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

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

Adherence with statins in a real-life setting is better when associated cardiovascular risk factors increase: a cohort study

<p>Abstract</p> <p>Background</p> <p>While the factors for poor adherence for treatment with statins have been highlighted, the impact of their combination on adherence is not clear.</p> <p>Aims</p> <p>To estimate adherence for statins and whether it differs according to the number of cardiovascular risk factors.</p> <p>Methods</p> <p>A cohort study was conducted using data from the main French national health insurance system reimbursement database. Newly treated patients with statins between September 1 and December 31, 2004 were included. Patients were followed up 15 months. The cohort was split into three groups according to their number of additional cardiovascular risk factors that included age and gender, diabetes mellitus and cardiovascular disease (using co-medications as a <it>proxy</it>). Adherence was assessed for each group by using four parameters: <it>(i) </it>proportion of days covered by statins, <it>(ii) </it>regularity of the treatment over time, <it>(iii) </it>persistence, and <it>(iv) </it>the refill delay.</p> <p>Results</p> <p>16,397 newly treated patients were identified. Of these statin users, 21.7% did not have additional cardiovascular risk factors. Thirty-one percent had two cardiovascular risk factors and 47% had at least three risk factors. All the parameters showed a sub-optimal adherence whatever the group: days covered ranged from 56% to 72%, regularity ranged from 23% to 33% and persistence ranged from 44% to 59%, but adherence was better for those with a higher number of cardiovascular risk factors.</p> <p>Conclusions</p> <p>The results confirm that long-term drug treatments are a difficult challenge, particularly in patients at lower risk and invite to the development of therapeutic education.</p

Whiting–related sediment export along the Middle Miocene carbonate ramp of Great Bahama Bank.

International audienc

A Characterization Theorem for the Canonical Basis of a Closure Operator.

The purpose of this paper is to provide a characterization result for the canonical basis of an arbitrary closure operator. This theorem strengthens the result of Burosch, Demetrovics and Katona, who propose in [1] a characterization of the generating system of a closure operator defined by the quasi-closed sets of the closure operator.MATHEMATICAL ANALYSIS

A characterization Theorem for all Interval Doubling Schemes of the Lattice of Permutations.

We have recalled that all interval doubling schemes of a bounded lattice L are in bijection with all different ways to conduct L starting from the two-element lattice by doublings of convex sets.MATHEMATICS

The Lattice of Permutations is Bounded

The purpose of this paper is to show that the lattice Sn of permutations on a n-element set is bounded. This result strengthens the semi-distributive nature of the lattice Sn. To prove this property, we use a characterization of the class of bounded lattices in terms of arrows relations defined on the join-irreductible elements of a lattice or, more precisely, in terms of the A-table of a lattice.MATHEMATICS