Remark. The general upper bound proven above is unfortunately not tight. Consider the (non-sparse) minimum bisection problem with d = n/2. Under log 2σ the same general assumptions for the weights, it can be shown that a tighter bound holds for β^ ≤ √ 1 lim n→∞ E[log Z(β, X)] + βµ log m √N log m β2σ2 ^ ^
Remark. The idea of the proof is that the minimum bisection problem is a constrained version of the Xxxxxxxxxxx-Xxxxxxxxxxx model, which is a spin model where all the spins are independent (cf. Xxxxxxxxxxx and Xxxxxxxxxxx, 1975). In the minimum bisection problem, it is required that the partition of the graph be balanced, or equivalently rephrased in spin model terms, it is required that there is the same number of up-spins as down-spins. Therefore, the only difference between the two problems is the solution space. n n More precisely, we have CMBP ⊂ CSK. Hence n n ZMBP(β) = Σ e−βR(c,X) ≤ Σ e−βR(c,X) = ZSK(β), (A.2) c∈CMBP
Remark. The notion of merge edges for (extended) reduction graphs is more closely related to the notion of reality edges for breakpoint graphs in the theory of sorting by reversal [17] compared to the notion of reality edges for (extended) reduction graphs. Thus in a way it would be more natural to call the merge edges reality
Remark. After Step 3 the construction of the ephemeral part of the private key of the HD, which consists of C, Cj, β, βj, σ, is complete. The T-values and the set of conjugates α are also part of the private key of the HD and must be treated as confidential information.
Remark. Recall that sig(i) and sig(i) for each i ∈ S, is of the form
Remark. 6.5.1. If the structural layout of the engine is such that the test cannot be performed by the methods described in paragraphs 6.1. to 6.4. of this annex, the measurements shall be effected by that method modified as follows:
Remark. We will present in Section 5, the design of networks that allow Byzantine agreement with at most 1 (3t − k − 1)(t + k + 1)(k + 1) 3-hyperedges, where n = 3t − k, for 0 ≤ k < t. Implication 3 There are several scenarios (networks) for which no known pro- tocol can achieve Byzantine agreement while our protocol succeeds. } { } { }} {{ } { } { } { } { } { { } Remark: For example, consider the network H(P, E) on five nodes two of which may be faulty and contains eight 3-hyperedges, P = p1, p2, p3, p4, p5 and Ebasis = p1, p2, p3 , p1, p2, p4 , p2, p3, p4 , p3, p4, p5 , p4, p5, p1 , p1, p2, p5 , p2, p3, p5 , p1, p3, p5 . Note that H satisfies the conditions1 of Theo- rem 1; hence our protocol of Section 4 achieves agreement while all the extant protocols fail.
Remark. In the event of extenuating circumstances, the Superintendent of Schools, at the request of the Director or Coordinator, has the discretion to modify or suspend a Director or Coordinator's participation in the evaluation process for one year.
Remark. For any given d, there are only finitely many n not covered by the Xxxxx-Xxxxxxx conjecture, and a simple argument shows that ∆d(n) ≥ 0 for these n. In essence, Xxxxx’x conjecture asks us to relate the coefficients of Σ Q (n)qn = Y 1 ∞ ∞ and d n=0 (1 − qn(d+3)—(d+2))(1 − qn(d+3)—1) ∞ ∞ d(n)+n d
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