In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets.

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

We study the stability of an interconnected system of Euler−Bernoulli beam and heat equation with boundary coupling, where the boundary temperature of the heat equation is fed as the boundary moment of the Euler−Bernoulli beam and, in turn, the boundary angular velocity of the Euler−Bernoulli beam is fed into the boundary heat flux of the heat equation. We show that the spectrum of the closed-l...

We prove that if G is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action ofG on Π g∈G (X0, μ0)g for (X0, μ0) an arbitrary probability space, has first cohomology group isomorphic to the character group of G.

x∈Λ px where px are prescribed and uniformly bounded above and below away from 0 and 1. Poincare inequalities are proved for the Glauber and Kawasaki dynamics, with constants of the same order as in the homogeneous case. 1. Inhomogeneous Bernoulli measures. Let hx ∈ [−K,K], x ∈ ZZ d be given and let px = ex 1 + ehx , x ∈ ZZ . For any Λ ⊂ ZZ d the inhomogeneous Bernoulli measure μΛ(η) on {0, 1} ...

— The so-called first fundamental transformation provides a natural combinatorial link between statistics involving cycle lengths of random permutations and statistics dealing with runs on Bernoulli sequences.

Let (b n) n≥0 be the binomial transform of (a n) n≥0. We show how a binomial transformation identity of Chen proves a symmetrical Bernoulli number identity attributed to Carlitz. We then modify Chen's identity to prove a new binomial transformation identity.

We prove that a Z d-action by automorphisms of a compact, abelian group is Bernoulli if and only if it has completely positive entropy. The key ingredients of the proof are the extension of certain notions of asymptotic block independence from Z-actionsto Z d-action and their equivalence with Bernoullicity, and a surprisingly close link between one of these asymptotic block independence propert...