In this thesis, we discuss several phenomena exhibiting `limiting behaviour' in physics. This includes the duality principle, delegated quantum computation, and super-resolution. The duality principle places a limit on the coexistence of wave and particle behaviours. We develop a framework that explains apparent violations of this principle while staying within the scope of quantum mechanics. In addition, we relate the duality principle to the sub-fidelity and weak-values. We also show that the maximum recoverable coherence of a qubit has a sharp transition from 0 to 1 when we have access to half of the environment to which the qubit is correlated. Delegated quantum computation consists of a computational weak client who wishes to delegate a complex quantum computation to a powerful quantum server. We develop a new protocol for delegated quantum computation requiring less quantum power than its predecessor. Finally, we develop and test a new theory for eigenmode super-resolution.

This thesis consists of four papers and an introduction.  In Paper I we calculate the second order derivatives of the Ronkin function of an affine polynomial in three variables. This gives an expression for the real Monge-Ampére measure associated to the hyperplane amoeba. The measure is expressed in terms of complete elliptic integrals and hypergeometric functions.  In Paper II and III we prove that a certain semi-explicit cohomological residue associated to a Cohen-Macaulay ideal or more generally an ideal of pure dimension, respectively, is annihilated precisely by the given ideal. This is a generalization of the local duality principle for the Grothendieck residue and the cohomological residue of Passare. These results follow from residue calculus, due to Andersson and Wulcan, but the point here is that our proof is more elementary. In particular, it does not rely on the desingularization theorem of Hironaka. In Paper IV we prove a global uniform Artin-Rees lemma for sections of ample line bundles over smooth projective varieties. We also prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proofs are based on multidimensional residue calculus. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 3: Manuscript. Paper 4. Manuscript.</p>

Submitted by (lucia.rodrigues@ufrpe.br) on 2016-11-23T13:00:41Z No. of bitstreams: 1 Paulo Fernando Lima de Souza.pdf: 1161641 bytes, checksum: 9ee330df7fe48bfd1dc20c6a5b98183a (MD5) / Made available in DSpace on 2016-11-23T13:00:41Z (GMT). No. of bitstreams: 1 Paulo Fernando Lima de Souza.pdf: 1161641 bytes, checksum: 9ee330df7fe48bfd1dc20c6a5b98183a (MD5) Previous issue date: 2009-02-25 / The objective of this work was to investigate if transdisciplinarity can contribute to the understanding of the wave-particle duality principle. The investigation was conducted with students from the Physics teaching course at UFRPE. The methodology used was Kelly’s Experience Cycle, in which the objective was to involve the students in a learning process through three experiments: the wave tank; the Young experiment and Crookes’ radiometer. These three experiments motivated a discussion about wave-particle duality. A transdisciplinary didactic workshop was conducted aiming to promote a discussion about third included logic, an adequate logic to be applied to wave-particle duality. The data analysis allowed the following conclusions: (i) the students from the Physics teaching course at UFRPE had a previous concept about the wave-particle duality principle, but without a logic subjacent to this principle; (ii) the three experiments and the transdisciplinary workshop conducted during Kelly’s Experience Cycle contributed for a better comprehension about the wave-particle duality principle. / O objetivo deste trabalho foi investigar, numa turma de alunos da Licenciatura em Física da UFRPE, se o Pensamento Transdisciplinar pode contribuir para a compreensão do Princípio da Dualidade da Luz. A metodologia empregada foi o Ciclo da Experiência de Kelly, que teve a perspectiva de engajar os alunos em um processo de aprendizagem, com o auxílio de três experimentos didáticos: a cuba de onda, a dupla fenda de Young e o radiômetro de Crookes. A partir desses experimentos foi possível motivar uma discussão sobre as propriedades ondulatórias e corpusculares da luz. Uma oficina sobre a Transdisciplinaridade foi também realizada com o intuito de discutir a lógica do terceiro incluído, uma lógica que se adequa ao fato da luz se comportar como onda e como partícula. As análises dos dados permitiram as seguintes conclusões: (i) as concepções prévias dos estudantes da Licenciatura em Física da UFRPE acerca do Princípio da Dualidade da Luz eram desprovidas de uma lógica que respaldasse tal Princípio e (ii) os três experimentos propostos e a oficina sobre a Transdisciplinaridade, realizadas durante o Ciclo da Experiência Kellyana, contribuíram para que os estudantes tivessem uma maior compreensão do Princípio da Dualidade da Luz.

Made available in DSpace on 2016-08-17T14:52:48Z (GMT). No. of bitstreams: 1 Carlos Augusto Pereira Braga.pdf: 1145854 bytes, checksum: 8898d3084e223e4e9ad52a8aa7363b01 (MD5) Previous issue date: 2008-04-07 / A tunning model for Kalman filter based on QR duality principle is presented to measure the state variables of the electrolytic bath in aluminium production cells. The main goal is to establish a set of filter gains that better represents the percentual of alumina in the bath. The filter bandwidth tunning is performed by increasing or decreasing the filter bandpass from the Q and R variations. The design and analysis of the Q and R covariance matrices are performed to find a pattern of the resistance variations that could be associated with the alumina concentration in the bath. The technical solution encloses on-line evaluation of the Kalman filter in order to prove its capableness of response when used to control real production cells. The Standard Kalman is coded upon a scalar form to reduce the use of computing resources when the filter is processed. The line current and pot voltage are directly read from the hardware interface and then converted in a third variable, the resistance, which is used to infer the alumina concentration in the bath. Moreover, the filter implementation goes in the direction of practical aspects limits of the indirect measurement system implementations, its robustness is appraised by observability, roundo® and modeling errors. / O controle da concentração de alumina no banho de cubas eletroliticas é de vital importância para um rendimento eficiente do processo de produção de aluminio. Este controle tem por objetivo promover, em uma visão macro, um retorno justificável perante ao planejamento global das metas de produção de uma fábrica de aluminio. O desenvolvimento de uma metodologia baseada no ajuste dos ganhos de um filtro tipo Kalman Padrão é o principal enfoque cientifico e tecnológico desta pesquisa. A sintese da metodologia é a implementação do filtro de Kalman Padrão e Escalar em um computador que controla a quantidade de alumina no banho eletrolitico. Considera-se as restrições de software e de hardware na implantação do algoritmo de Kalman no sistema que executa o controle de processo por computador. Desenvolve-se um modelo de sintonia do filtro de Kalman utilizando-se o principio da dualidade das matrizes Q e R. O objetivo é estabelecer um conjunto de ganhos que representem da melhor forma a quantidade de alumina dentro da cuba. Assim, faz-se uma análise considerando-se mudanças nas matrizes Q e R, e apresenta-se os resultados obtidos. Analisa-se a largura de banda do filtro, verificando-se diminuição ou aumento de banda na medida que as matrizes Q e R sofrem alterações em seus elementos. As estratégias de monitoramento são verificadas para diversas situações de operação(dinâmica) da planta, e as variações paramétricas do modelo são levadas em consideração para garantirmos uma operação robusta e estável do filtro. O desempenho do filtro é verificado em paralelo com um sistema de filtragem e de controle historicamente utilizados na estimação do percentual de alumina no banho eletrolitico.