In this thesis, we study the contractivity preserving, high order, time discretization methods for solving non-stiff ordinary differential equations. We construct a class of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite-Birkhoff-Taylor methods of order p=5,6, ..., 15, that we denote by CPHBT, with nonnegative coefficients by casting s-stage Runge-Kutta methods of order 4 and 5 with Taylor methods of order p-3 and p-4, respectively. The constructed CPHBT methods are implemented using an efficient variable step algorithm and are compared to other well-known methods on a variety of initial value problems. The results show that CPHBT methods have larger regions of absolute stability, require less function evaluations and hence they require less CPU time to achieve the same accuracy requirements as other methods in the literature. Also, we show that the contractivity preserving property of CPHBT is very efficient in suppressing the effect of the propagation of discretization errors when a long-term integration of a standard N-body problem is considered. The formulae of 49 CPHBT methods of various orders are provided in Butcher form.

This dissertation revolves around various mathematical aspects of nonlinear wave motion in viscoelasticity and free surface flows. The introduction is devoted to the physical derivation of the stress-strain constitutive relations from the first principles of Newtonian mechanics and is accessible to a broad audience. This derivation is not necessary for the analysis carried out in the rest of the thesis, however, is very useful to connect the different-looking partial differential equations (PDEs) investigated in each subsequent chapter. In the second chapter we investigate a multi-dimensional scalar wave equation with memory for the motion of a viscoelastic material described by the most general linear constitutive law between the stress, strain and their rates of change. The model equation is rewritten as a system of first-order linear PDEs with relaxation and the well-posedness of the Cauchy problem is established. In the third chapter we consider the Euler equations describing the evolution of a perfect, incompressible, irrotational fluid with a free surface. We focus on the Hamiltonian description of surface waves and obtain a recursion relation which allows to expand the Hamiltonian in powers of wave steepness valid to arbitrary order and in any dimension. In the case of pure gravity waves in a two-dimensional flow there exists a symplectic coordinate transformation that eliminates all cubic terms and puts the Hamiltonian in a Birkhoff normal form up to order four due to the unexpected cancellation of the coefficients of all fourth order non-generic resonant terms. We explain how to obtain higher-order vanishing coefficients. Finally, using the properties of the expansion kernels we derive a set of nonlinear evolution equations for unidirectional gravity waves propagating on the surface of an ideal fluid of infinite depth and show that they admit an exact traveling wave solution expressed in terms of Lambert’s W-function. The only other known deep fluid surface waves are the Gerstner and Stokes waves, with the former being exact but rotational whereas the latter being approximate and irrotational. Our results yield a wave that is both exact and irrotational, however, unlike Gerstner and Stokes waves, it is complex-valued.

yes / The properties of quantum probabilities are linked to the geometry of quantum mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Möbius (or non-additivity) operators which are defined through Möbius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice Λd, causes deviations from the law of the total probability (which is central in Kolmogorov’s probability theory). Projectors which quantify the lack of distributivity in Λd, and also deviations from the law of the total probability, are introduced. All these operators, are observables and they can be measured experimentally. Constraints for the Möbius operators, which are based on the properties of the Birkhoff-von Neumann lattice (which in the case of finite quantum systems is a modular lattice), are derived. Application of this formalism in the context of coherent states, generalizes coherence to multi-dimensional structures.

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

L'anneau $\Diff(n)$ des opérateurs différentiels $\h$-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels $\h$-déformés de type $\gl$. Contrairement aux espaces vectoriels $q$-déformés pour lequel l'anneau des opérateurs différentiels est unique \`a isomorphisme pr\`es, l'anneau généralisé des opérateurs différentiels $\h$-déformés $\Diffs(n)$ est indexée par une fonction rationnelle $\sigma$ en $n$ variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de $\Diffs(n)$ est un anneau des polynômes en $n$ variables. Nous construisons un isomorphisme entre des localisations de l'anneau $\Diffs(n)$ et de l’algèbre de Weyl $\text{W}_n$ l’étendue par $n$ indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de $\Diffs(n)$. Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau $\Diff(n,N)$ correspondant \`a $N$ copies de $\Diff(n)$. / The ring $\Diff(n)$ of $\h$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\h$-deformed vector spaces of $\gl$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\h$-deformed differential operators $\Diffs(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of $\Diffs(n)$ is a ring of polynomials in $n$ variables. We construct an isomorphism between certain localizations of $\Diffs(n)$ and the Weyl algebra $\W_n$ extended by $n$ indeterminates. We present some conditions for the irreducibility of the finite dimensional $\Diffs(n)$-modules. Finally, we discuss difficulties for finding analogous constructions for the ring $\Diff(n, N)$ formed by several copies of $\Diff(n)$.

[pt] Os resultados fundamentais da teoria ergódica – o teorema de Birkhoff e o teorema de Kingman – se referem a convergência em quase todo ponto de um processo ergódico aditivo e subaditivo, respectivamente. É bem conhecido que dado um sistema unicamente ergódico e um observável contínuo, as médias de Birkhoff correspondentes convergem em todo ponto e uniformemente. Desta forma, é natural também se perguntar o que acontece com o teorema de Kingman quando o sistema é unicamente ergódico. O primeiro objetivo desta dissertação é responder a essa pergunta utilizando o trabalho de A. Furman. Mais ainda, apresentamos algumas extensões e aplicações desse resultado para cociclos lineares, que foram obtidas por S. Jitomirskaya e R. Mavi. Nosso segundo objetivo é provar um novo resultado sobre taxas de convergências de médias de Birkhoff, para um certo tipo de processo unicamente ergódico: uma translação diofantina no toro com um observável Holder contínuo. / [en] The fundamental results in ergodic theory – the Birkhoff theorem and the Kingman theorem – refer to the almost everywhere convergence of additive and respectively subadditive ergodic processes. It is well known that given a uniquely ergodic system and a continuous observable, the corresponding Birkhoff averages converge everywhere and uniformly. It is therefore natural to ask what happens with Kingman s theorem when the system is uniquely ergodic. The first objective of this dissertation is to answer this question following the work of A. Furman. Moreover, we present some extensions and applications of this result for linear cocycles, which were obtained by S. Jitomirskaya and R. Mavi. Our second objective is to prove a new result regarding the rate of convergence of the Birkhoff averages for a certain type of uniquely ergodic process: a Diophantine torus translation with Holder continuous observable.

Esta tesis doctoral se enmarca dentro de la teoría de integración de funciones con valores en espacios de Banach. Analizamos con detalle la integral de Birkhoff de funciones vectoriales, así como sus correspondientes versiones dentro de los contextos de la integración respecto de medidas vectoriales y la integración de multi-funciones. Comparamos estos métodos de integración con otros bien conocidos (integrales de Bochner, Pettis, McShane, Debreu, etc.). Caracterizamos, en términos de integración vectorial, algunas propiedades de los espacios de Banach donde las (multi-) funciones toman valores. / The general framework of this memoir is the theory of integration of functions with values in Banach spaces. We analyze in detail the Birkhoff integral of vector-valued functions, as well as its corresponding versions within the settings of integration with respect to vector measures and integration of multi-valued functions. We compare these methods of integration with others which are well known (Bochner, Pettis, McShane, Debreu, etc.). We characterize, in terms of vector integration, some properties of the Banach spaces where the (multi-) functions take their values.

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés. / Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf $\mathcal{H}_\mathcal{T}$ sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs $\mathcal{T}$. Nous définissons encore un dédoublement $\wt\mathcal{D}_\mathcal{T}$ de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonctions de Schwartz, et qu'elles s'étendent en une fonction méromorphe dans le cas des fonctions de types Feynman. Nous pouvons alors déterminer les parties finies de ces intégrales en utilisant l'algorithme BPHZ après avoir appliqué la procédure de régularisation dimensionnelle. / In this thesis, we study the renormalization of Connes-Kreimer in the contex of specified Feynman graphs Hopf algebra. We construct a Hopf algebra structure $\mathcal{H}_\mathcal{T}$ on the space of specified Feynman graphs of a quantum field theory $\mathcal{T}$. We define also a doubling procedure for the bialgebra of specified Feynman graphs, a convolution product and a group of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.We recall the definition of Feynman integrals associated with a graph. We prove that these integrals are holomorphic in a complex variable D in the case oh Schwartz functions, and that they extend in a meromorphic functions in the case of a Feynman type functions. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure.

Cette thèse est consacrée à l’étude mathématique de divers systèmes de particules classiques et quantiques, en milieu désordonné. Elle comprend quatre travaux publiés ou soumis. Dans le premier nous fournissons une nouvelle formule permettant de prouver la localisation d’Anderson en une dimension d’espace et de caractériser la décroissance des fonctions propres à l’infini. Le second contient l’une des premières preuves de la localisation pour une infinité de particules en intéraction, dans l’approximation d’Hartree-Fock. Le troisième est dédié au modèle d’Anderson soumis à une perturbation périodique en temps. Sous certaines conditions sur la fréquence d’oscillation nous prouvons l’absence de diffusion. Dans le dernier travail nous montrons la décroissancedes corrélations pour le modèle du Jellium en une dimension dans un fond inhomogène, en utilisant la distance de Hilbert sur les cônes et le théorème de Birkhoff-Hopf. / This thesis is devoted to the mathematical study of some systems of classical and quantum particles, in a disordered medium. It comprises four published or submitted works. In the first one we provide a new formula allowing to prove Anderson localisation in one space dimension and to characterise the decay at infinity of the eigenfunctions. The second contains one of the first proofs of localisation for infinitely many particles in interaction, in the Hartree-Fock approximation. The third work is dedicated to the Anderson model in a time-periodic perturbation. Under certain conditions on the oscillation frequency we prove the absence of diffusion. In the last work we show the decay of correlations for the one-dimensional Jellium model in an inhomogeneous background, using the Hilbert distance on cones and the Birkhoff-Hopf theorem