A Note on Shortest Developments

De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simple and intuitive principle of duality to de Vrijer's approach one arrives at a proof that some developments are finite which in addition yields an effective reduction strategy computing shortest developments as well as a simple formula computing their length. The duality fails for general beta-reduction. Our results simplify previous work by Khasidashvili

Developments in Collectively Agreed Working Time 2012

In 2012, average collectively agreed weekly working time in the European Union, including Croatia, stood at 38.1 hours, the same as for the EU27 in 2011. The working week was on average 30 minutes shorter in the pre-2004 EU15 countries and over 1 hour and 30 minutes longer in the new Member States. Agreed normal annual working time averaged nearly 1,712 hours in the European Union as a whole – 1,678 hours in the EU15 and 1,824 hours in the new Member States. Of the three sectors examined, banking recorded the shortest average agreed normal weekly working hours with 37.6 hours, although this represents an increase of 0.2 hours on 2011. In the local government sector the average was 37.8 hours and in metalworking it was 37.9 hours. When collectively agreed paid annual leave entitlements are accounted for, average annual leave stood at 25.3 days across the EU, including Croatia, being slightly higher in the EU15 countries (26.7 days) and considerably lower in the new Member States (20.8 days)

Conical Existence of Closed Curves on Convex Polyhedra

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of "source unfolding" of the entire surface of P to one non-overlapping piece, as reported in a companion paper.Comment: 24 pages, 15 figures, 6 references. Version 2 includes a solution to one of the open problems posed in Version 1, concerning quasigeodesic loop

Developments in Collectively Agreed Working Time 2013

This annual report covers several issues related to the length of working time in the European Union and Norway in 2013. It is based mainly on contributions from the national correspondents to Eurofound’s European Industrial Relations Observatory (EIRO). This edition includes data from Croatia, which became a Member State on 1 July 2013. The report looks specifically at: average weekly working hours set by collective agreements, both economy-wide and for three specific sectors: chemicals, the retail trade and the civil service; statutory limits on weekly and daily working time; average actual weekly working hours; annual leave entitlements, as set by collective agreements and law; estimates of average collectively agreed annual working time

Scaling and Universality in City Space Syntax: between Zipf and Matthew

We report about universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf's law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.Comment: 24 pages, 5 *.eps figure