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Closure conditions of commutativity

Václav J. Havel, Josef Klouda (1994)

Archivum Mathematicum

There are investigated some closure conditions of Thomsen type in 3-webs which gurantee that at least one of coordinatizing quasigroups of a given 3-web is commutative.

Clusters in middle-phase percolation on hyperbolic plane

Jan Czajkowski (2011)

Banach Center Publications

I consider p-Bernoulli bond percolation on transitive, nonamenable, planar graphs with one end and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i.e. 0 < p c ( G ) < p u ( G ) < 1 , where p c is the critical probability and p u -the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in ∂ℍ². This result is similar to some results in [Lal].

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