semiprognostic Test To Elucidate Key Model Errors Of Warm Rain Processes Within A Unified Parameterization Of Clouds And Turbulence, Justin Kyle Weber

Theses and Dissertations

The representation of clouds and turbulence remains one of the foremost challenges in modeling earth's climate system and continues to remain one of the greatest sources of uncertainty in future climate projections. Increased attention has been given to unifying cloud and turbulence parameterizations in order to avoid the artificial categorization of cloud and turbulence regimes. One such unified parameterization is known as the Cloud Layers Unified by Binormals (CLUBB). CLUBB is a single column model of clouds and turbulence that assumes subgrid scale variability can be represented by a joint probability density function (PDF) of temperature, moisture, momentum, and hydrometeors. …

The L^2-Cohomology Of Discrete Groups, Kevin David Schreve

Theses and Dissertations

Given a space with a proper, cocompact group action, the L^2-cohomology groups are a particularly interesting invariant that incorporates the topology of the space and the geometry of the group action. We are interested in both the algebraic and geometric aspects of these invariants. From the algebraic side, the Strong Atiyah Conjecture claims that the L^2-Betti numbers assume only rational values, with certain prescribed denominators related to the torsion subgroups of the group. We prove this conjecture for the class of virtually cocompact special groups. This implies the Zero Divisor Conjecture holds for such groups. On the geometric side, the …

The Fattened Davis Complex And The Weighted L^2-(Co)Homology Of Coxeter Groups, Wiktor Jerzy Mogilski

Theses and Dissertations

Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given a positive real multiparameter $\Q$ indexed by $S$, one can define the weighted $L^2$--(co)homology groups of $\Sigma$ and associate to them a nonnegative real number called the weighted $L^2$--Betti number. Unfortunately, not much is known about the behavior of these groups when $\Q$ lies outside a certain restricted range, and weighted $L^2$--Betti numbers have proven difficult to compute. We propose a program to compute the weighted $L^2$--(co)homology of $\Sigma$ by introducing a thickened …

An Investigation Into Vaccination Behavior: Parametrization Of A Samoan Vaccine Scare, Amanda Ruth Spink

Theses and Dissertations

Vaccination behavior can be influenced by many factors. Some examples are vaccine scares, evolutionary game theory, social learning such as media coverage, feedback in the form of infectious cases, and herd immunity. We investigated a previously published model that attempts to explain vaccination behavior based on a game theoretic point of view. The model was applied to a large vaccine scare in the country of Samoa, and a parameter estimation problem was solved for different risk perception scenarios. It was found that the model fit best in the case of no social learning and no feedback. However, adding in these …

Some Results On Pseudo-Collar Structures On High-Dimensional Manifolds, Jeffrey Joseph Rolland

Theses and Dissertations

In this dissertation we outline a partial reverse to Quilen's plus construction in the high-dimensional manifold categor. We show that for any orientable manifold N with fundamental group Q and any fintely presented superperfect group S, there is a 1-sided s-cobordism (W, N, N-) with the fundamental group G of N- a semi-direct product of Q by S, that is, with G satisying 1 -> S -> G -> Q -> 1 and actually a semi-direct product.

We then use a free product of Thompson's group V with itself to form a superperfect group S and start with an orientable …

Infinitely Generated Clifford Algebras And Wedge Representations Of Gl∞|∞, Bradford J. Schleben

Theses and Dissertations

The goal of this dissertation is to explore representations of $\mathfrak{gl}_{\infty|\infty}$ and associated Clifford superalgebras. The machinery utilized is motivated by developing an alternate superalgebra analogue to the Lie algebra theory developed by Kac. In an effort to establish a natural mathematical analogue, we construct a theory distinct from the super analogue developed by Kac and van de Leur. We first construct an irreducible representation of a Lie superalgebra on an infinite-dimensional wedge space that permits the presence of infinitely many odd parity vectors. We then develop a new Clifford superalgebra, whose structure is also examined. From here, we extend …

Results On N-Absorbing Ideals Of Commutative Rings, Alison Elaine Becker

Theses and Dissertations

Let R be a commutative ring with n≥0. In his paper On 2-absorbing Ideals of Commutative Rings, Ayman Badawi introduces a generalization of prime ideals called 2-absorbing ideals, and this idea is further generalized in a paper by Anderson and Badawi to a concept called n-absorbing ideals. A proper ideal I of R is said to be an n-absorbing ideal if whenever x_1…x_(n+1) ∈I for x_1,…,x_(n+1 )∈R then there are n of the x_i's whose product is in I. This paper will provide proofs of several properties in Badawi’s paper which are stated without proof, and will study how several …