On the weak order of Coxeter groups

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

The complexity of weighted and unweighted #CSP

We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that a recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP.Comment: 11 page

A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs

We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, particularly, when degrees of input instances are bounded from above by a fixed constant. All degree-1 counting CSPs are obviously solvable in polynomial time. When the instance's degree is more than two, we present a dichotomy theorem that classifies all counting CSPs admitting free unary constraints into exactly two categories. This classification theorem extends, to complex-weighted problems, an earlier result on the approximation complexity of unweighted counting Boolean CSPs of bounded degree. The framework of the proof of our theorem is based on a theory of signature developed from Valiant's holographic algorithms that can efficiently solve seemingly intractable counting CSPs. Despite the use of arbitrary complex weight, our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited T-constructibility. For the remaining degree-2 problems, in contrast, they are as hard to approximate as Holant problems, which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary version published under a slightly different title in the Proceedings of the 4th International Conference on Combinatorial Optimization and Applications (COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I), pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201

Fit to practise? Processes for dealing with misconduct among pharmacists in Australia, Canada, the UK and US

© 2018 Elsevier Inc.In many countries with legal systems based on English common law, pharmacy regulators have a responsibility to protect, promote and maintain the health and safety of patients. Where there is a potential risk to patient safety, or where the public's confidence in pharmacy could be adversely affected by the actions of a pharmacist, these regulators have a statutory duty to investigate concerns. The legal provisions underpinning each jurisdiction's disciplinary processes depict distinctive outlooks from the different authorities, as each works towards the same goal. Legal statues, regulations, rules, and guidance affecting the disciplinary process in Great Britain, Australia, New York and New Brunswick were collated, and the processes they describe were attached to a common process flow diagram for step-by-step evaluation of their respective legal provisions. The initial stages of the respective investigation process are broadly similar in all the jurisdictions examined; however, each process has subtle differences that afford some level of advantage or disadvantage over its comparators. Factors including: how matters of discipline are framed; the existence of a separate process for minor and uncontested violations; the ability to effect an interim suspension of a practitioner's license; threshold criteria for escalation of complaints; the membership of disciplinary panels; and the perceived independence of these panels all philosophically affect the public safety remit of each regulator. This work constitutes the first comparison of international regulatory frameworks for the profession of pharmacy. Of the four jurisdictions examined, Great Britain most clearly acts in the interest of the public and the profession – rather than the respondent pharmacist – at every step of its process.Peer reviewe

Trade, urban hinterlands and market integration, 1300-1600: a summing up

Paper given at a conference organised by the Centre for Metropolitan History and supported by the Economic and Social Research Council, 7 July 199