Approved for public release; distribution is unlimited / Reissued 30 May 2017 with Second Reader’s non-NPS affiliation added to title page. / The scalar reciprocity equation commonly stated in underwater acoustics relates pressure fields and monopole sources. It is often used to predict the pressure measured by a hydrophone for multiple source locations by placing a source at the hydrophone location and calculating the field everywhere for that source. That method, however, does not work when calculating the orthogonal components of the velocity field measured by a fixed receiver. This thesis derives a vector-scalar reciprocity equation that accounts for both monopole and dipole sources. This equation can be used to calculate individual components of the received vector field by altering the source type used in the propagation calculation. This enables a propagation model to calculate the received vector field components for an arbitrary number of source locations with a single model run for each received field component instead of requiring one model run for each source location. Application of the vector-scalar reciprocity principle is demonstrated with analytic solutions for a range-independent environment and with numerical solutions for a range-independent and a range-dependent environment using a parabolic equation model. / Electronics Engineer, Naval Undersea Warfare Center

The increasing number and precision of measurements of neutron star masses, radii, and, in the near future, moments of inertia offer the possibility of precisely determining the neutron star equation of state (EOS). One way to facilitate the mapping of observables to the EOS is through a parametrization of the latter. We present here a generic method for optimizing the parametrization of any physically allowed EOS. We use mock EOS that incorporate physically diverse and extreme behavior to test how well our parametrization reproduces the global properties of the stars, by minimizing the errors in the observables of mass, radius, and the moment of inertia. We find that using piecewise polytropes and sampling the EOS with five fiducial densities between similar to 1-8 times the nuclear saturation density results in optimal errors for the smallest number of parameters. Specifically, it recreates the radii of the assumed EOS to within less than 0.5 km for the extreme mock EOS and to within less than 0.12 km for 95% of a sample of 42 proposed, physically motivated EOS. Such a parametrization is also able to reproduce the maximum mass to within 0.04 M-circle dot and the moment of inertia of a 1.338 M-circle dot. neutron star to within less than 10% for 95% of the proposed sample of EOS.

No description available.

In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

This paper revisits Grunfeld's well-known investment data, one of the most widely used data sets in all of econometrics, on the occasion of their 50th anniversary. It presents, apparently for the first time after the publication of the original Chicago Ph.D. thesis, the full data set and points out errors and inconsistencies in several currently available versions. It also revisits a number of empirical studies from the literature of the last five decades. / Series: Research Report Series / Department of Statistics and Mathematics

The chemical master equation (CME) represents the accepted stochastic description of chemical reaction kinetics in mesoscopic systems. As its exact solution – which gives the corresponding probability density function – is possible only in very simple cases, there is a clear need for approximation techniques. Here, we propose a novel perturbative three-step approach which draws heavily on graph theory: (i) we expand the eigenvalues of the transition state matrix in the CME as a series in a non-dimensional parameter that depends on the reaction rates and the reaction volume; (ii) we derive an analogous series for the corresponding eigenvectors via a graph-based algorithm; (iii) we combine the resulting expansions into an approximate solution to the CME. We illustrate our approach by applying it to a reversible dimerization reaction; then, we formulate a set of conditions, which ensure its applicability to more general reaction networks. We follow attempting to apply the results to a more complicated system, namely push-pull, but the problem reveals too complex for a complete solution. Finally, we discuss the limitations of the methodology.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Warren N. White / There are a number of different types of mechanical systems which can be termed as underactuated. The degrees of freedom (DOF) of a system are defined by the system’s number of independent movements. Underactuated mechanical systems have fewer actuators than DOF. Some examples such as satellites, air craft, overhead crane loads, and missiles have at least one unactuated DOF. The work presented here develops a nonlinear control law for the asymptotic stabilization of underactuated systems. This is accomplished by finding the solution of matching conditions that arise from Lyapunov’s second method, analogous to the dissipation of energy. The direct Lyapunov approach (DLA) offers a wide range of applications for underactuated systems due to the fact that the algebraic equations, ordinary differential equations, and partial differential equations stemming from the matching conditions are more tractable than those appearing in other approaches. Two lemmas of White et al. (2007) are applied for the positive definiteness and symmetry condition of the KD matrix which is used to define an analogous kinetic energy for the system. The defined KD matrix and the Lyapunov candidate function are developed to ensure stability. The KD matrix is analogous to the mass matrix of the dynamic system. The candidate Lyapunov function, involving the analogous kinetic energy and an undefined potential of the generalized position coordinates, is presented. By computing the time derivative of the Lyapunov candidate function, three equations called matching conditions emerge and parts of their solution provide the nonlinear control law that stabilizes the system. This dissertation presents the derivation of the DLA, provides a new method to solve the first matching condition (FMC), and shows the tools for the control law design. The stability is achieved from the proper shape of the potential, the positive definiteness of the KD matrix, and the non-positive rate of change of the Lyapunov function. The ball and beam, the inverted pendulum cart, and, a more complicated system, the ball and arc are presented to demonstrate the importance of the results because the methods to solve the matching equations, emerging from the system examples, are simple and easier. The presented controller design formulation satisfies the FMC exactly without introducing control law terms that are quadratic in the velocities or approximations. This methodology allows the development of the first nonlinear stabilizing control law for the ball and arc system, a simple and effective formulation to find a control law for the inverted pendulum cart, and a stabilizing control of the ball and beam apparatus without the necessity of approximations to solve the FMC. To illustrate the formulation, the derivation is performed using the symbolic manipulation program Maple and it is simulated in the Matlab/Simulink environment. The dissertation on the solvability of the first matching condition for stabilization is organized into six different chapters. The introduction of the problem and the previous approaches are presented in Chapter 1. Techniques for solving of the first matching condition, as well as the limitations, are provided in Chapter 2. The application of this general strategy to the ball and beam system appears in Chapter 3. Chapter 4 and 5 present the application of the method to the ball and arc apparatus and to the inverted pendulum cart, respectively. The difficulties for each application are also presented. Particularly, Chapter 5 shows the application of the produced material to obtain an easier formulation for the inverted pendulum cart compared to previous published controller examples. Finally, some conclusions and recommendations for future work are presented.

There is an increasing amount of evidence in scientific research and industrial engineering indicating that the graphic processing unit (GPU) has a higher efficiency and a stronger ability over CPUs to process certain computations. The heat equation is one of the most well-known partial differential equations with well-developed theories, and application in engineering. Thus, we chose in this report to use the heat equation to numerically solve for the heat distributions at different time points using both GPU and CPU programs. The heat equation with three different boundary conditions (Dirichlet, Neumann and Periodic) were calculated on the given domain and discretized by finite difference approximations. The programs solving the linear system from the heat equation with different boundary conditions were implemented on GPU and CPU. A convergence analysis and stability analysis for the finite difference method was performed to guarantee the success of the program. Iterative methods and direct methods to solve the linear system are also discussed for the GPU. The results show that the GPU has a huge advantage in terms of time spent compared with CPU in large size problems.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013. / In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is analytically considered using Fourier and Laplace transformations. The main focus of the dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction Implicit method. The one-dimensional parabolic mixed derivative diffusion equation is extended to a two-dimensional analog. In order to do this, the two-dimensional analog is solved using a Crank-Nicholson method and implemented according to the Peaceman- Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour, subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion equation is then implemented with a high-order method to unveil more accurate solutions. An error analysis is executed to show the accuracy differences between the numerical solutions of the general ADI and high-order compact methods. Each of the methods implemented in this dissertation are investigated via the von-Neumann stability analysis to prove stability under certain conditions.

Exclusive breastfeeding for the first six months of life has been shown to decrease morbidity and mortality of women and infants. Organizations such as the United Nations Children’s Fund (UNICEF, 2018), American Academy of Pediatrics (AAP, 2012), and the World Health Organization (WHO, 2017a) have universally endorsed exclusive breastfeeding for the first six months of life, and then continuation of breastfeeding for a minimum of one to two years, with only supplementation of other liquid or solid food sources. Breastfeeding rates in the United States have not met the minimum goals set forth by Healthy People 2020 (n.d.). Although 81% of U.S. mothers initiated breastfeeding after the birth of their infant, only 22% of mothers were found to be exclusively breastfeeding at six months postpartum (Centers for Disease Control and Prevention [CDC], 2016a). This prospective, longitudinal, structural equation modeling study examined millennial-aged, exclusively breastfeeding women within one month postpartum who were followers of at least one of 17 social media breastfeeding support groups. Relationships of the conceptual constructs within Pender’s (1996) revised health promotion model (RHPM); House’s (1981) dimensions of social support; and the added constructs of breastfeeding knowledge, breastfeeding confidence, and breastfeeding attitude were analyzed in an effort to better understand the variables that lead to sustained exclusive breastfeeding to six months. Data supported the use of the integrated model for breastfeeding women. The normed referenced chi-square (2) of 1.9 (CFI =.94, IFI =.94, NFI =.89, RMSEA =.06, CFI [PCFI] >.5) indicated a good model fit. Additionally, there were statistically significant gains in the confidence, knowledge, and attitude scores from pretest to follow-up at six months. Exclusive breastfeeding to six months was reported to be three times (66%) higher than the U.S. national average (22%) (CDC, 2016a). Future use of the integrated model has great potential to impact public health by the exploration of variables that promote exclusive breastfeeding to six months. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection