640 research outputs found

    On the weak order of Coxeter groups

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    This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).Comment: 37 pages, submitte

    On rigidity of abstract root systems of Coxeter systems

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    We introduce and study a combinatorially defined notion of root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems, and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.Comment: 34 page

    Semidirect product decomposition of Coxeter groups

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    Let (W,S)(W,S) be a Coxeter system, let S=IâˆȘ˙JS=I \dot{\cup} J be a partition of SS such that no element of II is conjugate to an element of JJ, let J~\widetilde{J} be the set of WIW_I-conjugates of elements of JJ and let W~\widetilde{W} be the subgroup of WW generated by J~\widetilde{J}. We show that W=W~⋊WIW=\widetilde{W} \rtimes W_I and that (W~,J~)(\widetilde{W},\widetilde{J}) is a Coxeter system.Comment: 28 pages, one table. We have added some comments on parabolic subgroups, double cosets representatives, finite and affine Weyl groups, invariant theory, Solomon descent algebr

    The nature of the observed free-electron-like state in a PTCDA monolayer on Ag(111)

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    A free-electron like band has recently been observed in a monolayer of PTCDA (3,4,9,10-perylene tetracarboxylic dianhydride) molecules on Ag(111) by two-photon photoemission [Schwalb et al., Phys. Rev. Lett. 101, 146801 (2008)] and scanning tunneling spectroscopy [Temirov et al., Nature 444, 350 (2006)]. Using density functional theory calculations, we find that the observed free-electron like band originates from the Shockley surface state band being dramatically shifted up in energy by the interaction with the adsorbed molecules while it acquires also a substantial admixture with a molecular band

    Imaginary cones and limit roots of infinite Coxeter groups

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    Let (W,S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropy cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots (see arXiv:1112.5415). In this article we study the close relations of the imaginary cone (see arXiv:1210.5206) with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.Comment: v1: 63 pages, 14 figures. v2: Title changed; abstract and introduction expanded and a few typos corrected. v3: 71 pages; some further corrections after referee report, and many additions (most notably, relations with geometric group theory (7.4) and Appendix on links with Benoist's limit sets). To appear in Mathematische Zeitschrif

    Garside families in Artin-Tits monoids and low elements in Coxeter groups

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    We show that every finitely generated Artin-Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order

    Cost-effectiveness of a European preventive cardiology programme in primary care: A Markov modelling approach

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    This is an open-access article distributed under the terms of the Creative Commons Attribution Non-commercial License, which permits use, distribution, and reproduction in any medium, provided the original work is properly cited, the use is non commercial and is otherwise in compliance with the license.OBJECTIVE: To investigate the longer-term cost-effectiveness of a nurse-coordinated preventive cardiology programme for primary prevention of cardiovascular disease (CVD) compared to routine practice from a health service perspective. DESIGN: A matched, paired cluster-randomised controlled trial. SETTING: Six pairs of general practices in six countries. PARTICIPANTS: 1019 patients were randomised to the EUROACTION intervention programme and 1005 patients to usual care (UC) and who completed the 1-year follow-up. OUTCOME MEASURES: Evidence on health outcomes and costs was based on patient-level data from the study, which had a 1-year follow-up period. Future risk of CVD events was modelled, using published risk models based on patient characteristics. An individual-level Markov model for each patient was used to extrapolate beyond the end of the trial, which was populated with data from published sources. We used an 11-year time horizon and investigated the impact on the cost-effectiveness of varying the duration of the effect of the intervention beyond the end of the trial. Results are expressed as incremental cost per quality-adjusted life-year gained. RESULTS: Unadjusted results found the intervention to be more costly and also more effective than UC. However, after adjusting for differences in age, gender, country and baseline risk factors, the intervention was dominated by UC, but this analysis was not able to take into account the lifestyle changes in terms of diet and physical activity. CONCLUSIONS: Although the EUROACTION study achieved healthier lifestyle changes and improvements in management of blood pressure and lipids for patients at high risk of CVD, compared to UC, it was not possible to show, using available risk equations which do not incorporate diet and physical activity, that the intervention reduced longer-term cardiovascular risk cost-effectively. Whether or not an intervention such as that offered by EUROACTION is cost-effective requires a longer-term trial with major cardiovascular events as the outcome.This study is sponsored solely by AstraZeneca through the provision of an unconditional educational grant