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2 Oct 2009: algebra. 1.2 Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a, b) : a < b}, (b) {(a, b] : a < b}, (c) {(, b] : b ... Show. that E(X̄n) = 0 for all n, but that X̄n 1 almost surely as n.

1 Jul 2009: b) if all 4 neighbours of i do not fail, then i obtains R, R > r. ... I All banks low risk is not an equilibrium.Bank i benefits by switching to high risk strategy since.

21 Dec 2009: a b). ab. which holds for all a,b 0, yields. log E(Stτ 1). [ ... lim supτ. Σt(τ,k1) Σs(τ,k2) 0 a.s. was shown in [7] to hold for all all k1,k2 R and 0 s t.

8 Jul 2009: Theorem 3. (a) The constant word of length n is M-seen in Y if and only if τk Mfor all 1 k n.(b) The alternating word An of length ... As Tn mn by Proposition 6, this impliesτi M for all i n.

21 Oct 2009: K.So the optimal choice of horizontal rates has all horizontal rates equal. ... 9.1). That is for all Borel measurable D RS+ we have that.

9 Aug 2009: infinite periodic word π, P(π M Y ) = 0 for all M 2. ... 1998. Almost all words areseen in critical site percolation on the triangular lattice.

24 Jul 2009: 1)κE(Spτ ). for all k [κ, κ], where we have used the simple inequality (a b) apb1p which holdsfor all a, b > 0 and p > 1. ... κE(Spτ 1{Sτ >0})12. for all k [κ, κ], where we have used the inequality (b a) apb1p which holds for alla, b > 0 and p <

25 Mar 2009: Moreprecisely:. (a) if β βc, there is πβ-probability 1 that all spin-clusters are finite,(b) if β > βc, there is πβ-probability 1 that there exists a unique infinite ... concentrated on even subgraphs of Z2 and satisfies:. (a′) if p pc, there

24 Jun 2009: We claim that. (3.20) µ̃(Ψ(B,G,D) = A) = sMB,G(ψA(B,G)),. for all B, G, where s is a constant, and the expectation MB,G is ... B′,G′).) There are 2|Q|2r. possible outcomes of the above choices, and each is equally likely.The process is