8 research outputs found
The strong thirteen spheres problem
The thirteen spheres problem is asking if 13 equal size nonoverlapping
spheres in three dimensions can touch another sphere of the same size. This
problem was the subject of the famous discussion between Isaac Newton and David
Gregory in 1694. The problem was solved by Schutte and van der Waerden only in
1953.
A natural extension of this problem is the strong thirteen spheres problem
(or the Tammes problem for 13 points) which asks to find an arrangement and the
maximum radius of 13 equal size nonoverlapping spheres touching the unit
sphere. In the paper we give a solution of this long-standing open problem in
geometry. Our computer-assisted proof is based on a enumeration of the
so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag
An evolutive approach for the delineation of local labour markets
This paper presents a new approach to the delineation of local labour markets based on evolutionary computation. The main objective is the regionalisation of a given territory into functional regions based on commuting flows. According to the relevant literature, such regions are defined so that (a) their boundaries are rarely crossed in daily journeys to work, and (b) a high degree of intra-area movement exists. This proposal merges municipalities into functional regions by maximizing a fitness function that measures aggregate intra-region interaction under constraints of inter-region separation and minimum size. Real results are presented based on the latest database from the Census of Population in the Region of Valencia. Comparison between the results obtained through the official method which currently is most widely used (that of British Travel-to-Work Areas) and those from our approach is also presented, showing important improvements in terms of both the number of different market areas identified that meet the statistical criteria and the degree of aggregate intra-market interaction.José M. Casado-DÃaz has received financial support from the Spanish Department of Education and Science (ref. BEC2003-02391) through a program partly funded by the European Regional Development Fund (ERDF). Lucas MartÃnez-Bernabeu acknowledges financial support from the Spanish Dept. of Education and Science, the European Social Fund (ESF) and the University of Alicante