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The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices.

Le travail de thèse proposé concerne la conception intégrée des algorithmes de surveillance des processus chimiques à base de modèles bond graph. Cette recherche est réalisée dans le cadre de la thématique «Bond graphs pour la supervision des procédés énergétiques» développée entre l’Université de Gabés (Tunisie) et l’Ecole Polytechnique Universitaire de Lille (France). Les processus chimiques sont des procédés polluants et à risque. Ils nécessitent alors pour la protection de l’environnement et du personnel des systèmes de surveillance en ligne pour la détection précoce et l’identification des défaillances. Un processus chimique est le siège de phénomènes de nature diverse : chimiques et ou biochimiques, thermo-fluidique, …. Leur modélisation nécessite alors une approche unifiée. L’outil bond graph par son caractère multidisciplinaire est bien adapté à cette tâche. D’autre part, cet outil peut aussi être utilisé pour la conception d’algorithmes de diagnostic grâce à ses propriétés comportementales, causales et structurelles et déterminer ainsi les conditions de surveillabilité structurelle des équipements pertinents sans calcul numérique. Les principales scientifiques contributions peuvent être résumées comme suit : (i) élaboration d’une bibliothèque de modèles Bond Graph de composants et phénomènes chimiques (ii) méthodologie de diagnostic à base de bond graphs pour la génération d’indicateurs de fautes sensibles à l’apparition des réactions secondaires sources de pollution et d’explosion, (iii) diagnostic robuste aux incertitudes paramétriques du modèle Bond Graphs couplés, (iv) informatisation des procédures d’analyse de surveillabilité d’une classe de procédés chimiques, (v) application à un procédé réel (installation d’estérification). / The proposed Ph.D. thesis deals with integrated design of bond graph model based for health monitoring of chemical processes. This research work is performed within the framework of the topic “Bond graphs for supervision of energetic processes" developed between the University of Gabés (Tunisia) and “Polytech’Lille”, the Engineering School within the University of Science and Technology of Lille. The chemical processes are polluting processes and with risk. They need, for staff safety and environmental protection, online surveillance for early detection and the identification of the failures. chemical processes occur phenomena of various nature: chemical and or biochemical, thermo-fluidic… Their modeling requires a unified approach. The Bond graph as a multidisciplinary tool is well suited to this task.Furthermore, this tool can be used also for the design of diagnosis algorithms thanks to its behavioral, causal and structural properties, and allows providing structural diagnosability conditions of the pertinent equipment without numerical calculation. The main scientific contributions of this research can be summarized as follows: (i)elaboration of a data base of dynamic bond graph models of components and chemical phenomena, (ii) methodology of bond graph model based diagnosis for systematic generation of fault indicators sensitive to the appearance of the secondary reactions source of pollution and explosion, (iii) robust diagnosis based on coupled uncertain bond graph model, (iv) computerization of the diagnosability analysis procedures applied to a class of a chemical process, (v) application to a real process (installation of esterification).

El presente artículo presenta los resultados del análisis de los gráficos estadísticos según las directrices curriculares y su implementación en dieciocho libros de texto de matemática de Educación Primaria en Perú, los que corresponden a tres series completas y de diferentes editoriales. En ellos se analizan, mediante análisis de contenido, las secciones en las que aparecen estas representaciones, identificado el tipo de actividad que se plantea, los gráficos involucrados, el nivel de lectura y el nivel de complejidad semiótica involucrado. Los libros de texto se adecúan parcialmente a las directrices curriculares en cuanto a la presentación de los gráficos por nivel educativo, el número de actividades propuestas por las tres editoriales es similar. La principal actividad que se solicita en los libros es de calcular y construir. Se observa un predominio del gráfico de barras, un nivel de lectura básico y la representación de una distribución de datos en el gráfico. / Revisión por pares

The usual graph parameters related to independent and dominating sets can be adapted to broadcasts on graphs. We examine some possible definitions for an inde- pendent broadcast. We determine the minimum maximal and the maximum broad- cast weight for all our independence parameters on both paths and grids. For graphs in general, we examine the relationships between these broadcast independence pa- rameters and the existing minimum and maximum minimal broadcast domination weight (or cost). We also determine upper and lower bounds for maximum boundary independent broadcasts and a new upper bound for hearing independent broadcasts. / Graduate

A Dual-Eulerian graph is a plane multigraph G that contains an edge list which is simultaneously an Euler tour in G and an Euler tour in the dual of G. Dual-Eulerian tours play an important role in optimizing CMOS layouts of Boolean functions. When circuits are represented by undirected multigraphs the layout area of the circuit can be optimized through finding the minimum number of disjoint dual trails that cover the graph. This paper presents an implementation of a polynomial time algorithm for determining whether or not a plane multigraph is Dual-Eulerian and for finding the Dual-Eulerian trail if it exists.

A d-dimensional (bar-and-joint) framework is a pair (G; p) where G = (V;E) is a graph and p : V > Rd is a function which is called the realisation of the framework (G; p). A motion of a framework (G; p) is a continuous function P : [0; 1] x V > Rd which preserves the edge lengths for all t 2 [0; 1]. A motion is rigid if it also preserves the distances between non-adjacent pairs of vertices of G. A framework is rigid if all of its motions are rigid motions. An in nitesimal motion of a d-dimensional framework (G; p) is a function q : V > Rd such that [p(u) - p(v)] ~ [q(u) - q(v)] = 0 for all uv 2 E. An in nitesimal motion of the framework (G; p) is rigid if we have [p(u) - p(v)] . [q(u) - q(v)] = 0 also for non-adjacent pairs of vertices. A framework (G; p) is in nitesimally rigid if all of its in nitesimal motions are rigid in nitesimal motions. A d-dimensional framework (G; p) is generic if the coordinates of the positions of vertices assigned by p are algebraically independent. For generic frameworks rigidity and in nitesimal rigidity are equivalent. We construct a matrix of size |E| xd|V| for a given d-dimensional framework (G; p) as follows. The rows are indexed by the edges of G and the set of d consecutive columns corresponds to a vertex of G. The entries of a row indexed by uv 2 E contain the d coordinates of p(u) - p(v) and p(v) - p(u) in the d consecutive columns corresponding to u and v, respectively, and the remaining entries are all zeros. This matrix is the rigidity matrix of the framework (G; p) and denoted by R(G; p). Translations and rotations of a given framework (G; p) give rise to a subspace of dimension d+1 2 of the null space of R(G; p) when p(v) affinely spans Rd. Therefore we have rankR(G; p) djV j�� d+1 2 if p(v) affinely spans Rd, and the framework is in infinitesimally rigid if equality holds. We construct a matroid corresponding to the framework (G; p) from the rigidity matrix R(G; p) in which F E is independent if and only if the rows of R(G; p) indexed by F are linearly independent. This matroid is called the rigidity matroid of the framework (G; p). It is clear that any two generic realisations of G give rise to the same rigidity matroid. In this thesis we will investigate rigidity properties of some families of frameworks. We rst investigate rigidity of linearly constrained frameworks i.e., 3- dimensional bar-and-joint frameworks for which each vertex has an assigned plane to move on. Next we characterise rigidity of 2-dimensional bar-and-joint frameworks (G; p) for which three distinct vertices u; v;w 2 V (G) are mapped to the same point, that is p(u) = p(v) = p(w), and this is the only algebraic dependency of p. Then we characterise rigidity of a family of non-generic body-bar frameworks in 3-dimensions. Finally, we give an upper bound on the rank function of a d-dimensional bar-and-joint framework for 1 < d < 11.

Les programmes sont partout dans notre vie quotidienne : les ordinateurs et les téléphones, mais aussi les frigo, les avions et ainsi de suite. L'acteur principal dans la création de ces programmes est humain les êtres. Aussi minutie qu'ils peuvent être, les humains sont connus pour faire des erreurs involontaires sans leur conscience. Ainsi, une fois une phase déjà difficile d'écriture d'un programme, ils doivent faire face à la phase de maintenance sur laquelle ils doivent faire face aux erreurs qu'ils ont eu précédemment réalisé. Toute la durée de leur tâche de développement, les développeurs doivent faire face continuellement leurs erreurs (ou leurs collègues). Cette observation clé soulève la nécessité d'aider les développeurs dans leurs tâches de développement / maintenance. / Programs are everywhere in our daily life: computers and phones but also fridges, planes and so on. The main actor in the process of creating these programs is human beings. As thorough as they can be, humans are known to make involuntary errors without their awareness. Thus, once finished an already hard phase of writing a program. they have to face the maintenance phase on which they have to deal with errors they had previously made. All long their development task, developers have to continuously face their (or their colleagues) errors. This key observation arises the need of aiding developers in their development/maintenance tasks.