Academic literature on the topic 'Christoffel'

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&gt;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<br>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 &gt; 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

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.

The purpose of this study is to determine whether a child with specific learning disabilities also experiences emotional problems like a poor selfimage. Special reference is made to the aid class pupil. Theoretical background knowledge of the child with specific learning disabilities has been compiled: the etiology and manilestation of learning problems as well as certain characteristic behaviour patterns of such a child. The concept of selfimage and factors that influence it have been attended to. The function and purpose of the aid class have also been dealt with. The questionnaire as emperical research method is explained and implemented in this study. Questions drawn from the above-mentioned study of literature formed the basis of the questionnaire. Questionnaires were sent to everyone of the Afrikaans speaking aid class teachers in Transvaal, totalling a number of 120. The data collected from the questionnaires returned, was analised by means of the S.A.S. computer programme. It is very clear form the frequency and percentage tables compiled from the data analysis, that most aid class pupils have a poor self-image. The questionnaire confirmed what was indicated in the literature: perceptual and academic problems, certain behaviour patterns and the bearing of these factors on the self-image of the pupil in the aid class. Likewise, similar confirmation was obtained concerning the influence of the home and school on the self-image of such a child. It can thus be stated from the study undertaken that an adequate selfimage is a pre-requisite for optimal learning and the actualisation of potential. For this reason the initial orthodidactical assistance to the aid class pupil should include the acquirement of a positive self-image in order to have the child affectively and emotionally prepared in a sufficient way for the intensified leaming process in the aid class.<br>Skripsie (MEd)--PU vir CHO, 1989

Perception and inquiries have revealed that not all vocational guidance teachers possess the necessary interviewing skills. The aim of the research was to identify generally applicable interviewing skills, as well as those specific interviewing skills which are characteristic of the vocational guidance provided to a pupil leaving school. The aim was furthermore to determine to what extent vocational guidance teachers in Transvaal secondary schools are grounded in these skills. It has been indicated that vocational guidance actually culminate in the interview. After the interview has been analysed from an educational perspective, attention has been given to the validity and reliability of the interview. Further attention has been given to the trainability in interviewing of vocational guidance teachers. The following most important, generally applicable interviewing skills were then identified. * Communication as interviewing skill - There is a distinction between verbal and non-verbal communication. * Listening as interviewing skill - A distinction is drawn between marginal listening, evaluative listening and projective listening. * Establishment of report as interviewing skill - A favourable emotional climate is a requirement for a successful interview. * "Silence" as interviewing skill - The following distinction applies: that between silence initiated by the interviewer, and silence initiated by the client. * Perception as interviewing skill - For the interviewer to correctly interpret that which he percepts is equally important, if not more important, than sensory perception. * Empathy as interviewing skill - It is essential for the interviewer to place himself in the position of the client without becoming emotionally involved. * Attitude as interviewing skill - Success will be determined by the interviewer's attitude towards the client. * Questioning as interviewing skill - There is a distinction between an openended and closed question. Questions should be put with utmost circumspection because it could influence the outcome of the interview. * The use of synopsis as interviewing skill - Through synopsis the client gets the opportunity to verify the information revealed to the interviewer. The specific interviewing skill characteristic to the vocational guidance situation has been identified as: The skilful interpretation of reports. The vocational guidance teacher has a lot of information about the school leaver from the guardian's file, accumulative card and vocational guidance file. The planning and application of an empirical research has been done. The research was mainly aimed at determining how important the vocational guidance teachers regard the interviewing skills and to what extent these teachers are grounded in interviewing skills. From the research it seemed that the training vocational guidance teachers received in interviewing, was inadequate. The most important conclusions were: * The importance of these interviewing skills to the interviewer can't be stressed enough. * The majority of vocational guidance teachers don't possess the basic qualifications to act as such. * Some vocational guidance teachers tend to underestimate the importance of the interviewing skills and they furthermore don't have the ability to handle• these interviewing skills adequately. The most important recommendations are: * That high priority be given to the training of interviewing skills to vocational guidance teachers. * That the Transvaal Education Department only appoint qualified vocational guidance teachers at secondary schools. * That attention should be given to the practical exercise of these skills. * That universities, in co-operation with the Transvaal Education Department present further courses in interviewing to acting vocational guidance teachers.<br>Thesis (MEd)--PU vir CHO, 1986

Consumer education as component of the Life Skill education programme in Guidance: Consumers are often deprived of their consumer rights. As a result of drastic changes in society and the market it has become more important to educate young consumers (at secondary level) regarding important consumer issues which have an effect on their daily lives. It is of great importance that the need for consumer education as part of the secondary school's guidance programme must be identified and recommendations in this regard should be made. In the course of this study several problem areas were identified. The researcher will endeavour to outline certain findings and make relevant recommendations in this regard. The objectives of this study intend to determine what has actually been done to implement consumer education at secondary level and if there is an existing need for secondary school consumer education. In order to obtain data available on the topic, a questionnaire was used for empirical research. It provided information regarding aspects of consumer knowledge, consumer behaviour and consumer rights. Demographic data for the representation of consumer groups questioned in this research was compiled, and std. 7 pupils representing the major population groups, Whites, Blacks, Indians and Coloureds, were all included in this survey. Scholars were addressed in Afrikaans and English. The majority of respondents emphasised the need for consumer education by indicating that consumer education as part of life skills could help all consumers. The majority of pupils indicated that children should have consumer rights and they would want to learn more about these rights. The majority of respondents were of the opinion that the school's guidance programme should include consumer education to make them more knowledgeable consumers. This study concludes with some recommendations.<br>Thesis (MEd (Voorligting))--PU vir CHO, 1995

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]

Un mot de christoffel fini de pente rationnelle r est une suite finie de 0 et de 1 construite a partir de la fraction continue de r: les mots de christoffel infinis sont les mots de pente reelle construits par le meme procede. Nous decrivons des algorithmes permettant de calculer le mot de christoffel dont la pente est la somme de celles de deux mots donnes ou bien celle d'un autre mot multipliee par un entier. Etant donne le lien fort entre les fractions continues et les mots de christoffel l'interet de ces algorithmes est de mieux comprendre le mecanisme des operations correspondantes sur les fractions continues. De tels algorithmes existent deja pour les fractions continues: la nouveaute est ici l'utilisation des mots de christoffel qui en permet une nouvelle interpretation. Les mots de christoffel peuvent etre definis de plusieurs facons: de maniere geometrique, recursive ou par le biais des mots de lyndon. L'algorithme d'addition utilise une fonction recursive: on construit une suite de mots de lyndon dont le dernier terme est le mot de christoffel. La multiplication consiste en la recherche de formules de recurrence, ainsi ces algorithmes fournissent une conception geometrique et arborescente des nombres et des fractions continues. L'interet de ces algorithmes est theorique ; ils sont avant tout une description des operations sur les fractions continues, au profit de laquelle a ete negligee la rapidite. C'est un point de depart pour trouver d'autres algorithmes plus performants utilisant les mots de christoffel comme nouvelle representation des nombres

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.<br>Master of Science

Tableau d'honneur de la Faculté des études supérieures et postdoctorales, 2015-2016<br>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.