From local to global in quasiconformal structures.

We exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure. A sufficient condition for this to happen is described in terms of a Poincaré-type inequality.

A metric map of humans: 23,500 loci in 850 bands

High-resolution maps integrated with the enhanced location data base software (ldb+) give improved estimates of genetic parameters and reveal characteristics of cytogenetic bands. Chiasma interference is intermediate between Kosambi and Carter–Falconer levels, as in Drosophila and the mouse. The autosomal genetic map is 2832 and 4348 centimorgans in males and females, respectively. Telomeric T-bands are strikingly associated with male recombination and gene density. Position and centromeric heterochromatin have large effects, but nontelomeric R-bands are not significantly different from G-bands. Several possible reasons are discussed. These regularities validate the maps, despite their high resolution and inevitable local errors. No other approach has been demonstrated to integrate such a large number of loci, which are increasing at about 45% per year. The maps and the data and software from which they are constructed are available through the Internet (http://cedar.genetics.soton.ac.uk/public_html). Successive versions of this location data base may also be accessed on CD-ROM.

Nonatomic games on Loeb spaces

In the setting of noncooperative game theory, strategic negligibility of individual agents, or diffuseness of information, has been modeled as a nonatomic measure space, typically the unit interval endowed with Lebesgue measure. However, recent work has shown that with uncountable action sets, for example the unit interval, there do not exist pure-strategy Nash equilibria in such nonatomic games. In this brief announcement, we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces, hyperfinite Loeb spaces. We also emphasize other desirable properties of games on hyperfinite Loeb spaces, and present a synthetic treatment, embracing both large games as well as those with incomplete information.