Mathematics Commons^™

Superconvergence Of Iterated Solutions For Linear And Nonlinear Integral Equations: Wavelet Applications, Boriboon Novaprateep

Mathematics & Statistics Theses & Dissertations

In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern …

A Forward-Backward Fluence Model For The Low-Energy Neutron Boltzmann Equation, Gary Alan Feldman

Mathematics & Statistics Theses & Dissertations

In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target …

Multi-Symplectic Integrators For Nonlinear Wave Equations, Alvaro Lucas Islas

Mathematics & Statistics Theses & Dissertations

Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as …

Protocols For Disease Classification From Mass Spectrometry Data, Michael Wagner, Dayanand Naik, Alex Pothen

Mathematics & Statistics Faculty Publications

We report our results in classifying protein matrix-assisted laser desorption/ionizationtime of flight mass spectra obtained from serum samples into diseased and healthy groups. We discuss in detail five of the steps in preprocessing the mass spectral data for biomarker discovery, as well as our criterion for choosing a small set of peaks for classifying the samples. Cross-validation studies with four selected proteins yielded misclassification rates in the 10-15% range for all the classification methods. Three of these proteins or protein fragments are down-regulated and one up-regulated in lung cancer, the disease under consideration in this data set. When cross-validation studies …

Nearly Balanced And Resolvable Block Designs, Brian Henry Reck

Mathematics & Statistics Theses & Dissertations

One of the fundamental principles of experimental design is the separation of heterogeneous experimental units into subsets of more homogeneous units or blocks in order to isolate identifiable, unwanted, but unavoidable, variation in measurements made from the units. Given v treatments to compare, and having available b blocks of k experimental units each, the thoughtful statistician asks, “What is the optimal allocation of the treatments to the units?” This is the basic block design problem. Let n_ij be the number of times treatment i is used in block j and let N be the v x b matrix N …

Four Anharmonic Oscillators On A Circle, J. N. Boyd, R. G. Hudepohl, P. N. Raychowdhury

Virginia Journal of Science

Four identical, uniformly separated particles interconnected by ideal anharmonic springs are constrained to move on a fixed, frictionless circular track. The Lagrangian for the system is written and then transformed by matrix operations suggested by the symmetry of the arrangement of springs and particles. The equations of motion derived from the transformed Lagrangian yield four natural frequencies of motion.

Environmental Factors Contributing To The Disaggregation Of A Colonial Cyanoprokaryote And Its Influence On Picoplankton Abundance Within Lake Joyce, Virginia, Lewis F. Affronti Jr., B. Thomas Duquette

Virginia Journal of Science

A colonial cyanoprokaryote, Aphanocapsa holsatica and autotrophic picoplankton abundance were monitored weekly over a two year period in Lake Joyce, Virginia. Significant differences were observed in both the cyanoprokaryote and picoplankton abundance over the study period and an inverse relationship was observed between these two plankton groups. Disaggregation of colonies was shown to contribute to picoplankton populations where water temperature and precipitation input apparently trigger colony dispersion. This relationship is suggested to occur in other aquatic habitats. Results of this work and its implications for ecosystem dynamics are discussed.

Geometric Integrators For Hamiltonian Pdes, Dmitry Karpeev

Computer Science Theses & Dissertations

We consider methods for systematic construction of algorithms for a class of time-dependent PDEs with Hamiltonian structure. These systems possess phase space geometry and constants of the motion that need to be preserved by the integration algorithm to reflect the qualitative features of the system.

We exploit the structure of Hamiltonian systems, in particular their variational formulation based on a Lagrangian, and the dual covariant formulation, to expose the geometric features of the system that have natural analogs when discretized. We emphasize the local space-time approach to the constructions, making them amenable to parallelization and preconditioning using domain decomposition methods, …

Like A Bridge Over Colored Water: A Mathematical Review Of The Rainbow Bridge: Rainbows In Art, Myth, And Science, John A. Adam

Mathematics & Statistics Faculty Publications

Commenting on a recent book, the author discusses various views of the rainbow: its role in culture, its scientific description, and its mathematical theory.

A New Error Analysis For Brun's Constant, Thomas R. Nicely

Virginia Journal of Science

Enumeration of the twin primes, and the sum of their reciprocals, is extended to 3 x 10¹⁵, yielding the count π₂(3 x 10¹⁵) = 3310517800844. A more accurate estimate is obtained for Brun’s constant, B2 = 1.90216 05823 ± 0.00000 00008 . Error analysis is presented to support the contentioli that this estimate produces a 95 % confidence interval for B₂. In addition, published values of the count π(x) of primes, obtained previously by indirect means, are verified by direct coiiIit to x = 3 x 10¹⁵

Unsteady Flow Model For An Irrigation Canal Network, Talaat Taher El Gamel

Civil & Environmental Engineering Theses & Dissertations

A mathematical model is developed to route unsteady flow through an irrigation canal network. The model is designed to handle branches, lateral inflow and outflow, sluice gates, and various boundary conditions. The model is developed for use on a personal computer. A specific irrigation canal network in Egypt is used as a case study to verify the model, and the results are presented. Full documentation is included in the appendix.

Nested Balanced Incomplete Block Designs, J. P. Morgan, D. A. Preece, D. H. Rees

Mathematics & Statistics Faculty Publications

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b₁;r;k₁) are each partitioned into sub-blocks of size k₂, and the b₂ =b₁k₁=k2 sub-blocks themselves constitute a BIBD with parameters (v; b₂;r;k₂), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k₁ =2k₂= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, …

Efficient Algorithms For Graphs With Few P-4’S, Luitpold Babel, Ton Kloks, Jan Kratochvíl, Dieter Kratsch, Kaiko Müller, Stephan Olariu

Computer Science Faculty Publications

We show that a large variety of NP-complete problems can be solved efficiently for graphs with 'few' P₄'s. We consider domination problems (domination, total domination, independent domination. connected domination and dominating clique), the Steiner tree problem, the vertex ranking problem, the pathwidth problem, the path cover number problem, the hamiltonian circuit problem, the list coloring problem and the precoloring extension problem. We show that all these problems can be solved in linear time for the class of (q,q - 4)-graphs, for every fixed q. These are graphs for which no set of at most q. vertices induces more …

Mathematical Models Of Quiescent Solar Prominences, Iain Mckaig

Mathematics & Statistics Theses & Dissertations

Magnetic fields in the solar atmosphere suspend and insulate dense regions of cool plasma known as prominences. The convection zone may be the mechanism that both generates and expels this magnetic flux through the photosphere in order to make these formations possible. The connection is examined here by modeling the convection zone as both one-dimensional, then more realistically, two-dimensional.

First a Dirichlet problem on a semi-infinite strip is solved using conformal mapping and the method of images. The base of the strip represents the photosphere where a current distribution can be given as a boundary condition, and the strip extends …

Fabrication And Characterization Of Silicon Field Emitter Arrays, Aaron M. Brock

Electrical & Computer Engineering Theses & Dissertations

Field emission is a process through which the application of a large electric field to the surface of a material causes electron emission into vacuum. The electron emission current is a strong function of the geometry of the emitting material. The focus of this thesis is to investigate the effect of geometry on the emission current. Specifically, silicon field emitter arrays were fabricated using two separate fabrication processes termed the standard process and the new process. The uniformity of the arrays was measured experimentally. The arrays were used in field emission tests from which parameters controlling field emission were extracted. …

The Fermat Point And The Steiner Problem, J. N. Boyd, P. N. Raychowdhury

Virginia Journal of Science

We revisit the convex coordinates of the Fermat point of a triangle. We have already computed these convex coordinates in a general setting. In this note, we obtain the coordinates in the context of the Steiner problem. Thereafter, we pursue calculations suggested by the problem.

Diffusion Problems In Wound Healing And A Scattering Approach To Immune System Interactions, Julia Suzanne Arnold

Mathematics & Statistics Theses & Dissertations

A theoretical model for the existence of a Critical Size Defect (CSD) in certain animals is the focus of the majority of this dissertation. Adam [1] recently developed a one-dimensional model of this phenomenon, and chapters I–V address the exist the CSD in a two-dimensional model and a three-dimensional model. The two dimensional (or 1-d circular) model is the more appropriate for a study of CSD's. In that model we assume a circular wound of uniform depth and develop a time-independent form of the diffusion equation relevant to the study of the CSD phenomenon. It transpires that the range of …

An Object-Oriented Algorithmic Laboratory For Ordering Sparse Matrices, Gary Karl Kumfert

Computer Science Theses & Dissertations

We focus on two known NP-hard problems that have applications in sparse matrix computations: the envelope/wavefront reduction problem and the fill reduction problem. Envelope/wavefront reducing orderings have a wide range of applications including profile and frontal solvers, incomplete factorization preconditioning, graph reordering for cache performance, gene sequencing, and spatial databases. Fill reducing orderings are generally limited to—but an inextricable part of—sparse matrix factorization.

Our major contribution to this field is the design of new and improved heuristics for these NP-hard problems and their efficient implementation in a robust, cross-platform, object-oriented software package. In this body of research, we (1) examine …

A Numerical Solution Of Low-Energy Neutron Boltzmann Equation, Martha Sue Clowdsley

Mathematics & Statistics Theses & Dissertations

A multigroup method using a straight ahead approximation is created to calculate low energy neutron fluence due to the elastic scattering of evaporation neutrons produced in interactions of high energy particles with target nuclei. This multigroup method is added to NASA Langley Research Center's HZETRN particle transport code. This new code is used to calculate the energy spectra of the neutron fluence in several different materials. The multigroup method is found to be an efficient way of calculating low energy neutron fluence in multiple atom materials as well as single atom materials. Comparisons to results produced by Monte Carlo methods …

The Solution Of Hypersingular Integral Equations With Applications In Acoustics And Fracture Mechanics, Richard S. St. John

Mathematics & Statistics Theses & Dissertations

The numerical solution of two classes of hypersingular integral equations is addressed. Both classes are integral equations of the first kind, and are hypersingular due to a kernel containing a Hadamard singularity. The convergence of a Galerkin method and a collocation method is discussed and computationally efficient algorithms are developed for each class of hypersingular integral equation.

Interest in these classes of hypersingular integral equations is due to their occurrence in many physical applications. In particular, investigations into the scattering of acoustic waves by moving objects and the study of dynamic Griffith crack problems has necessitated a computationally efficient technique …

Coupled Electrodynamic-Monte Carlo Simulations Of Nanoscale Gaas Terahertz Optical Mixers, Jiang Li

Electrical & Computer Engineering Theses & Dissertations

The concept of mixing or heterodyning has traditionally been used for microwaves and for radio frequency communications. However, the concept can easily be extended into the optical frequency regime. By doing so, the photomixing process can serve as a very versatile tool for both the generation of ultrahigh frequency (terahertz) and the detection of weak optical signals.

The aim of this thesis is to perform a theoretical study of the photomixing process inside GaAs devices as the non-linear elements. A coupled approach which combines the Monte Carlo simulation scheme for the carrier transport, with Maxwell's equation for the electrodynamics, has …

Superconvergence In Iterated Solutions Of Integral Equations, Peter A. Padilla

Mathematics & Statistics Theses & Dissertations

In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained …

Sparse Equation-Eigen Solvers For Symmetric/Unsymmetric Positive-Negative-Indefinite Matrices With Finite Element And Linear Programming Applications, Hakakizumwami Birali Runesha

Civil & Environmental Engineering Theses & Dissertations

Vectorized sparse solvers for direct solutions of positive-negative-indefinite symmetric systems of linear equations and eigen-equations are developed. Sparse storage schemes, re-ordering, symbolic factorization and numerical factorization algorithms are discussed. Loop unrolling techniques are also incorporated in the coding to enhance the vector speed. In the indefinite solver, which employs various pivoting strategies, a simple rotation matrix is introduced to simplify the computer implementation. Efficient usage of the incore memory is accomplished by the proposed "restart memory management" schemes. A sparse version of the Interior Point Method, IPM, has also been implemented that incorporates the developed indefinite sparse solver for linear …

Mathematical Models Of Tumors And Their Remote Metastases, Carryn Bellomo

Mathematics & Statistics Theses & Dissertations

Clinical observations and indications in the literature have led us to investigate several models of tumors. For example, it has been shown that a tumor has the ability to send out anti-growth factors, or inhibitors, to keep its remote metastases from growing. Thus, we model the depleting effect of such a growth inhibitor after the removal of the primary tumor (thus removing the source) as a function of time t and distance from the original tumor r.

It has also been shown clinically that oxygen and glucose are nutrients critical to the survival and growth of tumors. Thus, we model …

Reaction-Diffusion Models Of Cancer Dispersion, Kim Yvette Ward

Mathematics & Statistics Theses & Dissertations

The phenomenological modeling of the spatial distribution and temporal evolution of one-dimensional models of cancer dispersion are studied. The models discussed pertain primarily to the transition of a tumor from an initial neoplasm to the dormant avascular state, i.e. just prior to the vascular state, whenever that may occur. Initiating the study is the mathematical analysis of a reaction-diffusion model describing the interaction between cancer cells, normal cells and growth inhibitor. The model leads to several predictions, some of which are supported by experimental data and clinical observations $\lbrack25\rbrack$. We will examine the effects of additional terms on these characteristics. …

Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups, Iem H. Heng

Mathematics & Statistics Theses & Dissertations

Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.

The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, …

Characterization Of Generalized Haar Spaces, M. Bartelt, W. Li

Mathematics & Statistics Faculty Publications

We say that a subset G of C₀(T, ℝ^k) is rotation-invariant if [Qg: g ∈ G]=G for any k x k orthogonal matrix Q. Let G be a rotation-invariant finite-dimensional subspace of C₀(T, ℝ^k) on a connected, locally compact, metric space T. We prove that G is a generalized Haar subspace if and only if P_G(ƒ) is strongly unique of order 2 whenever P_G(ƒ) is a singleton.

High-Order Finite-Difference Schemes And Their Application To Computational Acoustics, Joe Leo Manthey

Mathematics & Statistics Theses & Dissertations

The primary focus of this study is upon the numerical stability of high-order finite-difference schemes and their application to duct acoustics. Since acoustic waves are known to be non-dissipative and non-dispersive, high-order schemes are favored for their low dissipation and low dispersion relative to the low-order schemes. The primary obstacle to the the development of explicit high-order finite-difference schemes is the construction of boundary closures which simultaneously maintain the formal order of accuracy and the numerical stability of the overall scheme. In this thesis a hybrid seven-point, fourth-order stencil for computing spatial derivatives is presented and the time stability is …

Reverse Engineering Of Aircraft Wing Data Using A Partial Differential Equation Surface Model, Jacalyn M. Huband

Mathematics & Statistics Theses & Dissertations

Reverse engineering is a multi-step process used in industry to determine a production representation of an existing physical object. This representation is in the form of mathematical equations that are compatible with computer-aided design and computer-aided manufacturing (CAD/CAM) equipment. The four basic steps to the reverse engineering process are data acquisition, data separation, surface or curve fitting, and CAD/CAM production. The surface fitting step determines the design representation of the object, and thus is critical to the success or failure of the reverse engineering process. Although surface fitting methods described in the literature are used to model a variety of …

A Dual Approach To Constrained Interpolation From A Convex Subset Of Hilbert Space, Frank Deutsch, Wu Li, Joseph D. Ward

Mathematics & Statistics Faculty Publications

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y …