A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. August 2015. / This work investigates the impact of the extragalactic background light fluctuations on very high energy !-ray spectra from distant blazars. We calculate the extragalactic background light spectral energy distribution using a model that extends those proposed by Razzaque et al. (2009ApJ.697.483R) and Finke et al. (2010ApJ.712.238F). We introduce a model for fluctuations in the extragalactic background light based on fluctuations in the star formation rate density, since these two fluctuations can reasonably be expected to be correlated. Fluctuations in the star formation rate are estimated from the semi-analytical galaxy catalogue of Guo et al. (2013MNRAS.428.1351G), we use his model to derive the resulting opacities for !-rays from distant sources. We determine the mean, lower and upper limits for the scatter of the star formation rate density, which then allow us to compute corresponding limits on the extragalactic background light spectrum. We then calculate the impact of these fluctuations limits on the !-ray optical depth. This appears to be the first detailed analytical model that aims to account for the impact of extragalactic background light fluctuations on the !-ray opacity. The model predicts relatively high variations ( 15%) on the opacity in the energy range less than 100 GeV for nearby sources. The impact is found to be smaller (⇠ 5%) for very high energy !-rays from distant sources.

A Thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg for the Degree of Doctor of Philosophy. Johannesburg, 1979

Demonstramos a Lei de Weyl sobre o comportamento assintótico dos autovalores do operador Laplaciano com condições de contorno de Dirichlet em domínios limitados e suaves com o auxílio do núcleo do calor. Para isso, fazemos um estudo dos operadores não-limitados, semigrupos e da transformada de Fourier. Por fim, expomos alguns resultados posteriores motivados pelo artigo de Mark Kac \"Can one hear the shape of a drum?\". / We prove the Weyl Law on the asymptotic behavior of eigenvalues of the Laplace operator with Dirichlet boundary conditions in smooth bounded domains with the help of the heat kernel. To that end, we study unbounded operators, semigroups and the Fourier transform. Lastly, we mention some further results motivated by Mark Kac\'s article \"Can one hear the shape of a drum?\".

Através da técnica do Grupo de Renormalização desenvolvida por Wilson em 1915, calcula-se o espectro de fotoemissão de elétrons (XPS) de um sistema constituído de dois níveis profundos separados pela distância R, acoplados entre si e adsorvidos num metal. A simetria de inversão desse sistema permite que se definam dois espectros de correntes, dos elétrons fotoemitidos do orbitalligante e do anti-ligante do adsorvido, respectivamente. Próximo ao limiar de fotoemissão (ωT) esses espectros seguem lei de potência; os expoentes correspondentes são determinados e expressos (com leis universais) em termos das defasagens dos elétrons de condução. Se a separação R entre os sítios for nula, o espectro associado ao orbital ligante terá uma singularidade em ω - ωT = Δ, onde Δ é a taxa de tunelamento eletrônico entre os níveis profundos. Com o aumento dessa separação a singularidade é arrendondada em razão de decaimentos não radiativos envolvendo os orbitais profundos, o que dá um tempo de vida finito ao buraco criado pelo raio-x. Este trabalho calcula pela primeira vez a renormalização da taxa de tunelamento, devido aos processos não radiativos. / The Renormalization Group technique developed by Wilson in 1975 is used to calculate photoemission spectra (XPS) for a dimer adsorbed on a metal; our model for this system comprises two deep levels, separated by a distance R, coupled to each other, and interacting electrostatically with the sorbate. The inversion symmetry is used to define two electronics currents, coming from the bonding and anti-bonding orbitals of the dimer, respectively. Near threshold each spectrum follow a power law; the exponents are calculated and expressed (by universal laws) in terms of the phase shifts of the conduction electrons. If the distance R between the levels is zero, the spectrum associated to the bonding orbital has a singularity at ω - ωT = Δ, where Δ is the electronic tunneling rate between the deep levels. For increasing R, the singularity is smoothed out, due to the non-radiative decay of an electron from the anti-bonding to the bonding orbital, which makes finite the lifetime of the hole created by the x-ray. This work presents the first accurate calculation of the renormalization of the tunneling rate by this non-radiative decay.

As sequências espectrais foram criadas por Jean Leray num campo de concentração durante a Segunda Guerra Mundial motivado por problemas inerentes à Topologia Algébrica. Num primeiro momento, surge como uma ferramenta para auxiliar no cálculo da cohomologia de um feixe. Porém, Jean-Louis Koszul apresenta uma formulação puramente algébrica para tais sequencias, que consiste basicamente no cálculo da homologia de um complexo total associado a um complexo duplo. Concentraremos nosso trabalho nas definições e resultados que nos permitem demonstrar os seguintes resultados conhecidos da Álgebra usando sequências espectrais: o Lema dos Cinco, o Lema da Serpente, Balanceamento para o Funtor Tor, Mudança de Base para o Funtor Tor e o Teorema dos Coeficientes Universais. Apresentamos, ao final do trabalho, uma generalização que nos permite entender melhor os funtores derivados à esquerda: as Sequências Espectrais de Grothendieck. / Spectral sequences were created by Jean Leray in a concentration camp during World War II motivated by problems of Algebraic Topology. At first, it appears as a tool to assist in calculating the cohomology of a sheaf. However, Jean-Louis Koszul presents a purely algebraic formulation for these sequences, which basically consists in calculating a total of homology complex associated with a double complex. We will focus our work on the definitions and results that allow us to demonstrate known results of algebra using spectral sequences: The Five Lemma, The Snake Lemma, Balancing of functor Tor, Base Change for Tor and Universal Coefficient Theorem. We present, at the end of this work, a generalization that allows us to better understand the left derivative functors: the Spectral Sequence of Grothendieck.

A dinâmica em meso-escala nos oceanos globais é dominada por sinais propagantes para oeste. Estudos pioneiros a utilizar medidas de altimetria por satélites associaram este sinal a ondas de Rossby longas do primeiro modo baroclínico. Com o recente aumento de resolução nos dados altimétricos, estudos mais recentes sugerem que vórtices de meso-escala não-lineares seriam os principais responsáveis pelo sinal propagante observado, em detrimento às ondas de Rossby. O objetivo do presente trabalho é identificar estruturas coerentes associadas a vórtices de meso-escala e distingui-las das ondas de Rossby longas do primeiro modo baroclínico. Mapas de anomalia da altura da superfície do mar (η) foram filtrados através da análise de ondaletas e um algoritmo de identificação e extração de estruturas vorticais. Os vórtices extraídos foram caracterizados através do ajuste de um paraboloide elíptico. O algoritmo demonstrou-se capaz de identificar e extrair as estruturas associadas a vórtices de meso-escala. Os resultados indicam predominância de anéis vorticais anti-ciclônicos. Os espectros de potência zonais-temporais de η indicam que a maior parte da variância distribui-se na região não-dispersiva do espectro teórico de ondas de Rossby lineares. A propagação observada nos componentes resultantes do filtro indicam coexistência de ondas de Rossby lineares e vórtices de meso-escala, comprovando assim a hipótese científica estabelecida para este trabalho. / The mesoscale dynamics in the global oceans is dominated by westward propagating signals. Pioneering studies using satellite altimetry measurements associated these observations with first-mode baroclinic Rossby waves. With the increase in altimetry data resolution, recent studies suggest that nonlinear mesoscale eddies are responsible for the westward propagating signal rather than Rossby waves. The objective of this study is to identify coherent structures associated with mesoscale eddies and distinguish them from long first-mode baroclinic Rossby waves. Sea surface height anomaly maps (η) were filtered throught wavelet analysis and an algorithm for identifying and extracting vortical structures. The extracted vortices were characterized by adjusting an elliptic paraboloid. The algorithm proved to be able to identify and extract the structures associated to mesoscale eddies. The results indicate a predominance of anti-cyclonic rings. Zonal-temporal power spectral density of η indicate that most of the variance is located at the non-dispersive region of the theoretical linear Rossby wave spectra. The observed propagation of the filtered components indicate coexistence of linear Rossby waves and mesoscale eddies, thus proving the scientific hypothesis of this study.

by He Zhaokui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 81-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- The Laplacian of a compact manifold --- p.8 / Chapter 2.2 --- The Laplacian of a graph --- p.9 / Chapter 2.3 --- Some basic facts about the eigenvalues of a graph --- p.13 / Chapter 3 --- Bound of the first non-zero eigenvalue in terms of Cheeger constant --- p.18 / Chapter 3.1 --- The Cheeger constant --- p.18 / Chapter 3.2 --- The Cheeger inequality of a compact manifold --- p.19 / Chapter 3.3 --- The Cheeger inequality of a graph --- p.23 / Chapter 4 --- Diameters and eigenvalues --- p.27 / Chapter 4.1 --- Some facts --- p.27 / Chapter 4.2 --- Estimate the eigenvalues of graphs --- p.29 / Chapter 4.3 --- The heat kernel of compact manifolds --- p.34 / Chapter 4.4 --- Estimate the eigenvalues of manifolds --- p.35 / Chapter 5 --- Harnack inequality and eigenvalues on homogeneous graphs --- p.40 / Chapter 5.1 --- Preliminaries --- p.40 / Chapter 5.2 --- The Neumann eigenvalue of a subgraph --- p.41 / Chapter 5.3 --- The Harnack inequality --- p.44 / Chapter 5.4 --- A lower bound of the first non-zero eigenvalue --- p.52 / Chapter 6 --- Harnack inequality and eigenvalues on compact man- ifolds --- p.54 / Chapter 6.1 --- Gradient estimate --- p.54 / Chapter 6.2 --- Lower bounds for the first non-zero eigenvalue --- p.59 / Chapter 7 --- Heat kernel and eigenvalues of graphs --- p.63 / Chapter 7.1 --- The heat kernel of a graph --- p.54 / Chapter 7.2 --- Lower bounds for eigenvalues --- p.70 / Chapter 8 --- Estimate the eigenvalues of a compact manifold --- p.73 / Chapter 8.1 --- An isoperimetric constant --- p.75 / Chapter 8.2 --- A lower estimate for the (m + l)-st eigenvalue --- p.77

Interpretation of ground penetrating radar (GPR) signals can be a key point in the overall operability of a GPR system. In stepped-frequency and Frequency-Modulated Continuous-Wave (FMCW)GPR systems in particular, the target or object of interest is often located by analysis of Fast Fourier Transform (FFT) derived data. Increasing the GPR system bandwidth can improve resolution, but at the cost of reduced penetrating depth. The challenge is to develop high-resolution signal processing strategies for GPR.A number of Fourier based methods are investigated. However, the main response over a target's position can make it difficult to recognise closely spaced targets. The Least-Suare method is found to be the best autoregression-based estimator. However the method requires high Signal-to-Noise ratio to achieve high- resolution. Furthermore a number of subspace-based methods are investigated. Although the MUItiple Signal Classification (MUSIC) method can theoretically offer infinite resolution, they must be seeded with the number of targets actually present. A superimposed MUSIC technique is proposed to suppress false targets. A novel windowed MUSIC (W-MUSIC) algorithm is developed, and it offers high resolution while still able to minimise spurious responses. Since the performance of any FMCW GPR is critically linked to the linearity of the sweep frequency, the non-linearity in the target range estimation is studied. A Novel Short-Time MUSIC method is proposed and higher time and frequency resolution is achieved than the conventional Short-Time Fourier Transform method. In addition a modified Adaptive Sampling method is proposed to solve the non-linear problem by utilising a reference channel in a GPR system.

Spectral rendering, or the synthesis of images by taking into account the wavelengths of light, allows effects otherwise impossible with other methods. One of these effects is dispersion, the phenomenon that creates a rainbow when white light shines through a prism. Spectral rendering has previously remained in the realm of off-line rendering (with a few exceptions) due to the extensive computation required to keep track of individual light wavelengths. Caustics, the focusing and de-focusing of light through a refractive medium, can be interpreted as a special case of dispersion where all the wavelengths travel together. This thesis extends Adaptive Caustic Mapping, a previously proposed caustics mapping algorithm, to handle spectral dispersion. Because ACM can display caustics in real-time, it is quite amenable to be extended to handle the more general case of dispersion. A method is presented that runs in screen-space and is fast enough to display plausible dispersion phenomena in real-time at interactive frame rates.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 48-49. / by David Jay Anick. / Ph.D.