Abstract The focus of this thesis is on two functions: the exponential function and Euler’s factorial series. By constructing explicit Padé approximations, we are able to improve lower bounds for linear forms in the values of these functions. In particular, the dependence on the height of the coefficients of the linear form will be sharpened in the lower bound. The first chapter contains some necessary definitions and auxiliary results needed in later chapters.We give precise definitions for a transcendence measure and Padé approximations of the second type. Siegel’s lemma will be introduced as a fundamental tool in Diophantine approximation. A brief excursion to exterior algebras shows how they can be used to prove determinant expansion formulas. The reader will also be familiarised with valuations of number fields. In Chapter 2, a new transcendence measure for e is proved using type II Hermite-Padé approximations to the exponential function. An improvement to the previous transcendence measures is achieved by estimating the common factors of the coefficients of the auxiliary polynomials. The exponential function is the underlying topic of the third chapter as well. Now we study the common factors of the maximal minors of some large block matrices that appear when constructing Padé-type approximations to the exponential function. The factorisation of these minors is of interest both because of Bombieri and Vaaler’s improved version of Siegel’s lemma and because they are connected to finding explicit expressions for the approximation polynomials. In the beginning of Chapter 3, two general theorems concerning factors of Vandermonde-type block determinants are proved. In the final chapter, we concentrate on Euler’s factorial series which has a positive radius of convergence in p-adic fields. We establish some non-vanishing results for a linear form in the values of Euler’s series at algebraic integer points. A lower bound for this linear form is derived as well.

Spectrum sharing is known as a key solution to accommodate the increasing number of users and the growing demand for throughput in wireless networks. While spectrum sharing improves the data rate in sparse networks, it suffers from interference of concurrent links in dense networks. In fact, interference is the primary barrier to enhance the overall throughput of the network, especially in the medium and high signal-to-noise ratios (SNR’s). Managing interference to overcome this barrier has emerged as a crucial step in developing efficient wireless networks. This thesis deals with optimum and sub-optimum interference management-cancelation in non-cooperative networks. Several techniques for interference management including novel strategies such as interference alignment and structural coding are investigated. These methods are applied to obtain optimum and sub-optimum coding strategies in such networks. It is shown that a single strategy is not able to achieve the maximum throughput in all possible scenarios and in fact a careful design is required to fully exploit all available resources in each realization of the system. This thesis begins with a complete investigation of the capacity region of the two-user Gaussian interference channel. This channel models the basic interaction between two users sharing the same spectrum for data communication. New outer bounds outperforming known bounds are derived using Genie-aided techniques. It is proved that these outer bounds meet the known inner bounds in some special cases, revealing the sum capacity of this channel over a certain range of parameters which has not been known in the past. A novel coding scheme applicable in networks with single antenna nodes is proposed next. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher multiplexing gains can be achieved. Tools from the field of Diophantine approximation in number theory are used to show that the proposed coding scheme in fact mimics the traditional schemes used in MIMO systems where each data stream is sent along a direction and alignment happens when several streams are received along the same direction. Two types of constellation are proposed for the encoding part, namely the single layer constellation and the multi-layer constellation. Using single layer constellations, the coding scheme is applied to the two-user $X$ channel. It is proved that the total Degrees-of-Freedom (DOF), i.e. $\frac{4}{3}$, of the channel is achievable almost surely. This is the first example in which it is shown that a time invariant single antenna system does not fall short of achieving this known upper bound on the DOF. Using multi-layer constellations, the coding scheme is applied to the symmetric three-user GIC. Achievable DOFs are derived for all channel gains. It is observed that the DOF is everywhere discontinuous (as a function of the channel gain). In particular, it is proved that for the irrational channel gains the achievable DOF meets the upper bound of $\frac{3}{2}$. For the rational gains, the achievable DOF has a gap to the known upper bounds. By allowing carry over from multiple layers, however, it is shown that higher DOFs can be achieved for the latter. The $K$-user single-antenna Gaussian Interference Channel (GIC) is considered, where the channel coefficients are NOT necessarily time-variant or frequency selective. It is proved that the total DOF of this channel is $\frac{K}{2}$ almost surely, i.e. each user enjoys half of its maximum DOF. Indeed, we prove that the static time-invariant interference channels are rich enough to allow simultaneous interference alignment at all receivers. To derive this result, we show that single-antenna interference channels can be treated as \emph{pseudo multiple-antenna systems} with infinitely-many antennas. Such machinery enables us to prove that the real or complex $M \times M$ MIMO GIC achieves its total DOF, i.e., $\frac{MK}{2}$, $M \geq 1$. The pseudo multiple-antenna systems are developed based on a recent result in the field of Diophantine approximation which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds. As a byproduct of the scheme, the total DOFs of the $K\times M$ $X$ channel and the uplink of cellular systems are derived. Interference alignment requires perfect knowledge of channel state information at all nodes. This requirement is sometimes infeasible and users invoke random coding to communicate with their corresponding receivers. Alternative interference management needs to be implemented and this problem is addressed in the last part of the thesis. A coding scheme for a single user communicating in a shared medium is proposed. Moreover, polynomial time algorithms are proposed to obtain best achievable rates in the system. Successive rate allocation for a $K$-user interference channel is performed using polynomial time algorithms.

Spectrum sharing is known as a key solution to accommodate the increasing number of users and the growing demand for throughput in wireless networks. While spectrum sharing improves the data rate in sparse networks, it suffers from interference of concurrent links in dense networks. In fact, interference is the primary barrier to enhance the overall throughput of the network, especially in the medium and high signal-to-noise ratios (SNR’s). Managing interference to overcome this barrier has emerged as a crucial step in developing efficient wireless networks. This thesis deals with optimum and sub-optimum interference management-cancelation in non-cooperative networks. Several techniques for interference management including novel strategies such as interference alignment and structural coding are investigated. These methods are applied to obtain optimum and sub-optimum coding strategies in such networks. It is shown that a single strategy is not able to achieve the maximum throughput in all possible scenarios and in fact a careful design is required to fully exploit all available resources in each realization of the system. This thesis begins with a complete investigation of the capacity region of the two-user Gaussian interference channel. This channel models the basic interaction between two users sharing the same spectrum for data communication. New outer bounds outperforming known bounds are derived using Genie-aided techniques. It is proved that these outer bounds meet the known inner bounds in some special cases, revealing the sum capacity of this channel over a certain range of parameters which has not been known in the past. A novel coding scheme applicable in networks with single antenna nodes is proposed next. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher multiplexing gains can be achieved. Tools from the field of Diophantine approximation in number theory are used to show that the proposed coding scheme in fact mimics the traditional schemes used in MIMO systems where each data stream is sent along a direction and alignment happens when several streams are received along the same direction. Two types of constellation are proposed for the encoding part, namely the single layer constellation and the multi-layer constellation. Using single layer constellations, the coding scheme is applied to the two-user $X$ channel. It is proved that the total Degrees-of-Freedom (DOF), i.e. $\frac{4}{3}$, of the channel is achievable almost surely. This is the first example in which it is shown that a time invariant single antenna system does not fall short of achieving this known upper bound on the DOF. Using multi-layer constellations, the coding scheme is applied to the symmetric three-user GIC. Achievable DOFs are derived for all channel gains. It is observed that the DOF is everywhere discontinuous (as a function of the channel gain). In particular, it is proved that for the irrational channel gains the achievable DOF meets the upper bound of $\frac{3}{2}$. For the rational gains, the achievable DOF has a gap to the known upper bounds. By allowing carry over from multiple layers, however, it is shown that higher DOFs can be achieved for the latter. The $K$-user single-antenna Gaussian Interference Channel (GIC) is considered, where the channel coefficients are NOT necessarily time-variant or frequency selective. It is proved that the total DOF of this channel is $\frac{K}{2}$ almost surely, i.e. each user enjoys half of its maximum DOF. Indeed, we prove that the static time-invariant interference channels are rich enough to allow simultaneous interference alignment at all receivers. To derive this result, we show that single-antenna interference channels can be treated as \emph{pseudo multiple-antenna systems} with infinitely-many antennas. Such machinery enables us to prove that the real or complex $M \times M$ MIMO GIC achieves its total DOF, i.e., $\frac{MK}{2}$, $M \geq 1$. The pseudo multiple-antenna systems are developed based on a recent result in the field of Diophantine approximation which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds. As a byproduct of the scheme, the total DOFs of the $K\times M$ $X$ channel and the uplink of cellular systems are derived. Interference alignment requires perfect knowledge of channel state information at all nodes. This requirement is sometimes infeasible and users invoke random coding to communicate with their corresponding receivers. Alternative interference management needs to be implemented and this problem is addressed in the last part of the thesis. A coding scheme for a single user communicating in a shared medium is proposed. Moreover, polynomial time algorithms are proposed to obtain best achievable rates in the system. Successive rate allocation for a $K$-user interference channel is performed using polynomial time algorithms.

Orientador: Edson Moschim / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-04T21:57:49Z (GMT). No. of bitstreams: 1 DomingosNeto_Adriano_D.pdf: 1544823 bytes, checksum: f09f4717b59d1cce526b7c8746e53efb (MD5) Previous issue date: 2005 / Resumo: Nesta tese, propõe se três novas construções de códigos ortogonais ópticos (OOC), do tipo congruentes, tendo como base a estrutura algébrica do grupo multiplicativo do corpo de Galois GF(p), para aplicação em sistemas de comunicação utilizando a técnica de acesso múltiplo por divisão de códigos ópticos (OCDMA). Os códigos ópticos primos e códigos quadráticos são, pela primeira vez na literatura, gerados a partir de códigos de Slepian (códigos esféricos) e, códigos de resíduos quadráticos, respectivamente. Através do algoritmo da d-cadeia fechada, são obtidos os códigos de primos, como caso particular dos códigos de Slepian. Os códigos quadráticos ópticos são representados por números inteiros quadráticos binários na forma de equações de Diofanto com duas variáveis, de modo que, o reticulado Z2 ou reticulado Â2 fornecem as palavra do código quadrático. O desempenho dos códigos é avaliado usando o critério da probabilidade de erro para situações em que o receptor óptico incorpora um limitador óptico e um fotodiodo APD. O desempenho do sistema é obtido considerando os efeitos da interferência de acesso múltiplo, o ruído balístico do fotodiodo e o ruído térmico do receptor. O desempenho dos códigos propostos é comparado ao desempenho de códigos amplamente divulgados em literatura técnica. Mostra-se ainda que os códigos propostos apresentam desempenho semelhante aos códigos divulgados, tendo como vantagem uma estrutura algébrica de simples implementação e melhor sincronismo / Abstract: This thesis presents a study of optical orthogonal codes (OOe) for application in communication systems using the technique of fiber-optics code division multiple access (OCDMA). The Prime Sequence codes and Quadratic codes are, for the first time in literature, characterized as Slepian group codes (spherical codes) and Quadratic Residues codes, respectively. Through the algorithm of the closed d-chain the Prime Sequence codes are obtained, as a particular case of the Slepian codes. The Quadratic codes are represented by binary quadratic integers in the form of Diophantine equations with two variables, so that, Z2 lattice or Â3 lattice supplies the codeword of the quadratic code. Furthermore, this thesis presents three new constructions of optical orthogonal codes (OOC), construed via congruences having as base the algebraic structure of the multiplicative group of the GaloisField GF(p). The performance of the codes is evaluated using the criterion of the error probability, for situations where the optic receiver incorporates a fiber-optic limiter and a APD photodiode. The performance of the system is evaluated considering the effect of the interference of multiple access, the ballistic noise of the photodiode and the thermal noise of the receiver. The performance of the considered codes is compared with the performance of other codes found in the technical literature. It is observed that the codes considered in this thesis, in this thesis, present similar performance to the reported codes, having as advantage an algebraic structure of simple implementation and better synchronism / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Nous traitons ici de questions d’effectivité dans les problèmes de Mordell-Lang et de Schanuel où la notion de hauteur algébrique joue un rôle central.Dans un premier temps nous revisitions la méthode de Vojta-Faltings dans un cadre général, en y incluant notamment un procédé de descente uniforme qui permet d’optimiser le nombre de recours au pesant mécanisme d’approximation diophantienne. Nous proposons ensuite une application de ce résultat au problème de Mordell-Lang plus Bogomolov dans le tore, qui consiste à décrire un sousensemble algébrique X comme réunion de translatés de sous-tores inclus dans X moyennant de se restreindre à un sous-groupe de rang fini épaissi. Nous nous appuyons en particulier sur un énoncé d’Amoroso et Viada concernant le problème de Bogomolov dans ce contexte et améliorons les bornes antérieures obtenues par Rémond.Dans un second temps, nous établissons une version du théorème de Schanuel dans le cadre d’un espace adélique hermitien sur un corps de nombres. Nous donnons une estimation asymptotique du nombre de points projectifs de hauteur bornée pour une hauteur définie par une famille de normes sur les complétés en chaque place, vérifiant certaines conditions mais sans hypothèse de pureté dans le cas ultramétrique. Le terme reste obtenu est totalement explicite et linéaire en le régulateur du corps de nombres grâce au recours à une méthode introduite par Schmidt. Nous traitons également plusieurs applications de ce résultat, notamment aux problèmes de Dedekind-Weber et de Loher-Masser. / We are dealing here with effectiveness matters about the Mordell-Lang and Schanuel problems where algebraic heights play a central role.At the first time, we modify the Vojta-Faltings method in a general context by including some uniform descending process which has the advantage to optimize the number of iterations of the heavy Diophantine approximation mechanism. We then propose an application to the toric Mordell-Lang plus Bogomolov problem whose aim is to describe an algebraic subset X as the union of translates of closed, irreducible subgroups included in X when restricted to some enlarged, finite rank subgroup. In particular we use a theorem of Amoroso and Viada about the Bogomolov problem in this context and we improve the previous bound given by Rémond.At the second time, we prove a version of the theorem of Schanuel in the setting of a Hermitian adelic vector bundle over a number field. We give an asymptotic estimate for the number of projective points of bounded height for heights given by a family of norms over the completions at each place, satisfying several conditions but no purity hypothesis in the ultrametric case. The error term is totally explicit and linear with respect to the regulator of the number field through the use of Schmidt’s method. We finally give some applications of our result in particular to the Dedekind-Weber and Loher-Masser problems.

Soit X un sous-schéma fermé d'une variété abélienne A sur un corps de nombres K. L'ancienne conjecture de Mordell-Lang nous assure que X(K) est une réunion finie de sous-ensembles a_i+Bj(K) où a_i est un point de X(K) et B_i est une sous-variété abélienne de A de sorte que le translaté aj+Bj soit contenu dans X. Dans cette thèse, nous montrerons un résultat permettant de majorer la hauteur des a_i en fonctions de la hauteur de X. On en déduira une majoration pour la hauteur des solutions d'une équation aux unités. En utilisant les mêmes méthodes, on obtiendra une majoration de la même forme pour la hauteur des points entiers d'une variété abélienne(plongé dans un espace projectif) privé d'un hyperplan. / Let be X a closed subscheme of an abelian variety on a number field K. Faltings proved the Mordell-Lang conjecture: there are points a_1 , ... ,a_n in X(K) and abelian subvarieties 8_1 , ... ,B_nin A such that a_i+B_i is in X and X(K) is equal to the union of aj+B_i(K). In this thesis, we proof a result wich gives a bound for the height of the point a_i with the height of X. We obtain a bound for the solutions of an unit equation. With the same method, we proof a similar result for the height of the integers points on an abelian variety (embedded in a projective space) minus a hyperplane.

Dans cette thèse, nous appliquons des outils issus de la théorie d’Arakelov à l’étude de problèmes de géométrie diophantienne. Une notion centrale dans notre étude est la théorie des pentes des fibrés vectoriels hermitiens, introduite par Bost dans les années 90. Nous travaillons plus précisément avec sa généralisation dans le cadre adélique, inspirée par Zhang et développée par Gaudron. Ce mémoire s’articule autour de deux axes principaux. Le premier consiste en l’étude d’un remarquable théorème de géométrie diophantienne dû à Faltings etWüstholz, qui généralise le théorème du sous-espace de Schmidt. Nous commencerons par retranscrire la démonstration de Faltings et Wüstholz dans le langage de la théorie des pentes. Dans un second temps, nous établirons des variantes effectives de ce théorème, que nous appliquerons pour démontrer une généralisation effective du théorème de Liouville valable pour les points fermés d’une variété projective fixée. Ce résultat fournit en particulier une majoration explicite de la hauteur des points satisfaisant une inégalité analogue à celle du théorème de Liouville classique. Dans la seconde partie de cette thèse, nous établirons de nouvelles mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif, dans le cas dit rationnel.Notre approche utilise les arguments de la méthode de Baker revisitée par Philippon et Waldschmidt, combinés avec des outils de la théorie des pentes. Nous y intégrons un nouvel argument, inspiré par des travaux antérieurs de Bertrand et Philippon, nous permettant de contourner les difficultés introduites par le cas périodique. Cette approche évite le recours à une extrapolation sur les dérivations à la manière de Philippon et Waldschmidt. Nous parvenons ainsi à supprimer une hypothèse technique contraignante dans plusieurs théorèmes de Gaudron, tout en précisant les mesures obtenues. / In this thesis, we study diophantine geometry problems on projective varieties and commutative algebraic groups, by means of tools from Arakelov theory. A central notion in this work is the slope theory for hermitian vector bundles, introduced by Bost in the 1990s. More precisely, we work with its generalization in an adelic setting, inspired by Zhang and developed by Gaudron. This dissertation contains two major lines. The first one is devoted to the study of a remarkable theorem due to Faltings and Wüstholz, which generalizes Schmidt’s subspace theorem. We first reformulate the proof of Faltings and Wüstholz using the formalism of adelic vector bundles and the adelic slope method. We then establish some effective variants of the theorem, and we deduce an effective generalization of Liouville’s theorem for closed points on a projective variety defined over a number field. In particular, we give an explicit upper bound for the height of the points satisying a Liouville-type inequality. In the second part, we establish new measures of linear independence of logarithms over a commutative algebraic group. We focus our study on the rational case. Our approach combines Baker’s method (revisited by Philippon and Waldschmidt) with arguments from the slope theory. More importantly, we introduce a new argument to deal with the periodic case, inspired by previous works of Bertrand and Philippon. This method does not require the use of an extrapolation on derivations in the sense of Philippon-Waldschmidt. In this way, we are able to remove an important hypothesis in several theorems of Gaudron establishing lower bounds for linear forms in logarithms.

Pour un réel ξ qui n’est pas algébrique de degré ≤ 2, on peut définir plusieurs exposants diophantiens qui mesurent la qualité d’approximation du vecteur (1, ξ, ξ² ) par des sous-espaces de ℝ³ définis sur ℚ de dimension donnée. Cette thèse s’inscrit dans l’étude de ces exposants diophantiens et des questions relatives à la détermination de leur spectre. En utilisant notamment les outils modernes de la géométrie paramétrique des nombres, nous construisons une nouvelle famille de réels – appelés nombres de type sturmien – et nous déterminons presque complètement le 3-système qui leur est associé. Comme conséquence, nous en déduisons la valeur de leurs exposants diophantiens et certaines informations sur les spectres. Nous considérons également le problème plus général de l’allure d’un 3-système associé à un vecteur de la forme (1, ξ, ξ ²), en formulant entre autres certaines contraintes qui n’existent pas pour un vecteur (1, ξ, η) quelconque, et en explicitant les liens qu’il entretient avec la suite des points minimaux associée à ξ. Sous certaines conditions de récurrence sur la suite des points minimaux nous montrons que nous retrouvons les 3-systèmes associés aux nombres de type sturmien. / Given a real number ξ which is not algebraic of degree ≤ 2 one may defineseveral diophantine exponents which measure how “well” the vector (1, ξ, ξ ²) can be approximated by subspaces of fixed dimension defined over ℚ. This thesis is part of the study of these diophantine exponents and their spectra. Using the parametric geometry of numbers, we construct a new family of numbers – called numbers of sturmian type – and we provide an almost complete description of the associated 3-system. As a consequence, we determine the value of the classical exponents for numbers of sturmian type, and we obtain new information on their joint spectra. We also take into consideration a more general problem consisting in describing a 3-system associated with a vector (1, ξ, ξ²). For instance we formulate special constraints which do not exist for a general vector (1, ξ, η) and we also clarify connections between a 3-system which represents ξ and the sequence of minimal points associated to ξ. Under a specific recurrence relation hypothesis on the sequence of minimal points, we show that the previous 3-system has the shape of a 3-system associated to a number of sturmian type.

Cette thèse est dédiée à l'étude d'opérateurs de composition pondérés sur plusieurs espaces fonctionnels sous fond du problème du sous-espace invariant. Cet important problème ouvert pose la question de l'existence pour tout opérateur sur un espace de Hilbert, complexe, séparable de dimension infinie, d'un sous-espace fermé, non-trivial et invariant par cet opérateur. La première partie est consacrée à l'étude spectrale et à la caractérisation des vecteurs cycliques d'un opérateur de composition à poids particulier sur L^2([0,1]^d) : l'opérateur de type Bishop, introduit comme possible contre-exemple au problème du sous-espace invariant. Les seconde, troisième et quatrième parties abordent ce problème sous un autre aspect : celui de l'universalité d'un opérateur. Ces opérateurs universels possèdent la propriété d'universalité : la description complète des sous-espaces invariants d'un opérateur universel permettrait de répondre au problème du sous-espace invariant. Déterminer l'universalité d'un opérateur repose sur l'établissement de propriétés spectrales fines de l’opérateur considéré (théorème de Caradus). Dans ce but, nous établissons des propriétés spectrales ad-hoc de classes d’opérateurs de composition à poids sur L^2([0,1]), les espaces de Sobolev d’ordre n, sur les espaces de Hardy du disque unité et du demi-plan supérieur, permettant de déduire des résultats d’universalité. Nous obtenons aussi une généralisation du critère d’universalité. Dans la dernière partie, nous donnons une caractérisation des opérateurs de composition rsid16415432 inversibles et une caractérisation partielle des opérateurs de composition isométriques sur les espaces de Hardy de l’anneau / In this thesis, we study classes of weighted composition operators on several functional spaces in the background of the invariant subspace problem. This important open problem asks the question of the existence for every operator, defined on a complex and separable infinite dimensional Hilbert space, of a non trivial closed subspace invariant under the operator. The first part is dedicated to the establishment of the spectrum and the characterization of cyclic vectors of a special weighted composition operator defined on L^2([0,1]^d) : the Bishop type operator, introduced as possible counter-example of the invariant subspace problem. The second, third and fourth part approach the problem from the point of view of universal operators. More precisely, universal operators have the universal property in the sense of the complete description of all the invariant subspaces of a universal operator could solve the invariant subspace problem. A sufficient condition for an operator to be universal (Caradus’theorem) is given in terms of spectral properties. To this aim, we establish ad-hoc spectral properties of a class of weighted composition operators on L^2([0,1]) and Sobolev spaces of order n, of composition operator in the Hardy space of the unit disc and of the upper half-plane, which lead us to deduce universality results. We also obtain a generalization of the universality criteria mentioned above. In the last part, we give a characterization of invertible composition operators and a partial characterization of composition operators on the Hardy space of the annulus