Going down in (semi)lattices of finite Moore families and convex geometries

International audienceIn this paper we first study the changes occuring in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we show that the poset of all convex geometries that have the same poset of join-irreducible elements is a ranked join-semilattice, and we give an algorithm for computing it. Finally, we prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3.Dans ce texte, nous étudions d'abord les changements dans les ensembles ordonnés d'éléments irréductibles lorsqu'on passe d'une famille de Moore arbitraire (respectivement, d'une géométrie convexe) à l'une de ses couvertures inférieures dans le treillis de toutes les familles de Moore (respectivement, dans le demi-treillis des géométries convexes). Nous montrons ensuite que l'ensemble ordonné de toutes les géométries convexes ayant le même ensemble ordonné d'éléments sup-irréductibles est un demi-treillis rangé et nous donnons un algorithme pour le calculer. Enfin nous caractérisons les ensembles ordonnés P pour lesquels le treillis de leurs idéaux est l'unique géométrie convexe ayant son ensemble ordonné d'éléments sup-irréductibles isomorphe à P

On implicational bases of closure systems with unique critical sets

We show that every optimum basis of a finite closure system, in D.Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce a K-basis of a general closure system, which is a refinement of the canonical basis of Duquenne and Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with the unique criticals and some of its subclasses, where the K-basis is unique. A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be effectively recognized. While E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into plenary talk on conference Universal Algebra and Lattice Theory, June 2012, Szeged, Hungary 29 pages and 2 figure

Uptake of childhood influenza vaccine from 2012–2013 to 2014–2015 in the UK and the implications for high-risk children:a retrospective observational cohort study

OBJECTIVES: To evaluate changes in influenza vaccination rates in healthy and at-risk children following the implementation of the UK's childhood influenza immunisation programme. DESIGN: Observational cohort study before and after initiation of the UK's childhood influenza immunisation programme over three influenza seasons (2012–2013, 2013–2014 and 2014–2015) using data from the Clinical Practice Research Datalink (CPRD). SETTING: More than 500 primary care practices in the UK. POPULATION: All individuals aged 2–17 years on 1 September, with at least 12 months of medical history documented in CPRD were retained in the analysis. INTERVENTION: Starting in 2013–2014, all children aged 2 and 3 years were offered influenza vaccination through general practice, and primary school-aged children were offered influenza vaccination in selected counties in England (described as pilot regions). The vaccination programme was extended to all children aged 4 years in England in 2014–2015. MAIN OUTCOME MEASURE: Cumulative vaccination rate from 1 September to 28 February of the next calendar year as assessed by a time-to-event statistical model (vaccination uptake). Age group, sex, region and type of high-risk medical condition were assessed as predictors. RESULTS: Vaccination uptake increased considerably from 2012–2013 to 2013–2014 in targeted children aged 2–3 years, both in children with a high-risk medical condition (from 40.7% to 61.1%) and those without (from 1.0% to 43.0%). Vaccination rates increased also, though less markedly, in older children. In 2014–2015, vaccination rates remained higher than 40% in healthy children aged 2–3 years, although they decreased slightly from 2013–2014 (from 43.0% to 41.8%). Vaccination rates in older healthy children continued to increase, driven primarily by an increase in children aged 4 years to 31.3% in 2014–2015. CONCLUSIONS: The introduction of a universal childhood vaccination policy in the UK increased vaccination rates for targeted children, including those with high-risk conditions

CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Comment: 14 pages, 4 figure

Lattice congruences of the weak order

We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let \eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of \eta_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure

KP line solitons and Tamari lattices

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).Comment: 49 pages, 36 figures, second version: section 4 expande

A escola, a família e o Estado: uma aproximação histórica de suas relações

A história do ensino destaca, particularmente na França, o papel sucessivo de dois grandes atores institucionais: Igreja e Estado. Esse artigo sublinha, ao contrário, a ação das famílias, do século 16 ao 19, nos diferentes níveis de ensino: primário, secundário, técnico. Conclui pela necessidade de explorar as fontes disponíveis para reavaliar essa ação, a fim de lhe dar o justo lugar na dimensão sociocultural da história do ensino e, também, para melhor compreender as razões de alguns dos maiores problemas encontrados hoje no sistema educativo francês.Palavras-chaves: história sociocultural, sistema escolar, longa duração, França, educação familiar. SCHOOL, FAMILY AND THE STATE: A HISTORICAL APPROACH TO ITS RELATIONSAbstractThe historiography of education focused, particularly in France, the role of two successive major institutional players: the church and state. This article draws attention to the work of families, the sixteenth to the nineteenth century, the different levels of education: primary, secondary, technical. It concludes with the need to exploit the available sources to reassess this action to give its proper place in the socio-cultural dimension of the history of education, and also to better understand why some major problems today by the French educational system. Keywords: cultural studies, school system, long term, France, status, family education. ESCUELA, FAMILIA Y ESTADO: UMA APROXIMACIÓN HISTORICA DE SUS RELACIONESResumenLa historiografía de la educación se centró, sobre todo en Francia, el papel de dos períodos consecutivos de los actores institucionales principales: la Iglesia y el Estado. En este artículo se hace hincapié en la labor de las familias, el siglo 16 hasta el siglo 19, los diferentes niveles de educación: primaria, secundaria, técnica. Se concluye con la necesidad de explotar los recursos disponibles para reevaluar esta acción para dar a su propio lugar en la dimensión socio-cultural de la historia de la educación, y también para comprender mejor por qué algunos problemas importantes en la actualidad por el sistema educativo francés.Palabras clave: estudios culturales, sistema escolar, largo plazo, Francia, Estado, educación familiar. L’ECOLE, LA FAMILLE ET L’ETAT: UN APERÇU HISTORIQUE DE LEURS RAPPORTSRésumeL’historiographie de l’enseignement met l’accent, particulièrement en France, sur le rôle successif de deux grands acteurs institutionnels : l’Église et l’État. Cet article souligne au contraire l’action des familles, du 16ème au 19ème siècle, aux différents niveaux de l’enseignement : primaire, secondaire, technique. Il conclut sur la nécessité d’exploiter les sources disponibles pour réévaluer cette action, afin de donner sa juste place à la dimension socioculturelle de l’histoire de l’enseignement, et aussi de mieux comprendre les raisons de quelques problèmes majeurs rencontrés aujourd’hui par le système éducatif français.Mots-clés: histoire socioculturelle, système scolaire, longue durée, France, état, éducation familiale

TRÉVISI (Marion), Au cœur de la parenté. Oncles et tantes dans la France des Lumières

Depuis que la démographie historique les a mises au premier plan de ses préoccupations, la famille et les relations de parenté ont fait l’objet de recherches s’attachant à mettre en évidence la place, le rôle et les interrelations spécifiques de certaines des composantes familiales : époux, mères, pères, frères et sœurs, mères et filles, grands-parents, etc. C’est aux relations entre oncles et tantes, d’une part, et neveux et nièces, de l’autre, que Marion Trévisi a pour sa part consacré la t..

A characterization for all interval doubling schemes of the lattice of permutations

The lattice \textbfS_n of all permutations on a n-element set has been shown to be \emphbounded [CAS], which is a strong constructive property characterized by the fact that \textbfS_n admits what we call an \emph interval doubling scheme. In this paper we characterize all interval doubling schemes of the lattice \textbfS_n, a result that gives a nice precision on the bounded nature of the lattice of permutations. This theorem is a direct corollary of two strong properties that are also given with their proofs

AVANZINI (Guy), CAILLEAU (René), AUDIC (Anne-Marie), PÉNISSON (Pierre). – Dictionnaire historique de l’éducation chrétienne d’expression française

On ne peut que se réjouir de voir paraître ce Dictionnaire, qui enrichit la panoplie des outils de travail offerts aux historiens de l’éducation. Le principe retenu a été de mettre l’accent sur les acteurs et les institutions. Les premiers sont des hommes et des femmes, clercs ou laïcs, qui ont joué un rôle significatif dans l’éducation et l’enseignement chrétiens à la suite du Christ éducateur, à qui le Dictionnaire consacre une notice inaugurale, avant que l’ordre alphabétique proprement di..