Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function. Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so. Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions.

Chan Lai-Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 83-84). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Table of Contents --- p.iv / List of Abbreviations and symbols --- p.viii / List of Appendices --- p.x / List of Figures --- p.xi / List of Tables --- p.xii / Chapter Chapter 1: --- General Introduction --- p.1 / Chapter 1.1 --- Studies of Plant Constituents --- p.1 / Chapter 1.2 --- Extraction of Chemical Ingredients --- p.2 / Chapter 1.3 --- Separation and Purification of Chemical Ingredients --- p.2 / Chapter 1.3.1 --- Thin Layer Chromatography (TLC) --- p.3 / Chapter 1.3.2 --- Open Column Chromatography (OCC) --- p.4 / Chapter 1.4 --- Methods of Structural Identification --- p.4 / Chapter 1.4.1 --- Nuclear Magnetic Resonance (NMR) --- p.5 / Chapter 1.4.1.1 --- Proton NMR Spectroscopy (1H-NMR) --- p.5 / Chapter 1.4.1.2 --- Carbon-13 NMR Spectroscopy (13C NMR) --- p.5 / Chapter 1.4.1.3 --- Distortionless Enhancement by Polarization Transfer (DEPT) Experiment --- p.6 / Chapter 1.4.1.4 --- Two-dimensional Homonuclear Correlation Experiment --- p.6 / Chapter 1.4.1.5 --- Two-dimensional Heternuclear Chemical shift Correlation Experiment --- p.7 / Chapter 1.4.2 --- Mass Spectroscopy --- p.7 / Chapter 1.5 --- Literature Review --- p.8 / Chapter 1.5.1 --- Medical information on the genus Zanthoxylum --- p.8 / Chapter 1.5.2 --- Compounds found in the genus Zanthoxylum --- p.8 / Chapter 1.5.3 --- Compounds found in the Zanthoxylum acanthopodium --- p.9 / Chapter 1.5.4 --- Biological activities of the genus Zanthoxylum --- p.9 / Chapter 1.6 --- Research plan and Objectives of the present study --- p.16 / Chapter 1.7 --- References --- p.17 / Chapter Chapter 2: --- Materials and Methods --- p.20 / Chapter 2.1 --- General experimental procedures --- p.20 / Chapter 2.1.1 --- Solvents --- p.20 / Chapter 2.1.2 --- Chromatographic methods --- p.20 / Chapter 2.1.2.1 --- Normal phase chromatography --- p.20 / Chapter 2.1.2.2 --- Macro-reticular resinous adsorption chromatography --- p.20 / Chapter 2.1.3.3 --- Normal phase preparative thin layer chromatography --- p.21 / Chapter 2.1.3.4 --- Thin layer chromatography --- p.21 / Chapter 2.1.3 --- Determination of physical data --- p.21 / Chapter 2.1.3.1 --- Melting point determination --- p.21 / Chapter 2.1.3.2 --- Optical Rotation (OR) --- p.21 / Chapter 2.1.3.3 --- Ultraviolet (UV) absorption spectra --- p.22 / Chapter 2.1.3.4 --- Infra-red (IR) absorption spectra --- p.22 / Chapter 2.1.3.5 --- Nuclear Magnetic Resonance spectra (NMR) --- p.22 / Chapter 2.1.3.6 --- Mass Spectra (MS) --- p.22 / Chapter Chapter 3: --- Results / Chapter 3.1 --- "Procurement, extraction and initial fractionation of dried leaves and stem bark of Zahthoxylum acanthopodium" --- p.23 / Chapter 3.2 --- Chromatographic separation of the hexane and ethyl acetate extract --- p.24 / Chapter 3.2.1 --- Column chromatographic separation of fraction HEA-E --- p.24 / Chapter 3.2.1.1 --- Characterization of β -sitosterol --- p.24 / Chapter 3.2.2 --- Column chromatographic separation of fraction HEA-F --- p.25 / Chapter 3.2.2.1 --- Characterization of β -amyrin --- p.25 / Chapter 3.2.2.2 --- Characterization of 1 - octacosanol --- p.26 / Chapter 3.2.2.3 --- Characterization of (-)-sesamin --- p.26 / Chapter 3.2.3 --- Column chromatographic separation of fraction HEA-G --- p.27 / Chapter 3.2.3.1 --- Characterization of methyl hexacosanoate --- p.27 / Chapter 3.2.3.2 --- Characterization of methyl pluviatilol --- p.28 / Chapter 3.2.4 --- Column chromatographic separation of fraction HEA-H --- p.28 / Chapter 3.2.4.1 --- Characterization of (+)-methyl piperitol --- p.29 / Chapter 3.2.4.2 --- Characterization of (-)-epieudesmin --- p.29 / Chapter 3.2.5 --- Column chromatographic separation of fraction HEA-K --- p.29 / Chapter 3.2.5.1 --- Characterization of (+)-syringaresinol --- p.30 / Chapter 3.2.6 --- Column chromatographic separation of fraction HEA-M --- p.30 / Chapter 3.2.6.1 --- Characterization of daucosterol 6'-stearate --- p.31 / Chapter 3.2.7 --- Column chromatographic separation of fraction HEA-P --- p.31 / Chapter 3.2.7.1 --- Characterization of β -sitosterol- β -D-glucoside --- p.31 / Chapter 3.3 --- Chromatographic separation of the n-butanol extract --- p.32 / Chapter 3.3.1 --- Column chromatographic separation of fraction Bu-I --- p.32 / Chapter 3.3.1.1 --- Characterization of collettiside III --- p.32 / Chapter 3.3.1.2 --- Characterization of gracilline --- p.33 / Chapter Chapter 4: --- Discussion --- p.35 / Chapter 4.1 --- Isolated compounds --- p.35 / Chapter 4.1.1 --- Sterols --- p.35 / Chapter 4.1.1.1 --- Identification of -sitosterol --- p.36 / Chapter 4.1.1.2 --- Identification of β -sitosterol- β -D-glucoside --- p.41 / Chapter 4.1.1.3 --- Identification of daucosterol 6'-stearate --- p.45 / Chapter 4.1.2 --- Saponins --- p.50 / Chapter 4.1.2.1 --- Steroid glycosides --- p.51 / Chapter 4.1.2.1.1 --- Identification of collettiside III --- p.51 / Chapter 4.1.2.1.2 --- Identification of gracilline --- p.55 / Chapter 4.1.3 --- Terpenoid --- p.60 / Chapter 4.1.3.1 --- Identification of β -amyrin --- p.61 / Chapter 4.1.4 --- Lignans --- p.66 / Chapter 4.1.4.1 --- Identification of (-)-sesamin --- p.68 / Chapter 4.1.4.2 --- Identification of (-)-epieudesmin --- p.69 / Chapter 4.1.4.3 --- Identification of (+)-methyl piperitol --- p.70 / Chapter 4.1.4.4 --- Identification of methyl pluviatilol --- p.71 / Chapter 4.1.4.5 --- Identification of (+)-syringaresinol --- p.72 / Chapter 4.1.5 --- Alkanol --- p.78 / Chapter 4.1.5.1 --- Identification of 1 -octacosanol --- p.78 / Chapter 4.1.5.2 --- Identification of methyl hexacosanoate --- p.79 / Chapter 4.2 --- Conclusion --- p.81 / Chapter 4.3 --- References --- p.83

This thesis is devoted to the study of multiphase jet fragmentation using Smoothed Particle Hydrodynamics (SPH). The theoretical aspects of three hydrodynamic instabilities, namely the Kelvin-Helmholtz instability (KHI), Rayleigh-Taylor instability (RTI), and Rayleigh Plateau instability (RPI) are reviewed. The linear growth rate of the combined KHI and RTI are derived by means of linear perturbation in chapter 2. The linear growth rate of the multiphase RPI is presented in chapter 7. An overview of the Smoothed Particle Hydrodynamics is given in chapter 3. A pseudo-consistent SPH scheme is presented for the simulation of multiphase flow problems. Additionally, two interface stabilisation models are presented: quasi-buoyancy model and gas-repulsion model. When used in combination with the pseudo-consistent SPH scheme, these models are found to be superior than those presented in the weakly-compressible SPH literature and allows for the simulations for density ratio up to three-magnitudes. The development of an idealised KHI and a KHI subjected to constant gravitational acceleration (stratified shear instability) is examined in chapter 5. The extracted linear growth rate are compared with the theoretical growth rate presented both in the literature and in chapter 2 for the purpose of validation. The development of a single- and multi-mode RTI are studied by means of SPH in chapter 6. Chapter 7 presents the results for the three-dimensional RPI occurring between two fluids. Based on the knowledge acquired in chapter 5-7, the multiphase jet fragmentation driven by the previously mentioned hydrodynamic instabilities are presented in chapter 8. Finally, the major research findings and recommendations are summarised in chapter 9.

Thesis (M.S.)--Brigham Young University. Dept. of Mechanical Engineering, 2006. / Includes bibliographical references (p. 35-36).

We show that a 3-D microwave crossing waveguide can be solved by a 2-D scalar Helmholtz equation with combining boundary conditions for TE and TM modes. Furthermore the crossing waveguide possesses a symmetry along two diagonal axes passing through the origin. Computation of the EM wave fields is decomposed into four smaller tasks of computing reflection coefficient vector of a parallel plate waveguide terminated with a corner made of two perfectly electric or magnetic conducting walls (PECW/PMCW). In this thesis, we propose a mixed Cartesian and polar coordinate mode-matching method to solve this 2-D corner cube microwave reflection problem. The solution is obtained by applying the continuity condition of both the tangential field and its normal derivative along a given curve inside the overlapped region of the two coordinate systems. We are able to compute up to the third decimal place of the reflection, through and cross transmission coefficients. All results pass the energy conservation test and are verified and compared with those computed by Integral equation method simulation.

Thesis (doctoral)--Universität Jena, 1911. / Vita.

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1910. / Vita.

Thesis (doctoral)--Universität Bern, 1911.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1901. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Columbia University, 1922. / Vita.