Remark Sample Clauses

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 E[log Z(β, X)] + βµ log m √N log m β2σ2 ^ ^

Remark. If X(R) = ∅, then the canonical map RΓ^c (Xét, F ∗) → RΓc(Xét, F ∗) is the identity.

Remark. Recall that sig(i) and sig(i) for each i ∈ S, is of the form

Remark. If X(R) = ∅, then RΓc(Xét, Z(n)) is the same as RΓc(Xét, Z(n)) (see 0.9.2), so that in this case we have an isomorphism of distinguished tri- angles RHom(RΓ(Xét, Zc(n)), Q[−2]) id RHom(RΓ(Xét, Zc(n)), Q[−2]) RHom(RΓ(Xét, Zc(n)), Q/Z[−2]) ' RHom(RΓ(Xét, Zc(n)), Z[−1]) ' RΓc(Xét, Z(n)) RΓfg(X, Z(n)) RHom(RΓ(Xét, Zc(n)), Q[−1]) id RHom(RΓ(Xét, Zc(n)), Q[−1]) where the left column is the result of application of RHom(RΓ(Xét, Zc(n)), ) to an appropriate rotation of the triangle We conclude that RΓfg(X, Z(n)) ' RHom(RΓ(Xét, Zc(n)), Z[−1]). However, this holds only if X(R) = ∅. In what follows, we are not going to make such an assumption on X, even though it would save quite some technical work. It is still helpful to keep in mind the special case X(R) = ∅. The complex of sheaves Zc(n) is bounded from below, under the as- sumption that their cohomology groups are finitely generated (which is our conjecture Lc (Xét, n), stated in 1.1.1).

Remark. The idea of the proof is that the minimum bisection problem is a constrained version of the ▇▇▇▇▇▇▇▇▇▇▇-▇▇▇▇▇▇▇▇▇▇▇ model, which is a spin model where all the spins are independent (cf. ▇▇▇▇▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇▇▇▇▇, 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. More precisely, we have CMBP ⊂ CSK. Hence 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. One can also obtain information about HomD(A )(A•, B• ) using the following hyperext spectral sequence: Epq = ∏ Extp (Hi(A•), Hq+i(B•)) =⇒ Extp+q (A•, B•) D(A ) (see e.g. [▇▇▇▇▇▇▇-thèse, Chapitre III, §4.6.10] and [Wei1994, §5.7.9]). For a hereditary category Extp = 0, unless p = 0, 1, and this spectral sequence consists of two columns and therefore gives us short exact sequences 0 → ∏ Ext1 (Hi(A•), Hi−1(B•)) → HomD(A )(A•, B•) → ∏ HomA (Hi(A•), Hi(B•)) → 0 i∈Z However, one should be careful with boundedness of A• and B• to make sure that the spectral sequence exists. Recall that a complex of abelian groups C• is called perfect if it is quasi- isomorphic to a bounded complex of finitely generated free (= projective) abelian groups. This is the same as asking Hi (C• ) to be finitely generated abelian groups, and Hi (C• ) = 0 for all but finitely many i. In §1.5 we are going to construct certain complexes RΓfg(X, Z(n)) that are almost perfect, in the sense that their cohomology groups Hi (X, Z(n)) are finitely generated, vanish for i 0, and for i 0 they are finite 2-torsion (that is, killed by multiplication by 2). Let us introduce the following notion.

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. This means that for the Weil-étale cohomology with rational coefficients, we could take as the definition