On the Coefficients and the Growth of Gap Power Series

In this paper we are interested in the limiting case, in which not the behavior of f in an angle, but only on a radius, for example z = x > 0, is known. Fabry gaps no longer suffice to get information about the growth of m(r) = [...], since already Pólya pointed out [11, p. 636] that there exist entire functions with Fabry gaps (even [...]), which are bounded for x > 0

Analytic approach to confinement and monopoles in lattice SU(2)

We extend the approach of Banks, Myerson, and Kogut for the calculation of the Wilson loop in lattice U(1) to the non-abelian SU(2) group. The original degrees of freedom of the theory are integrated out, new degrees of freedom are introduced in several steps. The centre group $Z_2$ enters automatically through the appearance of a field strength tensor $f_{\mu \nu}$, which takes on the values 0 or 1 only. It obeys a linear field equation with the loop current as source. This equation implies that $f_{\mu \nu}$ is non vanishing on a two-dimensional surface bounded by the loop, and possibly on closed surfaces. The two-dimensional surfaces have a natural interpretation as strings moving in euclidean time. In four dimensions we recover the dual Abrikosov string of a type II superconductor, i.e. an electric string encircled by a magnetic current. In contrast to other types of monopoles found in the literature, the monopoles and the associated magnetic currents are present in every configuration. With some plausible, though not generally conclusive, arguments we are directly led to the area law for large loops.Comment: 18 pages, uses latexsym, to appear in The European Physical Journal

'Cette autre nécessité essentielle: 'l'urbanisation': electrification and the Urbanisation of the Nebular City

The advent of modern utility systems together with improved transport infrastructures and information technologies introduced new spatial arrangements and temporalities in the territory. In time, these reveal a notion of urbanisation that does not only takes place in or directly adjacent to the traditional (territorially bounded) city, but in which co-evolving processes lead to differentiated territorial arrangements. Belgium’s distributed urban condition – the ‘nebular city’ – emerged out of the interplay of such multiple territorial arrangements. Often, it is explained by a historical roots in policies of industrial dispersal, while historical efforts to actively accommodate and organise the territory from the broader perspective of urbanisation are assigned a secondary role only. This article, however, takes a close look at two projects from the 1930’s that took the emerging condition of dispersal as their starting point and which both reflect on the role of urbanisation in the reproduction of the conditions in which industrialisation, among other processes of modernisation, can take place. In particular aspects surrounding the Belgian electrification are examined. Although not one of their main drivers, the electrification is both intertwined with the rise of industrial production and the development of an urban modern lifestyle. Only in the 1930’s, however, Belgian spatial planners started to explore issues concerning the distribution of electricity and its spatial and economic consequences. Both projects are embedded within the international debate on the functional city and present Belgium as a particular case. They show the general delay and mismatch between the process of industrialisation and urbanisation because of the nation’s chosen development path, both in spatial and temporal terms