Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this paper, we use the forward recursion BCJR algorithm to estimate the symmetric information rate for Two Dimensional Magnetic Recording (TDMR) channels. In particular, we consider a TDMR read/write channel whose all components, including recording medium, write and readback processes are modeled in software. Since the primary source of noise in TDMR arises from irregularities in the recording medium and leads to highly colored and data-dependent jitter, the pattern dependent noise predictive (PDNP) algorithm is implemented to improve the accuracy and performance of SIR estimation. Furthermore, we study the performance gain of using the PDNP algorithm in SIR estimation through simulations over the Voronoi based media model for different TDMR channel configurations.

Multivariate normal distribution is commonly encountered in any field, a frequent issue is the missing values in practice. The purpose of this research was to estimate the parameters in three-dimensional covariance permutation-symmetric normal distribution with complete data and all possible patterns of incomplete data. In this study, MLE with missing data were derived, and the properties of the MLE as well as the sampling distributions were obtained. A Monte Carlo simulation study was used to evaluate the performance of the considered estimators for both cases when ρ was known and unknown. All results indicated that, compared to estimators in the case of omitting observations with missing data, the estimators derived in this article led to better performance. Furthermore, when ρ was unknown, using the estimate of ρ would lead to the same conclusion.

Apresentamos, nesta tese, uma investigação sobre a estabilização ontológica do fóton, um actante inicialmente articulado no contexto da primeira revolução da Física Quântica. Partindo dos Estudos da Ciência de Bruno Latour, entendemos a ciência como uma rede que se estende por diferentes setores da sociedade e que, portanto, lida com diferentes gêneros do discurso. Além de identificar a importância do gênero científico (artigos seminais) no processo de autonomização dos fatos científicos, também reconhecemos que, na comunidade da Física, os livros didáticos, ou manuais de instrução, tem papel crucial no estabelecimento dos problemas exemplares bem como das teorias hegemônicas, tanto que, hoje, é possível um físico se formar sem estudar os artigos originais sobre Física Quântica. Ademais, os livros didáticos de ciências da Educação Básica são, atualmente, responsáveis pela extensão da rede científica para além dos laboratórios, autonomizando actantes como o fóton em esferas mais amplas da sociedade, papel que Latour poderia chamar de representação pública. O objetivo desta pesquisa é investigar a interação entres esses três gêneros do discurso (artigos seminais, livros didáticos de ensino superior e livros didáticos de ensino médio) e interpretar como que a estabilização ontológica do fóton é mediada por cada um deles. Ao fazer isso, estamos estendendo o programa de pesquisa de Latour em, pelo menos, três sentidos. Primeiramente, incluímos o papel do contexto didático na rede da ciência. Segundo, nossos objetos de investigação estão separados por um intervalo temporal maior do que o usual (aproximadamente cem anos). E, por fim, enquanto Latour dedica-se, principalmente, ao estudo de ciências empíricas, nosso trabalho volta-se para assim chamada Física Teórica, levantando questões ontológicas não discutidas originalmente por Latour. Para dar conta dessa extensão, o presente trabalho está dividido em duas partes. A parte I (Referencial Teórico) traz três artigos sobre as ideias de Bruno Latour e as possibilidades de sua aplicação na pesquisa em Educação em Ciências, o que temos chamado de Sociologia Simétrica da Educação em Ciências ou Estudos da Educação em Ciência. A parte II (Estudos Empíricos) traz quatro artigos (dois sobre as relações entre artigos seminais e livros de ensino superior e dois sobre as relações entre artigos seminais, livros de ensino superior e livros de ensino médio). O primeiro estudo empírico traz uma análise metalinguística do artigo original de Einstein de 1905 em que o quantum é articulado pela primeira vez sem que se precisasse recorrer a nenhum laboratório. No segundo artigo, apresentamos uma análise conceitual e matemática dos artigos de de Broglie e usamos as ideias de Latour para discutir o papel dos livros didáticos na autonomização de sua teoria. No terceiro artigo, analisamos visões epistemológicas em livros didáticos de Física aprovados no Plano Nacional do Livro Didático do Ensino Médio de 2015. E, no quarto artigo, discutimos a narrativa sobre o fóton nesses livros a partir de uma articulação teórica entre ideias de Latour e Bakhtin. Nossos principais resultados apontam o papel articulador da matemática na estabilização de actantes físicos e o papel ativo dos autores de livros didáticos na estabilização do fóton a partir da hibridização de diferentes visões e do uso de mecanismos de causação reversa. / We present an investigation about the ontological stabilization of photon, an actant articulated initially in the context of the first Quantum Revolution. Departing from Bruno Latour’s Science Studies, we take Science as a network spread through different sectors of society and which deals with different speech genders. Besides of identifying the importance of the scientific gender (scientific papers) in the process of autonomation of scientific facts, we also recognize that, in the Physics community, textbooks play a crucial role in the establishment of traditional problems as well as hegemonic theories. Nowadays, someone can conclude an undergraduate course on Physics without having read any of the original papers about Quantum Physics. Furthermore, textbooks in Basic Education are responsible by the extension of the scientific network beyond laboratories, autotomizing actants like the photon in wider spheres of society, what Latour would call public representation. The goal of this research is to investigate the interaction among these three speech genders (original papers, undergraduate textbooks and high school textbooks) and to interpret how the ontological stabilization of photon is mediated by each of them. By doing so, we are extending Latour’s research program in at least three ways. First, we include the didactic context in the science network. Second, our objects of investigation are separated in time by a greater gap (about a hundred years). And third, while Latour studied chiefly the empirical sciences, our work is directed towards the Theoretical Physics, raising ontological questions that were not discussed by Latour. To allow this extension, the work is divided in two parts. Part I (Theoretical Framework) presents three papers about Bruno Latour’s ideas and the possibilities of their application in the research on Science Education, what we have called The Symmetric Sociology of Science Education. Part II (Empirical Studies) presents four papers (two about the relation between original papers and undergraduate textbooks and two about the relation between original papers and high school textbooks). The first empirical study presents a metalinguistic analysis of 1905 Einstein’s paper in which the quantum is articulated by the first time without requiring any laboratory trial. In the second paper, we present a conceptual and mathematical analysis of de Broglie’s papers and we use Latour’s ideas to discuss the role of textbooks in the atomization of his theory. In the third paper, we analyze the epistemological visions present in the Physics textbooks approved by the 2015 National Plan of Textbook. In the fourth paper, we discuss the narrative about the photon present in these books departing from a theoretical articulation between Latour’s and Bakhtin’s ideas. Our chief results point to the articulating role of mathematics in the stabilization of physical actants and the active role of textbooks authors in the stabilization of photon through the hybridization of different visions and the use of the mechanisms of reverse causation.

The main purpose of this thesis is to describe methods of constructing computer-free proofs of existence of finite groups and give useful techniques to perform double coset enumeration of groups with symmetric presentations over their control groups.

Cette thèse s’intéresse à deux aspects de la cryptologie symétrique liés à l’utilisation de matrices MDS dans les couches de diffusion linéaires de primitives. Une première partie se fonde sur les conceptions de couches de diffusion linéaires de schémas de chiffrement symétrique à partir de matrices MDS. Les associations entre matrices récursives, respectivement circulantes, et polynômes sont calquées pour construire de nouvelles associations entre d’autres structures de matrices et des éléments d’anneaux de polynômes non commutatifs de Ore. À l’instar des matrices récursives et circulantes, ces structures bénéficient d’implémentations matérielles légères. Des codes de Gabidulin dérivent des méthodes de construction directe de telles matrices, optimales en termes de diffusion, proches d’involutions pour l’implémentation. La seconde partie développe une attaque par différenciation de permutations dont l’architecture s’inspire de l’AES. L’utilisation d’une couche de diffusion linéaire locale avec une matrice MDS induit une description macroscopique de la propagation de valeurs de différences à travers les étapes du chiffrement. Des chemins différentiels tronqués apparaissent, qui servent de point de départ à la conception d’attaques rebond. Les travaux présentés généralisent les attaques rebond connues à l’exploitation de chemins différentiels tronqués structurés non issus d’avalanches libres. Cette structure permet de ne pas consommer tous les degrés de libertés au cours d’une seule étape algorithmique mais de les répartir en trois étapes. Une attaque sur 11 tours d’une permutation de Grostl-512 est alors déployée. / This thesis focuses on two aspects of symmetric cryptology related to the use of MDS matrices as building blocks of linear layers for symmetric primitives. A first part handles designs of linear layers for symmetric ciphers based upon MDS matrices. Associations between recursive, respectively circulant, matrices and polynomials are reproduced between other matrix structures and elements in non-commutative polynomial rings of Ore. As for recursive and circulant matrices, those structures come along with lightweight hardware implementations. From Gabidulin codes are derived direct constructions of MDS matrices with properties close to involution from hardware perspectives. The second part is about distinguishing attacks on an exemple of AES-like permutations. The use of some MDS matrix to build the linear layer induces a macroscopic description of differential trails through the different steps of the algorithm computing the permutation. Truncated differential path appears, from which rebound attack are built. Original work here generalizes rebound attack applied on permutations of GROSTL-512 from structured differential path not raised from free propagations of differences. This structure allows not to consume all degrees of freedom in a simple algorithmic step but to divide this comsumption into three algorithmic steps. An attack of a reduced-round version with 11 rounds of one permutation of GROSTL-512 can then be mounted.

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.

Le thème central de cette thèse cumulative est l’étude de l’intrication multi-partite quantique pour des systèmes de dimension finie. Nous avons developpé un algorithme numérique basé sur un problème d’optimisation semi-définie, qui permet de décider si un état est intriqué ou pas en un nombre fini d’itérations. Cet algorithme est une extension d’algorithmes déjà connus qui ne permettent pas de conclure lorsque l’état en question est séparable. Dans notre cas, si l’état est séparable, l’algorithme permet d’obtenir une décomposition de l’état en une mixture d’états séparables. Ces résultats ont été obtenus en exploitant la correspondance entre le problème de l’intrication et le problème des moments tronqués (truncated moment problem). Nous avons aussi développé une nouvelle manière d’exprimer l’état partiellement transposé d’un état symétrique de plusieurs qubits, simplifiant par la-même nombre de résultats bien connus en théorie de l’intrication.Cette nouvelle manière d’écrire le critère de transposée partielle unifie différentes interprétations et formulations alternatives dudit critère, et fait partie intégrante de notre algorithme d’optimisation semi-définie.Nous avons aussi étudié en détails les propriétés géométriques des états intriqués de deux qubits : nous avons pu répondre à la question de savoir à quelle distance un état pur est de l’enveloppe convexe des états symétriques et séparables, en donnant une formule explicite de l’état symétrique et séparable le plus proche — la distance étant celle de Hilbert-Schmidt. Pour les états mixtes nous avons pu obtenir et une borne supérieure numérique et une borne inférieure analytique pour cette distance. Pour un plus grand6nombre de qubits, nous nous sommes intéressés à la boule des états absolument classique,c’est à dire des états symétriques de plusieurs qubits qui restent séparables sous n’importe quelle transformation unitaire. Nous avons trouvé une borne inférieure analytique pour le rayon de cette boule autour de l’état maximallement mixte ainsi qu’une borne supérieure numérique, cette dernière ayant été obtenue en cherchant un état intriqué aussi proche que possible de l’état maximallement mixte.La représentation tensorielle d’un état symétrique de plusieurs qubits, autrement dit de l’état d’un spin j, nous a permis d’étudier des propriétés de l’intrication en nous basant sur le spectre du tenseur (valeurs propres du tenseur). Le caractère défini du tenseur est relié à l’intrication de l’état qu’il représente, donnant la possibilité de détecter la présence d’intrication à l’aide de la valeur propre minimale du tenseur. Toutefois, les valeurs propres du tenseur sont autrement plus compliquée à calculer que les valeurs propres matricielle, rendant l’analyse numérique plus délicate. La relation entre la valeur propre minimale du tenseur et la quantité d’intrication présente dans l’état a aussi été étudiée.Il en ressort que les deux quantités sont étroitement corrélées pour des systèmes de petite taille, c’est à dire jusqu’à six qubits. L’étude de ces corrélations a nécessité une méthode indépendante pour jauger de la quantité d’intrication présente dans un état. Pour cela nous avons amélioré des méthodes numériques pour déterminer la distance entre un état et l’ensemble composé des états symétriques et séparables, en utilisant une combinaison d’algorithmes d’optimisation quadratique et d’optimisation linéaire. La représentation tensorielle des états symétriques de plusieurs qubits a aussi été utilisée pour définir formellement une nouvelle classe de tenseurs, appellés "regularly decomposable tensors",qui correspond à l’ensemble des états symétriques et séparables de plusieurs qubits. / The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem.Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm.The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance.For a larger number of qubits we investigated the ball of absolutely classical states, i.e.~symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state.The tensor representation of a symmetric multi-qubit state, or spin-$j$ state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging.The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e.~for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming.The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.

Dans cette thèse, nous nous intéressons au problème de tester en grande dimension l'uniformité sur la sphère-unité $S^{p_n-1}$ (la dimension des observations, $p_n$, dépend de leur nombre, $n$, et être en grande dimension signifie que $p_n$ tend vers l'infini en même temps que $n$). Nous nous restreignons dans un premier temps à des contre-hypothèses ``monotones'' de densité croissante le long d'une direction ${\pmb \theta}_n\in S^{p_n-1}$ et dépendant d'un paramètre de concentration $\kappa_n>0$. Nous commençons par identifier le taux $\kappa_n$ auquel ces contre-hypothèses sont contiguës à l'uniformité ;nous montrons ensuite grâce à des résultats de normalité locale asymptotique, que le test d'uniformité le plus classique, le test de Rayleigh, n'est pas optimal quand ${\pmb \theta}_n$ est connu mais qu'il le devient à $p$ fixé et dans le cas FvML en grande dimension quand ${\pmb \theta}_n$ est inconnu.Dans un second temps, nous considérons des contre-hypothèses ``axiales'', attribuant la même probabilité à des points diamétralement opposés. Elles dépendent aussi d'un paramètre de position ${\pmb \theta}_n\in S^{p_n-1}$ et d'un paramètre de concentration $\kappa_n\in\R$. Le taux de contiguïté s'avère ici plus élevé et suggère un problème plus difficile que dans le cas monotone. En effet, le test de Bingham, le test classique dans le cas axial, n'est pas optimal à ${\pmb \theta}_n$ inconnu et $p$ fixé, et ne détecte pas les contre-hypothèses contiguës en grande dimension. C'est pourquoi nous nous tournons vers des tests basés sur les plus grande et plus petite valeurs propres de la matrice de variance-covariance et nous déterminons leurs distributions asymptotiques sous les contre-hypothèses contiguës à $p$ fixé.Enfin, à l'aide d'un théorème central limite pour martingales, nous montrons que sous certaines conditions et après standardisation, les statistiques de Rayleigh et de Bingham sont asymptotiquement normales sous l'hypothèse d'invariance par rotation des observations. Ce résultat permet non seulement d'identifier le taux auquel le test de Bingham détecte des contre-hypothèses axiales mais aussi celui auquel il détecte des contre-hypothèses monotones. / In this thesis we are interested in testing uniformity in high dimensions on the unit sphere $S^{p_n-1}$ (the dimension of the observations, $p_n$, depends on their number, and high-dimensional data are such that $p_n$ diverges to infinity with $n$).We consider first ``monotone'' alternatives whose density increases along an axis ${\pmb \theta}_n\in S^{p_n-1}$ and depends on a concentration parameter $\kappa_n>0$. We start by identifying the rate at which these alternatives are contiguous to uniformity; then we show thanks to local asymptotic normality results that the most classical test of uniformity, the Rayleigh test, is not optimal when ${\pmb \theta}_n$ is specified but becomes optimal when $p$ is fixed and in the high-dimensional FvML case when ${\pmb \theta}_n$ is unspecified.We consider next ``axial'' alternatives, assigning the same probability to antipodal points. They also depend on a location parameter ${\pmb \theta}_n\in S^{p_n-1}$ and a concentration parameter $\kappa_n\in\R$. The contiguity rate proves to be higher in that case and implies that the problem is more difficult than in the monotone case. Indeed, the Bingham test, the classical test when dealing with axial data, is not optimal when $p$ is fixed and ${\pmb \theta}_n$ is not specified, and is blind to the contiguous alternatives in high dimensions. This is why we turn to tests based on the extreme eigenvalues of the covariance matrix and establish their fixed-$p$ asymptotic distributions under contiguous alternatives.Finally, thanks to a martingale central limit theorem, we show that, under some assumptions and after standardisation, the Rayleigh and Bingham test statistics are asymptotically normal under general rotationally symmetric distributions. It enables us to identify the rate at which the Bingham test detects axial alternatives and also monotone alternatives. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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