On introduit une famille de modules, appelés modules de Marlcoff, engendrée par un procédé itératif semblable à la mutation des objects inclinants dans une catégorie amassée. On montre que ces modules ont une structure combinatoire similaire à celle des mots de Christoffel. En outre, on construit une bijection entre l'ensemble des triplets de modules de Markoff et l'ensemble des triplets de Markoff propres. Ceci nous permet de reformuler la conjecture d'unicité des nombres de Markoff dans un cadre algébrique. Dans la deuxième partie, on étudie les dimensions projectives de la restriction des foncteurs [Special characters omitted.] HomC (-, X ) à une sous-catégorie contravariantement finie et rigide d'une catégorie triangulée [Special characters omitted.] C . On montre que la dimension projective de [Special characters omitted.] HomC (-, X )[Special characters omitted.] T est au plus un si et seulement si il n'existe aucun morphisme non nul entre objets de [Special characters omitted.] T [1] qui se factorise par X , lorsque X appartient à une certaine sous-catégorie convenable de [Special characters omitted.] C . Par conséquent, on obtient une caractérisation des objets de dimension projective infinie dans la catégorie des foncteurs contravariants de présentation finie sur une sous-catégorie inclinante amassée de [Special characters omitted.] C . [symboles non conformes]

Les mots de Christoffel forment un sous-ensemble des mots de {x,y}*. Nous les présenterons dans ce mémoire de façon géométrique comme étant la discrétisation d'une droite allant de (0,0) à (a,b), avec a et b des entiers premiers entre eux, par un chemin dans IN2. Nous associerons ainsi les mots de Christoffel aux couples d'entiers premiers entre eux. Nous introduirons ensuite les triplets de Markoff comme étant les solutions de l'équation diophantienne a2+b2+c2 = 3abc. Un homomorphisme µ du monoïde libre {x,y}* dans SL2(Z) sera défini de la façon suivante : µx = (2 1 / 1 1) et µy = (5 2 / 2 1). Celui-ci nous permettra de définir la bijection suivante entre les mots de Christoffel et les triplets de Markoff : w = w1w2 → {⅓Tr(µw1), ⅓Tr (µw2), ⅓Tr(µw)}. Par la suite, nous introduirons l'arbre de Stern-Brocot, l'arbre de Christoffel et l’arbre de Markoff et nous montrerons l'équivalence entre tous ces arbres, et établirons des bijections canoniques entre eux. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Mots de Christoffel, triplets de Markoff, bijection, arbre de Christoffel, arbre de Markoff, arbre de Stern-Brocot.

Zugl.: Frankfurt (Main), Univ., Diss., 2007

In this thesis, we set out to provide an enhanced set of techniques for determining the eigenvalues of the Laplacian in polygonal domains. Currently, finite-element methods provide a numerical means by which we can approximate these eigenvalues with ease. However, we would like a more analytic method which may allow us to avoid a basic parameter sweep in finite-element software such as COMSOL to determine what could possibly be an "optimal" distribution of eigenvalues. The hope is that this would allow us to draw conclusions about the acoustic quality of a pentagonally-shaped room. First, we find the eigenvalues using a common finite-element method through COMSOL Multiphysics. We then examine another method which makes use of conformal maps and Schwarz-Christoffel transformations with the prospect that it might provide a more analytic understanding of the calculation of these eigenvalues and possibly allow for variation of certain parameters. This method, as far as we could find, had not yet been developed on the pentagon. We end up carrying this method through nearly all of the steps necessary in finding these eigenvalues. We find that the finite-element method is not only easier to use, but is also more efficient in terms of computing power. / Master of Science

Tableau d'honneur de la Faculté des études supérieures et postdoctorales, 2015-2016 / La formule de Schwarz–Christoffel permet de trouver une transformation conforme entre un domaine polygonal et un disque. Par contre, cette formule ne s’applique qu’aux domaines simplement connexes. Récemment, Darren Crowdy a obtenu une généralisation de cette formule pour les domaines multiplement connexes. Celle-ci envoie des domaines circulaires sur des domaines polygonaux. Ce mémoire vise à faire la démonstration de la formule développée par Crowdy. Pour ce faire, il faudra définir la fonction de Schottky–Klein ainsi que la fonction de Green modifiée. Il faudra aussi introduire les domaines canoniques.

Extensos estudos realizados nas últimas décadas sobre a propagação de ondas ultrassônicas em sólidos levaram ao desenvolvimento de técnicas não destrutivas para a avaliação da segurança e integridade de estruturas e componentes industriais. O interesse na aplicação de técnicas ultrassônicas para medição de tensões aplicadas e residuais decorre da mudança mensurável da velocidade das ondas ultrassônicas na presença de um campo de tensões, fenômeno conhecido como efeito acustoelástico. Uma teoria de acustoelasticidade fornece um meio atrativo e não destrutivo de medir a tensão média ao longo do caminho percorrido pela onda. O estudo da propagação das ondas ultrassônicas em meios homogêneos anisotrópicos sob tensão conduz a um problema não linear de autovalores dado pela equação de Christoffel generalizada. A característica não linear deste problema decorre da interdependência entre as constantes elásticas efetivas do material e as tensões atuantes. A medição experimental de tensões por técnicas ultrassônicas é um problema inverso da acustoelasticidade. Esta dissertação apresenta a implementação de um algoritmo numérico, baseado no método proposto por Degtyar e Rokhlin, para solução do problema inverso da acustoelasticidade em sólidos ortotrópicos sujeitos a um estado plano de tensões. A solução da equação de Christoffel generalizada apresenta dificuldades de natureza numérica e prática. A estabilidade e a precisão do algoritmo desenvolvido, bem como a influência das incertezas na medição experimental das velocidades das ondas ultrassônicas, foram então investigadas. Dados sintéticos para as velocidades das ondas ultrassônicas de incidência oblíqua em uma placa sujeita a um estado plano de tensões foram gerados pela solução direta da equação de Christoffel generalizada para ilustrar a aplicação do algoritmo desenvolvido. O objetivo maior desta dissertação é a disponibilização de uma nova ferramenta de cálculo para suporte às atividades experimentais de medição de tensões por ultrassom no país. / Extensive studies carried out in the last decades on the propagation of ultrasonic waves in solids led to the development of nondestructive techniques for the assessment of the safety and integrity of industrial structures and components. The interest in the application of ultrasound techniques for stress measurement for example comes from the measurable change in the speed of the ultrasonic elastic waves in the presence of a stress field, a phenomenon known as acoustoelastic effect. An acoustoelastic theory provides an attractive way of non-destructively measuring the average stress along the waves path. The study of the propagation of ultrasonic waves in homogenous anisotropic bodies under stress leads to a nonlinear eigenvalue problem given by the generalized Christoffel equation. The nonlinearity characteristic of the problem derives from the interdependence between the materials effective elastic constants and the acting stresses. The experimental measurement of stresses using ultrasound techniques is an inverse problem of acoustoelasticity. This dissertation presents the implementation of a numeric algorithm, based on the method proposed by Degtyar and Rokhlin, for solution of the inverse problem of acoustoelasticity in orthotropic solids subjected to a plane stress state. The solution of the generalized Christoffel equation poses difficulties of numerical and practical order. The stability and precision of the algorithm developed, as well as the influence of the experimental uncertainties in the measurement of the speed of the ultrasonic waves, were thus investigated. Synthetic data for the speeds of ultrasonic waves of oblique incidence in a plane-stress plate were generated to illustrate the application of the algorithm developed. The main objective of this dissertation is to make available in the country a new numerical tool to support the use of ultrasonic waves for experimental stress analysis.

Extensos estudos realizados nas últimas décadas sobre a propagação de ondas ultrassônicas em sólidos levaram ao desenvolvimento de técnicas não destrutivas para a avaliação da segurança e integridade de estruturas e componentes industriais. O interesse na aplicação de técnicas ultrassônicas para medição de tensões aplicadas e residuais decorre da mudança mensurável da velocidade das ondas ultrassônicas na presença de um campo de tensões, fenômeno conhecido como efeito acustoelástico. Uma teoria de acustoelasticidade fornece um meio atrativo e não destrutivo de medir a tensão média ao longo do caminho percorrido pela onda. O estudo da propagação das ondas ultrassônicas em meios homogêneos anisotrópicos sob tensão conduz a um problema não linear de autovalores dado pela equação de Christoffel generalizada. A característica não linear deste problema decorre da interdependência entre as constantes elásticas efetivas do material e as tensões atuantes. A medição experimental de tensões por técnicas ultrassônicas é um problema inverso da acustoelasticidade. Esta dissertação apresenta a implementação de um algoritmo numérico, baseado no método proposto por Degtyar e Rokhlin, para solução do problema inverso da acustoelasticidade em sólidos ortotrópicos sujeitos a um estado plano de tensões. A solução da equação de Christoffel generalizada apresenta dificuldades de natureza numérica e prática. A estabilidade e a precisão do algoritmo desenvolvido, bem como a influência das incertezas na medição experimental das velocidades das ondas ultrassônicas, foram então investigadas. Dados sintéticos para as velocidades das ondas ultrassônicas de incidência oblíqua em uma placa sujeita a um estado plano de tensões foram gerados pela solução direta da equação de Christoffel generalizada para ilustrar a aplicação do algoritmo desenvolvido. O objetivo maior desta dissertação é a disponibilização de uma nova ferramenta de cálculo para suporte às atividades experimentais de medição de tensões por ultrassom no país. / Extensive studies carried out in the last decades on the propagation of ultrasonic waves in solids led to the development of nondestructive techniques for the assessment of the safety and integrity of industrial structures and components. The interest in the application of ultrasound techniques for stress measurement for example comes from the measurable change in the speed of the ultrasonic elastic waves in the presence of a stress field, a phenomenon known as acoustoelastic effect. An acoustoelastic theory provides an attractive way of non-destructively measuring the average stress along the waves path. The study of the propagation of ultrasonic waves in homogenous anisotropic bodies under stress leads to a nonlinear eigenvalue problem given by the generalized Christoffel equation. The nonlinearity characteristic of the problem derives from the interdependence between the materials effective elastic constants and the acting stresses. The experimental measurement of stresses using ultrasound techniques is an inverse problem of acoustoelasticity. This dissertation presents the implementation of a numeric algorithm, based on the method proposed by Degtyar and Rokhlin, for solution of the inverse problem of acoustoelasticity in orthotropic solids subjected to a plane stress state. The solution of the generalized Christoffel equation poses difficulties of numerical and practical order. The stability and precision of the algorithm developed, as well as the influence of the experimental uncertainties in the measurement of the speed of the ultrasonic waves, were thus investigated. Synthetic data for the speeds of ultrasonic waves of oblique incidence in a plane-stress plate were generated to illustrate the application of the algorithm developed. The main objective of this dissertation is to make available in the country a new numerical tool to support the use of ultrasonic waves for experimental stress analysis.

Dans ce travail, nous présentons une méthode de résolution de l'équation de la courbe géodésique en utilisant le symbole de Christoffel. En effet, l'équation de la courbe géodésique contient une dérivée covariante.

Se presentan algunos conceptos básicos tensoriales, junto con el desarrollo de una forma práctica del tensor de curvatura de Riemann-Christoffel, tensor que es de mucha utilidad en diferentes aplicaciones.

De nombreux chercheurs se sont intéressés à la Combinatoire des mots aussi bien d'un point de vue théorique que pratique. Pendant plus de $100$ ans de recherche, de nombreuses familles de mots ont été découvertes, certaines sont infinies et d'autres sont finies. Dans cette thèse, on s'intéresse aux mots de Christoffel. On aborde aussi les mots de Lyndon et les mots Strumians standards. Dans cette thèse, nous donnons de nombreuses propriétés sur les mots de Christoffel et on approfondit l'étude de la notion d'équilibre. Il est connu que les mots de Christoffel sont des mots équilibrés sur un alphabet binaire et sont formés par la discrétisation de segments de droite de pente rationnelle. Les mots de Christoffel sont aussi retrouvés dans l'étude de la synchronisation de k processus dirigé par k mots équilibrés. Pour k=2, on retombe sur les mots de Christoffel, tandis que pour k>2, la situation est plus compliquée et nous amène à la conjecture de Fraenkel qui est ouverte depuis plus de 40 ans. Comme c'est difficile d'atteindre cette conjecture, alors nous avons cherché à construire des outils qui nous aide à s'approcher de cette conjecture. On introduit ainsi la matrice d'équilibre B_w où w est un mot de Christoffel et la valeur maximale de cette matrice est l'ordre d'équilibre du mot binaire utilisé. Comme les mots de Christoffel sont équilibrés alors la valeur maximale dans ce cas là sera égale à 1 et chaque ligne de cette matrice sera formée des mots binaires. Cela nous pousse à tester de nouveau l'ordre d'équilibre de chaque mot obtenu et une nouvelle matrice est obtenue qui s'appelle matrice d'équilibre du second ordre . Cette matrice admet de plusieurs propriétés et de symétries et a une forme particulière comme on est capable de la partager en $9$ blocs où c'est suffisant de savoir 3 parmi eux pour construire le reste. Ces trois blocs correspondent à des matrices de mots de Christoffel qui se trouvent dans des niveaux plus proches de la racine de l'arbre des mots de Christoffel. La valeur maximale de cette nouvelle matrice U_w est appelée équilibre du second ordre. En regardant les chemins qui minimisent cette valeur tout au long de l'arbre, on remarque que le chemin suivi par les fractions obtenues du rapport des nombres consécutifs de la suite de Fibonacci, appelé chemin de Zig-zag est l'un des chemins minimaux. On retrouve ces chemins géométriquement sur le chemin de Christoffel en introduisant une nouvelle factorisation pour les mots de Christoffel appelée la factorisation standard symétrique. Nous avons, également, pu trouver une relation directe entre la matrice U_w et le mot de Christoffel initial sans passer par la matrice B_w et cela en étudiant l'ensemble des vecteurs abéliens associés. Tout ce travail nous a permis de réfléchir au sujet initial qui est la synchronisation de k mots équilibrés. Ainsi, pour le cas de 3 générateurs, nous avons pu étudier tous les cas possibles de la synchronisation et une discussion bien détaillée est faite en utilisant un nouvel élément appelé la graine qui est la première colonne de la matrice de synchronisation. La matrice du second ordre d'équilibre, avec toutes ses propriétés va être un bon outil pour étudier la synchronisation de k générateurs et cela constitut mon projet de recherche dans le futur. Nous avons aussi utilisé toutes nos connaissances autour des mots de Christoffel pour avancer dans la reconstruction de polyominoes convexes. Comme le contour d'un tel polyomino est formé des mots de Christoffel de pentes décroissantes, on a introduit un nouvel opérateur qui modifie ce chemin tout en gardant la décroissance des pentes c'est-à-dire en conservant la convexité qui est un premier pas vers la reconstruction. / Many researchers have been interested in studying Combinatorics on Words in theoretical andpractical points of view. Many families of words appeared during these years of research some ofthem are infinite and others are finite. In this thesis, we are interested in Christoffel words andwe introduce the Lyndon words and Standard sturmian words. We give numerous properties forthis type of words and we stress on the main one which is the order of balancedness. Well, itis known that Christoffel words are balanced words on two letters alphabet, where these wordsare exactly the discretization of line segments of rational slope. Christoffel words are consideredalso in the topic of synchronization of k process by a word on a k letter alphabet with a balanceproperty in each letter. For k = 2, we retrieve the usual Christoffel words. While for k > 2, thesituation is more complicated and lead to the Fraenkel’s conjecture that is an open conjecturefor more than 40 years. Since it is not easy to solve this conjecture, we were interested in findingsome tools that get us close to this conjecture. A balance matrix B w is introduced, where wis a Christoffel word, and the maximal value of this matrix is the order of balancedness of thebinary word. Since Christoffel words are one balanced then the maximal value obtained in thismatrix is equal to 1 and all the rows of this matrix is made of binary words. Testing again thebalancedness of these rows, a new matrix arises, called second order balance matrix. This matrixhas lot of characteristics and many symmetries and specially the way it is constructed since it ismade of 9 blocks where three of them belong to some particular Christoffel words appearing insome levels closer to the root of the Christoffel tree. The maximal value of this matrix is calledthe second order of balancedness for Christoffel words. From this matrix and this new orderof balancedness, we were able to show that the path followed by the fractions obtained fromthe ratio of the consecutive elements of Fibonacci sequence is a minimal path in the growth ofthis second order. In addition to that, these blocks are geometrically found on the Christoffelpath, by introducing a new factorization for the Christoffel words, called Symmetric standardfactorization. Similarly, we worked on finding a direct relation between the second order balancematrix U w and the initial Christoffel word without passing by the balance matrix B w but bystudying the set of factors of abelian vectors. All this work allow us to think about the initialtopic of research which is the synchronization of k balanced words. A complete study for the casek = 3 is given and we have discussed all the possible sub-cases for the synchronization by givingits seed, which is the starting column of the synchronized matrix. The second order balancematrix, with all its properties and decompositions form a good tool to study the synchronizationfor k generators that will be my future project of research. We have tried to use all the knowledgewe apply them on the reconstruction of digital convex polyominoes. Since the boundary wordof the digital convex polyominoe is made of Christoffel words with decreasing slopes. Hencewe introduce a split operator that respects the decreasing order of the slopes and therefore theconvexity is always conserved that is the first step toward the reconstruction.