Measure Recognition Problem

This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra \algB and a property $\Phi$ of measures one can recognize by purely combinatorial means if \algB supports a strictly positive measure with property $\Phi$. The most famous instance of this problem is MRP(countable additivity), and in the first part of the article we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v zamonja and Plebanek (2006) gives an easy solution to the former of these problems, and gives some partial information about the latter. The long term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem which gives such a structure theorem for measure algebras

Some new directions in infinite-combinatorial topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.Comment: Small update

The clustering problem : some Monte Carlo results

I present a series of statistical results on the distribution of the size (i.e. the number of atoms) of clusters formed by atoms scattered randomly in a continuous medium of two or three dimensions. If the range at which they interact is r0, the critical densities (at which infinite clusters appear) are probably of 2.7 ± 0.1 atoms /sphere of radius r 0 (in 3 dimensions) and 4.4 ± 0.2 atoms/disc of radius r0 (in 2 dimensions).Je présente une série de résultats statistiques sur la distribution de la taille des amas formés par les atomes répartis aléatoirement dans un milieu continu de deux ou trois dimensions. Si la portée des interactions est r 0, les densités critiques (auxquelles apparaissent les amas infinis) sont probablement de 2,7 ± 0,1 atomes/sphère de rayon r0 (en 3 dimensions) et 4,4 ± 0,2 atomes/disque de rayon r0 (en 2 dimensions)