Resolving sets for Johnson and Kneser graphs

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl

Geometric Reasoning with polymake

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background

Paving the way for transitions --- a case for Weyl geometry

This paper presents three aspects by which the Weyl geometric generalization of Riemannian geometry, and of Einstein gravity, sheds light on actual questions of physics and its philosophical reflection. After introducing the theory's principles, it explains how Weyl geometric gravity relates to Jordan-Brans-Dicke theory. We then discuss the link between gravity and the electroweak sector of elementary particle physics, as it looks from the Weyl geometric perspective. Weyl's hypothesis of a preferred scale gauge, setting Weyl scalar curvature to a constant, gets new support from the interplay of the gravitational scalar field and the electroweak one (the Higgs field). This has surprising consequences for cosmological models. In particular it leads to a static (Weyl geometric) spacetime with "inbuilt" cosmological redshift. This may be used for putting central features of the present cosmological model into a wider perspective.Comment: 54 pp, 2 figs. To appear in D. Lehmkuhl (ed.) "Towards a Theory of Spacetime Theories", Einstein Studies, Basel: Birkhaeuser), revised version June 201

Historical collaborative geocoding

The latest developments in digital have provided large data sets that can increasingly easily be accessed and used. These data sets often contain indirect localisation information, such as historical addresses. Historical geocoding is the process of transforming the indirect localisation information to direct localisation that can be placed on a map, which enables spatial analysis and cross-referencing. Many efficient geocoders exist for current addresses, but they do not deal with the temporal aspect and are based on a strict hierarchy (..., city, street, house number) that is hard or impossible to use with historical data. Indeed historical data are full of uncertainties (temporal aspect, semantic aspect, spatial precision, confidence in historical source, ...) that can not be resolved, as there is no way to go back in time to check. We propose an open source, open data, extensible solution for geocoding that is based on the building of gazetteers composed of geohistorical objects extracted from historical topographical maps. Once the gazetteers are available, geocoding an historical address is a matter of finding the geohistorical object in the gazetteers that is the best match to the historical address. The matching criteriae are customisable and include several dimensions (fuzzy semantic, fuzzy temporal, scale, spatial precision ...). As the goal is to facilitate historical work, we also propose web-based user interfaces that help geocode (one address or batch mode) and display over current or historical topographical maps, so that they can be checked and collaboratively edited. The system is tested on Paris city for the 19-20th centuries, shows high returns rate and is fast enough to be used interactively.Comment: WORKING PAPE