L'analyse dynamique intègre des outils propres aux systèmes dynamiques (comme l'opérateur de transfert) au cadre de la combinatoire analytique, et permet ainsi l'analyse d'un grand nombre d'algorithmes et objets qu'on peut associer naturellement à un système dynamique. Dans ce manuscrit de thèse, nous présentons, dans la perspective de l'analyse dynamique, l'étude probabiliste de plusieurs problèmes qui semblent à priori bien différents : l'analyse probabiliste de la fonction de récurrence des mots de Sturm, et l'étude probabiliste de l'algorithme du “logarithme continu”. Les mots de Sturm constituent une famille omniprésente en combinatoire des mots. Ce sont, dans un sens précis, les mots les plus simples qui ne sont pas ultimement périodiques. Les mots de Sturm ont déjà été beaucoup étudiés, notamment par Morse et Hedlund (1940) qui en ont exhibé une caractérisation fondamentale comme des codages discrets de droites à pente irrationnelle. Ce résultat relie ainsi les mots de Sturm au système dynamique d'Euclide. Les mots de Sturm n'avaient jamais été étudiés d'un point de vue probabiliste. Ici nous introduisons deux modèles probabilistes naturels (et bien complémentaires) et y analysons le comportement probabiliste (et asymptotique) de la “fonction de récurrence” ; nous quantifions sa valeur moyenne et décrivons sa distribution sous chacun de ces deux modèles : l'un est naturel du point de vue algorithmique (mais original du point de vue de l'analyse dynamique), et l'autre permet naturellement de quantifier des classes de plus mauvais cas. Nous discutons la relation entre ces deux modèles et leurs méthodes respectives, en exhibant un lien potentiel qui utilise la transformée de Mellin. Nous avons aussi considéré (et c'est un travail en cours qui vise à unifier les approches) les mots associés à deux familles particulières de pentes : les pentes irrationnelles quadratiques, et les pentes rationnelles (qui donnent lieu aux mots de Christoffel). L'algorithme du logarithme continu est introduit par Gosper dans Hakmem (1978) comme une mutation de l'algorithme classique des fractions continues. Il calcule le plus grand commun diviseur de deux nombres naturels en utilisant uniquement des shifts binaires et des soustractions. Le pire des cas a été étudié récemment par Shallit (2016), qui a donné des bornes précises pour le nombre d'étapes et a exhibé une famille d'entrées sur laquelle l'algorithme atteint cette borne. Dans cette thèse, nous étudions le nombre moyen d'étapes, tout comme d'autres paramètres importants de l'algorithme. Grâce à des méthodes d'analyse dynamique, nous exhibons des constantes mathématiques précises. Le système dynamique ressemble à première vue à celui d'Euclide, et a été étudié d'abord par Chan (2005) avec des méthodes ergodiques. Cependant, la présence des puissances de 2 dans les quotients change la nature de l'algorithme et donne une nature dyadique aux principaux paramètres de l'algorithme, qui ne peuvent donc pas être simplement caractérisés dans le monde réel.C'est pourquoi nous introduisons un nouveau système dynamique, avec une nouvelle composante dyadique, et travaillons dans ce système à deux composantes, l'une réelle, et l'autre dyadique. Grâce à ce nouveau système mixte, nous obtenons l'analyse en moyenne de l'algorithme. / Dynamical Analysis incorporates tools from dynamical systems, namely theTransfer Operator, into the framework of Analytic Combinatorics, permitting the analysis of numerous algorithms and objects naturally associated with an underlying dynamical system.This dissertation presents, in the integrated framework of Dynamical Analysis, the probabilistic analysis of seemingly distinct problems in a unified way: the probabilistic study of the recurrence function of Sturmian words, and the probabilistic study of the Continued Logarithm algorithm.Sturmian words are a fundamental family of words in Word Combinatorics. They are in a precise sense the simplest infinite words that are not eventually periodic. Sturmian words have been well studied over the years, notably by Morse and Hedlund (1940) who demonstrated that they present a notable number theoretical characterization as discrete codings of lines with irrationalslope, relating them naturally to dynamical systems, in particular the Euclidean dynamical system. These words have never been studied from a probabilistic perspective. Here, we quantify the recurrence properties of a ``random'' Sturmian word, which are dictated by the so-called ``recurrence function''; we perform a complete asymptotic probabilistic study of this function, quantifying its mean and describing its distribution under two different probabilistic models, which present different virtues: one is a naturaly choice from an algorithmic point of view (but is innovative from the point of view of dynamical analysis), while the other allows a natural quantification of the worst-case growth of the recurrence function. We discuss the relation between these two distinct models and their respective techniques, explaining also how the two seemingly different techniques employed could be linked through the use of the Mellin transform. In this dissertation we also discuss our ongoing work regarding two special families of Sturmian words: those associated with a quadratic irrational slope, and those with a rational slope (not properly Sturmian). Our work seems to show the possibility of a unified study.The Continued Logarithm Algorithm, introduced by Gosper in Hakmem (1978) as a mutation of classical continued fractions, computes the greatest common divisor of two natural numbers by performing division-like steps involving only binary shifts and substractions. Its worst-case performance was studied recently by Shallit (2016), who showed a precise upper-bound for the number of steps and gave a family of inputs attaining this bound. In this dissertation we employ dynamical analysis to study the average running time of the algorithm, giving precise mathematical constants for the asymptotics, as well as other parameters of interest. The underlying dynamical system is akin to the Euclidean one, and was first studied by Chan (around 2005) from an ergodic, but the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying this system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the algorithm by Dynamical Analysis.

Relevance of this research is conditioned by necessity of increasingmetalworking machines efficiency in modern machinery building. One of the main constructive elements of gear milling machines is mechanical gear train (speed gear train, feed gear train, dividing gear train, differential gear train). The adjustment of all these shaping moves is important objective. Though mechanical gear trains do not provide high tooling precision.That’s why precision can be increased by developing ways of replacing mechanical gear trains by hydraulic and electronic analogs. These ways were poorly considered before. Goal of the work: analyzing of existing principal gear milling machine schemes, analyzing of possible ways of shaping moves electronic adjustment, descrying of various frequent divisor structures. Calculation unevennessof impulses following for divisors, that work in different codes and improving schemes of machines with electric step motor and adjustable electric motor by way of increasing tooling precision. Master’s dissertation consists of 6 chapters: Chapter 1: Modern gear milling machines principal schemes. This chapter describes function of gear trains, structure of cutting moves in gear milling and schemes of existing gear milling machines with mechanical and electronic gear trains. Chapter 2: Analysis of ways and tools of electronic (“Non-gear-train”) adjustment of machine’s shaping moves. This chapter describes ways of coordinating an adjustable electronic motor and electronic step motor, ways of coordinating two adjustable electronic motors. Chapter 3: modern constructions of impulse frequency divisors. Schemes andfunctioning of divisors. This chapter analyze various schemes of frequency divisors, ways of divisor selecting for gear milling machine, calculating the unevenness of impulses, transferring with divisors for various codes. Chapter 4: Improvement of gear milling machines with electric step motor Suggestion of improved kinematic schemes for electric step motor milling machines. Chapter 5: Improvement in schemes of gear milling machines with adjustable electric motor. Suggestion of improved kinematic schemes for adjustable electric motor milling machines. Chapter 6: Work out of technical proposal for gear milling machine with adjustable electric motor. Chapter contains selection of electric motor for chosen kinematic scheme with electronic gear train, kinematic and robustness calculation, comparison of mechanical and electronic gears’ precision. Master’s dissertation consist of 128 pages, containing demonstrational materials, calculations and text description, and 17 graphic documents. / Актуальность темы исследования обусловлена необходимостью повышения эффективности металлорежущих станков в современном машиностроении. Одними из основных конструктивных элементов зубофрезерного станка являются механические гитары (скоростей, подач, деления, дифференциала). Важной задачей является согласовать всеэти формообразующие движения. Но механические гитары не обеспечивают высокой точности обработки. Поэтому были разработаны способы повышения точности путем замены механических гитар гидравлическими и электронными. Ранее эти способы были рассмотрены поверхностно. Цель работы: проведение анализа существующих принципиальных схем зубофрезерных станков, анализ возможных способов электронного согласования формообразующих движений, рассмотрение различных конструкций делителей частоты, расчет неравномерности следования импульсов для делителей, работающих в различных кодах, и совершенствование схем станков с шаговым и регулируемым электроприводами для повышения точности обработки. Магистерская диссертация состоит из 6 глав: Глава 1: Принципиальные схемы современных зубофрезерных станков В главе рассматривается назначение гитар, структура рабочего движения при зубофрезеровании, а также существующие кинематические схемы зубофрезерных станков с механическими и электронными гитарами. Глава 2: Анализ способов и средств электронного («безгитарного») согласования формообразующих движений в станке В главе описываются способы согласования работы регулируемого и шагового приводов, и согласование работы двух регулируемых приводов. Глава 3: Современные конструкции делителей частоты импульсов, их устройство и работа В главе анализируются различные конструкции делителей частоты, также производится выбор делителя, применительно к зубофрезерному станку. Производится расчет неравномерности следования импульсов, передаваемых через делители для различных кодов. Глава 4: Совершенствование зубофрезерных станков с шаговым электроприводом Предлагаются усовершенствованные кинематические схемы станков с шаговыми электроприводами. Глава 5: Совершенствование схем зубофрезерных станков с регулируемыми электроприводами Предлагаются усовершенствованные кинематические схемы станков с регулируемыми электроприводами. Глава 6: Разработка технического предложения зубофрезерного станка с регулируемыми электроприводами В главе выбирается электродвигатель для выбранной кинематической схемы с электронной гитарой, производится кинематический и прочностной расчет и сравниваются по точности механическая и электронная гитары. Магистерская диссертация выполнена на 128 страницах, содержащих демонстрационные материалы, расчеты и описание, и включает 17 графических документов.

[pt] O ensaio de impulso para perfuração de isoladores é um ensaio normalizado descrito na IEC 61211:2004 e é fundamental para verificar em laboratório a qualidade de isoladores de vidro ou porcelana utilizados no sistema elétrico de potência. A norma IEC 60060-2:2010 recomenda que a calibração dos Sistemas de Medição utilizados em ensaios para Perfuração de Isoladores (SMIP) deve ser realizada por comparação com um padrão. Contudo, embora o ensaio de perfuração seja normalizado, ainda não existem padrões para rastrear esse sistema de medição na maioria dos institutos de pesquisas em âmbito global. Quantificar a amplitude de um impulso de perfuração, com todas as garantias metrológicas, não é uma tarefa trivial, por se tratar de um sinal impulsivo com amplitude de centenas de kilovolts e durações da ordem de 200 ns. Além disso, divisores de tensão com alto desempenho dinâmico para essa aplicação especifica também não estão disponíveis comercialmente. Deste modo, o objetivo desta dissertação foi projetar e desenvolver um SMIP. O divisor de tensão, integrante do sistema de medição, foi construído a fim de garantir uma mínima indutância e um alto desempenho dinâmico. Também foi desenvolvido um filtro para a remoção de interferências eletromagnéticas baseado na transformada wavelet, integrado a um software que controla aquisição, condicionamento do sinal, determinação e análise dos parâmetros do impulso de perfuração. O desempenho do sistema de medição foi avaliado em baixa tensão e em alta tensão, bem como foi estimada sua incerteza de medição, sendo então capaz de atender às exigências normativas e metrológicas do Laboratório de Referência em medição do Cepel. / [en] Impulse testing for puncture insulator is a standardized test described in IEC 61211:2004 and is essential for laboratory testing of the quality of glass or porcelain insulators used in the electrical power system. IEC 60060-2:2010 recommends that calibration of Measurement Systems used in Impulse Puncture Test on insulators (IPMS) should be performed by comparison against a standard. However, although the impulse puncture testing is standardized, there are still no standards for traceability of this measurement system at most research institutes globally. Quantifying the amplitude of an impulse puncture test, with all metrological guarantees, is not a trivial task, since it is an impulsive signal with amplitude of hundreds of kilovolts and durations of the order of 200 ns. In addition, high dynamic performance voltage dividers for this specific application are not commercially available either. Thus, the objective of this dissertation was to design and develop an IPMS. The voltage divider, which is part of the measuring system, is designed to ensure minimum inductance and high dynamic performance. A filter for electromagnetic interference removal based on the wavelet transform was also developed, integrated with software that controls acquisition, signal conditioning, determination and analysis of the impulse puncture testing parameters. The performance of the measuring system was evaluated at low voltage and high voltage, as well as its measurement uncertainty was estimated, being able to meet the normative and metrological requirements of the Cepel s Measurement Reference Laboratory.

La thèse concerne différents aspects de la répartition des fonctions arithmétiques.1. Deshouillers, Iwaniec et Luca se sont récemment intéressés à la répartition modulo 1 de suites qui sont des valeurs moyennes de fonctions multiplicatives, par exemple phi(n)/n où phi est la fonction d'Euler. Nous étendons leur travail à la densité modulo 1 de suites qui sont des valeurs moyennes sur des suites polynômiales, typiquement n^2+1.2. On sait depuis les travaux de Katai, il y a une quarantaine d'années que la fonction de répartition des valeurs de phi(p-1)/(p-1) (où p parcourt les nombres premiers) est continue, purement singulière, strictement croissante entre 0 et 1/2. On précise cette étude en montrant que cette fonction de répartition a une dérivée infinie à gauche de tout point phi(2n)/(2n). / Abstract