In this paper we discuss two different models of dependent percolation on the graph Z2. We show that they both exhibit phase transitions. This proves a conjecture of Jonasson, Mossel and Peres [6], who proved a similar result on Z3.

We construct a bireversible self-dual automaton with 3 states over an alphabet with 3 letters which generates the lamplighter group Z3 ≀ Z. 2010 Mathematics Subject Classification: 20F65, 20M35

Recent protest movements brought attention to the one percent, a segment of the population that is critical to understanding inequality and social mobility but that attracts relatively little research attention. In this article, I survey current research on the one percent in the United States. I distinguish income from wealth and show that both are very concentrated but that the concentration ...

We prove the existence of two symmetric designs with parameters (256,120,56) for which the full automorphism group is isomorphic to (Z17.Z8) x (Z7.Z3).

In this paper are characterized the polynomials, in terms of their coefficients, that have all their orbits dense in the set of 3-adic integers Z3.

There is a canonical homomorphism ψ : π1(bdyX)→ π∞ 1 (X) from the fundamental group of the visual boundary, here denoted by bdy X, of any non-positively curved geodesic space X into its fundamental group at infinity. In this setting, the latter group coincides with the first shape homotopy group of the visual boundary: π∞ 1 (X) ≡ π̌1(bdy X). The induced homomorphism φ : π1(bdy X)→ π̌1(bdy X) prov...