In this dissertation, we devise a quantization scheme to construct eigenfunctions by classical periodic orbits in both regular systems as well as chaotic systems. Our method is based on the principle that eigenfunctions can be resolved from a time-dependent wavefunction. This is different from the classical (or EBK) quantization scheme that constructs eigenfunction in the energy-domain. The advantage of our method is that it can be applied to more varieties of systems, including some chaotic systems. Three systems, the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator, are used as examples for our eigenfunction construction. The key to the constructions is a family (or families) of periodic orbits with a newly defined quantization rule, the resolving quantization rule. The eigenspectrum for the x⁴-potential oscillator is also computed. Furthermore, the classical Green's function is used to explain the relation between the resolving quantization rule and the classical quantization rule. This dissertation begins with an introduction in Chapter 1. The semiclassical theory for the eigenfunction construction by periodic orbits is developed in Chapter 2. In Chapter 3 and Chapter 4, eigenfunctions are constructed for the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator. The eigenspectrum for the x⁴-potential oscillator is computed in Chapter 5. Chapter 6 is devoted to discussions including the interpretation of the resolving quantization rule from the classical Green's function, the interpretation of the photoabsorption spectrum for a Rydberg atom in a magnetic field, and the comparison of our method with the EBK quantization scheme. Conclusions are made in Chapter 7. / text

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Spatial interaction models of the gravity type are widely used to model origindestination flows. They draw attention to three types of variables to explain variation in spatial interactions across geographic space: variables that characterise an origin region of a flow, variables that characterise a destination region of a flow, and finally variables that measure the separation between origin and destination regions. This paper outlines and compares two approaches, the spatial econometric and the eigenfunction-based spatial filtering approach, to deal with the issue of spatial autocorrelation among flow residuals. An example using patent citation data that capture knowledge flows across 112 European regions serves to illustrate the application and the comparison of the two approaches.(authors' abstract)

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The main topic of this research is the simulation of internal wave interaction by a two-dimensional numerical model developed by Lynett & Liu (2002) of Cornell University, then modified by Cheng et al. (2005). The governing equation includes two-dimensional momentum and continuity equation. The model uses constant upper and lower layer densities; hence, these factors as well as the upper layer thickness. Should be determined before the simulation. This study discusses the interface depth and the density according to the buoyancy frequency distribution, the EOF, and the eigen-value based on the measured density profile. Besides, a method based on the two-layer KdV equation and the KdV of continuously-stratified fluid. By minimize the difference of linear celeriy, nonlinear and dispersion terms, the upper layer thicknes can also be determined. However, the interface will be much deeper than the depth of max temperature drop in the KdV method if the total water depth is bigger than 500 meters. Thus, the idealization buoyancy frequency formula proposed by Vlasenko et al. (2005) or Xie et al. (2010) are used to modify the buoyancy frequency. The internal wave in the Luzon Strait and the South China Sea are famous and deserves detailed study. We use the KdV method to find the parameters in the two fluid model to speed up the simulation of internal wave phenomena found in the satellite image.

Because of the accuracy and ease of implementation, the Monte Carlo methodology is widely used in the analysis of nuclear systems. The estimated effective multiplication factor (keff) and flux distribution are statistical by their natures. In eigenvalue problems, however, neutron histories are not independent but are correlated through subsequent generations. Therefore, it is necessary to ensure that only the converged data are used for further analysis. Discarding a larger amount of initial histories would reduce the risk of contaminating the results by non-converged data, but increase the computational expense. This issue is amplified for large nuclear systems with slow convergence. One solution would be to use the convergence of keff or the flux distribution as the criterion for initiating accumulation of data. Although several approaches have been developed aimed at identifying convergence, these methods are not always reliable, especially for slow converging problems. This dissertation has attacked this difficulty by developing two independent but related methodologies. One aims to find a more reliable and robust way to assess convergence by statistically analyzing the local flux change. The other forms a basis to increase the convergence rate and thus reduce the computational expense. Eventually, these two topics will contribute to the ultimate goal of improving the reliability and efficiency of the Monte Carlo criticality calculations.

The Laplace transform analytic element method (LT-AEM), applies the traditionally steady-state analytic element method (AEM) to the Laplace-transformed diffusion equation (Furman and Neuman, 2003). This strategy preserves the accuracy and elegance of the AEM while extending the method to transient phenomena. The approach taken here utilizes eigenfunction expansion to derive analytic solutions to the modified Helmholtz equation, then back-transforms the LT-AEM results with a numerical inverse Laplace transform algorithm. The two-dimensional elements derived here include the point, circle, line segment, ellipse, and infinite line, corresponding to polar, elliptical and Cartesian coordinates. Each element is derived for the simplest useful case, an impulse response due to a confined, transient, single-aquifer source. The extension of these elements to include effects due to leaky, unconfined, multi-aquifer, wellbore storage, and inertia is shown for a few simple elements (point and line), with ready extension to other elements. General temporal behavior is achieved using convolution between these impulse and general time functions; convolution allows the spatial and temporal components of an element to be handled independently.Comparisons are made between inverse Laplace transform algorithms; the accelerated Fourier series approach of de Hoog et al. (1982) is found to be the most appropriate for LT-AEM applications. An application and synthetic examples are shown for several illustrative forward and parameter estimation simulations to illustrate LT-AEM capabilities. Extension of LT-AEM to three-dimensional flow and non-linear infiltration are discussed.

An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differential equations. A Fourier-Bessel expansion of the solution vector in terms of Bessel functions with real orders is employed, resulting in a generalized matrix eigenvalue problem. The method is applied to two particular examples. The first example is a prolate spheroidal billiard which is also treated by using an alternative method. The numerical results obtained by using both the methods are compared. The second exampleis a billiard family depending on a parameter. Numerical results concerning the second example include the statistical analysis of the eigenvalues.

The inverse eigenvalue problem for a given operator is to determine the coefficients by using knowledge of its eigenfunctions and eigenvalues. These are determined by the behavior of the solutions on the domain boundaries. In our problem, the Schrödinger operator acting on functions defined on the unit ball of $\mathbb{R}^3$ has a radial potential taken from $L^2_{\mathbb{R}}[0,1].$ Hence the set of the eigenvalues of this problem is the union of the eigenvalues of infinitely many Sturm-Liouville operators on $[0,1]$ with the Dirichlet boundary conditions. Each Sturm-Liouville operator corresponds to an angular momentum $l =0,1,2....$. In this research we focus on the uniqueness property. This is, if two potentials $p,q \in L^2_{\mathbb{R}}[0,1]$ have the same set of eigenvalues then $p=q.$ An early result of P\"oschel and Trubowitz is that the uniqueness of the potential holds when the potentials are restricted to the subspace of the even functions of $L_{\mathbb{R}}^2[0,1]$ in the $l=0$ case. Similarly when $l=0$, by using their method we proved that two potentials $p,q \in L^2_{\mathbb{R}}[0,1]$ are equal if their even extension on $[-1,1]$ have the same eigenvalues. Also we expect to prove the uniqueness if $p$ and $q$ have the same eigenvalues for finitely many $l.$ For this idea we handle the problem by focusing on some geometric properties of the isospectral sets and trying to use these properties to prove the uniqueness of the radial potential by using finitely many of the angular momentum.

Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.