3-uniform hypergraphs: modular decomposition and realization by tournaments

Let $H$ be a 3-uniform hypergraph. A tournament $T$ defined on $V(T)=V(H)$ is a realization of $H$ if the edges of $H$ are exactly the 3-element subsets of $V(T)$ that induce 3-cycles. We characterize the 3-uniform hypergraphs that admit realizations by using a suitable modular decomposition

ETUDE DIDACTIQUE D’ACTIVITÉS D’INVESTISSEMENT DES NOMBRES DANS DES MANUELS SCOLAIRES DU CYCLE COLLÉGIAL AU MAROC / DIDACTICAL STUDY OF INVESTMENT ACTIVITIES OF THE NUMBERS IN SOME SCHOOL TEXTBOOKS OF THE COLLEGE CYCLE IN MOROCCO

Ce travail vise à étudier les formes d’investissement du nombre dans des situations proposées dans des manuels scolaires du cycle collégial au Maroc. Il s’agit d’un concept d’une grande importance épistémologique et didactique. En effet, l’acquisition de plusieurs autres concepts et de plusieurs capacités spécifiques aux mathématiques ou transversales exige une maitrise des nombres et de leurs manipulations. De plus le nombre est l’un des concepts qui favorisent des jeux de cadres qui sont au même temps nécessaires pour mieux percevoir son sens. La place du nombre dans le curriculum scolaire marocain adressé au cycle collégial et les cadres dont il est traité, font l’objet de cet article. L’exploration est entreprise dans des activités proposées par des manuels scolaires destinés aux apprenants des niveaux du cycle collégial. Pratiquement, il s’agit de l’examen d’activités destinées à l’animation des deux phases du processus enseignement/apprentissage, conceptualisation et d’exercisation et relevant du domaine des équations du premier degré à une inconnue. L’analyse a porté sur les types de nombres mis en situation et sur les cadres où ils sont traités. Il a été déduit une présence d’une certaine diversification dans les activités proposées dans les manuels scolaires choisis et une nette corrélation entre les choix faits dans les deux phases. Néanmoins, des insuffisances ont été observées pour certains niveaux scolaires.This work aims to study the forms of investment of the number in situations proposed in school textbooks of the college cycle in Morocco. It is a concept with great epistemological and didactical importance. Indeed, the acquisition of several other concepts and several specific skills in mathematics requires a mastery of numbers and their manipulations. In addition, number is one of the concepts that allows the framework games, which are at the same time necessary to perceive very well its meaning. The place of the number in the Moroccan school curriculum addressed to the college cycle and the frameworks where it is treated, are the subject of this article. Exploration is undertaken in activities given by textbooks for learners at the college level. Practically, it is the examination of the activities pertaining to the two phases of the teaching / learning, conceptualization and exercise process and also pertaining to the domain of equations from the first degree with one unknown. The analysis focused on the types of numbers put in situation and on the frameworks where they are treated. It was deduced a diversification in the activities proposed in the selected textbooks and a clear correlation between the choices made in the two phases. However, shortcomings have been observed for certain educational levels.This work aims to study the forms of investment of the number in situations proposed in school textbooks of the college cycle in Morocco. It is a concept with great epistemological and didactical importance. Indeed, the acquisition of several other concepts and several specific skills in mathematics requires a mastery of numbers and their manipulations. In addition, number is one of the concepts that allows the framework games, which are at the same time necessary to perceive very well its meaning. The place of the number in the Moroccan school curriculum addressed to the college cycle and the frameworks where it is treated, are the subject of this article. Exploration is undertaken in activities given by textbooks for learners at the college level. Practically, it is the examination of the activities pertaining to the two phases of the teaching / learning, conceptualization and exercise process and also pertaining to the domain of equations from the first degree with one unknown. The analysis focused on the types of numbers put in situation and on the frameworks where they are treated. It was deduced a diversification in the activities proposed in the selected textbooks and a clear correlation between the choices made in the two phases. However, shortcomings have been observed for certain educational levels.  Article visualizations

On Some Inequalities Involving Three or More Means

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