Tesi sul tema "Calcul fractal"

La description aux échelles macroscopiques et gigascopiques des feux de forêts permet l'établissement de modèles physiques aptes à représenter l'évolution d'un feu avec une meilleure précision que les modèles empiriques de type Rothermel développés jusqu'alors. Cependant ces modèles nécessitent l'ajustement de paramètres dont la mesure directe est impossible, car les équations associées à ces modèles ne sont pas relatives à l'air et à la matière végétale mais aux milieux équivalents à la végétation pour l'échelle considérée. Les propriétés des milieux équivalents sont alors liées aux propriétés des milieux les constituant, mais la connaissance des propriétés des milieux constitutifs ne permet pas de connaître directement les propriétés du milieu équivalent. Ce travail consistera tout d'abord en la reconstruction du milieu végétal à l'aide d'outils issus de la géométrie fractale. Des méthodes de mesures de paramètres géométriques venant de la foresterie ont ensuite été utilisées pour valider nos modèles de végétation. Enfin, des expériences numériques ont été menées sur nos structures reconstruites afin d'identifier les paramètres macroscopiques qui nous intéressent. Ces expériences permettent également de valider ou non les hypothèses effectuées lors de l'établissement des équations du milieu équivalent. Les paramètres ajustés sont la viscosité du milieu équivalent, le coefficient d'échange convectif et le coefficient d'extinction
The macroscopic and gigascopic scale description of forest fires allows physical modelings of the propagation which can predict the fire evolution with a better accuracy than usually developed empirical Rothermel-like models. However, those models need fitting for their parameters which cannot be measured directly as the models equations are related to the equivalent media at the considered scale and not related to the air and the vegetal material. The equivalent media properties are related to the inner media properties, but the inner media properties knowledge does not allow directly the equivalent media properties knowledge. This work is then aiming on the vegetal medium reconstruction using fractal geometry. Geometrical parameters measurement methods used in forestry sciences are applied for the vegetal modeling validation. Numerical studies are finally done on the reconstructed structures to fit the relevant macroscopic scale parameters. Those studies also allow us to validate or invalidate the assumptions which have been done for the equivalent medium equation development. Those parameters are: the equivalent medium viscosity, the convective heat transfer coefficient and the extinction coefficient

Cette thèse s'inscrit dans l'étude d'un modèle de calcul géométrique : les machines à signaux. Nous y montrons comment tracer des graphes de fonctions à l'aide d'arbres unaire-binaires. Dans le monde des automates cellulaires, il est souvent question de particules ou signaux : des structures périodiques dans le temps et l'espace, autrement dit des structures qui se déplacent à vitesse constante. Lorsque plusieurs signaux se rencontrent, une collision a lieu, et les signaux entrant peuvent continuer, disparaître ou laisser place à d'autres signaux, en fonctions des règles de l'automate cellulaire. Les machines à signaux sont un modèle de calcul qui reprend ces signaux comme briques de base. Visualisées dans un diagramme espace-temps, l'espace en axe horizontal et le temps vertical s'écoulant vers le haut, ce modèle revient à calculer par le dessin de segments et demi-droites colorés. On trace, de bas en haut, des segments jusqu'à ce que deux ou plus s'intersectent, et l'on démarre alors de nouveau segments, en fonctions de règles prédéfinies. Par rapport aux automates cellulaires, les machines à signaux permettent l'émergence d'un nouveau phénomène : la densité des signaux peut être arbitrairement grande et même infinie, y compris en partant d'une configuration initiale de densité finie. De tels points du diagramme espace-temps, des points au voisinage desquels se trouvent une infinité de signaux, sont appelés points d'accumulation. Ce nouveau phénomène permet de définir de nouveau problèmes, géométriquement. Par exemple : quels sont les points d'accumulations isolés possibles en utilisant des positions initiales et des vitesses rationnelles ? Peut-on faire en sorte que l'ensemble des points d'accumulation forment un segment ? un ensemble de Cantor ? Dans cette thèse, nous nous attelons à caractériser des graphes de fonctions qu'il est possible de dessiner par un ensemble d'accumulation. Elle s'inscrit dans l'exploration de la puissance de calcul des machines à signaux, qui s'inscrit plus généralement dans l'étude de la puissance de calcul de modèles non standards. Nous y montrons que les fonctions d'un segment compact de la droite réelle dont le graphe coïncide avec l'ensemble d'accumulation d'une machine à signaux sont exactement les fonctions continues. Nous montrons plus généralement comment les machines à signaux peuvent dessiner n'importe quel fonction semi-continue inférieurement. Nous étudions aussi la question sous des contraintes de calculabilité, avec le résultat suivant : si un diagramme de machine à signaux calculable coïncide avec le graphe d'un fonction suffisamment lipschitzienne, cette fonction est limite calculable d'une suite croissante de fonctions en escalier rationnelles
This thesis studies a geometric computational model: signal machines. We show how to draw function graphs using-binary trees. In the world of cellular automata, we often consider particles or signals: structures that are periodic in time and space, that is, structures that move at constant speed. When several signals meet, a collision occurs, and the incoming signals can continue, disappear, or give rise to new signals, depending on the rules of the cellular automaton. Signal-machines are a computational model that takes these signals as basic building blocks. Visualized in a space-time diagram, with space on the horizontal axis and time running upwards, this model consists of calculating by drawing segments and half-lines. We draw segments upwards until two or more intersect, and then start new segments, according to predefined rules. Compared to cellular automata, signal-machines allow for the emergence of a new phenomenon: the density of signals can be arbitrarily large, even infinite, even when starting from a finite initial configuration. Such points in the space-time diagram, whose neighborhoods contain an infinity of signals, are called accumulation points.This new phenomenon allows us to define new problems geometrically. For example, what are the isolated accumulation points that can be achieved using rational initial positions and rational velocities? Can we make so the set of accumulation points is a segment? A Cantor set? In this thesis, we tackle the problem of characterizing the function graphs that can be drawn using an accumulation set. This work fits into the exploration of the computational power of signal-machines, which in turn fits into the study of the computational power of non-standard models. We show that the functions from a compact segment of the line of Real numbers whose graph coincides with the accumulation set of a signal machine are exactly the continuous functions. More generally, we show how signal machines can draw any lower semicontinuous function. We also study the question under computational constraints, with the following result: if a computable signal-machine diagram coincides with the graph of a Lipschitz-function of sufficiently small Lipschitz coefficient, then that function is the limit of a growing and computable sequence of rational step functions

Cette thèse se veut d'abord une étude mathématique qui donne des bases théoriques au calcul numérique des dimensions fractales d'un attracteur. Des nombreuses expériences numériques relient la théorie aux exigences qui apparaissent dans les applications. L'algorithme de calcul des dimensions fractales propose par Paladin et Vulpiani, que nous appelons algorithme des points centraux, nous semble être le plus puissant parmi les algorithmes proposés. Nous donnons la description détaillée de cet algorithme, et espérons qu'elle est aussi compréhensible pour un non-spécialiste. Une méthode d'estimation d'erreur pour cet algorithme est proposée et justifiée par des résultats des expériences numériques. Le cout de l'algorithme est calcule. La thèse est complétée par l'étude d'un système dynamique, qui modélise une réaction biochimique chaotique, qui intervient dans le cycle de vie d'un genre d'amibes

The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.

Cette thèse est consacrée à l’étude du transport branché, de problèmes variationnels qui y sont liés et de structures fractales qui peuvent y apparaître. Le problème du transport branché consiste à connecter deux mesures de même masse par le biais d’un réseau en minimisant un certain coût, qui sera pour notre étude proportionnel à mLα afin de déplacer une masse m sur une distance L. Plusieurs modèles continus ont été proposés pour formuler le problème, et on s’intéresse plus particulièrement aux deux grands types de modèles statiques : le modèle Lagrangien et le modèle Eulérien, avec une emphase sur le premier. Après avoir posé proprement les bases de ces modèles, on établit rigoureusement leur équivalence en utilisant une décomposition de Smirnov des mesures vectorielles à divergence mesure. On s’intéresse par la suite à un problème d’optimisation de forme lié au transport branché qui consiste à déterminer les ensembles de volume 1 les plus proches de l’origine au sens du transport branché. On démontre l’existence d’une solution, décrite comme un ensemble de sous-niveau de la fonction paysage, désormais standard en transport branché. La régularité Hölder de la fonction paysage, obtenue ici sans hypothèse de régularité a priori sur la solution considérée, permet d’obtenir une borne supérieure sur la dimension de Minkowski de son bord, qui est non-entière et dont on conjecture qu’elle en est la dimension exacte. Des simulations numériques, basées sur une approximation variationnelle à la Modica-Mortola de la fonctionnelle du transport branché, ont été effectuées dans le but d’étayer cette conjecture. Une dernière partie de la thèse se concentre sur la fonction paysage, essentielle à l’étude de problèmes variationnels faisant intervenir le transport branché en ce sens qu’elle apparaît comme une variation première du coût d’irrigation. Le but est d’étendre sa définition et ses propriétés fondamentales au cas d’une source étendue, ce à quoi l’on parvient dans le cas d’un réseau possédant un système fini de racines, par exemple pour des mesures à supports disjoints. On donne une définition satisfaisante de la fonction paysage dans ce cas, qui vérifie en particulier la propriété de variation première et on démontre sa régularité Hölder sous des hypothèses raisonnables sur les mesures à connecter
This thesis is devoted to the study of branched transport, related variational problems and fractal structures that are likely to arise. The branched transport problem consists in connecting two measures of same mass through a network minimizing a certain cost, which in our study will be proportional to mLα in order to move a mass m over a distance L. Several continuous models have been proposed to formulate this problem, and we focus on the two main static models : the Lagrangian and the Eulerian ones, with an emphasis on the first one. After setting properly the bases for these models, we establish rigorously their equivalence using a Smirnov decomposition of vector measures whose divergence is a measure. Secondly, we study a shape optimization problem related to branched transport which consists in finding the sets of unit volume which are closest to the origin in the sense of branched transport. We prove existence of a solution, described as a sublevel set of the landscape function, now standard in branched transport. The Hölder regularity of the landscape function, obtained here without a priori hypotheses on the considered solution, allows us to obtain an upper bound on the Minkowski dimension of its boundary, which is non-integer and which we conjecture to be its exact dimension. Numerical simulations, based on a variational approximation a la Modica-Mortola of the branched transport functional, have been made to support this conjecture. The last part of the thesis focuses on the landscape function, which is essential to the study of variational problems involving branched transport as it appears as a first variation of the irrigation cost. The goal is to extend its definition and fundamental properties to the case of an extended source, which we achieve in the case of networks with finite root systems, for instance if the measures have disjoint supports. We give a satisfying definition of the landscape function in that case, which satisfies the first variation property and we prove its Hölder regularity under reasonable assumptions on the measures we want to connect

Les modèles géométriques de calcul permettent d'effectuer des calculs à l'aide de primitives géométriques. Parmi eux, le modèle des machines à signaux se distingue par sa simplicité, ainsi que par sa puissance à réaliser efficacement de nombreux calculs. Nous nous proposons ici d'illustrer et de démontrer cette aptitude, en particulier dans le cas de processus massivement parallèles. Nous montrons d'abord à travers l'étude de fractales que les machines à signaux sont capables d'une utilisation massive et parallèle de l'espace. Une méthode de programmation géométrique modulaire est ensuite proposée pour construire des machines à partir de composants géométriques de base -- les modules -- munis de certaines fonctionnalités. Cette méthode est particulièrement adaptée pour la conception de calculs géométriques parallèles. Enfin, l'application de cette méthode et l'utilisation de certaines des structures fractales résultent en une résolution géométrique de problèmes difficiles comme les problèmes de satisfaisabilité booléenne SAT et Q-SAT. Ceux-ci, ainsi que plusieurs de leurs variantes, sont résolus par machines à signaux avec une complexité en temps intrinsèque au modèle, appelée profondeur de collisions, qui est polynomiale, illustrant ainsi l'efficacité et le pouvoir de calcul parallèle des machines à signaux.

Orientador: Vitor Baranauskas
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: Apresentamos um método baseado na Teoria dos Fractais que permite efetuar o cálculo do grau de irregularidade em cada ponto da superfície de uma imagem topográfica. O algoritmo proposto fornece valores que são função da escala de observação, de forma a ignorar irregularidades com tamanhos muito maiores ou muito menores que o valor da escala. Dessa forma, é possível implementar duas funcionalidades: calcular os graus de irregularidade para todos os pixels de uma imagem em uma determinada escala de observação e calcular os graus de irregularidade em diversas escalas para um determinado pixel da imagem. Com a primeira funcionalidade, podemos segmentar a imagem topográfica em regiões de maior ou menor irregularidade quando observada sob uma determinada escala. Com a segunda funcionalidade, podemos estudar a variação do grau de irregularidade de um ponto da imagem quando variamos a escala de observação. Mostramos que esse estudo permite identificar os tamanhos das irregularidades que ocorrem em superfícies topográficas como, por exemplo, os tamanhos médios dos grãos e as distâncias médias entre eles. Um ambiente gráfico foi desenvolvido com a concepção de Orientação a Objetos em C++ para estudo desse algoritmo e pode ser facilmente usado para análise de outros algoritmos similares
Abstract: We describe a method based on the Theory of Fractals to calculate a measure of the degree of irregularity in each surface point of any topographic image. The proposed algorithm gives values that are dependent on the scale of observation so as to ignore irregularities which sizes are much greater or lower than the scale value. Therefore, it is possible to implement two features: calculation of the degrees of irregularity for all the pixels of an image in a given scale of observation and calculation of the degrees of irregularity in many scales of observation for a given image pixel. With the first feature we can segment the topographic image in regions of different degrees of irregularity in a given scale of observation. With the second feature we can study the variation of the degree of irregularity measured for an image pixel while we change the scale of observation. We show that the proposed method allows the identification of the sizes of irregularities that occur in topographic surfaces, such as the mean sizes of the grains
Mestrado
Mestre em Engenharia Elétrica

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.

Les « cermets» sont des matériaux hétérogènes composés de grains métalliques de tailles manométriques figés dans un isolant, ou posés sur un substrat de verre. Au voisinage du seuil de percolation, ces matériaux sont caractérisés par une géométrie fractale et une absorption anormale dans la gamme de fréquences du proche infrarouge correspondant à. Des exaltations de champ électromagnétique localisées. Notre étude numérique concerne leurs propriétés optiques et morphologiques. En premier lieu, nous nous sommes attachés dans ce mémoire, à. Présenter la théorie de la percolation, ainsi que les grandeurs et les modèles essentiels qui nous ont permis d'étudier statistiquement les couches métalliques semi-continues. Nous présentons ensuite l'élaboration d'un nouveau programme, fondé sur un algorithme de Monte-Carlo cinétique. Ce programme tient compte des interactions entre grains à l'aide d'un potentiel inter-atomique semi-empirique et permet ainsi de simuler les différents stades de la croissance de films minces d'or. Cette simulation est, à. Notre connaissance, la seule à. Donner. Des résultats quasiment identiques aux. Clichés pris par microscopie électronique à transmission de ces couches. La seconde partie de cette thèse est, quant à elle, consacrée à l'étude optique des cermets. En modélisant les grains d'or par des sphères métalliques polarisables posées sur un réseau carré de sites et en nous plaçant dans l'approximation quasi-statique, nous avons calculé l'interaction d'une onde sonde avec ces surfaces en tenant compte des intéractions dipolaires. Enfin, nous avons étendu notre étude aux échantillons à. Trois dimensions,en les modélisant par des réseaux aléatoires de liens. Grâce à une nouvelle méthode numérique, nous avons rigoureusement résolu l'hamiltonien de Kirchhoff dans ces systèmes. Nos résultats montrent qu'i\. Deme dimensions, la localisation des champs exaltés à la surface des cermets dépend de la polarisation de l'onde incidente. Nous avons aussi montrés que lorsque l'épaisseur des réseaux 3D était faible devant la longueur d'onde de l'onde incidente, leur comportement était proche de celui des réseaux 2D
Cermets» are heterogeneous materials containing nanometric metallic grains embedded in an insulating substrate or deposited on a glass substrate. Near the percolation threshold, these materials are characterized by their fractal geometric morphology and by an anomalous beorption of electromagnetic waves all over the near-infrared range corresponding to g1ant localized fluctuations of electomagnetic fields. Our nurnerical study is based on their optical and orphoIogical properties. We first Cocus on the percolation theory and on funamental modeles which allowed us to statistically study these semi-continuous films. We present then the elaboration of a new program, based on a Kinetic Monte-Carlo algorithm. With the help of a semiempirical interatomic potential to take into account the grains interactions, this program allowed us to simulate the different steps of the growth of gold thin films. From our knowledge, this model is the only one ·to g1ve qua. Si-identical results to the micrographs obtained by transmission electron microscopy at every step. The second part of our study is concerned with the optical properties of « cermets». We have calulated, on these systems, local electric field intensities by repla. Cing the film's morphology by a square lattice of polarizable metallic spheres in the quasi-static approximation. Finally, we used random cubic lattices of bonds to simulate three dimensional samples. With the help of an exact and very efficient numerical method we rigorously solved the Kirchoff's Hamiltonian for these systems. Our results show that the surface position of the exalted fields strongly depends, for bidimensionalsamples, on the polarisation of the incident wave. When the thickness of the hree dimensional samples is small in front of the wavelength of the incicident wave, their behavior is very similar ta the bidimensionals ones

Ce travail comprend deux parties. La première traite du calcul de la dimension de Hausdorff de certains ensembles du plan dont les recouvrements naturels se font au moyen de rectangles qui s'aplatissent à mesure que leur diamètre tend vers zéro. Or on sait que les mesures et dimension de Hausdorff se définissent au moyen de recouvrements par des boules. Il se pose donc le problème de passer d'un recouvrement économique par des rectangles à un recouvrement économique par des boules. Les ensembles que nous étudions sont définis par des propriétés des développements dans des bases éventuellement différentes des coordonnées de leurs points. Dans certains cas nous savons déterminer la dimension de Hausdorff de ces ensembles, dans d'autres nous en obtenons seulement un encadrement. Cette étude tire son origine de résultats d'Eggleston qu'elle généralise. Nous étudions aussi des ensembles aléatoires obtenus en effectuant des partages aléatoires successifs à la manière de Cantor, et déterminons leur dimension de Hausdorff, généralisant et améliorant ainsi des résultats de J. Peyrière. Dans la seconde partie nous définissons et étudions une variante d'un modèle de turbulence dû à B. Mandelbrot et étudié par J. P. Kahane et J. Peyrière : une mesure aléatoire est définie par un produit infini de fonctions aléatoires. Nous donnons une condition nécessaire et suffisante de non-dégénérescence de ce processus. Nous déterminons aussi à quelle condition certains moments sont finis et donnons la dimension minimum des boréliens qui portent une partie de cette mesure
This thesis is divided into two parts. The first one deals with the determination of the Hausdorff dimension of some planar sets of which the natural coverings are made of rectangles which become thinner and thinner as their diameter tends to zero. But we know that measures and Hausdorff dimension are defined by the mean of cave­ rings by balls. So the problem to pass from economical coverings by rectangles to economical coverings by balls is posed. The sets we are studying are defined by properties of the expansions in two different bases of the coordinates of their points. In certain cases we determine the Hausdorff dimension of these sets, which in ether cases we only obtain lower and upper bounds for it. This study sterns results by Eggleston which we generalize. We also determine the dimension of sets obtained by Cantor like constructions, generalizing and improving results by Peyrière. In the second part we define and study a modification of a model of turbulence due to B. Mandelbrot and studied by J. P. Kahane and J. Peyrière: a random measure is defined by an infinite product of random functions. We give a necessary and sufficient condition of non-degeneracy of this process. We also determine under what conditions within moments are finite and get the minimum dimension of sets which carry a part of this measure

Orientador: Lee Luan Ling
Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: A modelagem multifractal generaliza os modelos de tráfego existentes na literatura e se mostra apropriada para descrever as características encontradas nos fluxos de tráfego das redes atuais. A presente tese investiga abordagens para alocação de banda, predição de tráfego e estimação de probabilidade de perda de bytes considerando as características multifractais de tráfego. Primeiramente, um Modelo Multifractal baseado em Wavelets (MMW) é proposto. Levando em consideração as propriedades deste modelo, são derivados o parâmetro de escala global, a função de autocorrelação e a banda efetiva para processos multifractais. A capacidade de atualização em tempo real do MMW aliada à banda efetiva proposta permite o desenvolvimento de um algoritmo de estimação adaptativa de banda efetiva. Através deste algoritmo é introduzido um esquema de provisão adaptativo de banda efetiva. Estuda-se também a alocação de banda baseada em predição de tráfego. Para este fim, propõe-se um preditor adaptativo fuzzy de tráfego, o qual é aplicado em uma nova estratégia de alocação de banda. O preditor fuzzy adaptativo proposto utiliza funções de base ortonormais baseadas nas propriedades do MMW. Com relação à probabilidade de perda para tráfego multifractal, derivase uma expressão analítica para a estimação da probabilidade de perda de bytes considerando que o tráfego obedece ao MMW. Além disso, uma caracterização mais completa do comportamento de fila é efetuada pela obtenção de limitantes para a probabilidade de perda e para a ocupação média do buffer em termos da banda efetiva do MMW. Por fim, é apresentado um esquema de controle de admissão usando o envelope efetivo proposto para o MMW oriundo do cálculo de rede estatístico, que garante que os fluxos admitidos obedeçam simultaneamente aos requisitos de perda e de retardo. As simulações realizadas evidenciam a relevância das propostas apresentadas
Abstract: Multifractal modeling generalizes the existing traffic models and is believed to be appropriate to describe the characteristics of traffic flows of modern communication networks. This thesis investigates some novel approaches for bandwidth allocation, traffic prediction and byte loss probability estimation, by considering the multifractal characteristics of the network traffic. Firstly, a Wavelet based Multifractal Model (WMM) is proposed. Taking into account the properties of this multifractal model, we derive the global scaling parameter, the autocorrelation function and the effective bandwidth for multifractal processes. The real time updating capacity of the WMM in connection with our effective bandwidth proposal allows us to develop an algorithm for adaptive effective bandwidth estimation. Then, through this algorithm, an adaptive bandwidth provisioning scheme is introduced. In this work, we also study a prediction-based bandwidth allocation case. For this end, we develop an adaptive fuzzy predictor, which is incorporated into a novel bandwidth allocation scheme. The proposed adaptive fuzzy predictor makes use of orthonormal basis functions based on the properties of the WMM. Additionally, we derive an analytical expression for the byte loss probability estimation assuming that the traffic obeys the MMW. Besides, a more complete characterization of the queuing behavior is carried out through the estimation of the bounds for the loss probability and mean queue length in buffer in terms of the WMM based effective bandwidth. Finally, an admission control scheme is presented that uses the WMM based effective envelope derived through the statistical network calculus, guaranteeing that the admitted flows simultaneously attend the loss and delay requirements. The computer simulation results confirm the relevance of the presented proposals
Doutorado
Telecomunicações e Telemática
Doutor em Engenharia Elétrica

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Atrav?s de estudos sobre percola??o, pode-se determinar se uma rede bidimensional percola, percorrendo apenas parte das fronteiras dos aglomerados, verificando se existem dois s?tios da fronteira conectados em lados opostos da rede, isto ?, sem a necessidade de preencher todos os s?tios que formam os aglomerados. O objeto desta tese ? um algoritmo para tal fim. Diante da velocidade que este algoritmo ter?, percorrendo apenas parte das fronteiras dos aglomerados, vimos que seria poss?vel estudar redes de tamanhos jamais alcan?ados (superiores a um trilh?o de s?tios), com complexidade menor que 1 e um baixo custo computacional em rela??o aos algoritmos j? desenvolvidos sobre o tema percola??o. Passamos, com isso, a querer estudar o comportamento do limiar de percola??o e da dimens?o fractal da fronteira em redes dos mais diversos tamanhos e com uma grande quantidade de simula??es, as quais os resultados permitiram fazer compara??es e confirmar as previs?es feitas atrav?s de leis de escalas j? conhecidas na literatura.
Through studies on percolation, can determine if a percolates dimensional network, covering only part of the borders of the agglomerates by checking if there are two border sites connected on opposite sides of the net, i.e. without the need to fill all the sites form agglomerates. The object of this thesis is an algorithm for this purpose. Given the speed that this algorithm will, covering only part of the borders of the clusters, we saw that it would be possible to study never reached sizes of networks (more than one trillion websites), with less complexity than 1 and a low computational cost compared to the algorithms already developed on the theme percolation. We pass, therefore, to want to study the behavior of percolation threshold and the fractal dimension of the border into networks of different sizes and with a lot of simulations, which results allowed comparisons and confirm the predictions made by laws scales known in the literature.

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The present study is on the field of Mathematics Education in higher education and is focused on the teaching and learning of Calculus concepts, specifically related to the concepts of classical derivative and weak derivative. The work, developed in the context of a qualitative research, aims to investigate how some students of the Master degree course in Teaching Mathematics of a community institution in Rio Grande do Sul comprehend the concepts of classical derivative and weak derivative. The APOS theory serves as a theoretical and methodological reference for the elaboration of the genetic decomposition in which the possible mental constructions used by the students were described in what the understanding of the concepts of classical derivative and weak derivative concern. We proposed some activities on the historical constructions of the classical concept, the derivative in the present times, and the passage from the classical derivative to the weak derivative. The teaching situations were developed in the classroom in the second semester of 2016, within the subject of Fundamentals of Differential and Integral Calculus. The basis for these activities was the ACE teaching cycle. The results obtained by analyzing the students' records in the proposed activities and the observations recorded in the field diary indicate that the students were able to coordinate actions and processes in order to obtain the derivative and verify if a function is differentiable. They could also coordinate the interpretations of the derivative, such as slope of the tangent line, instantaneous velocity and rate of variation, besides using mechanisms of generalization and reversibility in the analysis of the graphs of functions and their derivatives and encapsulating the processes necessary for a satisfactory understanding of the concept of the classical derivative. By means of the integral equation, the integration formula by parts, and the fundamental theorem of Calculus, the students were able to coordinate the function and intervals, the functions with compact support by means of internalizing actions and processes for the encapsulation of the mathematical object and weak derivative. Although there are some errors in these processes, there is evidence that the concepts of classical derivative and weak derivative have been understood by the students. These evidences developed mental mechanisms of reflective abstraction that allowed the construction of the mental structures of action, process, object and scheme present in the genetic decomposition that allowed them to understand the concepts. Moreover, the treatment with the historical context of the derivative and the collaborative work of the students were significant factors to obtain the results of the research.
O presente estudo se situa no campo da Educação Matemática no ensino superior e se insere na linha de investigação voltada ao ensino e aprendizagem de conceitos do Cálculo, especificamente ligados aos conceitos de derivada clássica e derivada fraca. O trabalho, desenvolvido no contexto de uma pesquisa qualitativa, teve como objetivo investigar como se dá a compreensão dos conceitos de derivada clássica e derivada fraca por estudantes de um curso de mestrado em Ensino de Matemática de uma instituição comunitária do Rio Grande do Sul. Tendo a teoria APOS como referencial teórico e metodológico, elaborou-se a decomposição genética, em que foram descritas as possíveis construções mentais utilizadas pelos estudantes para a compreensão dos conceitos de derivada clássica e derivada fraca. Foram organizadas situações de ensino compostas por atividades sobre as construções históricas do conceito clássico, a derivada nos tempos atuais e a passagem da derivada clássica para a derivada fraca. As situações de ensino foram desenvolvidas em sala de aula, no segundo semestre de 2016, na disciplina de Fundamentos de Cálculo Diferencial e Integral, tendo como base o ciclo de ensino ACE. Os resultados obtidos, por meio da análise dos registros dos alunos nas atividades propostas e das observações anotadas no diário de campo, indicam que os estudantes foram capazes de coordenar ações e processos para obter a derivada e verificar se uma função é diferenciável, coordenar as interpretações da derivada como inclinação da reta tangente, velocidade instantânea e taxa de variação, além de, utilizar mecanismos de generalização e reversibilidade na análise dos gráficos das funções e suas derivadas e de encapsular os processos necessários para a compreensão, de forma satisfatória, do conceito da derivada clássica. Por meio da equação integral, da fórmula de integração por partes e do teorema fundamental do Cálculo, os alunos coordenaram a função e os intervalos, funções com suporte compacto, interiorizando ações e processos para a encapsulação do objeto matemático, derivada fraca. Embora com alguns erros cometidos nesses processos, há evidências de que houve compreensão dos conceitos de derivada clássica e derivada fraca pelos estudantes. Estes evidenciaram desenvolver mecanismos mentais de abstração reflexionante que possibilitaram a construção das estruturas mentais de ação, processo, objeto e esquema presentes na decomposição genética que lhes permitiu compreender os conceitos. Além do mais, o trato com o contexto histórico da derivada e o trabalho colaborativo dos alunos foram fatores significativos para a obtenção dos resultados da pesquisa.

This work summarizes the research path done by Walter Arrighetti during his three years of Doctorate of Research in Electromagnetism at Università degli Studi di Roma “La Sapienza,” Rome, under the guidance of Professor Giorgio Gerosa. This work was mainly motivated by the struggle to find simpler and simpler models to introduce complex geometries (like fractal ones, for example, which are complicated but far from being ‘irregular’) in physical field theories like the Classical Electrodynamics, and which stand at the base of most contemporary applied research activities: from antennas (of any sizes, bandwidths and operational distances) to waveguides & resonators (for devices ranging from IC motherboards , to high-speed fibre channel links), to magnetic resonance (RMI) devices (for both diagnostic and research purposes), all the way up to particle accelerators. All of these models need not only a solid physical base, but also a specifically crafted ensemble of mathematical methods, in order to tackle with problems which “standard-geometry” models (both in the continuum and the discrete cases) are not best-suited for. During his previous years of study towards the Laurea degree in Electronic Engineering, the author used different approaches toward Fractal Electrodynamics, form purely-analytical, to computer-assisted numerical simulations of applied electromagnetic structures (both radiating and wave-guiding), down to algebraic-topological ones. The latter approaches, more often than not, proved to be the best way to start with, because the author found out that self-similarity (a property which many complicated geometries —even non-fractal ones— seem to, at least, tend to possess) can be easily interpreted as a topological symmetry, wonderfully described using “ad hoc” nontrivial algebraic languages. Whatever can be successfully described in the language of Algebra (either via numbers, symmetry groups, graphs, polynomials, etc.) is then always simplified (or “quotiented” — so to speak in a more strict mathematical language) and, when numerical computation takes the way towards the solution of a specific applied problem, those simplifications turn in handy to reduce the complexity of it. For example, the strict self-similarity possessed by some fractals (like those generated via an Iterated Function System — or IFS) allows to numerically store the geometrical data for a fractal object in a sequence of simpler and simpler data which are, for example, instantly recovered by a computer starting from the simplest data (like simplices, squares/cubes, circles/spheres and regular polygons/polytopes). For the same reason, all the physical properties that depend on the geometry (or the topology — i.e. basically the number of “holes” or inner connections) of the domain can be reduced, estimated or be even completely known a priori, even before a numerical simulation is performed. In this work, several of these methods (coming from apparently different branches of pure and applied Mathematics) are presented and finally joined with Electromagnetism equations to solve some more or less applied problems. Since many of the mathematical tools used to build the studied models and methods are advanced and generally not sufficiently known to experts in either such different fields, the first two Chapters are devoted to a brief introduction of some purely mathematical topics. In that context, the author found that the best way to accomplish this was to re-write all those different results from different branches of both pure and applied Mathematics in a formalism as more solid and unified as possible, with continuous links back and forth to different topics (and to the next more applied Chapters). That approach is seldom found in most graduate-level texts. For example, very similar mathematical objects may be even called or classified in different ways, according to the different mathematical contexts they are introduced in, which is exactly the opposite philosophy which has guided underneath in writing these first Chapters. On the other end, simpler and more trivial mathematical definitions, formalisms or electromagnetic problems, when not elsewhere referenced to, can be found in [9], Arrighetti W., Analisi di Strutture Elettromagnetiche Frattali, the author’s Laurea degree dissertation (currently only in Italian language). The most original part of the work is in the last three Chapters where —always using the same “language” and helping with cross-links, as well as to the Bibliography— methods are introduced and then applied to model some electromagnetic problems (previously either unsolved — or already-known, but here solved with a different, usually simpler, or at least more elegant approach).

This work summarizes the research path done by Walter Arrighetti during his three years of Doctorate of Research in Electromagnetism at Università degli Studi di Roma “La Sapienza,” Rome, under the guidance of Professor Giorgio Gerosa. This work was mainly motivated by the struggle to find simpler and simpler models to introduce complex geometries (like fractal ones, for example, which are complicated but far from being ‘irregular’) in physical field theories like the Classical Electrodynamics, and which stand at the base of most contemporary applied research activities: from antennas (of any sizes, bandwidths and operational distances) to waveguides & resonators (for devices ranging from IC motherboards , to high-speed fibre channel links), to magnetic resonance (RMI) devices (for both diagnostic and research purposes), all the way up to particle accelerators. All of these models need not only a solid physical base, but also a specifically crafted ensemble of mathematical methods, in order to tackle with problems which “standard-geometry” models (both in the continuum and the discrete cases) are not best-suited for. During his previous years of study towards the Laurea degree in Electronic Engineering, the author used different approaches toward Fractal Electrodynamics, form purely-analytical, to computer-assisted numerical simulations of applied electromagnetic structures (both radiating and wave-guiding), down to algebraic-topological ones. The latter approaches, more often than not, proved to be the best way to start with, because the author found out that self-similarity (a property which many complicated geometries —even non-fractal ones— seem to, at least, tend to possess) can be easily interpreted as a topological symmetry, wonderfully described using “ad hoc” nontrivial algebraic languages. Whatever can be successfully described in the language of Algebra (either via numbers, symmetry groups, graphs, polynomials, etc.) is then always simplified (or “quotiented” — so to speak in a more strict mathematical language) and, when numerical computation takes the way towards the solution of a specific applied problem, those simplifications turn in handy to reduce the complexity of it. For example, the strict self-similarity possessed by some fractals (like those generated via an Iterated Function System — or IFS) allows to numerically store the geometrical data for a fractal object in a sequence of simpler and simpler data which are, for example, instantly recovered by a computer starting from the simplest data (like simplices, squares/cubes, circles/spheres and regular polygons/polytopes). For the same reason, all the physical properties that depend on the geometry (or the topology — i.e. basically the number of “holes” or inner connections) of the domain can be reduced, estimated or be even completely known a priori, even before a numerical simulation is performed. In this work, several of these methods (coming from apparently different branches of pure and applied Mathematics) are presented and finally joined with Electromagnetism equations to solve some more or less applied problems. Since many of the mathematical tools used to build the studied models and methods are advanced and generally not sufficiently known to experts in either such different fields, the first two Chapters are devoted to a brief introduction of some purely mathematical topics. In that context, the author found that the best way to accomplish this was to re-write all those different results from different branches of both pure and applied Mathematics in a formalism as more solid and unified as possible, with continuous links back and forth to different topics (and to the next more applied Chapters). That approach is seldom found in most graduate-level texts. For example, very similar mathematical objects may be even called or classified in different ways, according to the different mathematical contexts they are introduced in, which is exactly the opposite philosophy which has guided underneath in writing these first Chapters. On the other end, simpler and more trivial mathematical definitions, formalisms or electromagnetic problems, when not elsewhere referenced to, can be found in [9], Arrighetti W., Analisi di Strutture Elettromagnetiche Frattali, the author’s Laurea degree dissertation (currently only in Italian language). The most original part of the work is in the last three Chapters where —always using the same “language” and helping with cross-links, as well as to the Bibliography— methods are introduced and then applied to model some electromagnetic problems (previously either unsolved — or already-known, but here solved with a different, usually simpler, or at least more elegant approach).

Tese de doutoramento em Engenharia Electrotécnica (Instrumentação e Controlo) apresentada à Fac. Ciências e Tecnologia da Univ. Coimbra
Os manipuladores robóticos apresentam vibrações indesejadas durante o seu funcionamento. Por um lado, estas vibrações resultam de numerosos factores, tais como, folgas, flexibilidades, atritos, não-linearidades e outras causas. Por outro lado, os robôs, ao interagirem com o meio ambiente, geram frequentemente impactos que produzem vibrações que se propagam através de toda a estrutura mecânica. Neste contexto, de modo a reduzir, ou eliminar, o efeito das vibrações e dos impactos, é fundamental estudar as variáveis envolvidas para se poderem definir estratégias adequadas. Nesta ordem de ideias, este trabalho estuda e desenvolve metodologias de análise para aplicações em estruturas de manipulação sujeitas a impactos e a vibrações. As experiências realizadas com o sistema robótico desenvolvido, na presença de impactos, vibrações e na movimentação de líquidos, evidenciaram o comportamento de ordem fraccionária de alguns sinais. A transformada de Fourier com janela, utilizada no estudo dos sinais robóticos, revelou-se uma ferramenta adequada para a análise dos sinais não estacionários, como é o caso dos sinais originados nos fenómenos referidos. Os robôs utilizam uma multiplicidade de sensores de forma a adaptarem-se a perturbações ou a mudanças inesperadas no espaço de trabalho. Os dados assim obtidos podem ser redundantes, uma vez que a mesma informação pode ser captada por dois ou mais sensores. Neste contexto, faz-se um estudo do comportamento do espectro dos sinais e apresenta-se um método de classificação dos sinais que pode contribuir para a optimização da instrumentação utilizada nos sistemas robóticos. No estudo dos sinais robóticos apresentam-se várias experiências suportadas por conceitos da teoria da informação e implementadas através de uma reconstrução do espaço de estados. Assim, determina-se, experimentalmente, uma relação entre os declives das linhas de tendência dos espectros com a dimensão fractal do espaço de estados reconstruído e o correspondente tempo de atraso. Propõem-se ainda dois índices para determinação do grau das folgas em sistemas mecânicos sujeitos a oscilações periódicas. Desenvolve-se também um novo método, baseado na informação mútua, para sintonia da transformada de Fourier com janela.
The operation of robotic manipulators reveals unwanted vibrations. On one hand, these vibrations occur due to several factors, such as, backlash, flexibilities, friction, non-linearities and other effects. On the other hand, the robots, interacting with the environment, generate often impacts that produce vibrations which are propagated through the mechanical structure. In this perspective, in order to adopt adequate strategies for reducing or eliminating the effect of vibrations and impacts, it is important to study the involved variables. Bearing these ideas in mind, this work studies and develops analysis methodologies for applying to mechanical manipulators structures subject to impacts and vibrations. Several experiments are performed with the developed robotic system in the presence of impacts, vibrations, or when carrying liquid containers. Some of the captured signals reveal a fractional order behavior. The windowed Fourier transform is applied in the study of the robotic signals and reveals to be an adequate tool to deal with this type of non stationary signals. The robots use a multiplicity of sensors necessary to deal with the perturbations or with unexpected changes in its work space. Therefore, the data obtained can be redundant because the same type of information can be obtained by two or more sensors. In this context, is established the study of the signal spectra. A sensor classification scheme is developed that can help in the design optimization of the robotic instrumentation. Several experiments are performed for analyzing the robotic signals, based on the information theory, and implemented through the pseudo phase space. An experimental relationship is determined between the slopes of the trendlines spectra, with the fractal dimension of the pseudo phase space and the corresponding time lag. Additionally, two indices are proposed to detect the backlash effect on mechanical systems with periodic oscillations. Finally, a new method based on the mutual information, for tuning the windowed Fourier transform, is presented.