By M. Aizenman (Chief Editor)

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The papers comprising Vol. I and Vol. II have been ready for and awarded on the foreign convention on details Networking 2002 (ICOIN 2002), which used to be held from January 30 to February 1, 2002 at Cheju Island, Korea. It was once geared up via the KISS (Korean details technology Society) SIGIN in Korea, IPSJ SIG DPE (Distributed Processing structures) in Japan, the ITRI (Industrial expertise learn Institute), and nationwide Taiwan college in Taiwan.

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**Example text**

For i ≤ n, an interval J ⊂ ˜ n and aˆ ∈ J , we say i,n (a) ˆ has a smooth continuation to J if there is a map g : i,n (a) ˆ × J → R0 such that – g(·, a) = i,n (a) for all a and ˆ a → g(z, a) is smooth. – for each z ∈ i,n (a), Likewise one has the notion of the critical regions C (i) deforming continuously as a ranges over J . 4. Let aˆ ∈ n and J = [aˆ − ρ 2n , aˆ + ρ 2n ]. Then J ⊂ ˜ n ; moreover, has a smooth continuation to J , and C (i) , i ≤ n, deform continuously on J . ˆ n,n (a) The structual stability of the critical regions comes from the fact that the components of C (i) are stacked together in a very rigid way, and their relations to the components of C (i−1) are equally rigid.

They must lie on the two horizontal boundaries of Q(k−i0 ) (zk ) because γk−i0 passes through zk and intersects no other point of ∂Rk−i0 . 3 is < ( DT 200 b)k−i0 < (Kb) 100 k . 1. By the Borel–Cantelli Lemma, it suffices to show that −k Z (k) ) < ∞. We estimate m(T −k Z (k) ) by k m(T m(T −k Z (k) ) = m(T −k (Q(k) ∩ Z (k) )) ≤ max m(T −k (Q(k) ∩ Z (k) )) m(T −k Q(k) ) m(T −k Q(k) ), where the summations and maximum are taken over all components Q(k) of C (k) . Note also that m(T −k Q(k) ) < 1. 2 and the regularity of det(DT ) in (**), we obtain (k) ∩ Z (k) ) m(T −k (Q(k) ∩ Z (k) )) 2k m(Q · ≤ K m(T −k Q(k) ) m(Q(k) ) (Kb) 100 k · b 20 k 99 ≤ K 2k · ( Kb )k+1 · ρ k 1 b 25 k · b ρk 1 1 ≤ K 4k which decreases geometrically in k as desired.

2) With the properties of θN in (1) having been established, we observe that continuing to use θN as the source of control, the material in Sects. 3 are now valid for times up to min(m, 3N ). (3) We are now ready to argue that (IA2)–(IA6) hold for all z0 ∈ 3θN . For each z0 ∈ 3θN , whether it is in θN or of generation > θ N, there exists z0 ∈ θN such that |z0 −z0 | = O(b e−β3N θN 4 ). This implies, for i ≤ 3N , that |zi −zi | < b θ 4 θN 4 DT 3N << 1 −β . 2e provided θ is chosen so that b DT < (IA2) follows immediately from the corresponding condition for z0 .

### Communications in Mathematical Physics - Volume 218 by M. Aizenman (Chief Editor)

by Steven

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