Applied Mathematics Commons^™

Modelling Locally Changing Variance Structured Time Series Data By Using Breakpoints Bootstrap Filtering, Rajan Lamichhane

Mathematics & Statistics Theses & Dissertations

Stochastic processes have applications in many areas such as oceanography and engineering. Special classes of such processes deal with time series of sparse data. Studies in such cases focus in the analysis, construction and prediction in parametric models. Here, we assume several non-linear time series with additive noise components, and the model fitting is proposed in two stages. The first stage identifies the density using all the clusters information, without specifying any prior knowledge of the underlying distribution function of the time series. The effect of covariates is controlled by fitting the linear regression model with serially correlated errors. In …

Optimal Control Modeling And Simulation, With Application To Cholera Dynamics, Chairat Modnak

Mathematics & Statistics Theses & Dissertations

The theory of optimal control, a modern extension of the calculus of variations. has found many applications in a wide range of scientific fields, particularly in epidemiology with respect to disease prevention and intervention. In this dissertation. we conduct optimal control modeling, simulation and analysis to cholera dynamics. Cholera is a severe intestinal infectious disease that remains a serious public health threat in developing countries. Transmission of cholera involves complex interactions between the human host, the pathogen, and the environment. The worldwide cholera outbreaks and their increasing severity, frequency and duration in recent years underscore the gap between the complex …

Multi-Objective Preliminary Design Optimization Of Planing Craft Using Differential Evolution And Artificial Networks, Robert Eugene Weidman

Mechanical & Aerospace Engineering Theses & Dissertations

The differential evolution (DE) algorithm is an evolutionary algorithm that has been utilized as a metaheuristic optimizer in the last few years. A recently developed multi-objective version of DE is utilized in this research in order to optimize the preliminary design of planing craft. The craft performance is evaluated using the planing hull preliminary design software PHAST. The optimization problem objectives are the craft acquisition cost and the overall measure of effectiveness; the latter comprises various performance aspects: ride quality, transportability, payload, and mobility. Static stability and operating constraints are also considered. Each of the aforementioned four measures of effectiveness …

Linear Programming Algorithm With Mixed Real-Integer Variables In Matlab Environments, Gelareh Bakhtyar

Civil & Environmental Engineering Theses & Dissertations

Efficient numerical procedures for solving general Linear Programming (LP) problems with mixed real-integer variables are developed in this work. The proposed algorithms employ the revised dual simplex with Branch and Bound (B&B) algorithms, with special procedures for limited search of subsequent branches. Computational time can be significantly reduced by incorporating the updated inverse formulas into the developed procedures. Both generic LP problems and deterministic pavement maintenance and rehabilitation (M&R) problems are used in this study to vaiidate the developed procedures. Medium to large-scale examples ( 11 pavement M&R) presented in this work have demonstrated that the developed numerical procedures consistently …

Novel Algorithms And Instrumentation For Vibrational Spectroscopic Methods Of Analysis, Mohamed F. Abdelkader

Chemistry & Biochemistry Theses & Dissertations

Raman spectroscopy is a form of vibrational spectroscopy that has been increasingly applied to qualitative analysis of chemicals, explosives, pharmaceuticals, and fuels due to its non-invasive and non-destructive nature; its ease of sampling; and its high molecular specificity. These characteristics of Raman spectroscopy also facilitate its use for both in-line and at-line analysis. The principle limitation of Raman spectroscopy is optical interference arising from both analyte and non-analyte fluorescence. In this dissertation, a solution to this problem is presented in the form of a novel spectrometer design which operates in a sequentially shifted excitation mode to eliminate fluorescence backgrounds, fixed …

Lattice Boltzmann Simulations Of Thermal Convective Flows In Two Dimensions, Jia Wang, Donghai Wang, Pierre Lallemand, Li-Shi Luo

Mathematics & Statistics Faculty Publications

In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics with the Boussinesq approximation. We use the MRT thermal LBE (TLBE) to simulate the following two flows in two dimensions: the square cavity with differentially heated vertical walls and the Rayleigh-Benard convection in a rectangle heated from below. For the square cavity, the flow parameters in this study are the Rayleigh number Ra = 10³-10⁶, and the Prandtl number Pr = 0.71; and for the Rayleigh-Benard convection in a rectangle, Ra = 2 ^. 10³, 10 …

Duality Of The Weak Parallelogram Laws On Banach Spaces, Raymond Cheng, Charles B. Harris

Mathematics & Statistics Faculty Publications

This paper explores a family of weak parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak parallelogram law if and only if its dual satisfies an upper weak parallelogram law, and vice versa. Connections are established between the weak parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.

Mesoscopic Methods In Engineering And Science, Jos Derksen, Dmitry Eskin, Li-Shi Luo, Manfred Krafczyk

Mathematics & Statistics Faculty Publications

(First paragraph) Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications a macroscopic or continuum description has usually been sufficient, because of the large disparity between the spatial and temporal scales relevant to these applications and the scales of the underlying molecular dynamics. In this case, the microscopic physics merely determines material properties such as the viscosity of a fluid or the elastic constants of a solid. These material properties cannot be derived within the macroscopic framework, but the qualitative nature of the …

Momcmc: An Efficient Monte Carlo Method For Multi-Objective Sampling Over Real Parameter Space, Yaohang Li

Computer Science Faculty Publications

In this paper, we present a new population-based Monte Carlo method, so-called MOMCMC (Multi-Objective Markov Chain Monte Carlo). for sampling in the presence of multiple objective functions in real parameter space. The MOMCMC method is designed to address the "multi-objective sampling" problem, which is not only of interest in exploring diversified solutions at the Pareto optimal front in the function space of multiple objective functions, but also those near the front. MOMCMC integrates Differential Evolution (DE) style crossover into Markov Chain Monte Carlo (MCMC) to adaptively propose new solutions from the current population. The significance of dominance is taken into …

Numerical Methods For Fluid-Structure Interaction - A Review, Gene Hou, Jin Wang, Anita Layton

Mechanical & Aerospace Engineering Faculty Publications

The interactions between incompressible fluid flows and immersed structures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical-methods based on conforming and non-conforming meshes that are currently available for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interactions

Wavelet Collocation Method And Multilevel Augmentation Method For Hammerstein Equations, Hideaki Kaneko, Khomsan Neamprem, Boriboon Novaprateep

Mathematics & Statistics Faculty Publications

No abstract provided.

On The Global Stability Of A Generalized Cholera Epidemiological Model, Yuanji Cheng, Jin Wang, Xiuxiang Yang

Mathematics & Statistics Faculty Publications

In this paper, we conduct a careful global stability analysis for a generalized cholera epidemiological model originally proposed in [J. Wang and S. Liao, A generalized cholera model and epidemic/endemic analysis, J. Biol. Dyn. 6 (2012), pp. 568-589]. Cholera is a water-and food-borne infectious disease whose dynamics are complicated by the multiple interactions between the human host, the pathogen, and the environment. Using the geometric approach, we rigorously prove the endemic global stability for the cholera model in three-dimensional (when the pathogen component is a scalar) and four-dimensional (when the pathogen component is a vector) systems. This work unifies the …

Use Of Optimal Control Models To Predict Treatment Time For Managing Tick-Borne Disease, Holly D. Gaff, Elsa Schaefer, Suzanne Lenhart

Biological Sciences Faculty Publications

Tick-borne diseases have been on the rise recently, and correspondingly, there is an increased interest in implementing control measures to decrease the risk. Optimal control provides an ideal tool to identify the best method for reducing risk while accounting for the associated costs. Using a previously published model, a variety of frameworks are assessed to identify the key factors influencing mitigation strategies. The level and duration of tick-reducing efforts are key metrics for understanding the successful reduction in tick-borne disease incidence. The results show that the punctuated nature of the tick's life history plays a critical role in reducing risk …

Stability Analysis And Application Of A Mathematical Cholera Model, Shu Liao, Jim Wang

Mathematics & Statistics Faculty Publications

In this paper, we conduct a dynamical analysis of the deterministic cholera model proposed in [9]. We study the stability of both the disease-free and endemic equilibria so as to explore the complex epidemic and endemic dynamics of the disease. We demonstrate a real-world application of this model by investigating the recent cholera outbreak in Zimbabwe. Meanwhile, we present numerical simulation results to verify the analytical predictions.

Perfectly Matched Layer Absorbing Boundary Conditions For The Discrete Velocity Boltzmann-Bgk Equation, Elena Craig

Mathematics & Statistics Theses & Dissertations

Perfectly Matched Layer (PML) absorbing boundary conditions were first proposed by Berenger in 1994 for the Maxwell's equations of electromagnetics. Since Hu first applied the method to Euler's equations in 1996, progress made in the application of PML to Computational Aeroacoustics (CAA) includes linearized Euler equations with non-uniform mean flow, non-linear Euler equations, flows with an arbitrary mean flow direction, and non-linear clavier-Stokes equations. Although Boltzmann-BGK methods have appeared in the literature and have been shown capable of simulating aeroacoustics phenomena, very little has been done to develop absorbing boundary conditions for these methods. The purpose of this work was …

Preliminary Analysis Of An Agent-Based Model For A Tick-Borne Disease, Holly Gaff

Biological Sciences Faculty Publications

Ticks have a unique life history including a distinct set of life stages and a single blood meal per life stage. This makes tick-host interactions more complex from a mathematical perspective. In addition, any model of these interactions must involve a significant degree of stochasticity on the individual tick level. In an attempt to quantify these relationships, I have developed an individual-based model of the interactions between ticks and their hosts as well as the transmission of tick-borne disease between the two populations. The results from this model are compared with those from previously published differential equation based population models. …

A Least Squares Closure Approximation For Liquid Crystalline Polymers, Traci Ann Sievenpiper

Mathematics & Statistics Theses & Dissertations

An introduction to existing closure schemes for the Doi-Hess kinetic theory of liquid crystalline polymers is provided. A new closure scheme is devised based on a least squares fit of a linear combination of the Doi, Tsuji-Rey, Hinch-Leal I, and Hinch-Leal II closure schemes. The orientation tensor and rate-of-strain tensor are fit separately using data generated from the kinetic solution of the Smoluchowski equation. The known behavior of the kinetic solution and existing closure schemes at equilibrium is compared with that of the new closure scheme. The performance of the proposed closure scheme in simple shear flow for a variety …

An Extensible Mathematical Model Of Glucose Metabolism, Caleb L. Adams

Mathematics & Statistics Theses & Dissertations

The American Diabetes Association reports that diabetes is the fifth leading cause of death by disease in the United States. An estimated 23.6 million individuals, or seven percent of the population, have diabetes. Nearly one-third are unaware that they have the disease. The total of the direct and indirect medical costs associated with diabetes in 2007 was projected to be $174 billion, or approximately one out of every ten health care dollars.

One must understand the glucose regulatory system of the healthy body to understand diabetes. Blood glucose concentration returns to a constant level after eating and is maintained during …

Acceleration Techniques By Post-Processing Of Numerical Solutions Of The Hammerstein Equation, Khomsan Neamprem, Hideaki Kaneko

Mathematics & Statistics Faculty Publications

No abstract provided.

Mesoscopic Methods In Engineering And Science, Chuguang Zheng, Jiding Lu, Zhaoli Guo, Li-Shi Luo, Manfred Krafczyk

Mathematics & Statistics Faculty Publications

Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications a macroscopic or continuum description has usually been sufficient, because of the large disparity between the spatial and temporal scales relevant to these applications and the scales of the underlying molecular dynamics. In this case, the microscopic physics merely determines material properties such as the viscosity of a fluid or the elastic constants of a solid. These material properties cannot be derived within the macroscopic framework, but the qualitative nature of the macroscopic dynamics …

Uniform L1 Behavior Of A Time Discretization Method For A Volterra Integrodifferential Equation With Convex Kernel; Stability, Charles B. Harris, Richard D. Noren

Mathematics & Statistics Faculty Publications

We study stability of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation u'(t) + ∫0 β (t - s)Au(s) ds = 0, t ≥ 0, u(0) = u0, which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space, and β (t) is locally integrable, nonnegative, nonincreasing, convex, and -β'(t) is convex. We establish stability of the method under these hypotheses on β(t). Thus, …

Parallel-Sparse Symmetrical/Unsymmetrical Finite Element Domain Decomposition Solver With Multi-Point Constraints For Structural/Acoustic Analysis, Siroj Tungkahotara, Willie R. Watson, Duc T. Nguyen, Subramaniam D. Rajan

Civil & Environmental Engineering Faculty Publications

Details of parallel-sparse Domain Decomposition (DD) with multi-point constraints (MPC) formulation are explained. Major computational components of the DD formulation are identified. Critical roles of parallel (direct) sparse and iterative solvers with MPC are discussed within the framework of DD formulation. Both symmetrical and unsymmetrical system of simultaneous linear equations (SLE) can be handled by the developed DD formulation. For symmetrical SLE, option for imposing MPC equations is also provided.

Large-scale (up to 25 million unknowns involving complex numbers) structural and acoustic Finite Element (FE) analysis are used to evaluate the parallel computational performance of the proposed DD implementation using …

Analytic Construction Of Periodic Orbits In The Restricted Three-Body Problem, Mohammed A. Ghazy

Mechanical & Aerospace Engineering Theses & Dissertations

This dissertation explores the analytical solution properties surrounding a nominal periodic orbit in two different planes, the plane of motion of the two primaries and a plane perpendicular to the line joining the two primaries, in the circular restricted three-body problem. Assuming motion can be maintained in the plane and motion of the third body is circular, Jacobi's integral equation can be analytically integrated, yielding a closed-form expression for the period and path expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane …

A Solution Of The Heat Equation With The Discontinuous Galerkin Method Using A Multilivel Calculation Method That Utilizes A Multiresolution Wavelet Basis, Robert Gregory Brown

Mathematics & Statistics Theses & Dissertations

A numerical method to solve the parabolic problem is developed that utilizes the Discontinuous Galerkin Method for space and time discretization. A multilevel method is employed in the space variable. It is shown that use of this process yields the same level of accuracy as the standard Discontinuous Galerkin Method for the heat equation, but with cheaper computational cost. The results are demonstrated using a standard one-dimensional homogeneous heat problem.

Post-Processing Techniques And Wavelet Applications For Hammerstein Integral Equations, Khomsan Neamprem

Mathematics & Statistics Theses & Dissertations

This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation.

In …

Probability Models For Blackjack Poker, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

For simplicity in calculation, previous analyses of blackjack poker have employed models which employ sampling with replacement. in order to assess what degree of error this may induce, the purpose here is to calculate results for a typical hand where sampling without replacement is employed. It is seen that significant error can result when long runs are required to complete the hand. The hand examined is itself of particular interest, as regards both its outstanding expectations of high yield and certain implications for pair splitting of two nines against the dealer's seven. Theoretical and experimental methods are used in order …

Mesoscopic Methods In Engineering And Science, Alfons Hoekstra, Li-Shi Luo, Manfred Krafczyk

Mathematics & Statistics Faculty Publications

(First paragraph) Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications a macroscopic or continuum description has usually been sufficient, because of the large disparity between the spatial and temporal scales relevant to these applications and the scales of the underlying molecular dynamics. In this case, the microscopic physics merely determines material properties such as the viscosity of a fluid or the elastic constants of a solid. These material properties cannot be derived within the macroscopic framework, but the qualitative nature of the …

Mathematical Models And Stability Analysis Of Cholera Dynamics, Shu Liao

Mathematics & Statistics Theses & Dissertations

In this dissertation, we present a careful mathematical study of several epidemic cholera models, including the model of Codeco [11] in 2001, that of Hartley, Morris and Smith [22] in 2006, and that of Mukandavire, Liao, Wang and Gaff et al. [60] in 2010. We formally derive the basic reproduction number R0 for each model by computing the spectral radius of the next generation matrix. We focus our attention on the stability analysis at the disease-free equilibrium which determines the short-term epidemic behavior, and the endemic equilibrium which determines the long-term disease dynamics. Particularly, we incorporate the Volterra-Lyapunov matrix …

An Adaptive Method For Calculating Blow-Up Solutions, Charles F. Touron

Mathematics & Statistics Theses & Dissertations

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as "blow-up." The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting …

A Two-Population Insurgency In Colombia: Quasi-Predator-Prey Models - A Trend Towards Simplicity, John A. Adam, John A. Sokolowski, Catherine M. Banks

Mathematics & Statistics Faculty Publications

A sequence of analytic mathematical models has been developed in the context of the "low-level insurgency" in Colombia, from 1993 to the present. They are based on generalizations of the two-population "predator-prey" model commonly applied in ecological modeling, and interestingly, the less sophisticated models yield more insight into the problem than the more complicated ones, but the formalism is available to adapt the model "upwards" in the event that more data becomes available, or as the situation increases in complexity. Specifically, so-called "forcing terms" were included initially in the coupled differential equations to represent the effects of government policies towards …