Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-12T12:27:38Z No. of bitstreams: 1 arquivototal.pdf: 987629 bytes, checksum: 073f4a58e014f0b46588edf144d4a42b (MD5) / Made available in DSpace on 2017-09-12T12:27:38Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 987629 bytes, checksum: 073f4a58e014f0b46588edf144d4a42b (MD5) Previous issue date: 2016-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we work with the Horava-Lifshitz-like Gross-Neveu model in (2+1) dimensions in the Large N expansion. Firstly we make an article revision [6] where it is shown that the Gross-Neveu Model in the 1/N expansion presents a dynamic mass generation by means of the introduction of an auxiliary field, which results in the dynamical parity broken. We calculate the gap equation where we will see the generated mass dependence with the coupling constant. After that, we will put a gauge field to the model and study the polarization tensor which will generate an induced Chern-Simons term in the Effective Lagrangian. As a novelty, we work with the Gross-Neveu Model in the context of Horava-Lifshitz, where anisotropic scaling is done, thus breaking the Lorentz invariance. We introduce an auxiliary field and we study the cases which the value of the critical dynamic exponent Z is even and when it is odd. In the case where z is even, there is no dynamic mass generation so the parity symmetry is conserved and we will not have the term induced of Chern-Simons either. In the case where z is odd, we will have the dynamic mass generation and the dynamic parity symmetry will occur. Finally we couple a gauge field in the model and find the Chern-Simons term, which clearly shows the anisotropy of space and time for values of z> 1 / Nesta dissertacao trabalhamos corn o modelo de Gross-Neveu ern (2+1) dimensoes na expansao 1/N no contexto de Horava-Lifshitz. Primeiro, faremos uma revisao do artigo [6], onde se mostra que o Modelo de Gross-Neveu na expansao 1/N apresenta uma geracao dinamica de massa mediante a introducao de urn campo auxiliar, o que traz como consequencia a quebra dinamica da simetria de paridade. Calculamos a equacao de gap, onde veremos a dependencia da massa gerada corn a constante de acoplamento. ApOs isso, acoplaremos urn campo de gauge ao modelo, estudamos o tensor de polarizacao, o qual vai gerar urn termo induzido de tipo Chern-Simons na lagrangiana efetiva. Como novidade, trabalhamos corn o Modelo de Gross-Neveu no contexto de Horava-Lifshitz, onde se faz urn escalonamento anisotrOpico, quebrando, assim, a invariancia de Lorentz. Introduzimos urn campo auxiliar e estudamos os casos ern que o valor do exponente dinamico critico z é par quando é Impar. No caso ern que z é par, nao ha geracao dinamica de massa pelo que a simetria de paridade é conservada e tambern nao teremos o termo induzido de Chern-Simons. No caso ern que z é impar, vamos ter a geracao dinamica de massa vai ocorrer a quebra dinamica de simetria de paridade. Finalmente, acoplamos urn campo de gauge no modelo e encontramos o termo tipo Chern-Simons, o qual mostra claramente a anisotropia do espaco tempo para valores de z > 1.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.