Lemma 2 Sample Clauses

Lemma 2. Let k = [k1, k2, k3, k4] be a key agreement scheme. Let α, β, γ, δ, δ′, η, η′, φ be functions, such that K1(a, b) = δ(k1(α(a), β(b))) K2(b, c) = η(k2(β(b), γ(c))) K3(a, e) = φ(k3(α(a), η′(e))) K4(d, c) = φ(k4(δ′(d), γ(c))). are well-defined. If δ′(δ(x)) = x and η′(η(x)) = x for all x in the appropriate domains, then K = [K1, K2, K3, K4] is a key agreement scheme.

Lemma 2. (The Singular Series). Let h be a non-zero even integer and Q0 be defined as in (2.3.5). Let A > 3 be fixed and let (log X)19+ε ≤ H ≤ X log−A X. Then, for all but at most O(HQ−1/3) values of 0 < |h| ≤ H we φ2(q) = S(h) + O(Q0 log H). Σ µ2(q)cq(−h) q≤Q0 −1/3 −q

Lemma 2. Let A ≥ 1 and ε ∈ (0, 1 ] be fixed. Let X ≥ 1, 1 ≤ Q ≤ ∆ and ∆ = Xθ with 1 + 2ε ≤ θ ≤ 1. Then we have that X1/6+ε x<n≤x+q∆ Σ Σ ∫ 2X Σ q≤Q χ(q) (Λ(n)χ(n) − δχ) Q3∆2X logA X , where we define δχ = 1 if χ = χ0 and δχ = 0 otherwise.

Lemma 2. Assume that {Th}h>0 is a shape regular family of affine mesh which ap- proximates a non empty, Lipschitz, compact subset Ω of Rd. We consider a quadrature rule with degree of exactness q > d − 1 (which is trivially true for d = 2, 3). Let {aˆK,l} and {ωK,l} be the nodes and the weights of the quadrature formula on each element K. Let f be a L1 function in Ω such that it belongs to Hq+1(K) for each element K. Then, there exists a constant c independent of h, such that .∫ Σ Σ  .  2 . Σ fdΩ − . Ω K∈Th ωK,lf (aˆK,l). ≤ chq+1  K∈Th |f |Hq+1

Lemma 2. There is a value of λ such that the contract is complete if and only if the arbiter is biased in favor of honest parties.

Lemma 2. If H is a normal subgroup of G of order prime to char(k), then one has + QuotG..V − V >HΣ/.V − V ≥HΣΣ = QuotG/H ..V ≥H − V >HΣ ∧ QuotG,H.Th.V /V ≥HΣΣΣ. QuotG = QuotG/H QuotG,H, it is sufficient to show that QuotG,H((V − V >H)/(V − V ≥H)) is isomorphic to (V ≥H − V >H)+ ∧ QuotG,H(Th(V /V ≥H)) as a G/H-space. Since the order of H is prime to char(k), there is an isomorphism V = V ≥H ⊕ (V /V ≥H ). Using this isomorphism, we get an isomorphism + .V − V >HΣ/.V − V ≥HΣ = .V ≥H − V >HΣ ∧ Th.V /V ≥HΣ. Since the action of H on V ≥H is trivial, we get + QuotG,H..V − V >HΣ/.V − V ≥HΣΣ = .V ≥H − V >HΣ Combining Lemmas 2.2 and 2.4, we get the following result. ∧ QuotG,H.Th.V /V ≥HΣΣ.

Lemma 2. Consider good transcript τ = (τcq, h) and denote by the system of q equations corresponding to (ϕτ , ν1 h(m1),..., νq h(mq)). This system of equations is (i) circle-free, (ii) (ξ + 1)-block-maximal, and (iii) relaxed non- degenerate with respect to partition {1,..., r} = R1 ∪ R2.

Lemma 2. 2. The set D defined in (2.27) is normal. Furthermore, the constraint set is compact, bounded, and connected. The program- ming problem in (2.25) corresponds exactly to the monotonic optimization problem. Therefore, we can apply the outer polyblock approximation algorithm described in [3] to solve all three problems, the weighted sum-rate maximization in (2.17), the proportional fair problem in (2.18), and the max-min problem in (2.19).

Lemma 2. Let A in Pn be a Qp-line. There exist P, Q two distinct Qp- rational points of it, such that P ƒ= Q. In that case, we shall call {sP + tQ | [s : t] ∈ P1(Qp)} a good parametrisation of A(Qp).