Mathematics Commons^™

Global Attractor Of Solutions Of A Rational System In The Plane, Miron B. Bekker, Martin Bohner, Hristo D. Voulov

Mathematics and Statistics Faculty Research & Creative Works

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

Generation And Validation Of Optimal Topologies For Solid Freeform Fabrication, Purnajyoti Bhaumik

Masters Theses

"The study of fabricating topologically optimized parts is presented hereafter. The mapping of topology optimization results for Standard Tessellation Language (STL) writing would enable the solid freeform fabrication of lightweight mechanisms. Aerospace leaders such as NASA, Boeing, Airbus, European Aeronautic Defense And Space Company (EADS), and GE Aero invest in topology optimization research for the production of lightweight materials. Certain concepts such as microstructural homogenization, discretization, and mapping are reviewed and presented in the context of topology optimization. Future biomedical applications of solid freeform fabrication such as organ printing stand to save millions of lives through the robust development of …

Day Of The Week Effect In Returns And Volatility Of The S&P 500 Sector Indices, Juan Liu

Masters Theses

"Previous studies have shown that returns associated with the stock market or foreign exchange's futures show variations across the day of the week. On such study, that employs a modified GARCH model for estimation, shows that returns associated with the S&P 500 stock index is highest on Wednesday and lowest returns on Monday. The same study shows that volatility is highest on Fridays and lowest on Wednesdays. In this study we investigate if this day-of-the-week effect on returns and volatility is present in the different sectors that constitute the S&P 500 index. The data set used provides daily returns from …

Discrete And Dynamic Population Models With Logistic Growth Rate, Sabrina Heike Streipert

Doctoral Dissertations

"The Beverton-Holt difference equation defines a discrete relation describing a population model. Considering periodic carrying capacity and periodic inherent growth rate, a population with seasonal changing life cycle and environment is reflected. The so-called periodically forced Beverton-Holt equation is investigated and its unique periodic solution is derived. This provides the first Cushing-Henson conjecture, while a counterexample proves that the classical second Cushing-Henson conjecture is not satisfied. Modifications of the conjecture are formulated. To extend the studies, the Beverton-Holt equation is investigated in the quantum calculus time setting. The existence of the globally attracting periodic solution of the Beverton-Holt q-difference …

On Testing Common Indices For Several Multi-Index Models: A Link-Free Approach, Xuejing Liu

Doctoral Dissertations

"To avoid the curse of dimensionality, and to help us better understand the structure of the high dimensional data, methods for dimension reduction are clearly called for. The common linear dimension reduction techniques for single population include principal component analysis (PCA) which is unsupervised in regression and supervised Partial Least Squares (PLS). Modern sufficient dimension reduction techniques, like the ones we consider, constitute a form of supervised linear dimension reduction which outperform PCA and PLS without the underlying model assumptions.

In practice, we often deal with situations where the same variables are being measured on objects from different groups, and …

Discrete Analogues Of Some Classical Special Functions, Thomas Joseph Cuchta

Doctoral Dissertations

"Analogues of special functions on time scales are studied with special focus on the time scale 𝕋 = hℤ. Functions investigated in particular include complex monomials, new trigonometric functions, Gaussian bell, Hermite and Laguerre polynomials, Bessel functions, and hypergeometric series"--Abstract, page iii.

Small Sample Saddlepoint Confidence Intervals In Epidemiology, Pasan Manuranga Edirisinghe

Doctoral Dissertations

"In section 1, we develop a novel method of confidence interval construction for directly standardized rates. These intervals involve saddlepoint approximations to the intractable distribution of a weighted sum of Poisson random variables and the determination of hypothetical Poisson mean values for each of the age groups. Simulation studies show that, in terms of coverage probability and length, the saddlepoint confidence interval (SP) outperforms four competing confidence intervals obtained from the moment matching (M8), gamma-based (G1,G4) and ABC bootstrap (ABC) methods.

In section 2, we first consider Brillinger's classical model for a vital rate estimate with a random denominator. We …

Small Sample Umpu Equivalence Testing Based On Saddlepoint Approximations, Renren Zhao

Doctoral Dissertations

"In the first section, we consider small sample equivalence tests for exponentiality. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require a much larger sample size, in comparison to classical "difference tests", to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. …

Volterra Difference Equations, Nasrin Sultana

Doctoral Dissertations

"This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the …

Essays On Unit Root Testing In Time Series, Xiao Zhong

Doctoral Dissertations

"Unit root tests are frequently employed by applied time series analysts to determine if the underlying model that generates an empirical process has a component that can be well-described by a random walk. More specifically, when the time series can be modeled using an autoregressive moving average (ARMA) process, such tests aim to determine if the autoregressive (AR) polynomial has one or more unit roots. The effect of economic shocks do not diminish with time when there is one or more unit roots in the AR polynomial, whereas the contribution of shocks decay geometrically when all the roots are outside …

A Domain Decomposition Method For The Steady-State Navier-Stokes-Darcy Model With Beavers-Joseph Interface Condition, Xiaoming He, Jian Li, Yanping Lin, Ju Ming

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In …

Some Combinatorial Applications Of Sage, An Open Source Program, Jessica Ruth Chowning

Masters Theses

"In this thesis, we consider the usefulness of Sage, an online and open-source program, in analyzing permutation puzzles such as the Rubik's cube and a specific combinatorial structure called the projective plane. Many programs exist to expedite calculations in research and provide previously-unavailable solutions; some require purchase, while others, such as Sage, are available for free online. Sage is asked to handle a small permutation puzzle called Swap, and then we explore how it calculates solutions for a Rubik's cube. We then discuss projective planes, Sage's library of functions for dealing with projective planes, and how they relate to the …

Oscillation Of Second-Order Emden–Fowler Neutral Delay Differential Equations, Ravi P. Agarwal, Martin Bohner, Tongxing Li, Chenghui Zhang

Mathematics and Statistics Faculty Research & Creative Works

We establish some new criteria for the oscillation of second-order Emden–Fowler neutral delay differential equations. We study the case of super linear and the case of sublinear equations subject to various conditions. The results obtained show that the presence of a neutral term in a differential equation can cause or destroy oscillatory properties. Several examples are provided to illustrate the relevance of new theorems.

Planarity Of Whitney Levels, Jorbe Bustamante, W. J. Charatonik, Raul Escobedo

Mathematics and Statistics Faculty Research & Creative Works

First, we characterize all locally connected continua whose all Whitney levels are planar. Second, we show by example that planarity is not a (strong) Whitney reversible property. This answers a question from Illanes-Nadler book [2].

On Second Order Elliptic Equations And Variational Inequalities With Anisotropic Principal Operators, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

This paper is about boundary value problems of the form where ƨ is a convex function of ξ.∈ RN rather than a function of the norm|ξ|. The problem is formulated appropriately in an anisotropic Orlicz-Sobolev space associated with ƨ. We study the existence of solutions and some other properties of the above problem and its corresponding variational inequality in such space.

Extreme Self-Adjoint Extensions Of A Semibounded Q-Difference Operator, Miron B. Bekker, Martin Bohner, Hristo Voulov

Mathematics and Statistics Faculty Research & Creative Works

For a certain q-difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self-adjoint extensions, i.e., the so-called Friedrichs and Kreǐn extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kreǐn extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial role in our investigation plays the fact that both the Friedrichs and the Kreǐn extensions are scale invariant. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Oscillation Theorems For Fourth-Order Half-Linear Delay Dynamic Equations With Damping, Ravi P. Agarwal, Martin Bohner, Tongxing Li, Chenghui Zhang

Mathematics and Statistics Faculty Research & Creative Works

This article is concerned with oscillatory behavior of a class of fourth-order half-linear delay dynamic equations with damping on a time scale. Some new oscillation criteria are established. © 2013 Springer Basel.

New Pod Error Expressions, Error Bounds, And Asymptotic Results For Reduced Order Model Of Parabolic Pdes, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

The derivations of existing error bounds for reduced order models of time varying partialdi erential equations (PDEs) constructed using proper orthogonal decomposition (POD) haverelied on bounding the error between the POD data and various POD projections of that data.Furthermore, the asymptotic behavior of the model reduction error bounds depends on theasymptotic behavior of the POD data approximation error bounds. We consider time varyingdata taking values in two di erent Hilbert spacesHandV, withVH, and prove exactexpressions for the POD data approximation errors considering four di erent POD projectionsand the two di erent Hilbert space error norms. Furthermore, the exact error expressions …

Qualitative Behavior Of Solutions Of Difference Equations With Several Oscillating Coefficients, Martin Bohner, George E. Chatzarakis, Ioannis P. Stavroulakis

Mathematics and Statistics Faculty Research & Creative Works

Sufficient conditions which guarantee the convergence of the non-oscillatory solutions or oscillation of all solutions of a difference equation with several deviating arguments and oscillating coefficients are presented. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given. [Media Object not available: see full text.]

Oscillatory Behavior Of Solutions Of Third-Order Delay And Advanced Dynamic Equations, Murat Adivar, Elvan Akin, Raegan Higgins

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scale T is a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results.

A Linear Iteration Algorithm For A Second-Order Energy Stable Scheme For A Thin Film Model Without Slope Selection, Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. the PDE, which is a nonlinear, fourth-order parabolic equation, is the L 2 gradient flow of the energy d x. the energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work (Shen et al. in SIAM J Numer Anal 50:105-125, 2012). the resulting scheme is highly nonlinear, and its implementation is non-trivial. in this paper, we propose a linear iteration algorithm to solve …

Oscillation Criteria For Second-Order Dynamic Equations On Time Scales, Ravi P. Agarwal, Martin Bohner, Tongxing Li, Chenghui Zhang

Mathematics and Statistics Faculty Research & Creative Works

Erbe's and Hassan's contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results. © 2014 Elsevier Ltd. All rights reserved.

Oscillation Of Second-Order P-Laplace Dynamic Equations With A Nonpositive Neutral Coefficient, Martin Bohner, Tongxing Li

Mathematics and Statistics Faculty Research & Creative Works

We establish some new oscillation criteria for a class of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient on a time scale. The results obtained supplement and improved those reported in the literature. © 2014 Elsevier Ltd. All rights reserved.

Multi-Valued Parabolic Variational Inequalities And Related Variational-Hemivariational Inequalities, Siegfried Carl, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

In this paper we study multi-valued parabolic variational inequalities involving quasilinear parabolic operators, and multi-valued integral terms over the underlying parabolic cylinder as well as over parts of the lateral parabolic boundary, where the multi-valued functions involved are assumed to be upper semicontinuous only. Note, since lower semicontinuous multi-valued functions allow always for a Carathéodory selection, this case can be considered as the trivial case and therefore will be omitted. Our main goal is threefold: First, we provide an analytical framework and an existence theory for the problems under consideration. Unlike in recent publications on multi-valued parabolic variational inequalities, the …

Asymptotic Behavior Of Nonoscillatory Solutions Of Higher-Order Integro-Dynamic Equations, Martin Bohner, Said Grace, Nasrin Sultana

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we establish some new criteria on the asymptotic behavior of Non oscillatory solutions of higher-order integro-dynamic equations on time scales. © AGH University of Science and Technology Press, Krakow 2014.

Some Dynamic Hardy-Type Inequalities With General Kernel, Martin Bohner, Ammara Nosheen, Josip Pečarić, Awais Younus

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we extend some Hardy-type inequalities with certain kernels to arbitrary time scales. Certain classical and some new integral and discrete inequalities are deduced in seek of applications. © Zagreb Paper JMI-08-12.

Safety Effect Of Missouri's Strategic Highway Safety Plan: Missouri's Blueprint For Safer Roadways, Mojtaba A. Mohammadi, V. A. Samaranayake, Ghulam H. Bham

Mathematics and Statistics Faculty Research & Creative Works

This study systematically evaluated the changes in motor vehicle crashes that occurred on the Missouri Interstate highway system following the implementation of the Missouri Strategic Highway Safety Plan (MSHSP) between 2004 and 2007. The MSHSP implemented crash injury reduction strategies in enforcement, education, engineering, and public policy. Empirical Bayesian methods were commonly used to evaluate the effects of any change in safety as a result of countermeasures. This paper presents a simple new approach to evaluating the effects of Missouri's safety plans on roadway crashes. For crash data associated with traffic and roadway characteristics, negative binomial regression models were developed …

On Some Inferential Problems With Recurrent Event Models, Withanage Ajith Raveendra De Mel

Doctoral Dissertations

"Recurrent events (RE) occur in many disciplines, such as biomedical, engineering, actuarial science, sociology, economy to name a few. It is then important to develop dynamic models for their modeling and analysis. Of interest with data collected in a RE monitoring are inferential problems pertaining to the distribution function F of the time between occurrences, or that of the distribution function G of the monitoring window, and their functionals such as quantiles, mean. These problems include, but not limited to: estimating F parametrically or nonparametrically; goodness of fit tests on an hypothesized family of distributions; efficient of tests; regression-type models, …

Adaptive Wavelet Discretization Of Tensor Products In H-Tucker Format, Mazen Ali

Masters Theses

"In previous work, the solution to a system of coupled parabolic PDEs, modeling the price of a CDO, was approximated numerically. Due to the nature of the problem, the system involved a large number of equations such that the parameters cannot be stored explicitly. The authors combined the data sparse H-Tucker storage format with the Galerkin method to approximate the solution, using wavelets for the space discretization together with time stepping (Method of Lines). The aforementioned approximation is of the linear kind, i.e., using a nonadaptive method. In this work, three methods for solving such systems adaptively are presented, together …

An Iterative Algorithm For Variational Data Assimilation Problems, Xin Shen

Masters Theses

"Data assimilation is a very powerful and efficient tool to use collected raw data for improving model prediction in numerical weather forecasting, hydrology, and many other areas of geosciences. In this thesis, an iterative algorithm [23] of variational data assimilation with finite element method is utilized to study different models. One motivation for this fundamental mathematical study is to provide a potential tool for simulation of CO₂ sequestration by extending it to more realistic and sophisticated models in the future. The basic idea of variational data assimilation is to utilize the framework of optimal control problems. We apply the …