Neste trabalho estuda-se a quantização de vários exemplos de teorias covariantes gerais do ponto de vista canônico e do ponto de vista do método de BRST. Os exemplos considerados são: partículas, bósons quirais, cordas nulas e membranas (e suas extensões supersimétricas). Demonstra-se para os exemplos estudados que é sempre possível obter teorias quânticas consistentes de ambos os pontos de vista. Apresentamos vários resultados mencionados a seguir: A) Propomos um procedimento para a quantização de bósons quirais baseado no método da raiz quadrada. O modelo possui uma estrutura tensorial de Lorentz explicita e permite quantizar consistentemente partículoas com spin superior em duas dimensões. B) Propomos uma nova corda nula com supersimetria local na superfície de evolução. Fazemos as formulações hamiltoniana e lagrangiana desta corda. Estes modelos de cordas são quantizados e demonstramos que as cordas nulas têm dimensões críticas 26 e 10 para os casos bosônicos e fermiônicos respectivamente. C) Propomos um modelo de p-branas nulas e as quantizamos via integral funcional. Obtemos uma expressão formal para o propagador destes objetos. O método proposto aqui poderia ser considerado como um ponto de partida para uma teoria de campos de p-branas nulas. Também propomos aqui um modelo de p-branas nulas supersimétricas e fazemos a formulação hamiltoniana deste modelo. D) Apresentamos um argumento que demonstra que o teorema de Fradkin-Vilkovisky é incompleto para quantizar teorias covariantes gerais. Propomos uma modificação deste teorema e obtemos expressões explícitas (e corretas) para os propagadores de partículas relativísticas (bosônicas e supersimétricas) e bósons quirais. / In this thesis, we study the quantization of several examples of the generally covariant theories using the canonical and the BRST methods. The examples studied are: particles, chiral bosons, null strings and null membranes (and their supersymmetric extensions). We show, for all the examples considered that it is always possible to obtain consistent quantum theories. We present several results: a) we propose a procedure to quantize chiral bosons based in the square root method. The resulting model has an explicit tensorial structure and it permits to quantize consistently particles with higher spin in two dimensions. B) We propose a new model of null strings with local supersymmetry in the world sheet. The null spinning string is constructed in the Lagrangian and the Hamiltonian formalism. The bosonic and fermionic null string are quantized and we shown that the critical dimensions are 26 and 10 respectively. C) We explore the path integral quantization of a null p-brane. A functional difusion equation is obtained for the free quantum propagator of these objects. Our method can be considered, probably, a first step in the formulation of the null p-brane field theory. Here, we present also a supersymmetric null p-branes model. d) We present an argument that shows that thw Fradkin-Vilkovisky theorem is incomplete to quantize generally covariant theories. We propose a modification of this theorem and we obtain explicit (and correct) expressions for the propagator of the bosonic relativistic particle, spinning relativistic particle and chiral bosons.

We investigate principles of rationality based on symmetry in Polyadic Pure Inductive Logic. The aim of Pure Inductive Logic (PIL) is to determine how to assign probabilities to sentences of a language being true in some structure on the basis of rational considerations. This thesis centres on principles arising from instances of symmetry for sentences of first-order polyadic languages. We begin with the recently introduced Permutation Invariance Principle (PIP), and find that it is determined by a finite number of permutations on a finite set of formulae. We test the consistency of PIP with established principles of the subject and show, in particular, that it is consistent with Super Regularity. We then investigate the relationship between PIP and the two main polyadic principles thus far, Spectrum Exchangeability and Language Invariance, and discover there are close connections. In addition, we define the key notion of polyadic atoms as the building blocks of polyadic languages. We explore polyadic generalisations of the unary principle of Atom Exchangeability and prove that PIP is a natural extension of Atom Exchangeability to polyadic languages. In the second half of the thesis we investigate polyadic approaches to the unary version of Constant Exchangeability as invariance under signatures. We first provide a theory built on polyadic atoms (for binary and then general languages). We introduce the notion of a signature for non-unary languages, and principles of invariance under signatures, independence, and instantial relevance for this context, as well as a binary representation theorem. We then develop a second approach to these concepts using elements as alternative building blocks for polyadic languages. Finally, we introduce the concepts of homomorphisms and degenerate probability functions in Pure Inductive Logic. We examine which of the established principles of PIL are preserved by these notions, and present a method for reducing probability functions on general polyadic languages to functions on binary languages.

We study mirror symmetry via Fourier-Mukai-type transformations, which we call SYZ mirror transformations, in view of the ground-breaking Strominger-Yau-Zaslow Mirror Conjecture which asserted that the mirror symmetry for Calabi-Yau manifolds could be understood geometrically as a T-duality modified by suitable quantum corrections. We apply these transformations to investigate a case of mirror symmetry with quantum corrections, namely the mirror symmetry between the A-model of a toric Fano manifold X¯ and the B-model of a Landau-Ginzburg model (Y, W). Here Y is a noncompact Kahler manifold and W : Y → C is a holomorphic function. We construct an explicit SYZ mirror transformation which realizes canonically the isomorphism QH*X&d1; ≅Ja cW between the quantum cohomology ring of X¯ and the Jacobian ring of the function W. We also show that the symplectic structure oX¯ of X¯ is transformed to the holomorphic volume form eWOY of ( Y, W). Concerning the Homological Mirror Symmetry Conjecture, we exhibit certain correspondences between A-branes on X¯ and B-branes on (Y, W) by applying the SYZ philosophy. / Chan, Kwok Wai. / Adviser: Nai Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3536. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-56). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

在約束滿足問題中，破壞指數量級數量的所有對稱通常過於昂貴。在實踐中，我們通常只有效地破壞對稱的一個子集。我們稱之為目標對稱。在靜態對稱破壞中，我們的目標是發佈一套約束去破壞這些目標對稱，以達到減少解集以及搜索空間的效果。一個問題中的所有對稱之間是互相交織的。一個旨在特定對稱的破壞對稱約束几乎總會產生副作用，而不僅僅破壞了預期的對稱。破壞相同目標對稱的不同約束可以有不同的副作用。傳統智慧告訴我們應該選擇一個破壞更多對稱從而有更多副作用的破壞對稱約束。雖然這樣的說法在許多方面上都是有效的，我們應該更加注意副作用發生的地方。 / 給與一個約束滿足問題，一個對稱被一個約束保留當且僅當該對稱仍然是新的約束滿足問題的對稱。這個新的約束滿足問題是有原問題加上該約束組成的。我們給出定律和例子，以表明發佈儘量保留目標對稱以及限制它的副作用發生在非目標對稱上的破壞約束是有利的。這些好處來自于被破壞的對稱數目以及一個對稱被破壞（或消除）的程度，并導致一個較小的解集和搜索空間。但是，對稱不一定會被保留。我們顯示，旨在一個已經被消除的目標對稱的破壞對稱約束仍然可以被發佈。我們建議根據問題的約束以及其他破壞對稱約束來選擇破壞對稱約束，以繼續消除更多的對稱。我們進行了廣泛的實驗來確認我們的建議的可行性與效率。 / Breaking the exponential number of all symmetries of a constraint satisfaction problem is often too costly. In practice, we often aim at breaking a subset of the symmetries efficiently, which we call target symmetries. In static sym-metry breaking, the goal is to post a set of constraints to break these target symmetries in order to reduce the solution set and thus also the search space. Symmetries of a problem are all intertwined. A symmetry breaking constraint intended for a particular symmetry almost always breaks more than just the intended symmetry as a side-effect. Different constraints for breaking the same target symmetry can have different side-effects. Conventional wisdom suggests that we should select a symmetry breaking constraint that has more side-effects by breaking more symmetries. While this wisdom is valid in many ways, we should be careful where the side-effects take place. / A symmetry σ of a CSP P =(V, D, C) is preserved by a set of symmetry breaking constraints C{U+02E2}{U+1D47} i σ is a symmetry of P¹ =(V, D, CU C{U+02E2}{U+1D47}). We give theorems and examples to demonstrate that it is beneficial to post symmetry breaking constraints that preserve the target symmetries and restrict the side-effects to only non-target symmetries as much as possible. The benefits are in terms of the number of symmetries broken and the extent to which a symmetry is broken (or eliminated), resulting in a smaller solution set and search space. However, symmetry preservation may not always hold. We illustrate that symmetry breaking constraints, which aim at a target symmetry that is already eliminated, can still be posted. To continue eliminating more symmetries, we suggest to select symmetry breaking constraints based on problem constraints and other symmetry breaking constraints. Extensive experiments are also conducted to confirm the feasibility and efficiency of our proposal empirically. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Jingying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 101-112). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Constraint Satisfaction Problems --- p.1 / Chapter 1.2 --- Motivation and Goals --- p.3 / Chapter 1.3 --- Outline of Thesis --- p.5 / Chapter 2 --- Background --- p.8 / Chapter 2.1 --- Constraint Satisfaction Problems --- p.8 / Chapter 2.1.1 --- Backtracking Search --- p.9 / Chapter 2.1.2 --- Consistency Techniques --- p.12 / Chapter 2.1.3 --- Local Consistencies with Backtracking Search --- p.15 / Chapter 2.2 --- Symmetry Breaking in CSPs --- p.16 / Chapter 2.2.1 --- Symmetry Classes --- p.18 / Chapter 2.2.2 --- Breaking Symmetries --- p.22 / Chapter 2.2.3 --- Variable and Value Symmetries --- p.23 / Chapter 2.2.4 --- Symmetry Breaking Constraints --- p.26 / Chapter 3 --- Effects of Symmetry Breaking Constraints --- p.29 / Chapter 3.1 --- Removing Symmetric Search Space --- p.29 / Chapter 3.1.1 --- Properties --- p.30 / Chapter 3.1.2 --- Canonical Variable Orderings --- p.31 / Chapter 3.1.3 --- Regenerating All Solutions --- p.33 / Chapter 3.1.4 --- Remaining Solution Set Sizes --- p.36 / Chapter 3.2 --- Constraint Interactions in Propagation --- p.43 / Chapter 4 --- Choices of Symmetry Breaking Constraints --- p.45 / Chapter 4.1 --- Side-Effects --- p.45 / Chapter 4.2 --- Symmetry Preservation --- p.50 / Chapter 4.2.1 --- De nition and Properties --- p.50 / Chapter 4.2.2 --- Solution Reduction --- p.54 / Chapter 4.2.3 --- Preservation Examples --- p.55 / Chapter 4.2.4 --- Preserving Order --- p.64 / Chapter 4.3 --- Eliminating Eliminated Symmetries --- p.65 / Chapter 4.3.1 --- Further Elimination --- p.65 / Chapter 4.3.2 --- Aggressive Elimination --- p.71 / Chapter 4.4 --- Interactions with Problem Constraints --- p.72 / Chapter 4.4.1 --- Further Simplification --- p.72 / Chapter 4.4.2 --- Increasing Constraint Propagation --- p.73 / Chapter 5 --- Experiments --- p.75 / Chapter 5.1 --- Symmetry Preservation --- p.75 / Chapter 5.1.1 --- Diagonal Latin Square Problem --- p.76 / Chapter 5.1.2 --- NN-Queen Problem --- p.77 / Chapter 5.1.3 --- Error Correcting Code - Lee Distance (ECCLD) --- p.78 / Chapter 5.2 --- Eliminating Eliminated Symmetries --- p.80 / Chapter 5.2.1 --- Equidistance Frequency Permutation Array Problem --- p.80 / Chapter 5.2.2 --- Cover Array Problem --- p.82 / Chapter 5.2.3 --- Sports League Scheduling Problem --- p.83 / Chapter 6 --- Related Work --- p.86 / Chapter 6.1 --- Symmetry Breaking Approaches --- p.86 / Chapter 6.2 --- Reducing Overhead and Increasing Propagation --- p.90 / Chapter 6.3 --- Selecting and Generating Choices --- p.91 / Chapter 6.3.1 --- Reducing Conflict with Search Heuristic --- p.92 / Chapter 6.3.2 --- Choosing the Subset of Symmetries --- p.93 / Chapter 6.4 --- Detecting Symmetries --- p.93 / Chapter 7 --- Conclusion and Remarks --- p.95 / Chapter 7.1 --- Conclusion --- p.95 / Chapter 7.2 --- Discussions --- p.97 / Chapter 7.3 --- Future Work --- p.99 / Bibliography --- p.101

Orientador: Claudemir Murari / Banca: Henrique Lazari / Banca: Marcos Luiz Lourenço / Resumo: Neste trabalho apresentamos uma estratégia de ensino utilizando caleidoscópios, jogos e o software Cabri Géomètre II, para aprendizagem de Geometria. Sob a perspectiva do método Resolução de Problemas, foram elaboradas atividades abordando temas como simetrias (reflexão, rotação e translação), polígonos regulares, construções geométricas, pavimentações do plano, seqüências numéricas, etc. Compilamos neste estudo um referencial teórico matemático detalhado, exemplos de várias atividades e relatos de nossa experiência com alunos do 2º ano do Ensino Médio. A utilização dos caleidoscópios e do software Cabri Géomètre II proporcionou uma interação entre o laboratório de ensino e o de informática, dinamizando o ambiente de aprendizagem. / Abstract: In this scientific paper we present a teaching strategy which uses kaleidoscopes, games and Cabri Géomètre II software for the geometry learning. Under the perspective of the problem-solving method, it was done activities which approach subjects such as symmetry (reflection, rotation and translation), regular polygons, geometric constructions, plan floor, numeric sequences, etc. In this study, it was gleaned a detailed theoretical mathematics reference, examples of many activities and reports of our experience with High School students of the second year. The use of kaleidoscopes and of the Cabri Géomètre II software gave the interaction between the teaching laboratory and the computer one, making the learning atmosphere dynamic. / Mestre

O espalhamento pion-nucleon para energias até 300 MeV é estudado por meio de um modelo teórico baseado na simetria quiral, implementada por lagrangianas efetivas. A interação entre o pion e o nucleon é mediada por apenas quatro partículas: nucleon, delta, rho e sigma. As amplitudes associadas ao delta e ao sigma contêm parâmetros livres que devem ser extraídos dos dados experimentais. Os conjuntos de valores obtidos em ajustes abaixo e acima do limiar são discrepantes indicando que, no co~ texto do modelo, não é possível uma descrição unificada das duas regiões. Os resultados acima do limiar são sensíveis ao método de unitarizaçâo empregado. O método adotado neste trabalho tem interpretação física simples e produz um ajuste muito bom da on da ressonante associada ao delta. / The pion-nucleon scattering for energies up to 300 MeV is studied by means of a theoretical model based on chiral symmetry, implemented by effective lagrangians. The interaction between the pion and the nucleon is mediated by only four particles: nucleon, delta, rho and sigma. The amplitudes associated to the delta and sigma contain free parameters, that must be extracted from experiment. The set of values obtained from fits below and above threshold disagree, indicating that, in the context of the model, a unif ied description in both regions is not possible. The results above threshold are sensitive to the method of unitarization employed. The method adopted in this work has a simple physical meaning and the ressonating wave associated to the delta is quite well reproduced.

Obtivemos, a partir do reconhecimento de propriedades de simetria em uma classe de funções de onda tipo BCS, expressões analíticas para energias de BCS projetadas e taxas de transição de reações entre estados coletivos de emparelhamento isovetorial em núcleos. Como consequência, efetuamos um tratamento simplificado de coordenadas geradoras. Nesta tese, o objetivo é mais o de estabelecer um instrumento simples e eficiente para descrever estes estados coletivos O+, do que apresentar novos resultados. Testamos a validade, as limitações e vantagens desta nossa aproximação em modelos simples e núcleos. Entre os diversos resultados, nossas amplitudes espectroscópicas obtidas por GCM são comparáveis aos métodos de diagonallização exata no modelo de camada. De maneira geral, podemos afirmar: o que foi obtido encoraja-nos sobremaneira, especialmente se considerarmos que elas foram conseguidos por uma aproximação bastante simples. Também apresentamos alguns novos resultados de tal forma que podemos colocar em bases sólidas a nossa abordagem dos estados coletivos de emparelhamento isovetoriais. Em vista da simplicidade de nossa aproximação, o presente modelo pode ser considerado um bom candidato para um estudo sistemático dos estados coletivos de emparelhamento isovetorial, que em núcleos semi-pesados, são fortemente populados por reações de transferência de um par. / Analytical expressions for the projected-BCS energies and reaction transition rates among the isovector pairing collective states are obtained by the recognition of symmetry properties in a class of BCS wave functions. As a consequence it was possible to employ a simplified Generator Coordinate Method calculation. The main purpose of the present work is to establish and develop a simple yet powerful tool for the less complicated nuclear models and for some real nuclei. Among the various results, our GCM values of spectroscopic amplitude are shown to be comparable to those of the Shell Model calculations. This is indeed encouraging especially in view of the fact that they were reached using a simple approximation such as the present one. The results also clearly demonstrate that our model is on a rather sound basis and can be applied immediately to the study of isovector collective motions. The great simplicity of the present method, as compared with their earlier complicated versions, suggests that they might prove useful in the study of isovector pairing collective states which are strongly populated by pair transfer reactions in the medium weight nuclei.

There are currently over 138, 000 known macromolecular structures deposited in the wwPDB (Worldwide Protein Data Bank) database. While all the macromolecular structure files contain information about a particular structure, the collection of these files also allows combining the macromolecular structures to obtain statistical information about macromolecules in general. This fact has been the basis for many structural biology methods including the molecular replacement method used in X-ray crystallography or homologous structure restraints in the refinement methods. With the success of methods based on prior information, it is feasible that novel methods could be developed and current methods improved using further prior information; more specifically, by using the structure density-map shape similarity instead of sequence or model similarity. Therefore, this project introduces a mathematical framework for computing three different measures of macromolecular three-dimensional shape similarity and demonstrates how these descriptors can be applied in symmetry detection and protein-domain clustering. The ability to detect cyclic (C), dihedral (D), tetrahedral (T), octahedral (O) and icosahedral (I) symmetry groups as well as computing all associated symmetry elements has direct applications in map averaging and reducing the storage requirements by storing only the asymmetric information. Moreover, by having the capacity to find structures with similar shape, it was possible to reduce the size of the BALBES protein domain database by more than 18.7% and thus achieve proportional speed-up in the searching parts of its applications. Finally, the development of the method described in this project has many possible applications throughout structural biology. The method could, for example, facilitate matching and fitting of protein domains into the density maps produced by the electron-microscopy techniques, or it could allow for molecular-replacement candidate search using shape instead of sequence similarity. To allow for the development of any further applications, software for applying the methods described here is also presented and released for the community.

The construction introduced by Gross, Hacking and Keel in [28] allows one to construct a mirror family to (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the n-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for D is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in [GHK]. In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to S being a rational elliptic surface and D a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family.

O espalhamento pion-nucleon para energias até 300 MeV é estudado por meio de um modelo teórico baseado na simetria quiral, implementada por lagrangianas efetivas. A interação entre o pion e o nucleon é mediada por apenas quatro partículas: nucleon, delta, rho e sigma. As amplitudes associadas ao delta e ao sigma contêm parâmetros livres que devem ser extraídos dos dados experimentais. Os conjuntos de valores obtidos em ajustes abaixo e acima do limiar são discrepantes indicando que, no co~ texto do modelo, não é possível uma descrição unificada das duas regiões. Os resultados acima do limiar são sensíveis ao método de unitarizaçâo empregado. O método adotado neste trabalho tem interpretação física simples e produz um ajuste muito bom da on da ressonante associada ao delta. / The pion-nucleon scattering for energies up to 300 MeV is studied by means of a theoretical model based on chiral symmetry, implemented by effective lagrangians. The interaction between the pion and the nucleon is mediated by only four particles: nucleon, delta, rho and sigma. The amplitudes associated to the delta and sigma contain free parameters, that must be extracted from experiment. The set of values obtained from fits below and above threshold disagree, indicating that, in the context of the model, a unif ied description in both regions is not possible. The results above threshold are sensitive to the method of unitarization employed. The method adopted in this work has a simple physical meaning and the ressonating wave associated to the delta is quite well reproduced.