On the superimposition of Christoffel words

Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a $k$-letter alphabet A, with a fixed $k \geq 3$, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words

A characterization of balanced episturmian sequences

It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly $(n+1)$ factors of length $n$. One possible generalization of Sturmian sequences is the set of infinite sequences over a $k$-letter alphabet, $k \geq 3$, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a $k$-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed $k> 2$, there is only one way to cover $\Z$ by $k$ Beatty sequences. The problem can be translated to combinatorics on words: for a $k$-letter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies

Sterilization Assembly Development Laboratory - Quality assurance program Plan

Quality control program plan for sterilization assembly development laborator

Is My Assignment Ready To Be Submitted? Five Essential Questions to Ensure my Work Respects Intellectual Property

Outil d'aide à la décision développé par le Service de développement pédagogique, des programmes et de la recherche du Cégep Marie-Victorin et présenté au colloque 2017 de l'AQPC

Les musées et les musées virtuels d’histoire : appréciation, utilisation et effet d’une formation sur la pratique enseignante

Une formation sur les ressources éducatives des musées et des musées virtuels d’histoire mises à profit dans l’enseignement-apprentissage est dispensée à de futurs enseignants depuis quelques années. L’article porte sur l’appréciation, l’utilisation et l’effet de cette formation sur la pratique enseignante. L’analyse fait ressortir que la majorité des nouveaux enseignants ont apprécié cette formation, même s’ils utilisent assez peu de telles ressources, en raison de leur situation de survie dans la profession. Pour beaucoup, le fait d’en avoir bénéficié les incite à vouloir éventuellement les utiliser, surtout les ressources en ligne des musées virtuels parmi ceux qui exercent en région ou qui sont compétents dans les technologies de l’information et des communications en éducation (TICÉ).Future teachers have, over several years, received training regarding the educational resources that museums and virtual historical museums offer for facilitating teaching and learning. This article describes the level of appreciation, the use, and the effect of this training on teachers’ practices. The analysis shows that the majority of new teachers appreciated this training, even if these resources are little used given their need to simply survive in the teaching profession. For many, the fact that they benefited from this training promotes a desire for eventual use. This is especially so for those resources linked to on-line virtual museums, and especially for those teaching in the regions or those skilled in using information technology for education.Una formación sobre los recursos educativos de los museos y de los museos virtuales de historia explotada en la enseñanza-aprendizaje es impartida a los docentes en prácticas desde hace algunos años. El artículo trata de la apreciación, la utilización y el efecto de esta formación sobre la práctica docente. El análisis hace resaltar que la mayoría de los nuevos docentes han apreciado esta formación, aunque utilizan relativamente poco este tipo de recursos, por encontrarse en una situación de supervivencia en la profesión. Para gran parte de ellos, en particular los que laboran fuera de la capital o que son competentes en las tecnologías de la información y de las comunicaciones en educación (TICE), el hecho de haber beneficiado de tales recursos les incita a querer eventualmente utilizarlos, sobre todo los recursos en línea de los museos virtuales

Regard sur les Technologies numériques immersives

Réflexions, enjeux et prospective portant sur la montée en puissance et la convergence des technologies numériques immersives à l’aube du 21e siècleReflections, challenges and future possibilities: the rise to power and convergence of immersive digital technologies at the dawn of the 21st century.Relexões, desaios e perspectivas sobre o crescimento e a convergência das na tecnologias digitais no início do século XXI