Este trabalho descreve as soluções de algumas equações diofantinas em duas e três variáveis. O objetivo é apresentar a análise de alguns casos simples e de outros mais difíceis relativos ao Último Teorema de Fermat. Primeiramente são apresentados os pré-requisitos necessários dentre os quais incluímos a noção de número primo, máximo divisor comum, congruência, o Algoritmo de Euclides e o Teorema Fundamental da Aritmética. Este material é desenvolvido primeiramente no anel dos inteiros racionais e posteriormente em duas extensões algébricas conhecidas como os inteiros de Gauss e de Eisenstein. A estrutura dos últimos é indispensável na resolução do primeiro caso não trivial do Último Teorema de Fermat, a saber, da equação diofantina x3 + y3 = z3. O último capítulo apresenta algumas aplicações de problemas diofantinos e do Algoritmo de Euclides que podem ser desenvolvidos em sala de aula com alunos do sexto e do oitavo ano. / This work describes the solutions to some diophantine equations in two and three variables. The objective is to present the analysis of some simple and other more difficult cases related to Fermats Last Theorem. First, we present the necessary prerequisites which include the notion of a prime number, the maximum common divisor, congruences, Euclids Algorithm and the Fundamental Theorem of Arithmetic. This material is first developed by using the rational integers and then presented for two algebraic extensions known as Gauss and Eisenstein integers. The structure of the latter is indispensable for the first non-trivial case of Fermats Last Theorem, namely, the diophantine equation x3 + y3 = z3. The last chapter presents some applications of simple diophantine equations and Euclids algorithm which can be developed in the classroom with sixth and eight grade students.

<p>We present a detailed study on application of factor<br />graphs and the belief propagation (BP) algorithm to the<br />power system state estimation (SE) problem. We start<br />from the BP solution for the linear DC model, for which<br />we provide a detailed convergence analysis. Using BPbased<br />DC model we propose a fast real-time state<br />estimator for the power system SE. The proposed<br />estimator is easy to distribute and parallelize, thus<br />alleviating computational limitations and allowing for<br />processing measurements in real time. The presented<br />algorithm may run as a continuous process, with each<br />new measurement being seamlessly processed by the<br />distributed state estimator. In contrast to the matrixbased<br />SE methods, the BP approach is robust to illconditioned<br />scenarios caused by significant differences<br />between measurement variances, thus resulting in a<br />solution that eliminates observability analysis. Using the<br />DC model, we numerically demonstrate the performance<br />of the state estimator in a realistic real-time system<br />model with asynchronous measurements. We note that<br />the extension to the non-linear SE is possible within the<br />same framework.<br />Using insights from the DC model, we use two different<br />approaches to derive the BP algorithm for the non-linear<br />model. The first method directly applies BP methodology,<br />however, providing only approximate BP solution for the<br />non-linear model. In the second approach, we make a key<br />further step by providing the solution in which the BP is<br />applied sequentially over the non-linear model, akin to<br />what is done by the Gauss-Newton method. The resulting<br />iterative Gauss-Newton belief propagation (GN-BP)<br />algorithm can be interpreted as a distributed Gauss-<br />Newton method with the same accuracy as the<br />centralized SE, however, introducing a number of<br />advantages of the BP framework. The thesis provides<br />extensive numerical study of the GN-BP algorithm,<br />provides details on its convergence behavior, and gives a<br />number of useful insights for its implementation.<br />Finally, we define the bad data test based on the BP<br />algorithm for the non-linear model. The presented model<br />establishes local criteria to detect and identify bad data<br />measurements. We numerically demonstrate that the<br />BP-based bad data test significantly improves the bad<br />data detection over the largest normalized residual test.</p> / <p>Glavni rezultati ove teze su dizajn i analiza novih<br />algoritama za re&scaron;avanje problema estimacije stanja<br />baziranih na faktor grafovima i &bdquo;Belief Propagation&ldquo; (BP)<br />algoritmu koji se mogu primeniti kao centralizovani ili<br />distribuirani estimatori stanja u elektroenergetskim<br />sistemima. Na samom početku, definisan je postupak za<br />re&scaron;avanje linearnog (DC) problema kori&scaron;ćenjem BP<br />algoritma. Pored samog algoritma data je analiza<br />konvergencije i predloženo je re&scaron;enje za unapređenje<br />konvergencije. Algoritam se može jednostavno<br />distribuirati i paralelizovati, te je pogodan za estimaciju<br />stanja u realnom vremenu, pri čemu se informacije mogu<br />prikupljati na asinhroni način, zaobilazeći neke od<br />postojećih rutina, kao npr. provera observabilnosti<br />sistema. Pro&scaron;irenje algoritma za nelinearnu estimaciju<br />stanja je moguće unutar datog modela.<br />Dalje se predlaže algoritam baziran na probabilističkim<br />grafičkim modelima koji je direktno primenjen na<br />nelinearni problem estimacije stanja, &scaron;to predstavlja<br />logičan korak u tranziciji od linearnog ka nelinearnom<br />modelu. Zbog nelinearnosti funkcija, izrazi za određenu<br />klasu poruka ne mogu se dobiti u zatvorenoj formi, zbog<br />čega rezultujući algoritam predstavlja aproksimativno<br />re&scaron;enje. Nakon toga se predlaže distribuirani Gaus-<br />Njutnov metod baziran na probabilističkim grafičkim<br />modelima i BP algoritmu koji postiže istu tačnost kao i<br />centralizovana verzija Gaus-Njutnovog metoda za<br />estimaciju stanja, te je dat i novi algoritam za otkrivanje<br />nepouzdanih merenja (outliers) prilikom merenja<br />električnih veličina. Predstavljeni algoritam uspostavlja<br />lokalni kriterijum za otkrivanje i identifikaciju<br />nepouzdanih merenja, a numerički je pokazano da<br />algoritam značajno pobolj&scaron;ava detekciju u odnosu na<br />standardne metode.</p>

Neste trabalho foi realizado um grande levantamento de técnicas de esparsidade relacionadas a sistemas estáticos de energia elétrica. Tais técnicas visam, do ponto de vista computacional, ao aumento da eficiência na solução de rede elétrica objetivando, além da resolução em si, a redução dos requisitos de memória, armazenamento e tempo de processamento. Para tanto, uma extensa revisão bibliográfica foi compilada, apresentando um posicionamento histórico e uma ampla visão do desenvolvimento teórico. Os testes comparativos realizados para sistemas de 14, 30, 57 e 118 barras, sobre a implantação de três das técnicas mais empregadas, apontou a Bi-fatoração como tendo o melhor desempenho médio. Para sistemas pequenos, a Eliminação Esparsa e Sintética de Gauss apresentou melhores resultados. Este trabalho fornecerá subsídios conceituais e metodológicos a técnicos e pesquisadores da área. / In this work a great survey of sparsity techniques related to static systems of electric power was accomplished. Such techniques seek, for of the computational point of view, the increase of the efficiency in the solution of the electric net aiming, besides the resolution of itself, the reduction of memory requirements, the storage and time processing. For that, an extensive bibliographic review was compiled providing a historic positioning and a broad view of theoretic development. The comparative tests accomplished for systems of 14,30, 57 and 118 buses, on the implementation of three of the most employed techniques, it pointed out an bi-factorisation as best medium performance. For small systems, the sparse symmetric Gaussian elimination showed the best results. This work will supply conceptual and methodological subsidies to technicians and researchers of the area.

Cette thèse est constituée de quatre chapitres. Le premier contient les notions de base qui permettent d'aborder les divers thèmes qui y sont étudiés. Le second est consacré à l'étude des sous-variétés lagrangiennes d'une variété presque kählérienne. J'y présente les résultats obtenus en collaboration avec Burcu Bektas, Joeri Van der Veken et Luc Vrancken. Dans le troisième, je m'intéresse à un problème de géométrie différentielle affine et je donne une classification des hypersphères affines qui sont isotropiques. Ce résultat a été obtenu en collaboration avec Luc Vrancken. Et enfin dans le dernier chapitre, je présente quelques résultats sur les surfaces de translation et les surfaces homothétiques, objet d'un travail en commun avec Rafael López. / Abstract in English not available

<p>U doktorskoj disertaciji je prikazan postupak optimizacije podzemne mreže<br />za potrebe izgradnje beogradskog metroa. U postupku optimizacije kori&scaron;ćen<br />je metod prethodne ocene tačnosti. Na osnovu građevinskih standarda<br />izvr&scaron;en je proračun zahtevane tačnosti proboja tunela, kao osnovnog<br />kriterijuma tačnosti za razvijanje podzemne tunelske mreže. U postupku<br />optimizacije analizirani su različiti planovi opažanja, kao i dobijeni rezultati<br />prethodne analize za svaki plan pojedinačno. Na osnovu zadatog kriterijuma<br />maksimalne poprečene gre&scaron;ke proboja tunela usvojen je konačan plan<br />opažanja.</p> / <p>The docotoral thesis presents an optimization method of the underground<br />network for the construction of the Belgrade metro. In the process of<br />optimization, method of preanalysis has been used. Based on the<br />construction standards, the calculation of the required breakthrough<br />accuracy, as the fundamental criteria of accuracy for the development of the<br />underground tunnel network, has been made. In the process of optimization<br />different plans of observations have been analyzed, as well as the results<br />obtained from the preanalysis for each plan individually. Based on the<br />required criteria of maximal transverse error of the tunnel breakthrough, the<br />final plan of observations has been adopted.</p>

Différents résultats de complexité sont présentés pour les communications et le calcul sur des machines à mémoire distribuée. Les topologies concernées sont le réseau linéaire, l'anneau, la grille, l'hypercube et le réseau complet. Un réseau systolique est présenté pour l'algorithme de diagonalisation de Jordan. Une étude sur l'accélération et une étude de l'allocation des données sont formulées dans le contexte des mémoires distribuées

Les deux principaux problèmes à résoudre pour concevoir un simulateur dynamique interactif sont la détection de collisions entre les objets virtuels, et le calcul de leur mouvement contraint. La plupart des méthodes de détection de collisions sont discrètes : elles détectent seulement des interpénétrations entre les objets virtuels à des instants discrets successifs. Afin de détecter efficacement des collisions entre objets polyédriques rigides en continu, c'est-à-dire de calculer l'instant de premier contact entre eux et éviter les problèmes inhérents aux méthodes discrètes, nous proposons d'utiliser un mouvement intermédiaire arbitraire pour remplacer le mouvement réel de l'objet et obtenir des équations de détection de collisions faciles à résoudre. Nous présentons deux approches, fondées sur l'exploitation de mouvements intermédiaires arbitraires, qui permettent de détecter des collisions en continu entre objets polyédriques complexes en temps réel. Nous proposons ensuite d'ajouter des informations géométriques aux volumes englobants afin d'exploiter le mouvement de recul relatif des objets et accélérer significativement la détection de collisions lorsque les objets sont proches les uns des autres. La plupart des méthodes classiques de calcul de mouvement contraint sont formulées dans l'espace des contacts. Grâce au principe des moindres contraintes de Gauss, il est possible d'obtenir une formulation équivalente des problèmes dynamiques sans friction dans l'espace des mouvements. Nous montrons que cette formulation est plus avantageuse sur le plan algorithmique. Ceci nous incite à proposer un modèle de friction dans l'espace des mouvements. Les algorithmes proposés ont été implantés et rassemblés dans une librairie C++, CONTACT Toolkit. Nous présentons plusieurs applications de cette librairie, notamment à des cas industriels fournis par Renault et Airbus-EADS, ainsi que son utilisation dans la simulation avec retour d'efforts.

Dans ce travail, nous nous intéressons à l'imagerie de diffraction dans des configurations à deux ou trois dimensions avec pour objectif la reconstruction d'une image (fonction contraste) d'un objet inconnu à l'aide de plusieurs mesures du champ qu'il diffracte. Ce champ résulte de l'interaction entre l'objet et un champ incident connu dont la direction de propagation et la fréquence peuvent varier. La difficulté de ce problème réside dans la non-linéarité du modèle direct et le caractère mal posé du problème inverse qui nécessite l'introduction d'une information a priori (régularisation). Pour cela, nous utilisons une approche bayésienne avec une estimation conjointe du contraste de l'objet, des courants induits et des autres paramètres du modèle. Le modèle direct est décrit par deux équations intégrales couplées exprimant les champs électriques observé et existant à l'intérieur de l'objet, dont les versions discrètes sont obtenues à l'aide de la méthode des moments. Pour l'inversion, l'approche bayésienne permet de modéliser notre connaissance a priori sur l'objet sous forme probabiliste. Les objets que nous étudions ici sont connus pour être constitués d'un nombre fini de matériaux homogènes répartis en régions compactes. Cette information a priori est introduite dans l'algorithme d'inversion à l'aide d'un mélange de gaussiennes, où chaque gaussienne représente une classe de matériaux, tandis que la compacité des régions est prise en compte au travers d'un modèle de Markov caché. La nature non linéaire du modèle direct et l'utilisation de cet a priori nous amènent à des estimateurs qui n'ont pas de formes explicites. Une approximation est donc nécessaire et deux voies sont possibles pour cela: une approche numérique, par exemple MCMC, et une approche analytique comme l'approche bayésienne variationnelle. Nous avons testé ces deux approches qui ont donné de bons résultats de reconstruction par rapport aux méthodes classiques. Cependant, l'approche bayésienne variationnelle permet de gagner énormément en temps de calcul par rapport à la méthode MCMC.

In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a&lt;1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a&lt;1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.

ABSTRACT DETERMINATION OF STOCHASTIC MODEL PARAMETERS OF INERTIAL SENSORS &Uuml / nver, Alper PhD, Department of Electric Electronic Engineering Supervisor: Prof. Dr. M&uuml / beccel Demirekler January 2013, 82 pages Gyro and accelerometer systematic errors due to biases, scale factors, and misalignments can be compensated via an on-board Kalman filtering approach in a Navigation System. On the other hand, sensor random noise sources such as Quantization Noise (QN), Angular Random Walk (ARW), Flicker Noise (FN), and Rate Random Walk (RRW) are not easily estimated by an on-board filter, due to their random characteristics. In this thesis a new method based on the variance of difference sequences is proposed to compute the powers of the above mentioned noise sources. The method is capable of online or offline estimation of stochastic model parameters of the inertial sensors. Our aim in this study is the estimation of ARW, FN and RRW parameters besides the quantization and the Gauss-Markov noise parameters of the inertial sensors. The proposed method is tested both on the simulated and the real sensor data and the results are compared with the Allan variance method. Comparison shows very satisfactory results for the performance of the method. Computational load of the new method is less than the computational load of the Allan variance on the order of tens. One of the usages of this method is the individual noise characterization. A noise, whose power spectral density has a constant slope, can be identified accurately by the proposed method. In addition to this, the parameters of the GM noise can also be determined. Another idea developed here is to approximate the overall error source as a combination of ARW and some number of GM sources only. The reasons of selecting such a structure is the feasibility of using these models in a Kalman filter framework for error propagation as well as their generality of modeling other noise sources.