In this thesis a set of recently proposed multigravity theories is analysed. In the special case of bimetric gravity, the theory has been conclusively shown to be ghost-free. On the other hand, for multigravity theories in general, the ghost-issue has not been settled conclusively. Motivated by this fact, the main object of this thesis is to clarify what has been proven so far and what issues that still needs to be addressed. We also provide new calculations and results pointing in the direction that the multigravity theories must be restricted to a set of bimetric Hassan-Rosen couplings in a tree-type structure in order to be consistent. In particular, we prove that for a multivielbein theory of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BN%7D" /> interacting vielbeins, the Lorentz equations of motion is a set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?(%5Cmathscr%7BN%7D-1)" /> Deser-van Nieuwenhuizen conditions if and only if the theory consists of bimetric Hassan-Rosen couplings in a tree-type structure.

Cette thèse se situe dans le domaine de la combinatoire algébrique et porte sur l'étude et les applications de trois ordres sur les permutations : les deux ordres faibles (gauche et droit) et l'ordre fort ou de Bruhat. Dans un premier temps, nous étudions l'action du groupe symétrique sur les polynômes multivariés. En particulier, les opérateurs de emph{différences divisées} permettent de définir des bases de l'anneau des polynômes qui généralisent les fonctions de Schur aussi bien du point de vue de leur construction que de leur interprétation géométrique. Nous étudions plus particulièrement la base des polynômes de Grothendieck introduite par Lascoux et Schützenberger. Lascoux a montré qu'un certain produit de polynômes peut s'interpréter comme un produit d'opérateurs de différences divisées. En développant ce produit, nous ré-obtenons un résultat de Lenart et Postnikov et prouvons de plus que le produit s'interprète comme une somme sur un intervalle de l'ordre de Bruhat. Nous présentons aussi l'implantation que nous avons réalisée sur Sage des polynômes multivariés. Cette implantation permet de travailler formellement dans différentes bases et d'effecteur des changements de bases. Elle utilise l'action des différences divisées sur les vecteurs d'exposants des polynômes multivariés. Les bases implantées contiennent en particulier les polynômes de Schubert, les polynômes de Grothendieck et les polynômes clés (ou caractères de Demazure).Dans un second temps, nous étudions le emph{treillis de Tamari} sur les arbres binaires. Celui-ci s'obtient comme un quotient de l'ordre faible sur les permutations : à chaque arbre est associé un intervalle de l'ordre faible formé par ses extensions linéaires. Nous montrons qu'un objet plus général, les intervalles-posets, permet de représenter l'ensemble des intervalles du treillis de Tamari. Grâce à ces objets, nous obtenons une formule récursive donnant pour chaque arbre binaire le nombre d'arbres plus petits ou égaux dans le treillis de Tamari. Nous donnons aussi une nouvelle preuve que la fonction génératrice des intervalles de Tamari vérifie une certaine équation fonctionnelle décrite par Chapoton. Enfin, nous généralisons ces résultats aux treillis de $m$-Tamari. Cette famille de treillis introduite par Bergeron et Préville-Ratelle était décrite uniquement sur les chemins. Nous en donnons une interprétation sur une famille d'arbres binaires en bijection avec les arbres $m+1$-aires. Nous utilisons cette description pour généraliser les résultats obtenus dans le cas du treillis de Tamari classique. Ainsi, nous obtenons une formule comptant le nombre d'éléments plus petits ou égaux qu'un élément donné ainsi qu'une nouvelle preuve de l'équation fonctionnelle des intervalles de $m$-Tamari. Pour finir, nous décrivons des structures algébriques $m$ qui généralisent les algèbres de Hopf $FQSym$ et $PBT$ sur les permutations et les arbres binaires / This thesis comes within the scope of algebraic combinatorics and studies problems related to three orders on permutations: the two said weak orders (right and left) and the strong order or Bruhat order.We first look at the action of the symmetric group on multivariate polynomials. By using the emph{divided differences} operators, one can obtain some generalisations of the Schur function and form bases of non symmetric multivariate polynomials. This construction is similar to the one of Schur functions and also allows for geometric interpretations. We study more specifically the Grothendieck polynomials which were introduced by Lascoux and Schützenberger. Lascoux proved that a product of these polynomials can be interpreted in terms of a product of divided differences. By developing this product, we reobtain a result of Lenart and Postnikov and also prove that it can be interpreted as a sum over an interval of the Bruhat order. We also present our implementation of multivariate polynomials in Sage. This program allows for formal computation on different bases and also implements many changes of bases. It is based on the action of the divided differences operators. The bases include Schubert polynomials, Grothendieck polynomials and Key polynomials. In a second part, we study the emph{Tamari lattice} on binary trees. This lattice can be obtained as a quotient of the weak order. Each tree is associated with the interval of its linear extensions. We introduce a new object called, emph{interval-posets} of Tamari and show that they are in bijection with the intervals of the Tamari lattice. Using these objects, we give the recursive formula counting the number of elements smaller than or equal to a given tree. We also give a new proof that the generating function of the intervals of the Tamari lattice satisfies some functional equation given by Chapoton. Our final contributions deals with the $m$-Tamari lattices. This family of lattices is a generalization of the classical Tamari lattice. It was introduced by Bergeron and Préville-Ratelle and was only known in terms of paths. We give the description of this order in terms of some family of binary trees, in bijection with $m+1$-ary trees. Thus, we generalize our previous results and obtain a recursive formula counting the number of elements smaller than or equal to a given one and a new proof of the functional equation. We finish with the description of some new $"m"$ Hopf algebras which are generalizations of the known $FQSym$ on permutations and $PBT$ on binary trees

The present thesis analyses the proposal of CubeHash with spe- cial emphasis on the following papers: "Inside the Hypercube" [1], "Sym- metric States and Their Improved Structure" [7] and "Linearisation Frame- work for Collision Attacks" [6]. The CubeHash algorithm is presented in a concise manner together with a proof that the CubeHash round function R : ({0, 1}32 )32 → ({0, 1}32 )32 is a permutation. The results of [1] and [7] con- cerning the CubeHash symmetric states are reviewed, corrected and substan- tiated by proofs. More precisely, working with a definition of D-symmetric state, based on [7], the thesis proves both that for V = Z4 2 and its linear subspace D, there are 22 |V | |D| D-symmetric states and an internal state x is D-symmetric if and only if the state R(x) is D-symmetric. In response to [1], the thesis presents a step-by-step computation of a lower bound for the num- ber of distinct symmetric states, explains why the improved preimage attack does not work as stated and gives a mathematical background for a search for fixed points in R. The thesis further points out that the linearisation method from [6] fails to consider the equation (A ⊕ α) + β = (A + β) ⊕ α (∗), present during the CubeHash iteration phase. Necessary and sufficient conditions for A being a solution to (∗) are...

Diversas aplicações de redes de sensores sem fio necessitam de serviços de segurança, como confidencialidade, integridade e autenticação de origem de dados. Contudo, dadas as limitações de processamento, memória e suprimento de energia dos dispositivos, os mecanismos de segurança tradicionais podem causar efeitos indesejáveis na rede, como atraso na comunicação e aumento no consumo de energia, impondo obstáculos para seu uso na tecnologia em questão. Muitas propostas de esquemas de segurança baseados em criptografia simétrica projetados especificamente para redes de sensores sem fio são encontradas na literatura. Contudo, essas soluções são focadas na segurança salto-a-salto. Tal abordagem é adequada para garantir a segurança dos enlaces deste tipo de rede, mas não garante a segurança na comunicação fim-a-fim. Neste trabalho são apresentados cenários e desafios de implementação de segurança neste tipo de rede, e a concepção, o projeto e a implementação de uma arquitetura de segurança para redes de sensores sem fio, que tem como objetivos: prover segurança na comunicação fim-a-fim; permitir a interoperabilidade entre diferentes sistemas; e possibilitar uma maior flexibilidade em relação à utilização de chaves criptográficas em diferentes cenários e topologias. Adicionalmente, a solução proposta suporta ativação e desativação de seus serviços em tempo de execução. O projeto da referida arquitetura, atuante na camada de aplicação da pilha de protocolos de rede, foi construído com base na análise das características de arquiteturas encontradas na literatura, bem como de estratégias adotadas por estas. Para a construção da implementação foram selecionados mecanismos e algoritmos criptográficos a partir da avaliação de desempenho que considerou assimétricas de uso de memória, tempo de execução e consumo de energia. Como resultados são apresentados a especificação da arquitetura, a avaliação qualitativa da mesma e a avaliação de desempenho da implementação desenvolvida como prova de conceito. Além disso, é apresentada uma análise do impacto de diferentes topologias e características de disposição na tarefa de distribuição de chaves criptográficas em redes de sensores sem fio. / Many wireless sensor networks applications need security services, such as confidentiality, data integrity and data source authentication. On the other hand, because of device limitations, security mechanisms may affect the network energy consumption and communication delay, which impose a great challenge for practical implementation of security mechanisms in such scenario. Many solutions based on symmetric cryptography were proposed for the specific challenges of wireless sensor networks. Nevertheless, they are focused on hop-by-hop security. Such approach is suited to provide link-layer security, but it cannot guarantee end-to-end security. This work presents scenarios and challenges to implement security in wireless sensor networks, and the conception, design and implementation of a security architecture, which aims to provide: security in end-to-end communication; interoperability between different systems, and enable greater flexibility in cryptographic keys distribution in different scenarios and topologies. Additionally, the proposed solution supports on-the-y adjustment of its security services. The architecture design, which targets the application layer of the network protocol stack, was based on the main properties of the architectures found in literature as well as adopted strategies. For the implementation, mechanisms and cryptographic algorithms were selected through the performance evaluation that considers memory usage, execution time and power consumption as metrics. The results were the architecture specification and its qualitative analysis, and the performance evaluation of the implementation developed as proof of concept. Furthermore, we present an analysis of topology and deployment impact on key distribution task.

Os modelos normais de regressão têm sido utilizados durante muitos anos para a análise de dados. Mesmo nos casos em que a normalidade não podia ser suposta, tentava-se algum tipo de transformação com o intuito de alcançar a normalidade procurada. No entanto, na prática, essas suposições sobre normalidade e linearidade nem sempre são satisfeitas. Como alternativas à técnica clássica, foram desenvolvidas novas classes de modelos de regressão. Nesse contexto, focamos a classe de modelos em que a distribuição assumida para a variável resposta pertence à classe de distribuições simétricas. O objetivo geral desse trabalho é a modelagem desta classe no contexto bayesiano, em particular a modelagem da classe de modelos não-lineares heterocedásticos simétricos. Vale ressaltar que esse trabalho tem ligação com duas linhas de pesquisa, a saber: a inferência estatística abordando aspectos da teoria assintótica e a inferência bayesiana considerando aspectos de modelagem e critérios de seleção de modelos baseados em métodos de simulação de Monte Carlo em Cadeia de Markov (MCMC). Uma primeira etapa consiste em apresentar a classe dos modelos não-lineares heterocedásticos simétricos bem como a inferência clássica dos parâmetros desses modelos. Posteriormente, propomos uma abordagem bayesiana para esses modelos, cujo objetivo é mostrar sua viabilidade e comparar a inferência bayesiana dos parâmetros estimados via métodos MCMC com a inferência clássica das estimativas obtidas por meio da ferramenta GAMLSS. Além disso, utilizamos o método bayesiano de análise de influência caso a caso baseado na divergência de Kullback-Leibler para detectar observações influentes nos dados. A implementação computacional foi desenvolvida no software R e para detalhes dos programas pode ser consultado aos autores do trabalho / The normal regression models have been used for many years for data analysis. Even in cases where normality could not be assumed, was trying to be some kind of transformation in order to achieve the normality sought. However, in practice, these assumptions about normality and linearity are not always satisfied. As alternatives to classical technique new classes of regression models were developed. In this context, we focus on the class of models in which the distribution assumed for the response variable belongs to the symmetric distributions class. The aim of this work is the modeling of this class in the bayesian context, in particular the modeling of the nonlinear models heteroscedastic symmetric class. Note that this work is connected with two research lines, the statistical inference addressing aspects of asymptotic theory and the bayesian inference considering aspects of modeling and criteria for models selection based on simulation methods Monte Carlo Markov Chain (MCMC). A first step is to present the nonlinear models heteroscedastic symmetric class as well as the classic inference of parameters of these models. Subsequently, we propose a bayesian approach to these models, whose objective is to show their feasibility and compare the estimated parameters bayesian inference by MCMC methods with the classical inference of the estimates obtained by GAMLSS tool. In addition, we use the bayesian method of influence analysis on a case based on the Kullback-Leibler divergence for detecting influential observations in the data. The computational implementation was developed in the software R and programs details can be found at the studys authors

Vasilhas de pedra-sabão vêm sendo produzidas e utilizadas em Minas Gerais desde 2000 a.C., sendo reconhecidas hoje como artigos da indústria e comércio cultural, representativos de uma identidade mineira fortemente ligada ao Barroco. Sua ocorrência em sítios arqueológicos e museus, bem como o volume, o alcance de sua produção e comércio atual, chamam a atenção para a importância destes artefatos na história de Minas. Apesar disso, evidencia-se a falta de estudos que contemplem esta categoria de artefatos, levando a uma grande confusão no que se refere às informações sobre as vasilhas de pedra-sabão, instituindo inclusive uma memória forjada. A problemática da presente dissertação está focada no acompanhamento das relações coletivas estabelecidas durante a longa história de vida das vasilhas de pedra-sabão em Minas Gerais, ontem e hoje. Através da convergência de um estudo etnoarqueológico (Cachoeira do Brumado, Mariana), com o estudo das coleções (Museu de Artes e Ofícios, Belo Horizonte, Museu da Inconfidência, Ouro Preto e sítio arqueológico Faz. do Padre Fraga, Mariana), documentação primária e registro arqueológico, é possível o acompanhamento de como essas relações se davam no passado. No presente, além do estudo etnoarqueológico também houve o do processo de musealização, enfatizando como se forjou uma memória de pedra-sabão em Minas Gerais que inclusive desconsidera atores e informações acerca desse coletivo. A partir dos pressupostos da Arqueologia Simétrica, partindo da materialidade, pretendemos dissertar sobre aspectos que vão desde a busca de matéria-prima até a exposição ao público. Como resultado, pretendemos dissertar sobre possíveis caminhos para programas de extroversão do conhecimento e valorização e gestão desse patrimônio. Salientamos o altíssimo potencial para aplicação de programas de gestão pública e participativa que tenham como parte essencial de seus objetivos o aumento da qualidade de vida das comunidades. Tratando humanos e materialidade de maneira simétrica, propomos a construção de uma abordagem que traga novas vozes ao debate, considerando novas entidades na dinâmica coletiva que, uma vez vivas, influenciariam no andamento do bom mundo comum. / Soapstone vessels have been used and produced in Minas Gerais since 2.000 b.C. Nowadays, they have being recognized as industrial and cultural commerce artifacts, representing the \"mineira\" identity, strongly linked to Baroque. Their occurrence in museums and archaeological sites, as well as its volume, production ranges and actual commerce, draw attention to the importance of these artifacts to \"Minas Gerais\" history. Nevertheless, it is shown up a lack of studies that contemplates these artifacts category, causing confusion information referring to soapstone vessels, including an institutional forged memory. The problematic point of this essay is focused on the accompaniment of the collective relationship established during the soapstone vessels long history of life in \"Minas Gerais\", from the past to nowadays. Through a convergence of a Etnoarchaeology study (Cachoeira do Brumado, Mariana), with a collection study (Museu de Artes e Ofícios, Belo Horizonte, Museu da Inconfidência, Ouro Preto e sítio arqueológico Faz. do Padre Fraga, Mariana), primary documentation and archaeological record, enable the accompaniment of how the relationships happened in the past. In the present, beyond the Etnoarchaeology study, occurred the musealization process, as well, emphasizing how the soapstone vessels memory was forged in \"Minas Gerais\", disregarding actors and some information related to this collective. From these Symmetric Archeology presuppositions, starting in the materiality, we intend to discourse about the aspects that go from the raw material search to the public exposition. As a result, discourse about the possible ways to knowledge extroversion, appreciation and heritage management. Stress the high potential of applying participative and public managements, as an essential objective to increase the community quality of life. Treating people and materiality in a symmetric way, proposing the development of an approach that brings new voices to the debate, considering new entities to the dynamic collective, that once alive, would influence in the good common world course.

Neste trabalho foi realizado um grande levantamento de técnicas de esparsidade relacionadas a sistemas estáticos de energia elétrica. Tais técnicas visam, do ponto de vista computacional, ao aumento da eficiência na solução de rede elétrica objetivando, além da resolução em si, a redução dos requisitos de memória, armazenamento e tempo de processamento. Para tanto, uma extensa revisão bibliográfica foi compilada, apresentando um posicionamento histórico e uma ampla visão do desenvolvimento teórico. Os testes comparativos realizados para sistemas de 14, 30, 57 e 118 barras, sobre a implantação de três das técnicas mais empregadas, apontou a Bi-fatoração como tendo o melhor desempenho médio. Para sistemas pequenos, a Eliminação Esparsa e Sintética de Gauss apresentou melhores resultados. Este trabalho fornecerá subsídios conceituais e metodológicos a técnicos e pesquisadores da área. / In this work a great survey of sparsity techniques related to static systems of electric power was accomplished. Such techniques seek, for of the computational point of view, the increase of the efficiency in the solution of the electric net aiming, besides the resolution of itself, the reduction of memory requirements, the storage and time processing. For that, an extensive bibliographic review was compiled providing a historic positioning and a broad view of theoretic development. The comparative tests accomplished for systems of 14,30, 57 and 118 buses, on the implementation of three of the most employed techniques, it pointed out an bi-factorisation as best medium performance. For small systems, the sparse symmetric Gaussian elimination showed the best results. This work will supply conceptual and methodological subsidies to technicians and researchers of the area.

Dans cette thèse, j’étudie trois sujets reliés : les modèles de tenseurs, les nombres de Hurwitz et les polynômes de Macdonald-Koornwinder. Les modèles de tenseurs généralisent les modèles de matrices en tant qu’une approche à la gravité quantique en dimension arbitraire (les modèles de matrices donnent une version bidimensionnelle). J’étudie un modèle particulier qui s’appelle le modèle quartique mélonique. Sa spécialité est qu’il s’écrit en termes d’un modèle de matrices qui est lui-même aussi intéressant. En utilisant les outils bien établis, je calcule les deux premiers ordres de leur 1=N expansion. Parmi plusieurs interprétations, les nombres de Hurwitz comptent le nombre de revêtements ramifiés de surfaces de Riemann. Ils sont connectés avec de nombreux sujets en mathématiques contemporaines telles que les modèles de matrices, les équations intégrables et les espaces de modules. Ma contribution principale est une formule explicite pour les nombres doubles avec 3-cycles complétées d’une part. Cette formule me permet de prouver plusieurs propriétés intéressantes de ces nombres. Le dernier sujet de mon étude est les polynôme de Macdonald et Koornwinder, plus précisément les identités de Littlewood. Ces polynômes forment les bases importantes de l’algèbre des polynômes symétriques. Un des problèmes intrinsèques dans la théorie des fonctions symétriques est la décomposition d’un polynôme symétrique dans la base de Macdonald. La décomposition obtenue (notamment si les coefficients sont raisonnablement explicites et compacts) est nommée une identité de Littlewood. Dans cette thèse, j’étudie les identités démontrées récemment par Rains et Warnaar. Mes contributions incluent une preuve d’une extension d’une telle identité et quelques progrès partiels vers la généralisation d’une autre. / In this thesis, I study three related subjects: tensor models, Hurwitz numbers and Macdonald-Koornwinder polynomials. Tensor models are generalizations of matrix models as an approach to quantum gravity in arbitrary dimensions (matrix models give a 2D version). I study a specific model called the quartic melonic tensor model. Its specialty is that it can be transformed into a multi-matrix model which is very interesting by itself. With the help of well-established tools, I am able to compute the first two leading orders of their 1=N expansion. Among many interpretations, Hurwitz numbers count the number of weighted ramified coverings of Riemann surfaces. They are connected to many subjects of contemporary mathematics such as matrix models, integrable equations and moduli spaces of complex curves. My main contribution is an explicit formula for one-part double Hurwitz numbers with completed 3-cycles. This explicit formula also allows me to prove many interesting properties of these numbers. The final subject of my study is Macdonald-Koornwinder polynomials, in particular their Littlewood identities. These polynomials form important bases of the algebra of symmetric polynomials. One of the most important problems in symmetric function theory is to decompose a symmetric polynomial into the Macdonald basis. The obtained decomposition (in particular, if the coefficients are explicit and reasonably compact) is called a Littlewood identity. In this thesis, I study many recent Littlewood identities of Rains and Warnaar. My own contributions include a proof of an extension of one of their identities and partial progress towards generalization of one another.

Ocean submesoscales are the subject of increasing focus in the oceanographic literature; with instrumentation now more capable of observing them in situ and numerical models now able to reach the resolution required to more fully capture them. Submesoscales are typified by horizontal spatial scales of O(1 − 10) km, vertical scales O(100) m and time-scales of O(1) day and are known to be associated with regions of high vertical velocity and vorticity. Occurring most commonly at density fronts at the ocean surface they can control mixed layer restratification and provide an important control on fluxes between the atmosphere and the deep ocean. This thesis sets out to better understand the fundamental physical processes underpinning submesoscale instabilities using a number of idealised process models. Linear stability analysis complemented by non-linear, high-resolution simulations will be used initially to explore the ways in which submesoscale instabilities in the mixed layer may compete and interact with one another. In particular, we will investigate the way in which symmetric and ageostrophic baroclinic instabilities interact when simultaneously present in a flow, with focus on the growth rates and energetic pathways of previously unexplored dynamic instabilities that arise in this paradigm; three-dimensional, mixed symmetric-baroclinic instabilities. Further, these non-linear simulations will allow us to investigate the transition to dissipative scales that can occur in the classical Eady model via a multitude of small-scale secondary instabilities that result from primary submesoscale instabilities. Finally, observational data, taken aboard the SMILES project cruise to the Southern Ocean, helps to motivate the consideration of a new dynamical paradigm; the Eady model with superimposed high amplitude barotropic jet. Non-linear simulations investigate the extent to which the addition of such a jet is capable of damping submesoscale growth. The causes of this damping are then investigated using linear analysis. With this approach eventually demonstrated as being unable to fully explain growth rate reductions, we introduce a new framework combining potential vorticity mixing by submesoscale instabilities with geostrophic adjustment, which relaxes the flow back to a geostrophic balanced state. This framework will help to explain, conceptually, how non-linear eddies control the linear stability of the flow.

Made available in DSpace on 2016-04-27T16:57:40Z (GMT). No. of bitstreams: 1 Cleusiane Vieira Silva.pdf: 6400182 bytes, checksum: 6c0d40d69c8f2b844eb2ae51a0dac0bf (MD5) Previous issue date: 2015-12-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This thesis aimed to investigate how an environment of action and reflection that involves the pre-analysis , reflections on the pre-analysis , experimentation with students from Elementary School II , post-analysis and reflections on the post-analysis that is related to a didactic sequence on orthogonal symmetric , interferes in the Mathematics teachers knowledge , in the mentioned level. Nevertheless, this research aimed to answer the following question: how can an environment of action and reflection that is constituted in the times for the Complementary Activities, influence the Mathematics teachers (from Elementary School II) knowledge on orthogonal symmetric? The methodology that was used in this study was based on the presupposition of Didactical Engineering according to Artigue (1995) and on the contributions by Schön (1995; 2000). The theoretical referential had its basis on the Theory of Didactical Situations - Brousseau (1997) - to do a study on the influence of the didactical variables that were chosen in the Math teachers (Elementary School II) procedures and answers, as well as their students , and Margolinas (2002; 2004) to hold an analysis on the teacher s activity, in view of understanding how he/she develops the teaching practice and how it can influence students learning. Yet, the theoretical referential was based in Parzysz Picture of Geometric Paradigms (2001; 2006), in the analysis of the nature of the geometric work that is developed by teachers in the moments of construction and analysis of problem situations, and by students in the moments of interaction with these problems. The studies by Grenier (1988) were used as a reference in order to observe the students conceptions of Elementary School II according to orthogonal symmetric. The analysis of the registers that had been provided by the students made it possible the identification of conceptions that are related to the orthogonal symmetric some of them corroborate with the results from studies done by Grenier (1988); other ones seem to be specific in the group of investigated students. The analysis of registers of Math teachers also showed some concepts about orthogonal symmetric, and these conceptions seem to be related to the way this concept is presented in the course books. During the investigation, teachers had the opportunity to evaluate their own practice and reflected on the teaching methods they were using, to really know if they were working or not with their students. It was noted that an environment of action and reflection that is constituted at school, have influence on Math teachers knowledge, but this influence is limited / Esta tese teve por objetivo investigar como um ambiente de ação e reflexão, que envolve a pré-análise, reflexões sobre a pré-análise, experimentação com alunos do Ensino Fundamental II, pós-análise e reflexões sobre a pós-análise relacionadas a uma sequência didática sobre a simetria ortogonal, interfere nos saberes docentes de professores de Matemática desse mesmo nível de ensino. Portanto, foi pretensão desta pesquisa responder à seguinte questão: como um ambiente de ação e reflexão constituído nos horários destinados às Atividades Complementares (A.C.) pode influenciar os saberes docentes de professores de Matemática do Ensino Fundamental II, sobre a simetria ortogonal? A metodologia utilizada para este estudo apoiou-se nos pressupostos da Engenharia Didática, segundo Artigue (1995) e nas contribuições de Schön (1995; 2000). O referencial teórico baseou-se na Teoria das Situações Didáticas de Brousseau (1997), para fazer um estudo sobre a influência das variáveis didáticas escolhidas nos procedimentos e respostas de professores de Matemática do Ensino Fundamental II e de seus alunos, e Margolinas (2002; 2004), para realizar a análise da atividade do professor no sentido de compreender como esse profissional desenvolve sua prática docente e como esta influência na aprendizagem dos alunos. O referencial teórico baseou-se ainda no quadro dos Paradigmas Geométricos de Parzysz (2001; 2006) na análise da natureza do trabalho geométrico desenvolvido por professores nos momentos de resolução e análise das situações-problema e por alunos nos momentos de interação com essas mesmas situações-problema. Foram utilizados, como trabalho de referência, os estudos de Grenier (1988) para observar as concepções de alunos do Ensino Fundamental II quanto à simetria ortogonal. A análise nos registros fornecidos pelos alunos propiciou a identificação de concepções relativas à simetria ortogonal, algumas corroboram os resultados obtidos nos estudos realizados por Grenier (1988), outras parecem específicas do grupo de alunos investigado. A análise nos registros de professores de Matemática também expôs algumas concepções acerca da simetria ortogonal, cujas concepções parecem estar relacionadas à forma como esse conceito é apresentado nos livros didáticos. Durante a investigação, os professores avaliaram a própria prática e ponderaram sobre os métodos de ensino adotados por eles, no sentido de observar se tais métodos estão ou não surtindo efeito na aprendizagem de seus alunos. Constatou-se que um ambiente de ação e reflexão, constituído na escola, influencia nos saberes docentes de professores de Matemática, embora sua influência seja limitada