Abstract We study in this thesis functional spaces involved in crack problems in unbounded domains. These spaces are defined by closing spaces of Sobolev H1 regularity functions (or vector fields) of bounded support, by the L2 norm of the gradient. In the case of linear elasticity, the closure is done under the L2 norm of the symmetric gradient. Our main result states that smooth functions are in this closure if and only if their gradient, (respectively symmetric gradient for the elasticity case), is in L2. We provide examples of functions in these newly defined spaces that are not in L2. We show however that some limited growth in dimension 2, or some decay in dimension 3 must hold for functions in those spaces: this is due to Hardy's inequalities.

In modelling the energetics of molecules and solids, the need for practical electron density functionals that seamlessly include the van der Waals interaction is growing. Such functionals are still in their infancy, and there is yet much experimentation to be performed in the formulation and numerical testing of the requisite approximations. A ground-state density functional approach that uses the exact relations of the adiabatic connection formula and the fluctuation-dissipation theorem to obtain the xc energy from the density-density response function seems promising, though a direct local density approximation for the interacting susceptibility will fail to yield the vdW interaction. Significant nonlocality can be built into the interacting susceptibility by screening a 'bare' susceptibility, for which a carefully chosen constraint-obeying local approximation is sufficient to yield a non-trivial van der Waals energy [6]. The constraints of charge conservation, and no response to a constant potential, are guaranteed by expressing the bare susceptibility in terms of the double gradients of a nonlocal bare polarisability. for which it should be easier to make an approximation based on physical principles than it would be for the susceptibility. The 'no-flow' condition is also deemed important. In this work, a simple delta-function approximation for the nonlocal polarisability is fully constrained by a new version of a recently-discovered force theorem (sum rule), requiring the additional input of the independent-electron Kohn-Sham potential. This constrained polarisability cannot be used as input for the seamless vdW scheme, which requires a non-delta-function bare polarisability, and is instead applied to systems containing spherical fragments in a perturbative/asymptotic fashion for calculation of the widely-separated van der Waals interaction. The main thrust of this work is an investigation of the efficacy of the force theorem to constrain simple approximations for response quantities. Many recent perturbative vdW density functionals are based on response functions that are electron-hydrodynamical approximations to the response of the uniform electron gas. These schemes require their response functions to be 'cut off' at low density and high density-gradient, where the approximation overestimates the true response. The imposition of the cut-off is crucial to the success of such schemes. Here, we replace the cut-off with an exact theorem (the force theorem) which naturally 'ties down' the response, based on the potential- and density-functions of the system. This is the first time that the force theorem has been directly applied as a constraint upon a model response function (its original use, by Vignale and Kohn (7), was as an exact identity in time-dependent DFT). Also new in this work is the orbital-by-orbital Kohn-Sham version of the force theorem, and its proof (differing significantly from Vignale's original derivation (8) of the interacting theorem) by directly appealing to the Kohn-Sham orbitals makes its first appearance here. For quantum dots, our constrained response-approximation exactly recovers the net linear dipole response, due mainly to the force theorem's ideal applicability to harmonically confined systems. For angularly-averaged atoms, reasonable static dipole polarisabilities are obtained for the independent-electron Kohn-Sham (bare) case. The results are poor for the fully-interacting case, attributable to the local nature of the approximation. This lends weight to the assertion that it is better to approximate a bare quantity, then screen it, than it is to directly approximate a fully-interacting quantity. Dynamic net polarisabilities constrained by the force theorem are guaranteed to have the correct high-frequency asymptotic convergence to the free electron response. It is seen that the calculated dynamic polarisabilities for atoms are too small at intermediate frequencies, since the calculated vdW C6 coefficients (Hamaker constants) of atomic dimers are up to an order of magnitude too small, even without the use of a low-density cutoff. It is seen that our constrained local model response is non-analytic along the imaginary-frequency axis, and this is very detrimental to the C6 calculations, even though the integrated net polarisability is analytic. Improvement of the polarisability ansatz is indicated, perhaps to a non-deltafunction uniform-gas-based approximation. The use of pseudopotentials may improve the force theorem results, by softening the extreme nature of the bare Coulomb potential.

No description available.

The main focus of my dissertation is the Modified Stochastic Sine-Gordon Equation: utt = 2uxx − ut − sin(|u|^(&gamma)) + b(u, du/dt)dW/dt where &gamma > 0 is the parameter of the power of non-linearity, &delta &ge 0 is the magnitude of non-linearity, &alpha> 0 be the damping parameter, and &sigma the diffusion intensity, on one dimensional domain. We analyze the properties of the solution of the SPDE by the eigenfunctions approach allowing us to truncate the infinite-dimensional stochastic system (i.e the SDEs of Fourier coefficients related to the SPDE), to control its energy, existence, uniqueness, continuity and stability. The analysis relies on the investigation of expected Lyapunov functional of the energy in terms of all system-parameters. We simulate the model with respect to all system-parameters to visualize our conclusions.

No description available.

A partir d’un ensemble de données interprétées et issues d’une analyse préalable par un opérateur expert (horizons, failles), l’objectif de la thèse est de proposer une segmentation d’une image sismique sous-jacente en parfaite cohérence avec les lois de la géologie. L’originalité de la démarche consistera à développer des techniques de segmentation d’images sismiques, entre autres basées sur des approches de type contours actifs, contraintes par des données interprétées en supplément de propriétés intrinsèques calculées par des procédés automatiques à partir de la donnée traitée sans nécessiter une quelconque supervision contrairement aux travaux existants. Un deuxième axe consistera à ordonnancer automatiquement les horizons (surfaces) interprétés et analyser finement chaque intervalle (le lieu existant entre deux horizons), en prenant en compte son contenu (amplitude, orientation, etc.). Tout cela aboutissant à la reconstruction du pseudo-temps géologique. / The objective of the thesis is to propose a segmentation of an underlying seismic image in perfect coherence with the results of a preliminary analysis by an expert (horizons, faults). laws of geology. The originality of the approach will be to develop techniques for segmenting seismic images, among others based on active contour type approaches, constrained by data interpreted in addition to intrinsic properties calculated by automatic processes from the data processed without requiring any supervision in contrast to existing work. A second axis will be to automatically schedule the horizons (surfaces) interpreted and to analyze each interval (the place between two horizons) finely, taking into account its content (amplitude, orientation, etc.). All this resulted in the reconstruction of the geological pseudo-time.

Submitted by Geyciane Santos (geyciane_thamires@hotmail.com) on 2015-06-18T14:37:14Z No. of bitstreams: 1 Dissertação - Felipe Gomes de Oliveira.pdf: 8063130 bytes, checksum: c3a3271d66779ae934aca4f3e28b999b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-19T21:07:20Z (GMT) No. of bitstreams: 1 Dissertação - Felipe Gomes de Oliveira.pdf: 8063130 bytes, checksum: c3a3271d66779ae934aca4f3e28b999b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-19T21:10:24Z (GMT) No. of bitstreams: 1 Dissertação - Felipe Gomes de Oliveira.pdf: 8063130 bytes, checksum: c3a3271d66779ae934aca4f3e28b999b (MD5) / Made available in DSpace on 2015-06-19T21:10:24Z (GMT). No. of bitstreams: 1 Dissertação - Felipe Gomes de Oliveira.pdf: 8063130 bytes, checksum: c3a3271d66779ae934aca4f3e28b999b (MD5) Previous issue date: 2011-07-25 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research presents a visual data fusion approach to recover dense depth map from sequences of images. The conventional methods to estimate depth map have many drawbacks with respect to environment illumination changes and camera positioning. We propose a Global optimization data fusion strategy to improve the measurements from stereo and focus depth maps. Di erent from typical stereo and focus fusion techniques, we use a single pair of stereo cameras to acquire series of images scenes without occlusion and illumination constraints. Then, we use Energy Functional fusion to associate the geometric coherence with multiple frames. In order to evaluate the results we de ned a metric using similarity measurements between traditional stereo and the proposed approach. The experiments are performed in real scene images, and the estimated mapping was superior than those found using traditional stereo methods, which demonstrates the good performance and robustness of our approach. / Este trabalho propõe uma abordagem de Fusão de Dados Visuais para estimar a estrutura tridimensional de uma cena a partir de sequências de imagens obtidas por meio de duas ou mais câmeras. Os métodos convencionais para estimar mapas de profundidade apresentam desvantagens relacionadas a mudanças na iluminação do ambiente e posicionamento de câmeras. Por essa razão, foi proposta uma estrategia de Fusão de Dados baseada em minimiza c~ao de energia para aprimorar as medições proporcionadas pela disparidade entre pixels de uma imagem e pela variação de foco. A abordagem proposta faz uso de uma rede distribuída de sensores visuais utilizando um par de câmeras estéreo sem restrições de oclusão ou iluminação no processo de captura de imagens. A função de energia foi usada para integrar múltiplos frames e inferir a coerência geométrica contida na cena. Para avaliar os resultados obtidos foram utilizadas métricas da literatura através de medições de similaridade entre técnicas de estéreo tradicionais e a estrategia desenvolvida. Os experimentos foram conduzidos a partir de imagens de cenas reais, e as informações de profundidade estimadas foram qualitativamente superior que os resultados obtidos pelos métodos tradicionais. Tais informações demonstram a qualidade dos resultados alcançados pela técnica proposta.

<p><strong>ABSTRACT </strong> Approximating the kinetic energy as a functional of the electron density is a daunting, but important, task. For molecules in equilibrium geometries, the kinetic energy is equal in magnitude to the total electronic energy, so achieving the exquisite accuracy in the total energy that is needed for chemical applications requires similar accuracy for the kinetic energy functional. For this reason, most density functional theory (DFT) calculations use the Kohn-Sham method, which provides a good estimate for the kinetic energy. But the computational cost of Kohn-Sham DFT calculations has a direct dependence on the total number of electrons because the Kohn-Sham method is based on the orbital picture, with one orbital per electron. Explicit density functionals, where the kinetic energy is written explicitly in terms of the density, and not in terms of orbitals, are much faster to compute. Unfortunately, the explicit density functionals in the literature had disappointing accuracy. This dissertation introduces several new approaches for orbital-free density functional methods. One can try to include information about the Pauli principle using the exchange hole. In the weighted density approximation (WDA), a model for the exchange hole is used to approximate the one-electron density matrix, which is then used to compute the kinetic energy. This thesis introduces a symmetric, normalized, weighted density approximation using the exchange hole of the uniform electron gas. Though the key results on kinetic energy are not accurate enough, an efficient algorithm is introduced which, with a more sophisticated hole model, might give better results. The effects of electron correlation on the kinetic energy can be modeled by moving beyond the one-electron distribution function (the electron density) to higherorder electron distributions (k-electron DFT). For example, one can model electron correlation directly using the pair electron density. In this thesis, we investigated two different functionals of the pair density, the Weizsäcker functional and the March-Santamaria functional. The Weizsäcker functional badly fails to describe the accurate kinetic energy due to the N-representability problem. The March-Santamaria functional is exact for a single Slater determinant, but fails to adequately model the effects of electron correlation on the kinetic energy. Finally, we established a relation between Fisher information and Weizsäcker kinetic energy functional. This allowed us to propose generalisations of the Weizsäcker kinetic energy density functional. It is hoped that the link between information theory and kinetic energy might provide a new approach to deriving improved kinetic energy functionals. <strong> Keywords: </strong><em>Kinetic energy functional, Density functional theory (DFT), von-Weizsäcker</em> <em> functional, March-Santamaria functional, Thomas-Fermi model, density matrix, Twopoint normalization, Pair-density functional theory (PDFT). </em></p> / Doctor of Science (PhD)

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T18:45:49Z No. of bitstreams: 1 HILDÊNIO JOSÉ MACEDO - DISSERTAÇÃO PPGMAT 2011..pdf: 310824 bytes, checksum: ce96943d42ca2ee474b2fd99f6612b5c (MD5) / Made available in DSpace on 2018-07-25T18:45:49Z (GMT). No. of bitstreams: 1 HILDÊNIO JOSÉ MACEDO - DISSERTAÇÃO PPGMAT 2011..pdf: 310824 bytes, checksum: ce96943d42ca2ee474b2fd99f6612b5c (MD5) Previous issue date: 2011-05 / CNPq / Na presente dissertação, estudamos a existência e unicidade de solução para o problema de Cauchy associado a equação de evolução não local (Baixar arquivo para ver a equação). Exibimos um funcional energia, associado a esta equação, e verificamos que ele satisfaz a propriedade de Lyapunov. Além disso, usamos este funcional para mostrar a existência e estabilidade local de uma solução de equilíbrio referida na literatura como instanton. / In this work we prove existence and uniqueness of solution for the Cauchy problem corresponding to nonlocal evolution equation (Download file to see the equation). We exhibit an energy functional associated to this equation, and verify that it satisfies the Lyapunov property. Moreover, use this function to show the existence and local stability of a equilibrium solution reported in the literature as instanton.

On étudie les déformations des applications harmoniques $f$ tordues par rapport à une représentation. Après avoir construit une application harmonique tordue "universelle", on donne une construction de toute déformations du premier ordre de $f$ en termes de la théorie de Hodge ; on applique ce résultat à l'espace de modules des représentations réductives d'un groupe de Kähler, pour démontrer que les points critiques de la fonctionnelle de l'énergie $E$ coïncident avec les représentations de monodromie des variations complexes de structures de Hodge. Ensuite, on procède aux déformations du second ordre, où des obstructions surviennent ; on enquête sur l'existence de ces déformations et on donne une méthode pour le construire. En appliquant ce résultat à la fonctionnelle de l'énergie comme ci-dessus, on démontre (pour n'importe quel groupe de présentation finie) que la fonctionnelle de l'énergie est strictement pluri sous-harmonique sur l'espace des modules des représentations. En assumant de plus que le groupe soit de Kähler, on étudie les valeurs propres de la matrice hessienne de $E$ dans les points critiques. / We study the deformations of twisted harmonic maps $f$ with respect to a representation. After constructing a continuous ``universal'' twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a K\"ahler group, to show that the critical points of the energy functional $E$ coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is strictly pluri sub-harmonic on the moduli space of representations; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of $E$ at critical points.