Proposition 5. The r-DHI assumption holds for Qr A under the assumption that the AS scheme is IND-CPA secure for parameters (x, k, t, r + 1).
Proposition 5. .1. Suppose B is a functor satisfying the hypotheses of Theorem 5.
1. Let (CB¯ , s) be the codensity monad of B¯, with distributive law τ and monad structure (CB¯ , η, μ). If Bk+1,k is an isomorphism for some k, then
1. sk : CB¯ Bk → Bk is the algebra induced by τ on the final coalgebra;
2. if C has an initial object 0 then sk is isomorphic to μ0.
5.1 Codensity and the Companion of a Monotone Function Throughout this section, let b : L L be a monotone function on a complete lattice. By Theorem 5.1, the companion of a monotone function b (viewed as a functor on a poset category) is given by the right Kan extension of the final sequence ¯b : Ordop L along itself. Using Lemma 4.1, we obtain the characteri- sation of the companion given in the Introduction (5). .
Theorem 5.2. The companion t of b is given by t : x ›→ bi x≤bi .
Proposition 5. .1. Suppose B is a functor satisfying the hypotheses of Theorem 5.1. Let (CB¯ , ϵ) be the codensity monad of B¯, with distributive law τ and monad structure (CB¯ , η, μ). If Bk+1,k is an isomorphism for some k, then
1. ϵk : CB¯ Bk → Bk is the algebra induced by τ on the final coalgebra;
2. if C has an initial object 0 then ϵk is isomorphic to μ0.
5.1 Codensity and the Companion of a Monotone Function Throughout this section, let b : L L be a monotone function on a complete lattice. By Theorem 5.1, the companion of a monotone function b (viewed as a functor on a poset category) is given by the right Kan extension of the final sequence ¯b : Ordop L along itself. Using Lemma 4.1, we obtain the characteri- sation of the companion given in the Introduction (5).
Theorem 5.2. The companion t of b is given by t : x '→ bi x≤bi Proof. By Lemma 4.1, the codensity monad C¯b can be computed by C¯b(x)= (Ran¯b¯b)(x)= bi , x≤bi a limit that exists since L is a complete lattice. We apply Theorem 5.1 to show that C¯b is the companion of b. The preservation condition of the theorem amounts to the equality b ◦ Ran¯b¯b = Ran¯b(b ◦ ¯b) which, by Lemma 4.1, in turn amounts to b( bi)= b(bi) x≤bi x≤bi for all x L. The sequence (bi)i∈Ord is decreasing and stagnates at some ordinal ϵ; therefore, the two intersections collapse into their last terms, say bδ and b(bδ) (with δ the greatest ordinal such that x /≤ bδ+1, or ϵ if such an ordinal does not exist). The equality follows. H In fact, the category (b) defined in Sect. 4 instantiates to the following: an object is a monotone function f : L L such that f (bi) bi for all i Ord, and an arrow from f to g exists iff f g. The companion t is final in this category. This yields the following characterisation of functions below the companion.
Proposition 5. 3.1. Let V be a free, finite index, subgroup of E+ of rank n − 1. Then σ∈G u2(V ) = ∑ u2(σ)[V ∶E+] ⊗ σ−1, u2 = ∑ u2(σ) ⊗ σ−1. σ∈G
Proposition 5. 9. Let A be a Noetherian ring and B = A[[x1, . . . , xn]] be the A-algebra of formal power series in n variables. Then B is faithfully flat over A.
Proposition 5. For some constant probability p, any domain V , and any integer itr, V,p,itr can be UC-realized with statistical security in the a-smt-hybrid model, in constant rounds and in the presence of an adaptive and malicious t-adversary, provided t < n/3.
Proposition 5. Let (nn : Z× → &× )n∈Z be any sequence of weights satisfying np ≡ nn mod p, then inside &K[[q]], If moreover nb : Z× → &× (En )p ≡ En mod p. n+1 n Kb is the corresponding t-adic weight, then E b corresponds to the sequence (En )n under the isomorphism M + (0) = xxx X + (0)/p of Theorem 5.3.3. n ∈Z≥0 nb ←− n
Proposition 5. Let M be a submodule of W. Then M has a generating set X with the following property: No subset Y of X, whose image τ (Y ) in T is a chain with respect to the partial order ™, can have more than min ad − am + am, χ RF(Kj) elements, where χ = χ(K, V ∗) denotes the number of orbits of K on the nonzero elements of V . Before proving Proposition 5.4.9, we need a preliminary lemma.
Proposition 5. The closing of the demerger deal and legal independence creates a sense of “being done” with the demerger and moving on to normal business. However, IT dis- integration most commonly will not be completed at this time and requires complete attention and priority within the company in order to succeed.
Proposition 5. The function f (n, m) is multiplicative in m. Proof. Let m1 and m2 be relatively prime positive integers and let P be the set of all prime divisors of m1m2. For i = 1, 2, let Ki be a number field that attains the maximum value for fK(mi). For each prime p in P with p | mi, define Ep = Ki⊗Qp. According to lemma 5.17, there exists a number field K such that K ⊗ Qp Ep for all p ∈ P . For i = 1, 2 and pk | mi, we have fK(pk) = fK⊗Q (k) = fK ⊗Q (k) = fK (pk) by proposition 5.19. By lemma 5.21, we get fK(mi) = fKi (mi) = f (n, mi) and therefore f (n, m1m2) ≥ fK(m1m2) = f (n, m1)f (n, m2). On the other hand let K′ be a number field of degree n that attains the maximum value for fK(m1m2). Then we also have the bound f (n, m1m2) = fK′ (m1m2) = fK′ (m1)fK′ (m2) ≤ f (n, m1)f (n, m2). Now we prove proposition 5.3, restated here for convenience. Proposition 5.3. The following equality holds for all x. xxx sup log f (n, m) = lim sup log fEt(n, p, k) log m pk →∞ k log p log fEt(n, p, k) log f (n, pk) lim sup pk →∞ = lim sup k log p pk →∞ log pk = x. The set of prime powers is a subset of the integers, so we can bound ≤ log f (n, m) log m Hence, if x is infinite, we are done. Assume x is finite. Then for all ǫ > 0 we have lim f (n,pk) pk →∞ pk(x+ǫ)
