Medians of discrete sets according to a linear distance

l'URL de l'article publié est http://www.springerlink.com/link.asp?id=9rukhuabxp8abkweIn this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions

Contributions à l'analyse de figures discrètes en dimension quelconque

Les polyominos sont souvent représentés par des mots de quatre lettres ou des mots de changements de direction décrivant leur contour. La combinatoire des mots classique y joue donc un rôle descriptif important, particulièrement dans le choix d'un représentant canonique. Les mots de Lyndon fournissent, de façon naturelle, un tel représentant. Une approche systématique pour le calcul de propriétés des polyominos, basée sur une version originale d'une discrétisation du théorème de Green classique en calcul bivarié, est élaborée. Ceci nous a naturellement amené à analyser les propriétés géométriques d'ensembles du réseau discret de rondeur maximale. Pour une taille donnée, ces ensembles minimisent le moment d'inertie par rapport à un axe passant par leur centre de gravité. Nous introduisons la notion de quasi-disque et montrons entre autres que ces ensembles minimaux sont des poIyominos\ud fortement-convexes. Nous développons également un algorithme permettant de les engendrer systématiquement. Un autre aspect concerne des propriétés sur les contours d'ensembles discrets donnant lieu à une nouvelle démonstration d'un résultat de Daurat et Nivat sur les points dits saillants et rentrants d'un polyomino. Nous présentons également une généralisation de ce résultat aux réseaux hexagonaux et montrons que le résultat est faux pour les autres réseaux semi-réguliers. Nous poursuivons par l'introduction d'opérations de mélange spéciaux sur des mots décrivant des chemins discrets selon la suite de leurs changements de direction. Ces opérations de mélange permettent d'engendrer des courbes fractales du type courbe de dragon et d'analyser\ud certains de leurs invariants. Finalement, une généralisation aux dimensions supérieures des algorithmes précédents basés sur le théorème de Green discret, est présentée. Plus particulièrement, nous développons une version discrète du théorème de Stokes basée sur des familles de poids sur les hypercubes de dimension k dans l'espace discret Zn, k ≤ n. Quelques applications sont également décrites. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Géométrie discrète, Combinatoire des mots, Ensembles discrets, Polyominos, Quasi-disques, Chemins polygonaux, Courbes de dragon, Théorème de Green discret, Théorème de Stokes discret, Algorithmes

Visualizing Set Relations and Cardinalities Using Venn and Euler Diagrams

In medicine, genetics, criminology and various other areas, Venn and Euler diagrams are used to visualize data set relations and their cardinalities. The data sets are represented by closed curves and the data set relationships are depicted by the overlaps between these curves. Both the sets and their intersections are easily visible as the closed curves are preattentively processed and form common regions that have a strong perceptual grouping effect. Besides set relations such as intersection, containment and disjointness, the cardinality of the sets and their intersections can also be depicted in the same diagram (referred to as area-proportional) through the size of the curves and their overlaps. Size is a preattentive feature and so similarities, differences and trends are easily identified. Thus, such diagrams facilitate data analysis and reasoning about the sets. However, drawing these diagrams manually is difficult, often impossible, and current automatic drawing methods do not always produce appropriate diagrams. This dissertation presents novel automatic drawing methods for different types of Euler diagrams and a user study of how such diagrams can help probabilistic judgement. The main drawing algorithms are: eulerForce, which uses a force-directed approach to lay out Euler diagrams; eulerAPE, which draws area-proportional Venn diagrams with ellipses. The user study evaluated the effectiveness of area- proportional Euler diagrams, glyph representations, Euler diagrams with glyphs and text+visualization formats for Bayesian reasoning, and a method eulerGlyphs was devised to automatically and accurately draw the assessed visualizations for any Bayesian problem. Additionally, analytic algorithms that instantaneously compute the overlapping areas of three general intersecting ellipses are provided, together with an evaluation of the effectiveness of ellipses in drawing accurate area-proportional Venn diagrams for 3-set data and the characteristics of the data that can be depicted accurately with ellipses

Analyse de quelques algorithmes probabilistes à délais aléatoires

Dans la première partie de cette étude, nous proposons et analysons des algorithmes probabilistes d’élection uniforme dans des graphes de types arbres, les k-arbres et les polyominoïdes. Ces algorithmes utilisent des durées de vie aléatoires associées aux sommets découverts (sommets feuilles ou simpliciaux). Ces durées sont des variables aléatoires indépendantes et sont localement engendrées au fur et à mesure que les sommets sont découverts. Dans la seconde partie, nous analysons un algorithme probabiliste de synchronisation pour le problème de rendez-vous avec agendas dynamiques. L’objectif est de trouver un couplage maximal dans un graphe donné. Ensuite, nous proposons et étudions un modèle de diffusion à délai aléatoire pour la transmission d’un message dans un réseau. Finalement, dans la dernière partie, nous exposons les outils utilisés pour implémenter la simulation des algorithmes distribués.In the first part of this study, we propose and analyze a probabilistic algorithms of uniform election in graphs of structures of the trees type, k-trees and polyominoids. These algorithms use random delay associated to discovered vertices (leaf vertices or simplicial vertices). These delays are independent random variables and are locally generated as and when the vertices are discovered. In the second part, we analyze a probabilistic algorithm of synchronization for the problem of rendezvous with dynamic agendas. The goal is to find a maximal matching in a given graph. Then, we propose and study a model of diffusion with random delay for the transmission of a message in a network. Finally, in the last part, we expose the tools used to implement the simulation of the distributed algorithms

Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany

This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future