In der Dissertation werden Schranken für Abschätzungen des Kommutators in verschiedenen Normen gegeben. Den Ausgangspunkt bildet die Frobenius-Norm, für die eine überraschend kleine Schranke bewiesen werden kann. Auf diesem Resultat aufbauend lassen sich über eine spezielle Adaption der Interpolationsmethode von Riesz-Thorin scharfe Schranken bei Verwendung von Schatten- und Vektornormen weitestgehend bestimmen. Es werden ferner die Fälle untersucht, in denen die obere Abschätzung erreicht wird (sog. Maximalität). Eine wichtige Rolle spielen verschiedene Darstellungen der Ungleichung, welche vielfältige Interpretationsmöglichkeiten eröffen und Verbindungen der algebraischen Abschätzung zu einem wichtigen Satz der Differentialgeometrie über die Krümmung von Mannigfaltigkeiten aufzeigen.

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No. of bitstreams: 2 Dissertação - Jonas Augusto Kunzler - 2015.pdf: 5934373 bytes, checksum: a736817202816bba60673f1a39184580 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-04-14 / New technologies that have emerged from the transistor advent enabled digital signal processing techniques were aggregated to systems operated or partially in analog improving performance of these systems. Added to this the use of sensor arrangements which made it possible to identify the directions of signals of interest and are used in critical areas of knowledge, such as in tracking radar systems; in astronomy; in sonar systems; mobile communications; the estimation of direction of arrival; in seismology and medical diagnosis and treatment. This work aims to study specific features of an estimation method of direction of arrival based on a linear array of sensors with special attention to mobile communications. Some methods have been proposed in order to get the position of a source of electromagnetic waves and can cite the MUSIC, the MODEX and SEAD, the latter of fundamental importance to this work because it is the basis of conducted research and because it is a new method, lacked further clarification with regard to the differential spectrum, their origin, meaning and importance, as well as obtaining an analytical expression to describe their conduct on the variables that compose the system. Based on eigenvalue decomposition of the correlation matrix has been observed that the differential spectrum is basically a matrix norm calculation, with the largest eigenvalue of the matrix is ​​also standard 2-induced vector. It was proposed to use the Frobenius norm which is simpler to be computed, and consequently requires less computational effort. Moreover, the behavior of the angular spectrum calculated using the Frobenius norm is fully described by the sum of cosines with the formulation described for each part which composes the calculation of the standard. Through this outcome was possible to analyze aspects related to angular resolution, the number of signal sources, the number of sensors, the influence of noise and the correlation between sources. / Novas tecnologias que surgiram a partir do advento transistor permitiram técnicas de processamento digital de sinais fossem agregadas a sistemas que operavam ou parcialmente de forma analógica aprimorando o desempenho desses sistemas. Soma-se a isto a utilização de arranjos de sensores que possibilitaram a identificação das direções dos sinais de interesse e são empregados em importantes áreas do conhecimento, como por exemplo, em sistemas de rastreamento por radar; na Astronomia; em Sistemas sonares; nas comunicações móveis; na estimação de direção de chegada; na sismologia e no diagnóstico e tratamento médico. Este trabalho tem como objetivo estudar características específicas de um método de estimação de direção de chegada baseado em um arranjo linear de sensores com atenção especial em comunicações móveis. Alguns métodos foram propostos com o fim de obter a posição de uma fonte de ondas eletromagnéticas, podendo-se citar o MUSIC, o MODEX e o SEAD, este último de fundamental importância para este trabalho, pois, ele é a base da investigação conduzida e por se tratar de um método novo, carecia de mais esclarecimentos no que diz respeito ao espectro diferencial, sua origem, significado e importância, como também a obtenção de uma expressão analítica que descrevesse seu comportamento em função das variáveis que compões o sistema. Baseado na decomposição em autovalores da matriz de correlação observou-se que o espectro diferencial é basicamente um cálculo de norma matricial, sendo que o maior autovalor da matriz é também a norma 2 induzida por vetor. Propôs-se a utilização da norma de Frobenius que é mais simples de ser calculada e, consequentemente, exige menos esforço computacional. Além disso, o comportamento do espectro angular calculado com a norma de Frobenius é totalmente descrito através da soma de cossenos com a formulação descrita para cada parcela que compõe o cálculo da norma. Através deste resultado foi possível analisar aspectos referentes à resolução angular, ao número de fontes de sinal, ao número de sensores, à influência do ruído e à correlação entre as fontes.

Cette thèse s'intéresse aux fonctions L, à leurs aspects explicites et à leurs applications Dans le premier chapitre, nous donnons une définition précise de ce que nous appelons une fonction L ainsi que leurs principales propriétés, notamment concernant les invariants appelés paramètres locaux. Ensuite, nous traitons le cas des fonctions L d'Artin. Pour celles-ci, nous avons créé un programme dans le logiciel PARI/GP donnant les coefficients et les invariants d'une fonction L d'Artin lorsque le corps de base est Q.Le deuxième chapitre explicite un théorème dû à Henryk Iwaniec et Emmanuel Kowalski permettant de différencier deux fonctions L générales en considérant leurs paramètres locaux pour tous les premiers jusqu'à une certaine borne théorique.Dans la suite, nous constaterons que distinguer la somme des paramètres locaux de fonctions L d'Artin revient à séparer les caractères associés par les automorphismes de Frobenius. Ce sera l'objet du troisième chapitre qui est à relier au théorème de Chebotarev. En appliquant notre résultat à des caractères conjugués du groupe alterné, on obtient une borne sur un nombre premier p donnant l'écriture de la factorisation modulo p d'un polynôme répondant à certains critères. Ce travail est à comparer avec un résultat de Joël Bellaïche (2013). Nous illustrons enfin numériquement nos résultats en étudiant l'évolution de la borne sur des polynômes de la forme X^n+uX+v avec n=5, 7 et 13. / This thesis focuses on L-functions, their explicit aspects and their applications.In the first chapter, we give a precise definition of L-functions and their main properties, especially about the invariants called local parameters. Then, we deal with Artin L-functions. For them, we have created a computer program in PARI/GP which gives the coefficients and the invariants for an Artin L-function above Q.In the second chapter, we make explicit a theorem of Henryk Iwaniec and Emmanuel Kowalski, which distinguishes between two L-functions by considering their local parameters for primes up to a theoretical bound.Actually, distinguishing between sums of local parameters of Artin L-functions is the same as separating the associated characters by the Frobenius automorphism. This is the subject of the third chapter, that can be related to Chebotarev Theorem. By applying the result to conjugate characters of the alternating group, we get a bound for a prime p giving the factorization modulo $p$ of a certain polynomial. This work has to be compared with a result from Joël Bellaïche (2013).Finally, we numerically illustrate our results by studying the evolution of the bound on polynomials X^n+uX+v, for n=5, 7 and 13.

Cette thèse est composée de deux parties. Nous étudions dans la première la conjecture de Wilf pour les semi-groupes numériques et la résolvons dans certains cas. Dans la seconde nous considérons une classe de semi-groupes presque arithmétiques et donnons pour ces semi-groupes des formules explicites pour la base d’Apéry, le nombre de Frobenius, et les nombres de pseudo-Frobenius. Nouscaractérisons aussi ceux qui sont symétriques (resp. pseudo-symétriques). / The thesis is made up of two parts. We study in the first part Wilf’s conjecture for numerical semigroups. We give an equivalent form of Wilf’s conjecture in terms of the Apéry set, embedding dimension and multiplicity of a numerical semigroup. We also give an affirmative answer for the conjecture in certain cases. In the second part, we consider a class of almost arithmetic numerical semigroups and give for this class of semigroups explicit formulas for the Apéry set, the Frobenius number, the genus and the pseudo-Frobenius numbers. We also characterize the symmetric (resp. pseudo-symmetric) numerical semigroups for this class of numerical semigroups.

La cryptologie consiste en l’étude des techniques utilisées par deux entités pour communiquer en secret en présence d’une troisième. Les propriétés mathématiques qui sous-tendent ces techniques garantissent que leur attaque reste infaisable en pratique par un adversaire malveillant. Ainsi, les protocoles s’appuient sur diverses hypothèses, comme la di fficulté présumée de factoriser des entiers ou de calculer le logarithme discret d’un élément arbitraire dans certains groupes. Cette thèse qui porte sur le problème du logarithme discret dans les corps finis s’articule autour de trois volets.Nous exposons les résultats théoriques associés au problème sans considération du groupe cible, détaillant ainsi les classes de complexité auxquelles il appartient ainsi que di fférentes approches pour tenter de le résoudre.L’étude du problème dans les corps finis commence en tant que telle par les corps présentant une caractéristique de petite taille relativement à l’ordre total du corps en question. Cette seconde partie résulte sur l’exposition d’un algorithme par représentation de Frobenius dont une application a aboutit au record actuel de calcul de logarithme discret en caractéristique 3.Pour les corps de moyenne ou grande caractéristiques, une autre méthode est requise. Le crible par corps de nombres (NFS) multiples obtient les complexités asymptotiques les plus basses pour un corps arbitraire. Un dernier chapitre introduit la notion de matrice presque creuse. L’élaboration d’un nouvel algorithme spécifique qui explicite le noyau d’une telle matrice facilite en pratique l’étape d’algèbre sous-jacente à toute variante de NFS. / Cryptography is the study of techniques for secure communication in the presence of third parties, also called adversaries. Such techniques are detailed in cryptosystems, explaining how to securely encode and decode messages. They are designed around computational hardness assumptions related to mathematical properties, making such algorithms hard to break in practice by any adversary. These protocols are based on the computational difficulty of various problems which often come from number theory, such as integer factorization or discrete logarithms computations. This manuscript focuses on the discrete logarithm problem in finite fields and revolves around three axes.First we detail classical results about the problem without any consideration to the target group. We deal with complexity classes and some general methods that do not need any information on the group.The study of the discrete logarithm problem in finite fields starts with small characteristic ones. The aim is to present a Frobenius representation algorithm that leads to the current discrete logarithm record in characteristic 3.For medium or large characteristics finite fields, another approach is required. The multiple number field sieve reaches the best asymptotic heuristic complexities for this double range of characteristics. We also introduce the notion of nearly sparse matrices. Designing a new algorithm dedicated to explicitly give the kernel of such a matrix eases in practice the linear algebra step of any variant of the number field sieve.

In der Dissertation werden Schranken für Abschätzungen des Kommutators in verschiedenen Normen gegeben. Den Ausgangspunkt bildet die Frobenius-Norm, für die eine überraschend kleine Schranke bewiesen werden kann. Auf diesem Resultat aufbauend lassen sich über eine spezielle Adaption der Interpolationsmethode von Riesz-Thorin scharfe Schranken bei Verwendung von Schatten- und Vektornormen weitestgehend bestimmen. Es werden ferner die Fälle untersucht, in denen die obere Abschätzung erreicht wird (sog. Maximalität). Eine wichtige Rolle spielen verschiedene Darstellungen der Ungleichung, welche vielfältige Interpretationsmöglichkeiten eröffen und Verbindungen der algebraischen Abschätzung zu einem wichtigen Satz der Differentialgeometrie über die Krümmung von Mannigfaltigkeiten aufzeigen.

In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid.

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, <em>i. e. </em> in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, <em>i. e. </em> develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2-Calabi–Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, it is natural to try to model the cluster combinatorics via a Frobenius category, with the indecomposable projective-injective objects corresponding to these special variables. Amiot–Iyama–Reiten show how Frobenius categories admitting (d-1)-cluster-tilting objects arise naturally from the data of a Noetherian bimodule d-Calabi–Yau algebra A and an idempotent e of A such that A/< e > is finite dimensional. In this work, we observe that this phenomenon still occurs under the weaker assumption that A and A^op are internally d-Calabi–Yau with respect to e; this new definition allows the d-Calabi–Yau property to fail in a way controlled by e. Under either set of assumptions, the algebra B=eAe is Iwanaga–Gorenstein, and eA is a cluster-tilting object in the Frobenius category GP(B) of Gorenstein projective B-modules. Geiß–Leclerc–Schröer define a class of cluster algebras that are, by construction, modelled by certain Frobenius subcategories Sub(Q_J) of module categories over preprojective algebras. Buan–Iyama–Reiten–Smith prove that the endomorphism algebra of a cluster-tilting object in one of these categories is a frozen Jacobian algebra. Following Keller–Reiten, we observe that such algebras are internally 3-Calabi–Yau with respect to the idempotent corresponding to the frozen vertices, thus obtaining a large class of examples of such algebras. Geiß–Leclerc–Schröer also attach, via an algebraic homogenization procedure, a second cluster algebra to each category Sub(Q_J), by adding more frozen variables. We describe how to compute the quiver of a seed in this cluster algebra via approximation theory in the category Sub(Q_J); our alternative construction has the advantage that arrows between the frozen vertices appear naturally. We write down a potential on this enlarged quiver, and conjecture that the resulting frozen Jacobian algebra A and its opposite are internally 3-Calabi–Yau. If true, the algebra may be realised as the endomorphism algebra of a cluster-tilting object in a Frobenius category GP(B) as above. We further conjecture that GP(B) is stably 2-Calabi–Yau, in which case it would provide a categorification of this second cluster algebra.

Cette thèse se compose de deux parties. Partie 1 étudie la décomposition des groupes de cohomologie pour une famille de courbes non hyperelliptiques de genre 3 avec une involution, et le bénéfice d'une telle décomposition dans le calcul de Frobenius utilisant l'algorithme de Kedlaya. L'involution d'une telle courbe C induit un morphisme de degré 2 vers une courbe elliptique E, ce qui donne une décomposition de Jac(C) en E et en une surface abélienne A, à partir desquelles le Frobenius sur C peut être récupérée. En E, le polynôme caractéristique du Frobenius peut être calculé en utilisant un algorithme efficace et rapide en pratique. En travaillant avec le sous-groupe V de $H^1_{MW}(C)$, on obtient une meilleure constante que l'application directe de la méthode de Kedlaya à C. À ma connaissance, ceci est la première utilisation de la décomposition de la cohomologie induite par une décomposition (à isogénie près) de la jacobienne en l'algorithme de Kedlaya. Dans partie 2, je propose une nouvelle approche aux distributions de Frobenius et aux groupes de Sato-Tate, qui utilise les relations d'orthogonalité des caractères irréductibles du groupe de Lie USp(2g) et ses sous-groupes. Dans ce but, je présente d'abord une méthode simple pour calculer les caractères irréductibles de USp(2g), et puis je développe un algorithme basé sur la formule de Brauer-Klimyk. Les avantages de cette nouvelle approche sont examinés en détail. J'utilise aussi la famille de courbes dans partie 1 comme une étude de cas. Les analyses et les comparaisons montrent que l'approche par la théorie des caractères est un outil plus intrinsèque et très prometteur pour l'étude des groupes de Sato-Tate. / This thesis consists of two parts. Part 1 studies the decomposition of cohomology groups induced by automorphisms for a family of non-hyperelliptic genus 3 curves with involution, and I investigate the benefit of such decomposition in the computation of Frobenius using Kedlaya's algorithm. The involution of a curve C in this family induces a degree 2 map to an elliptic curve E, which gives a decomposition of the Jacobian of C into E and an abelian surface A, from which the Frobenius on C can be recovered. On E, the characteristic polynomial of the Frobenius can be computed using an efficient and fast algorithm. By working with the cohomology subgroup V of $H^1_{MW}(C)$, we get a constant speed-up over a straightforward application of Kedlaya's method to C. To my knowledge, this is the first use of decomposition of the cohomology induced by an isogeny decomposition of the Jacobian in Kedlaya's algorithm. In Part 2, I propose a new approach to Frobenius distributions and Sato-Tate groups, which uses the orthogonality relations of the irreducible characters of the compact Lie group USp(2g) and its subgroups. To this purpose, I first present a simple method to compute the irreducible characters of USp(2g), then I develop an algorithm based on the Brauer-Klimyk formula. The advantages of this new approach to Sato-Tate groups are examined in detail. The results show that the error grows slowly. I also use the family of genus 3 curves studied in Part 1 as a case study. The analyses and comparisons show that the character theory approach is a more intrinsic and very promising tool for studying Sato-Tate groups.