In this thesis, we study non-relativistic scalar fields in (3+1) space-time subjected to an inverse-square potential. We use a point-particle effective field theory (PPEFT) framework to describe the scalar fields coupling to a point-particle in different cases of interest. In Chapter 3, we encode particle conversion for both Schrodinger and Klein-Gordon fields in a two-species toy model and find that the point-particle couplings all must be renormalized with respect to the radial cut-off near the origin. In addition to this, we find that cross sections have an interesting dependence on the ratio k_{out}/k_{in} of outgoing and incoming momenta. In certain regimes at low energies, we found inelastic behaviour \sigma_{S}^{(in)} ~ O(1) and \sigma_{KG}^{(in)} ~ 1/k_{in} for Schrodinger and Klein-Gordon fields respectively. In Chapter 4, we study the case of a single-particle non-self-adjoint PPEFT whose formulation is taken to next-to-leading order. We find that the point-particle couplings continue to require renormalization and present a series of relevant computations such as field equations, boundary conditions and renormalization runnings, concluding with an exposition of bound state energies, scattering lengths and cross sections. Similar to what was found in Chapter 3, a 1/k_{\text{in}} enhancement is observed in a particular regime of the PPEFT in Chapter 4. In addition, we find that the observables computed therein are modified from what was found in other papers [3][13], where only the leading PPEFT term was kept. These results may provide relevance for future calculations in more complex reactions such as baryon number violation and monopole catalysis. / Thesis / Master of Science (MSc)

We consider the Cauchy problem for the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of the kinetic energy and potential energy, is conserved for all time. Due to the focusing nature, solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or even have mixed behaviors. Our goal in this thesis is to fully characterize the solution when the energy is below or at the level of the energy of the ground state solution $W_a$. Our main result contains two parts. First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters. Second, we show a rigidity result at the level of ground state solution. We prove that among all solutions with the same energy as the ground state solution, there are only two (up to symmetries) solutions $W_a^+, W_a^-$ that are exponential close to $W_a$ and serve as the threshold of scattering and blow-up. All solutions with the same energy will blow up both forward and backward in time if they go beyond the upper threshold $W_a^+$; all solutions with the same energy will scatter both forward and backward in time if they fall below the lower threshold $W_a^-$. In the case of NLS with no potential, this type of results was first obtained by Kenig-Merle \cite{R: Kenig focusing} and Duyckaerts-Merle \cite{R: D Merle}. However, as the potential has the same scaling as $\Delta$, one can not expect to extend their results in a simple perturbative way. We develop crucial spectral estimates for the operator $-\Delta+a/|x|^2$, we also rely heavily on the recent understanding of the operator $-\Delta+a/|x|^2$ in \cite{R: Harmonic inverse KMVZZ}.

Surface-generalized ambient noise in a shallow ocean waveguide with a sediment layer possessing a specific class of density and sound speed distributions capable of describing a realistic seabed environment is considered in this analysis. This class of non-uniform sediment layer has the density and sound speed distributions varying with respect to depth as a genearlized-exponential and an inverse-square function, respectively. The study invokes a formulation developed by Kuperman and Ingenito for surface noise generation, in conjunction with the analytical solutions for the Helmholtz equation corresponding to the sediment layer, to arrive at an analytical expression convenient for numerical implementation. The intensity and spatial correlation of the noise sound field are analyzed with respect to the variations of the system parameters, including frequency, sediment layer thickness, sound speed gradient, with emphasis on the effects of sediment properties on the ambient noise field. The results have demonstrated that the intensity of the noise field is relatively sensitive to the variations of the paramters, while the spatial correlation is not, suggesting that the energy distribution, rather than the spatial structure, of the noise field is susceptible to the environmental variation.

The objective of this thesis is to develop and implement an algorithm for inverting the sound speed profile via estimation of the parameters embedded in a geoacoustic model. The environmental model inscribes a continuously-varying marine sediment layer with density and sound speed distributions represented by the generalized-exponential and inverse-square functions, respectively. Based upon a forward problem of plane-wave reflection from a non-uniform sediment layer overlying a uniform elastic basement, an inversion procedure for estimating the sound speed profile from the reflected sound field under the influence of noise is established and numerically implemented. The inversion invokes a probabilistic approach quantified by the posterior probability density for measuring the uncertainties of the estimated parameters from synthetic noisy data. Preliminary analysis on the solution of the forward problem and the sensitivity of the model parameters is first conducted, leading to a determination of the parameters chosen for inversion in the ensuing study. The parameter uncertainties referenced 1-D and 2-D marginal posterior probability densities are then examined, followed by the statistical estimation for the sound speed profile in terms of 99 % credibility interval. The effects of, the signal-to-noise ratio (SNR), the dimension of data vector, the region in which the data sampled, on the statistical estimation of sound speed profile are demonstrated and discussed.

Acoustic plane wave intearctions with a rough seabed with a continuously varying density and sound speed in a fluid-like sediment layer overlying an elastic basement is considered in this thesis. The acoustic properties in the sediment layer possess an exponential type of variation in density and one of the three classes of sound speed profiles, which are constant, k^2-linear, or inverse-square variations. Analytical solutions for the Helmholtz equation in the sediment layer, combined with a formulation based upon boundary perturbation theory, facilitate numerical implementation for the solution of coherent field. The coherent reflection coefficients corresponding to the aformentioned density and sound speed profiles for various frequencies, roughness parameters, basement stiffness, are numerically generated and analyzed. Physical interpretations are provided for various results. This simple model characterizes three important features of an realistic sea floor, including seabed roughness, sediment inhomogenieties, and basement shear property,%Two dimensions is considered in the seafloor environment and the random roughness is belong to one dimension space.% , therefore, provides a canonical model for the study of seabed acoustics. The variation of the acoustic properties takes such a form that it is not only geologically realistic, but also renders analytical solutions for the Helmholtz equation, thus facilitating the formulation of the problem. The computational algorithm for the spatial spectrum of the scattered field due to random seabed has been developed based upon a boundary perturbation method. %About scattering field, only one time reflection from the sediment is taked account of, because the higher numerical order is, the lower scattering energy exist.% The results have shown that, while the coherent field mainly depends upon the gross structure of the rough seabed represented by the RMS roughness, the scattered field heavily depends upon the details of the roughness structure specialized by the roughness power spectrum and the spatial correlation length of the rough surface. The dependence of the spatial spectrum on the sediment stratification is also carefully examined.

Submitted by Neusa Fagundes (neusa.fagundes@unioeste.br) on 2017-09-05T12:35:02Z No. of bitstreams: 1 Jianice_Frigo2017.pdf: 1119086 bytes, checksum: 87f159fe89d2840c9e6c7be155e2a65c (MD5) / Made available in DSpace on 2017-09-05T12:35:02Z (GMT). No. of bitstreams: 1 Jianice_Frigo2017.pdf: 1119086 bytes, checksum: 87f159fe89d2840c9e6c7be155e2a65c (MD5) Previous issue date: 2017-04-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The infiltration process understanding and its relationship with soil properties is essential for the efficient management of soil and water. Thus, this paper aimed at evaluating water infiltration in an area of Oxisol, using single ring infiltrometers with different diameters. In addition, it also aimed at comparing the performance of 7 cm (A7) and 15 cm (A15) cylinders, matching soil moisture and bulk density to water infiltration and evaluating spatial distribution of infiltration. For this purpose, 24 data points were evaluated, placed in 8 plots of a circular area prepared for permaculture cropping, named as Mandala. Shewhart graphs and process capability index were used to compare cylinders and the inverse of distance square for data interpolation. It was observed that the average stable infiltration rate, determined with A7 cylinder, was 196.79 mm h-1 and with A15 cylinder was 187.30 mm h-1, consequently, there was a difference of 5.6%. The process capacity was estimated at 0.61 for A7 and 1.62 for A15, considered adequate for A15 and inadequate for A7 cylinder. Thus, it was concluded that the infiltration rates that were stabilized and determined by the cylinders were compatible with the rates determined by other methods in other papers. Regarding the performance of single-ring infiltrometers, it is concluded that, although both cylinders presented controlled processes, and did not differ statistically, cylinder A15 presented less variability on data, and may be considered adequate due to its process capability index. / O entendimento do processo de infiltração e das suas relações com as propriedades do solo é fundamental para o eficiente manejo do solo e da água. Assim, este trabalho teve como objetivo avaliar a infiltração de água em uma área de latossolo vermelho, para tanto, foram utilizados infiltrômetros de único cilindro, de diferentes diâmetros. Além disso, objetivou-se comparar o desempenho desses cilindros de 7 (A7) e de 15 cm (A15), relacionar a umidade e a densidade aparente do solo à infiltração da água e avaliar a distribuição espacial da infiltração. Foram avaliados 24 pontos de coleta de dados, alocados em 8 parcelas de um talhão circular preparado para plantio em regime de permacultura, denominado mandala. Gráficos de Shewhart e o índice de capacidade de processo foram utilizados para a comparação entre os cilindros e inverso do quadrado da distância para interpolação de dados. Verificou-se que a taxa de infiltração estável média, determinada com o cilindro A7, foi de 196,79 mm h-1 e com o cilindro A15 foi de 187,30 mm h-1, cuja diferença foi de 5,6 %. A capacidade de processo foi estimada em 0,61 para A7 e 1,62 para A15, a qual foi considerada adequada para o cilindro A15, porém inadequada para cilindro A7. Desta forma, neste trabalho, concluiu-se que as taxas de infiltração estáveis e determinadas pelos cilindros foram compatíveis com as taxas determinadas por outros métodos em outros trabalhos. Em relação ao desempenho dos infiltrômetros de cilindros únicos, conclui-se que, embora ambos os cilindros tenham apresentado processos controlados e não tenham diferido estatisticamente, o cilindro A15 apresentou menos variabilidade dos dados, e pode ser considerado adequado em função do seu índice de capacidade de processo.