Mathematics Commons^™

An Invariance Property Of Common Statistical Tests, N. Rao Chaganty, A. K. Vaish

Mathematics & Statistics Faculty Publications

Let A be a symmetric matrix and B be a nonnegative definite (nnd) matrix. We obtain a characterization of the class of nnd solutions Σ for the matrix equation AΣA = B. We then use the characterization to obtain all possible covariance structures under which the distributions of many common test statistics remain invariant, that is, the distributions remain the same except for a scale factor. Applications include a complete characterization of covariance structures such that the chisquaredness and independence of quadratic forms in ANOVA problems is preserved. The basic matrix theoretic theorem itself is useful in other characterizing …

Scattering From Stellar Acoustic-Gravity Potentials: Ii. Phase Shifts Via The First Born Approximation, J. A. Adam, I. Mckaig

Mathematics & Statistics Faculty Publications

Using the first Born approximation, properties of the scattering phase shift are investigated for waves that are scattered by a schematic representation of a large-scale “stellar potential,” i.e., one for which the star itself is viewed as the potential inducing a phase shift in an incoming wave. In particular, the phase shift properties are examined as functions of the relative wavenumber (α) and the azimuthal wavenumber (l), high l-values being of interest in helioseismology.

Analysis Of Repeated Measures Data Under Circular Covariance, Andrew Montgomery Hartley

Mathematics & Statistics Theses & Dissertations

Circular covariance is important in modelling phenomena in epidemiological, communications and numerous physical contexts. We introduce and develop a variety of methods which make it a more versatile tool. First, we present two classes of estimators for use in the presence of missing observations. Using simulations, we show that the mean squared errors of the estimators of one of these classes are smaller than those of the Maximum Likelihood (ML) estimators under certain conditions. Next, we propose and discuss a parsimonious, autoregressive type of circular covariance structure which involves only two parameters. We specify ML and other types of estimators …

Note: On The Degree Of Lp Approximation With Positive Linear Operators, J. J. Swetits, B. Wood

Mathematics & Statistics Faculty Publications

The degree of approximation in L_p-spaces by positive linear operators is estimated in terms of the integral modulus of smoothness. It is shown that the conjectured optimal degree of approximation is not attained in the class of functions having a second derivative belonging to L_p.

Exact Solutions For Orthogonal And Non-Orthogonal Magnetohydrodynamic Stagnation-Point Flow, Shahrooz Moosavizadeh

Mathematics & Statistics Theses & Dissertations

The viscous plane flow of an electrically conducting fluid towards an infinite wall is solved in the presence of a magnetic field which is aligned with the flow far from the wall. The problem has two dimensionless parameters-- ε, the magnetic Prandtl number, and β, the square of the ratio of Alfven velocity to fluid velocity far from the wall. The problem has a similarity solution which reduces the governing equations to a system of coupled ordinary differential equations which can be solved numerically. For extreme values of ε, both large and small, singular perturbation techniques are used to derive …

Data Compression Based On The Cubic B-Spline Wavelet With Uniform Two-Scale Relation, S. K. Yang, C. H. Cooke

Mathematics & Statistics Faculty Publications

The aim of this paper is to investigate the potential artificial compression which can be achieved using an interval multiresolution analysis based on a semiorthogonal cubic B-spline wavelet. The Chui-Quak [1] spline multiresolution analysis for the finite interval has been modified [2] so as to be characterized by natural spline projection and uniform two-scale relation. Strengths and weaknesses of the semiorthogonal wavelet as regards artificial compression and data smoothing by the method of thresholding wavelet coefficients are indicated.

Studies Of Mixing Processes In Gases And Effects On Combustion And Stability, Frank Paul Kozusko Jr.

Mathematics & Statistics Theses & Dissertations

Three physical models of laminar mixing of initially separated gases are studied. Two models study the effects of the mixing dynamics on the chemical reactions between the gases. The third model studies the structure and stability of a laminar mixing layer in a binary gas. The three models are:

1. Two ideal and incompressible gases representing fuel and oxidizer are initially at rest and separated across an infinite linear interface in a two dimensional system. Combustion, expected as the gases mix, will lead to a rapid rise in temperature in a localized area, i.e. ignition. The mixing of the gases …

Thermal Ignition Analysis In The Laminar Boundary Layer Behind A Propagating Shock Front, Mushtaq Ahmed Khan

Mathematics & Statistics Theses & Dissertations

Asymptotic analysis in the limit of large activation energy is performed to investigate the ignition of a reactive gas in the laminar boundary layer behind a propagating shock front. The study is based on a one-step, irreversible Arrhenius reaction of a premixed gas; therefore, the ignition phenomenon is thermally induced. The boundary layer consists of a thin, diffusive, reaction region at the point where the temperature is maximum and diffusive-convective non-reacting regions adjacent to the reacting region. Both adiabatic and isothermal boundary conditions are examined. For the adiabatic wall, the reaction zone is near the insulated boundary. The reaction zone …

Elimination Of Edge Effects Using Spline Wavelets Which Maintain A Uniform Two-Scale Relation, Sang Kyu Yang

Mathematics & Statistics Theses & Dissertations

Use of the compactly supported B-spline wavelet of Chui and Wang is hindered by loss of accuracy on decomposition, through truncation of weight sequences which are countably infinite. Adaptations to finite intervals often encounter significant problems with error near boundaries, called edge effects. For multiresolution analysis on a finite interval which employ the piecewise linear B-wavelet the present research provides a frontal approach to decomposition which avoids truncation of weight sequences, experiences no error at boundaries, and which exhibits a factor of three increase in computational efficiency, over the usual approach characterized by truncation of infinite weight sequences. As a …

Fermi Problems: Educated Guesses, John A. Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Nozzle Flow With Vibrational Nonequilibrium, John Gary Landry

Mathematics & Statistics Theses & Dissertations

Flow of nitrogen gas through a converging-diverging nozzle is simulated. The flow is modeled using the Navier-Stokes equations that have been modified for vibrational nonequilibrium. The energy equation is replaced by two equations. One equation accounts for energy effects due to the translational and rotational degrees of freedom, and the other accounts for the affects due to the vibrational degree of freedom. The energy equations are coupled by a relaxation time which measures the time required for the vibrational energy component to equilibrate with the translational and rotational energy components. An improved relaxation time is used in this thesis. The …

A Mathematical Model Of Cycle Chemotherapy, J. C. Panetta, J. Adam

Mathematics & Statistics Faculty Publications

A mathematical model is used to discuss the effects of cycle-specific chemotherapy. The model includes a constraint equation which describes the effects of the drugs on sensitive normal tissue such as bone marrow. This model investigates both pulsed and piecewise-continuous chemotherapeutic effects and calculates the parameter regions of acceptable dose and period. It also identifies the optimal period needed for maximal tumor reduction. Examples are included concerning the use of growth factors and how they can enhance the cell kill of the chemotherapeutic drugs.

Rational Cubic B-Spline Interpolation And Its Applications In Computer Aided Geometric Design, Kotien Wu

Mathematics & Statistics Theses & Dissertations

Because of the flexibility that the weights and the control points provide, NURBS have recently become very popular tools for the design of curves and surfaces. If the weights are positive then the NURB will lie in the convex hull of its control points and will not possess singularities. Thus it is desirable to have positive weights.

In utilizing a NURB a designer may desire that it pass through a set of data points {x_i} This interpolation problem is solved by the assigning of weights to each data point. Up to now little has been known regarding the …

Some Sampling Designs And Estimation Problems, Hassan Lakkis

Mathematics & Statistics Theses & Dissertations

In the first chapter we review some standard estimators in sampling from a finite population, and some design-based estimators in sampling from a continuous universe.

In concert with the theory initiated by professor Douglas Robson (personal communication) and later presented by Cordy (1993), we consider design-based variance estimation for probability sampling from a continuous and spatially distributed universe. Using this theory in chapter two, the sampling design of one random point from each cell of a translated grid is investigated and the problem of edge effects on estimation is illustrated with examples. Also in chapter four, standard systematic sampling methods …

Characterization Of The Local Lipschitz Constant, M. W. Bartelt, J. J. Swetits

Mathematics & Statistics Faculty Publications

A characterization, using polynomials introduced by A. V. Kolushov, is given for the local Lipschitz constant for the best approximation operator in Chebyshev approximation from a Haar set. The characterization is then used to study the existence of uniform local Lipschitz constants.

Invariant Manifolds Of A Toy Climate Model, Michael Toner

Mathematics & Statistics Theses & Dissertations

According to astronomical theory, ice ages are caused by variations in the Earth's orbit. However, ice core data shows strong fluctuations in ice volume at a low frequency not significantly present in orbital variations. To understand how this might occur, the dynamics of a two dimensional nonlinear differential equation representing glacier/temperature interaction of an idealized climate was studied. Self sustained oscillation of the autonomous equation was used to model the internal mechanisms that could produce these fluctuations. Periodic parametric modulation of a damped internal oscillation was used to model periodic climate response at double the external modulation period. Both phenomena …

Prediction Of The Stochastic Behavior Of Nonlinear Systems By Deterministic Models As A Classical Time-Passage Probabilistic Problem, L. M. Ivanov, A. D. Kirwan Jr., O. V. Melnichenko

CCPO Publications

Assuming that the behaviour of a nonlinear stochastic system can be described by a Markovian diffusion approximation and that the evolution equations can be reduced to a system of ordinary differential equations, a method for the calculation of prediction time is developed. In this approach, the prediction time depends upon the accuracy of prediction, the intensity of turbulence, the accuracy of the initial conditions, the physics contained in the mathematical model, the measurement errors, and the number of prediction variables. A numerical application to zonal channel flow illustrates the theory. Some possible generalizations of the theory are also discussed.

Multigrid Acceleration Of Time-Dependent Solutions Of Navier-Stokes Equations, Sarafa Oladele Ibraheem

Mechanical & Aerospace Engineering Theses & Dissertations

Recent progress in Computational Fluid Dynamics is encouraging scientists to look at fine details of flow physics of problems in which natural unsteady phenomena have hitherto been neglected. The acceleration methods that have proven very successful in steady state computations can be explored for time dependent computations. In this work, an efficient multigrid methods is developed to solve the time-dependent Euler and Navier-Stokes equations. The Beam-Warming ADI method is used as the base algorithm for time stepping calculations. Application of the developed algorithm proved very efficient in selected steady and unsteady test problems. For instance, the inherent unsteadiness present in …

On Cohesion Stable Graphs, Virginia Rice, Richard D. Ringeisen

Mathematics & Statistics Faculty Publications

The cohesion of a graph was introduced to model vulnerability of a graph relative to the neighborhoods of its vertices. We are concerned in this paper with the changes in this parameter when an edge is deleted. In particular, after displaying some results on stability under edge destruction, we go on to display various infinite classes of cohesion stable graphs. Several ways in which graphs or parts of graphs may be combined to produce stable graphs are also presented, along with a look at what cannot be stated at this time.

Boundary Value Problems In Rectilinearly Anisotropic Thermoelastic Solids, Gilbert Kerr

Mathematics & Statistics Theses & Dissertations

The boundary value problems which are considered are the type that arise due to the presence of a Griffith crack (or cracks) in an anisotropic thermoelastic solid. The thermoelastic field, in such materials, when the infinitesimal theory is employed, is governed by a set of elliptic partial differential equations. The general solution of these equations is expressed in terms of arbitrary analytic functions whose real parts, in turn, are expressed in terms of Fourier type integrals or Fourier series. Integral transform techniques are then used to determine the stress intensity factors (and other pertinent information) for various crack geometries. In …

A Study Of The Interface Between A Turbulent Boundary Layer And The Irrotational Free Stream, Shih-Chao Lin

Mechanical & Aerospace Engineering Theses & Dissertations

A mathematical model has been developed to simulate the events which occur at the interface between a turbulent boundary layer and the irrotational free stream. By assuming the streamwise velocity component of this interface to be constant and that the interface is uncontorted initially, the solution to the equations of motion is used to derive the position functions of a point element on the interface. This analysis is restricted to a fully developed incompressible turbulent boundary layer with zero pressure gradient. Vertical velocity disturbances which are periodic in the spanwise direction and restricted to small streamwise domains are assumed.

The …

The Sharp Lipschitz-Constants For Feasible And Optimal-Solutions Of A Perturbed Linear Program, Wu Li

Mathematics & Statistics Faculty Publications

The purpose of this paper is to derive the sharp Lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right-hand-side perturbations. The Lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp.

Design And Implementation Of Fuzzy Logic Controllers, Osama A. Abihana

Electrical & Computer Engineering Theses & Dissertations

The main objectives of our research are to present a self-contained overview of fuzzy sets and fuzzy logic, develop a methodology for control system design using fuzzy logic controllers, and to design and implement a fuzzy logic controller for a real system. In this thesis we first present the fundamental concepts of fuzzy sets and fuzzy logic. Fuzzy sets and basic fuzzy operations are defined. In addition, for control systems, it is important to understand the concepts of linguistic variables, linguistic values, term sets, fuzzy rule base, inference methods, and defuzzification methods. Second, we introduce a four-step fuzzy logic control …

Some Solutions To A Lens Model With Applications To Warm-Core Eddies, Juping Liu

OES Theses and Dissertations

A model of lens-shaped anticyclonic eddies based on nonlinear shallow water equations is developed. The model is a three-layer fluid and allows for one asymmetric mode as well as specified environmental flows. The solution scheme is a polynomial expansion of the field variables. When inserted into the hydrographic equations, the expansion yields eight first-order differential equations for the time dependent amplitudes. This system of ordinary differential equations is numerically tractable. As long as the initial values meet the requirement of elliptical structure and the prescribed external force is tolerable for the initial values, the numerical solutions are stable. Numerical solutions …

Bounds On Constraint Weight Parameters Of Hopfield Networks For Stability Of Optimization Problem Solutions, Gursel Serpen

Electrical & Computer Engineering Theses & Dissertations

The purpose of the presented research is to study the convergence characteristics of Hopfield network dynamics. The relation between constraint weight parameter values and the stability of solutions of constraint satisfaction and optimization problems mapped to Hopfield networks is investigated. A theoretical development relating constraint weight parameter values to solution stability is presented. The dependency of solution stability on constraint weight parameter values is shown employing an abstract optimization problem. A theorem defining bounds on the constraint weight parameter magnitudes for solution stability of constraint satisfaction and optimization problems is proved. Simulation analysis on a set of optimization and constraint …

The Solution Of A Singular Integral Equation Arising From A Lifting Surface Theory For Rotating Blades, Mark H. Dunn

Mathematics & Statistics Theses & Dissertations

A technique is presented for the solution of a linear, two dimensional, singular, Volterra integral equation of the first kind. The integral equation, originally developed by Farassat and Myers, is derived from the basic equations of linearized acoustics and models the lifting force experienced by an infinitesimally thin surface moving tangent to itself. As a particular application, the motion of modern high speed aircraft propellers (Advanced Technology Propellers) is considered. The unknown propeller blade surface pressure distribution is approximated by a piecewise constant function and the integral equation is solved numerically by the method of collocation. Certain simplifying assumptions applied …

On Shock Capturing For Liquid And Gas Media, Tze Jang Chen

Mathematics & Statistics Theses & Dissertations

The numerical investigation of shock phenomena in gas or liquid media where a specifying relation for internal energy is absent poses special problems. Classically, for gas dynamics the usual procedure is to employ a splitting scheme to remove the source terms from the Euler equations, then up-wind biased shock capturing algorithms are built around the Riemann problem for the system which remains. However, in the case where the Euler equations are formulated in the term of total enthalpy, a technical difficulty associated with equation splitting forces a pressure time derivative to be treated as a source term. This makes it …

Indifference Graphs And The Single Row Routing Problem, Peter J. Looges

Computer Science Theses & Dissertations

This thesis investigates the subclass of interval graphs known as indifference graphs. New optimal algorithms for recognition, center, diameter, maximum matching, Hamiltonian path and domination in indifference graphs are presented. The recognition algorithm produces a linear order with properties which allow the solution of the other problems in linear time. Indifference graphs are further applied to the single row routing problem which results in both sequential,. and parallel routing algorithms.

Nonlinear-Interaction Of A Detonation Vorticity Wave, D. G. Lasseigne, T. L. Jackson, M. Y. Hussaini

Mathematics & Statistics Faculty Publications

The interaction of an oblique, overdriven detonation wave with a vorticity disturbance is investigated by a direct two-dimensional numerical simulation using a multidomain, finite-difference solution of the compressible Euler equations. The results are compared to those of linear theory, which predict that the effect of exothermicity on the interaction is relatively small except possibly near a critical angle where linear theory no longer holds. It is found that the steady-state computational results whenever obtained in this study agree with the results of linear theory. However, for cases with incident angle near the critical angle, moderate disturbance amplitudes, and/or sudden transient …

Activator-Inhibitor Control Of Tissue Growth, John A. Adam

Mathematics & Statistics Faculty Publications

This note develops a simple model for the competition between activator and inhibitor control mechanisms in one-dimensional tissue growth. The pedagogic usefulness of such a model is that it is easily accessible to undergraduate applied mathematicians and is suggestive of behavior known to occur in more realistic biological systems (e.g., some types of cancer). The limitations of the model are obvious and can provide a basis for discussion of the applicability of complementary levels of description in mathematical modeling.