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8 Jun 2017: Then. c>x c>x λ>(bAx)= x>(cA>λ) b>λ b>y. The conclusion now follows by taking the supremum over such all x and infimum over allsuch λ. ... Rm1 such that. L(x,z,λ,µ) > 0. for all x 0 and z 0.Letting x 0 be such that Ax b and setting z = bAx 0, we

18 Aug 2017: It turnsout that all clusters of the infinite-volume limit are almost surely finite. ... Each vertex has degreeeither 1 or 2, and therefore all components are either paths orcycles.

22 Dec 2017: and on the other hand since the balls B(xi, 4/3) are disjoint and are all contained in W4/3[0, t],. ... Wδr [0, t] :=. i:τδi t. B(Zδi ,r), for all r 0.

18 Aug 2017: less, all of whose vertices x, other thanits endvertices, satisfy h(u) < h(x) < h(v′). ... A, B] =〈[a, b] : a A, b B. 〉,. that is, the subgroup generated by all commutators [a, b] with a A, b B.

15 Aug 2017: all δ (0, 1), there exists a constant c(depending on δ, d and p) so that for large n and for all x,y B(0,n(1 δ)). ... Using this uniform bound over all η0 and all x0 B(x,r), we can perform log n/c′ independentexperiments to deduce.

17 Mar 2017: b) [infinitesimal definition] conditional on Xs = x the process (Xst)t0 is independent of(Xr)rs and uniformly in t as h 0 for all x, y. ... b)(c) For all x, y S we have uniformly for all t as h 0.

17 Feb 2017: Wehave that log Zn(a,b,c,Bn) log Zn2(a,b,c,B0n2) |Vn Vn2|K,for some K = K(a,b,c) > 0 and all Bn. ... As in (5.1), a horizontal (respectively, NW, NE) bisector edgeof H is assigned weight a (respectively, b, c), and all the other edges are assignedweight 1