Since its development, the minimax framework has been one of the corner stones of theoretical statistics, and has contributed to the popularity of many well-known estimators, such as the regularized M-estimators for high-dimensional problems. In this thesis, we will first show through the example of sparse Gaussian sequence model, that the theoretical results under the classical minimax framework are insufficient for explaining empirical observations. In particular, both hard and soft thresholding estimators are (asymptotically) minimax, however, in practice they often exhibit sub-optimal performances at various signal-to-noise ratio (SNR) levels. To alleviate the descrepancy, we first demonstrate that this issue can be resolved if the signal-to-noise ratio is taken into account in the construction of the parameter space. We call the resulting minimax framework the signal-to-noise ratio aware minimaxity. Then, we showcase how one can use higher-order asymptotics to obtain accurate approximations of the SNR-aware minimax risk and discover minimax estimators. Theoretical findings obtained from this refined minimax framework provide new insights and practical guidance for the estimation of sparse signals. In a broader context, we investigated the same problem for sparse linear regression. We assume the random design and allow the feature matrix to be high dimensional as 𝑿 ∈ R^{𝑛 x 𝑝} and 𝑝 ⪢ 𝑛 . This adds an extra layer of challenge to the estimation of coefficients. Previous studies have largely relied on results expressed in rate-minimaxity, where estimators are compared based on minimax risk with order-wise accuracy, without specifying the precise constant in the approximation. This lack of precision contributes to the notable gap between theoretical conclusions of the asymptotic minimax estimators and empirical findings of the sub-optimality. This thesis addresses this gap by initially refining the classical minimax result, providing a characterization of the constant in the first-order approximation. Subsequently, by following the framework of SNR-aware minimaxity we introduced before, we derived improved approximations of minimax risks under different SNR levels. Notably, these refined results demonstrated better alignment with empirical findings compared to classical minimax outcomes. As showcased in the thesis, our enhanced SNR-aware minimax framework not only offers a more accurate depiction of sparse estimation but also unveils the crucial role of SNR in the problem. This insight emerges as a pivotal factor in assessing the optimality of estimators.

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by Ki-sing Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 30.

by Lo Yiu Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-90). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Main Idea --- p.5 / Chapter 1.2. --- Organization --- p.8 / Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9 / Chapter 2.1. --- A Review on Measure Theory --- p.9 / Chapter 2.2. --- Submanifolds in Rn --- p.11 / Chapter 2.3. --- Several Results from PDEs --- p.17 / Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20 / Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24 / Chapter 3.1. --- Some Preliminaries --- p.24 / Chapter 3.2. --- Approximate Graphical Decomposition --- p.30 / Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41 / Chapter 4.1. --- Willmore Functional and Area --- p.41 / Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45 / Chapter 4.3. --- Higher Regularity at Good Points --- p.54 / Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62 / Chapter 4.5. --- Regularity near Bad Points --- p.64 / Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83 / References --- p.89

Research on multi-agent systems performing cooperative tasks has received considerable attention in recent years. Because multiple agents perform cooperative tasks in close proximity, the coordination control of multiple agents to avoid collisions holds one of the critical keys to mission success. The potential function approach has been extensively employed for collision avoidance, but it has one inherent limitation of local minimum. This dissertation proposes a new avoidance strategy for the issue of local minimum. The primary objective of this research is to construct novel potential-function-based control schemes that drive agents from their initial to the goal configurations while avoiding collision with other agents and obstacles. The control schemes enable agents to avoid being trapped at a local minimum by forcing them to exit from the regions that may contain a local minimum. This dissertation consists of three studies, each of which has different technical assumptions. In the first study, all-to-all communication ability among agents is assumed. In addition, each agent is assumed to a priori know the location of all obstacles. In the second study, all-to-all communication ability is again assumed, but each agent is assumed to determine the location of obstacles using a sensor with a limited sensing range. In the third study, limited communication ability is assumed (i.e., each agent exchanges information only with agents within its limited communication range), and each agent is assumed to determine the location of the obstacles using its sensor with a limited sensing range. Relative to existing solutions, the new control schemes presented here have three distinct advantages. First, our avoidance strategy can provide cost-efficient solutions in applications because agents will never be trapped at a local minimum. Second, our control signals are continuous, which allows agents to change their speed in a realistic manner that is consistent with their natural motion traits. Finally, our control scheme allows for setting the upper bound of the velocity of each agent, which guarantees that the speed of agents will never exceed a maximum speed limit. / text

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[pt] A continuidade de operação das linhas de transmissão em Extra Alta Tensão é fundamental para o sistema de potência envolvido, especialmente em sistemas similares ao do Brasil. Torna-se então importante, realizaremos manutenção nas cadeias de isoladores com a linha energizada. A segurança do homem para a execução de serviços em linhas energizadas é definida através de distâncias que são influenciadas pela geometria da torre, por parâmetros meteorológicos e pela sobretensão que possa ocorrer no sistema. Para essa análise um método estatístico foi desenvolvido e implementado em um computador digital, a fim de se determinar o risco de falha do isolamento de uma torre frente a surtos de manobra. Como conclusão são recomendados procedimentos de cálculos para determinação das distâncias mínimas para Trabalhos em Linha Viva. / [en] The continuity of power supply in Extra-High Voltage transmission lines is one of major requirements to be achieved in electrical power systems specially in countries like Brazil. Therefore, it is of capital importance to perform maintenance work on insulator strings with energised line. The safety of personnel working o energise lines depends on the observance of clearances which are affected by geometrical characteristics of the tower, by meteorological parameters and by eventual occurrence of system overvoltages. To this purpose, a statistical method was developed and solved on a digital computer, in order to evaluate the probability of insulation failure in a tower under switching surges. On the basis of this analysis, minimum safety clearances are recommended for carrying out Live Line Works.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Procuramos desenvolver a nossa pesquisa no estudo da filosofia moral de Adorno, e desta perspectiva, buscar na Teoria CrÃtica da sociedade, da qual este filÃsofo fundamentou seu pensamento, e com ela pesquisaremos a temÃtica da vida danificada no mundo administrado e sua moral estabelecida. Para fomentar a metodologia desta pesquisa, cuja vida falsa no mundo administrado e sua moral danificada do indivÃduo no mundo administrado sÃo o nosso interesse, seguirà na leitura da Minima Moralia: reflexÃes a partir da vida danificada. Theodor W. Adorno analisa a respeito da condiÃÃo da vida humana no mundo contemporÃneo e tenta mostrar que a âvida boaâ praticamente desapareceu das relaÃÃes humanas. Dessa forma, apresentamos os argumentos que apontam no pensamento do filÃsofo para uma filosofia moral prÃpria. No sentido de uma âantimoralâ da moral estabelecida pelas formas condicionantes do sistema. Quando lemos Minima Moralia, percebemos o carÃter fragmentÃrio da obra, escrita por meio de aforismos, desvela em seu modo de exposiÃÃo os traÃos de uma realidade tambÃm fragmentada, a qual nÃo poderia ser tratada por Adorno de outro modo a nÃo ser como parte de uma totalidade que se desfez e do qual nÃo restam senÃo pequenos fragmentos. Assim, surge a necessidade de investigar nos detalhes mais banais da vida cotidiana dos indivÃduos esse problema. à uma crÃtica, uma denÃncia que Adorno realiza no mais intimo da sociedade e suas relaÃÃes interpessoais, quando ele apresenta de maneira desveladora o carÃter dominador da sociedade, na frieza e indiferenÃa das relaÃÃes e na barbÃrie cometida. O resultado do estudo nos conduziu para uma possÃvel proposta Ãtica de Adorno, ao falar desse assunto, ele traz uma perspectiva de um âdeverâ, exortando que lugares como Auschwitz nÃo voltem a acontecer. Para que isso nÃo retorne com novos aspectos da sociedade atual, o filÃsofo propÃe que nÃo deixemos esquecer o que ocorreu - o passado que ainda à o trauma das testemunhas, mas tambÃm, a resistÃncia que leva o ao conhecimento as novas geraÃÃes como forma de educar.

by Chun Gu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 69-71). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 2 --- Definitions --- p.7 / Chapter 3 --- Some Previous Work on OBDD --- p.13 / Chapter 3.1 --- The Work of Bryant --- p.13 / Chapter 3.2 --- Some Variations of the OBDD --- p.14 / Chapter 3.3 --- Previous Work on Variable Ordering of OBDD --- p.16 / Chapter 3.3.1 --- The FIH Heuristic --- p.16 / Chapter 3.3.2 --- The Dynamic Variable Ordering --- p.17 / Chapter 3.3.3 --- The Interleaving method --- p.19 / Chapter 4 --- Two Level Logic Function and OBDD --- p.21 / Chapter 5 --- DSCF Algorithm --- p.25 / Chapter 6 --- Thin Boolean Function --- p.33 / Chapter 6.1 --- The Structure and Properties of thin Boolean functions --- p.33 / Chapter 6.1.1 --- The construction of Thin OBDDs --- p.33 / Chapter 6.1.2 --- Properties of Thin Boolean Functions --- p.38 / Chapter 6.1.3 --- Thin Factored Functions --- p.49 / Chapter 6.2 --- The Revised DSCF Algorithm --- p.52 / Chapter 6.3 --- Experimental Results --- p.54 / Chapter 7 --- A Pattern Merging Algorithm --- p.59 / Chapter 7.1 --- Merging of Patterns --- p.60 / Chapter 7.2 --- The Algorithm --- p.62 / Chapter 7.3 --- Experiments and Conclusion --- p.65 / Chapter 8 --- Conclusions --- p.67