In this paper, the ramified Cauchy problem in C² for operator with multiple characteristics of constant multiplicity and second member ramified around some analytic set is studied.

In this thesis we study lower ramification numbers of power series tan- gent to the identity that are defined over fields of positive characteristics. Let f be such a series, then f has a fixed point at the origin and the corresponding lower ramification numbers of f are then, up to a constant, the multiplicity of zero as a fixed point of iterates of f. In this thesis we classify power series having ‘small’ ramification numbers. The results are then used to study ramification numbers of polynomials not tangent to the identity. We also state a few conjectures motivated by computer experiments that we performed.

In this research,we study iterations of non-pleasantly ramified polynomials over fields of positive characteristic and subsequently, their lower ramification numbers. Of particular interest for this thesis are polynomials for which both the multiplicity and  the degree of its iterates grow exponentially. Specifically we consider the family  of polynomials such that given a positive integer k the family is given by P(z) = z(1 + z (3^k-1)/2 + z3^k-1). The cubic polynomial z + z2 + z3 is a special case of this family and is particularly interesting.

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation with a specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Furthermore, this argument is generalized to surfaces of higher genus. Recently, Bousquet-M&eacute;lou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called <em>m</em>-Eulerian trees. In this thesis, we will give a new proof of the result by Bousquet-M&eacute;lou and Schaeffer, introducing a simple partial differential equation. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-M&eacute;lou and Schaeffer's <em>m</em>-Eulerian trees. Some partial results are also given for a refinement of this problem, in which the number of cycles in each factor is specified. This involves Lagrange's theorem in many variables.

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation with a specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Furthermore, this argument is generalized to surfaces of higher genus. Recently, Bousquet-M&eacute;lou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called <em>m</em>-Eulerian trees. In this thesis, we will give a new proof of the result by Bousquet-M&eacute;lou and Schaeffer, introducing a simple partial differential equation. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-M&eacute;lou and Schaeffer's <em>m</em>-Eulerian trees. Some partial results are also given for a refinement of this problem, in which the number of cycles in each factor is specified. This involves Lagrange's theorem in many variables.

This thesis consists of two papers with somewhat different flavours. In Paper I we compute the étale cohomology ring H^*(X,Z/nZ) for X the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. We also give examples of two distinct number fields whose rings of integers have isomorphic cohomology groups but distinct cohomology ring structures. In Paper II we define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2, 1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a stacky building datum comes from a ramified cover for a finite abelian group scheme over k, generalizing a result of Biswas–Borne. / Denna avhandling består av två artiklar som skiljer sig något i karaktär. I Artikel I beräknar vi den étala kohomologiringen H^*(X,Z/nZ) då X är ringen av heltal av en talkropp K. Som en tillämpning, ger vi ett kriterium i form av en formel för när en invariant definierad av Minhyong Kim är noll eller ej. Vi ger också exempel på två olika talkroppar vars ringar av heltal har isomorfa kohomologigrupper men olika kohomologiringstrukturer. I Artikel II definierar vi stackig byggnadsdata för stackiga övertäckningar i Pardinis anda och visar en ekvivalens av (2,1)-kategorier mellan kategorin av stackiga övertäckningar och kategorin av stackig byggnadsdata. Vi visar att varje stackig övertäckning är en platt rotstack i Olsson och Borne–Vistolis mening och vi ger en intrinsisk beskrivning av den som en rotstack med hjälp av stackig byggnadsdata. När basen S är definierad över en kropp, ger vi ett kriterium för när ett stackigt byggnadsdatum kommer från en ramifierad övertäckning för ett ändligt abelskt gruppschema över k. Detta generaliserar ett resultat av Biswas–Borne.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25092号 / 理博第4999号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 市野 篤史, 准教授 伊藤 哲史 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Este trabalho à dedicado ao estudo do processo de transporte de fluido e massa atravÃs de sistemas irregulares. Na primeira parte desta tese, abordamos a dinÃmica do escoamento ocorrendo atravÃs de canais rugosos auto-afins. Essencialmente, os aspectos relevantes na compreensÃo do escoamento em sistemas irregulares sÃo o {it estrutural}, intimamente associado à conformaÃÃo topolÃgica e morfolÃgica do meio, e o {it fenomenolÃgico}, que faz referÃncia aos mecanismos de transporte. Portanto, inicialmente descrevemos a topologia e a morfologia do sistema irregular. Consideramos que a geometria das interfaces que constituem o duto apresenta propriedades estatÃsticas invariantes sob transformaÃÃes de escala anisotrÃpicas, ou seja, possuem correlaÃÃes espaciais de longo alcance e podem, portanto, ser caracterizadas como superfÃcies fractais auto-afins. Mostramos que o carÃter irregular desta geometria adiciona um grau de complexidade ao problema do escoamento, refletindo-se nas propriedades dos campos de velocidade e pressÃo. Como complementaÃÃo deste estudo, investigamos o processo do transporte de partÃculas com massa arrastadas por um fluido escoando no interior das estruturas rugosas anteriormente mencionadas. Investigamos como a rugosidade da estrutura influencia fortemente a natureza deste fenÃmeno e estudamos o comportamento do tempo mÃdio de trÃnsito das partÃculas no interior destes dutos rugosos, em funÃÃo de parÃmetros como o nÃmero de Stokes e o coeficiente de restituiÃÃo. Mostramos que o transporte de partÃculas em dutos com geometria auto-afim à caracterizado pela existÃncia de quatro regimes bem distintos entre si e determinados pela competiÃÃo localizada entre dois mecanismos: as interaÃÃes partÃcula-estrutura (colisÃes) e partÃcula-fluido (escoamento). AtravÃs de simulaÃÃes numÃricas de partÃculas nÃo-Brownianas transportadas por um fluido em um meio poroso, investigamos a influÃncia da geometria e dos efeitos inerciais sobre a eficiÃncia de captura de uma matriz sÃlida. No caso de um arranjo periÃdico de cilindros e sob a aÃÃo da gravidade, nossos resultados revelam que $delta sim St$, onde $delta$ à a eficiÃncia de captura de partÃcula, e $St$ à o nÃmero de Stokes. Na ausÃncia de gravidade, observamos uma tÃpica transiÃÃo de segunda ordem entre a captura e a nÃo-captura de partÃculas, que pode ser expressa como $delta sim(St-St_{c})^{alpha}$, com um expoente $alpha approx 0.5$, onde $St_{c}$ à o nÃmero de Stokes crÃtico. TambÃm realizamos simulaÃÃes para o escoamento atravÃs de um meio poroso aleatÃrio e confirmamos que este comportamento para a captura de partÃculas à consistente com o modelo periÃdico simples. AlÃm disso, abordamos outro aspecto do processo de transporte de fluido e massa atravÃs de sistemas irregulares, nomeadamente o transporte e captura de partÃculas arrastadas por um escoamento hidrodinÃmico no interior de uma estrutura arborescente. Uma vez caracterizado o escoamento nestas estruturas complexas, passamos efetivamente à abordagem do processo de transporte de partÃculas com massa arrastadas por um fluido, integrando-se numericamente a equaÃÃo do movimento para cada partÃcula. O objetivo deste estudo à compreender, principalmente, a dinÃmica de captura de partÃculas e poluentes no interior das vias respiratÃrias. No entanto, a aproximaÃÃo realizada foi mais abrangente. Examinamos a influÃncia que certos parÃmetros fÃsicos e geomÃtricos, tais como os fatores de homotesia, os Ãngulos de ramificaÃÃo e o nÃmero de Reynolds, exercem sobre o processo citado. Esta abordagem nos permitiu pÃr em destaque uma universalidade notÃvel das leis de captura nas estruturas ramificadas. / This work is dedicated to the study of the transport process of fluid flow and mass through irregular systems. In the first part of this thesis, we approach the transport of fluids in self-affine fractured channels. Essentially, the important aspects to the comprehension of the fluid flow in an irregular geometry are the {it structural} one, associated to the topological and morphologic conformation of the fractured media, and the {it phenomenological} one, that refers to the transport mechanisms. Therefore, initially we describe the topology and the morphology of the fracture network. We consider that the geometry of the interfaces that constitutes the channel has statistical properties that are invariant under anisotropic transformations of scale, i.e. they possess long-range correlations in space and, than, can be characterized as self-affine fractals surfaces. We show that the irregular character of this geometry adds a degree of complexity to the problem of the fluid flow that substantially affects the statistical properties of the velocity and pressure fields. As a complement of this study, we investigate the process of the particle transport with mass dragged by a fluid that flows into the rough structures previously mentioned. We investigate how the roughness influences the nature of the transport of particles by studying the behavior of their average transit time, as a function of parameters like the Stokes number and the coefficient of restitution. We show that the particle transport in ducts with self-affine geometry displays a complex behavior that is characterized by the existence of four regimes and determined by the local competition between two mechanisms: the interaction between particle and structure (collisions) and between particle and fluid (flow). We investigate through numerical calculation of non-Brownian particles transported by a fluid in a porous medium, the influence of geometry and inertial effects on the capture efficiency of the solid matrix. In the case of a periodic array of cylinders and under the action of gravity, our results reveal that $delta sim St$, where $delta$ is the particle capture efficiency, and $St$ is the Stokes number. In the absence of gravity, we observe a typical second order transition between non-trapping and trapping of particles that can be expressed as $delta sim(St-St_{c})^{alpha}$, with an exponent $alpha approx 0.5$, where $St_{c}$ is the critical Stokes number. We also perform simulations for flow through a random porous structure and confirm that its capture behavior is consistent with the simple periodic model. Moreover, we inquire into another aspect of the transport process of fluid and mass through irregular systems, namely the transport and capture of particles dragged by a hydrodynamic flow into a ramified structure. After we solve the fluid flow through these complex structures, we pass effectively to the investigation of the transport process of massive particle dragged by a fluid, through numerical integration of the movement equation for each particle. The aim of this study is to understand, mainly, the dynamics of particle and pollutants capture into the airways. Nevertheless, the selected approach was broader. We examine the influence that certain physical and geometric parameters, such as the homothety factors, the bifurcation angles and the Reynolds number, exert on the mentioned process. This approach allowed us to clarify a remarkable universality of the laws of capture in the ramified structures.

L'objet de cette thèse est l'étude de revêtements ramifiés V' to V de variétés hyperboliques compactes V cubiques, c'est-à-dire dont le groupe fondamental pi_1(V) opère proprement et cocompactement sur un complexe cubique CAT(0). Notre première approche consiste à construire un complexe cubique localement CAT(0) comme revêtement ramifié du complexe obtenu par cubulation de V. La difficulté est alors de vérifier que ce complexe a le même groupe fondamental que V’. On réalise ce programme dans le cas ou V’ est une « variété de Gromov-Thurston ». Notre seconde approche concerne plus généralement le cas où le lieu de ramification du revêtement V' to V est contenu dans une sous-variété convexe de codimension 1. La préimage de cette variété dans V’ puis dans le revêtement universel X’ de V’ fournit un système naturel de « murs ». La difficulté consiste alors à montrer que ces murs séparent linéairement X’ afin d'utiliser les théorèmes classiques de cubulation. / The goal of this thesis is to study of branched covers V' to V of closed hyperbolic manifolds that can be cubulated, i.e. Whose fundamental group pi_1(V) acts properly and cocompactly on a CAT(0) cube complex. We give sufficient conditions for pi_1(V') to be cubic as well.We tackle this question in two different ways. In a first approach we build a negatively curved cubical complex as a ramified cover of a cubical complex obtained by cubulating V. Then the main issue is to check that the fundamental group of this complexe is isomorphic to the fundamental group of V'. We manage to do so when V' is so called “Gromov-Thurston manifold “. Our second approach deals with the more general case where the branched locus of V' to V is contained in a codimension 1 convex submanifold. The preimage of this submanifold on V' and on the universal cover X' of V' provides a natural system of “walls”. Then the main issue is to show that these walls linearly separate X'. This enables us to use classical cubulation theorems.

La synthèse de nanoparticules de fer zéro-valent, par le couplage des procédés d'ultrasonication et d'électrodéposition, est étudiée selon deux approches. La première partie de l'étude s'intéresse à l'influence du substrat, utilisé pour l'électrodéposition, sur la croissance des dépôts de fer et sur leur dispersion par ultrasonication. L'énergie interfaciale ainsi que l'énergie d'adhésion du dépôt sur le substrat (Y_(Fe/substrat) et W_(Fe/substrat) respectivement) étant reliées à l'énergie de surface et à la rugosité du substrat, un intérêt particulier a été porté à ces deux propriétés. Ainsi, deux matériaux présentant des énergies de surface différentes, l'or (Au) et le carbone vitreux (VC), ainsi que des rugosités différentes ont été testés. Un développement théorique basé sur les interactions de Van der Waals a permis de démontrer que Y_(Fe/VC)>Y_(Fe/Au) ce qui suggère une meilleure affinité du dépôt de fer avec l'or qu'avec le VC. Cette différence influence la morphologie (croissance 2D sur or et 3D sur le VC) mais aussi l'adhésion des dépôts. En effet, les expériences réalisées pour étudier l'effet des ultrasons sur le dépôt de fer révèlent une dispersion du dépôt progressive et complète pour le cas du VC alors qu'aucun détachement du dépôt n'est obtenu en utilisant l'or. La seconde partie de l'étude est consacrée à la synthèse de nanoparticules de fer par une nouvelle approche : l'électrodéposition de dépôts de fer ramifiés est étudiée dans une cellule de Hele-Shaw intégrant un élément vibrant (diaphragme piézoélectrique) permettant à la fois la formation de dépôts de fer et leur fragmentation. Les expériences menées révèlent que les bulles d'hydrogène, formées lors de la co-réduction des protons libres durant l'électrodéposition du fer, influencent fortement le processus de fragmentation. En utilisant des hautes fréquences et amplitudes de vibration du PZT, les bulles d'hydrogène oscillent avec des déformations de surface. Celles-ci génèrent des vitesses d'interface suffisamment hautes (˜ 4 m/s) pour permettre aux bulles de fragmenter des dépôts ramifiés en particules de fer, de tailles comprises entre 1 µm et 100 nm, et présentant une grande surface spécifique due à leur morphologie dendritique. Cette deuxième partie de l'étude permet d'ouvrir la voie à une nouvelle technologie de fabrication des nanoparticules. / This study concerns the coupling of the ultrasounds with the electrodeposition process for the synthesis of zero-valent iron nanoparticles; it is structured in two sections. The first focuses on the electrode substrate used for the iron electrodeposition and aims to determine its influence on both the deposit growth and its dispersion by ultrasonication. The interfacial and the adhesion energies of the deposit on the substrate (Y_(Fe/substrate) and W_(Fe/substrate) respectively) being related to the surface energy and the roughness of the substrate, a particular focus is put on the influence of these two properties. Thus, two materials of different surface energies, gold (Au) and vitreous carbon (VC), as well as various roughnesses, are tested. Considering only the Van der Waals interactions, a theoretical development has enabled to determine that Y_(Fe/VC)>Y_(Fe/Au) which suggests a better affinity of the iron deposit with the gold than with the VC substrate. This difference impacts the deposit morphology (2D growth on the gold and 3D growth on the VC substrate) but also the deposit adhesion. Indeed, experiments performed to study the effect of ultrasounds on the iron electrodeposit reveal its progressive and complete dispersion for the vitreous carbon case while no dispersion (no removal of the deposit from the electrode) is obtained with the gold substrate. The second section of the present study deals with the synthesis of iron nanoparticles; to this end, the electrodeposition of branched deposits has been investigated in a Hele-Shaw cell integrating a vibrating element (piezoelectric diaphragm), expected to allow both the deposit formation and its fragmentation. Experiments reveal that the hydrogen bubbles, formed by the co-reduction of free protons during the iron electrodeposition, strongly influence the fragmentation process. Using high vibration frequencies and high amplitudes, the bubbles oscillate with surface deformations, inducing interface velocity sufficiently high (˜ 4 m/s) to allow the fragmentation of the deposit into particles of sizes ranging between 1 µm and 100 nm and showing a high specific surface due to their dendritic morphology. Thus this work opens the way for a new particles manufacturing technology.