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. 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. (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. Xxx´asz-Xxxxxxxxxx Inequality). Let T ≥ 1, q ≥ 2. Let T ⊂ [−T, T ] be a well-spaced set, and S = T × {χ mod q}. With the same assumptions as Lemma 2.3.27, we have that (tΣ,χ)∈S F (it, χ) 2 φ(q)X + (qT )1/2 (log(2qT )) Σ| | ≪ |S| X<n 2X (n,q)=1 |an| .
Lemma 2. The morphism φ : X (p−nϵ) → X (p−(n−1)ϵ) has a canonical finite flat formal model φ : X(p−nϵ) → X(p−(n−1)ϵ). It reduces to the relative Frobenius map mod p1−є. We may interpret φ as a formal model XΓ0(pn)(ϵ)a → XΓ0(pn−1)(ϵ)a of the forgetful map.
Lemma 2. For any ε > 0 there exists a function fε ∈ L1(R) such that fε is continuous everywhere except at 0, fˆε(x) = x—1 for |x| ≥ ε, fε L (R) ≤ Cε—1, and
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. 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. For any 1 ≤ i < j ≤ n + 1 the following relations are fulfilled: [αGk, Eij] = [αGk, βFij ] = 0 (∀ k {i, j}, ∀ α, β ∈ {i, j, k}); [αGi, Eij] = √2αFij, [αGj, Eij] = −√2αFij (∀ α ∈ {i, j, k}); [αGi, αFij] = −√2Eij, [αGj, αFij ] = √2Eij (∀ α ∈ {i, j, k}); [αGi, βFij ] = [αGj, βFij ] = √2(α · β)Fij (∀ α, β ∈ {i, j, k}, β · The following lemma is very important for our goals.
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
