Proof of Theorem. 4.1.2 The proof of the result will be broken down in two Steps. For brevity, we de- ^ note by Γ (N)(ω) the operator τN ٨ Γ (ω). We also recall that Ω is the set of jump- discontinuities of the symbol ω and c is the function in (4.1.10).
Step 1. Finitely many jumps. Suppose that Ω is finite. Setting γz(v) = −iγ(zv), with Σ γ being the symbol defined in (4.1.4), write ω(v) = κz(ω)γz(v) + η(v) (4.4.46) z∈Ω Σ where η is continuous on T and let Φ denote the symbol Φ(v) = κz(ω)γz(v). z∈Ω Weyl’s inequality (2.4.18) shows that for 0 < s < t one has ^ ^ n(t; Γ (N)(ω)) ≤ n(t − s; Γ (N)(Φ^)) + n(s; Γ (N)(η)), n(t; Γ (N)(ω^)) ≥ n(t + s; Γ (N)(Φ^)) − n(s; Γ (N)(η^)). Since Γ (η^) is compact, Lemma 2.4.4 shows that n(s; Γ (N)(η)) = O (1), N → ∞,s ^ ^ and so, using the definition of the fu^nctionals LDτ , LDτ we deduce that for any t > 0 LDτ (t; Γ (ω)) ≤ LDτ (t − 0; Γ (Φ)), (4.4.47) ^ ^ LDτ (t; Γ (ω)) ≥ LDτ (t + 0; Γ (Φ)). (4.4.48) ^ Σ ^ Integration by parts shows that Φ(j) = κz(ω)γz(j) z∈Ω π(j + 1)
j + 1 = −i Σ κ (ω)zj = O 1 , j → ∞ (4.4.49) z∈Ω ^
Proof of Theorem. 4.1.3 ^ Just as in the proof of Theorem 4.1.2, we break the argument into two steps, and use the same notation as before for the operator τN ٨ Γ (ω) and for the symbols γz. We also set Ω+ = {z ∈ Ω | Im z > 0}.
Step 1. Finitely many jumps. Just as before, suppose that the symbol ω has finitely- many jump-discontinuities. Write
Proof of Theorem. In Lemma C.1, we prove succinctness, in Lemma C.2, we prove robustness, and in Lemma C.5, we prove unforgeability.
Proof of Theorem. 11 First, we introduce some alternate notations for the minimum bisection problem in order to ease the transition to the Sherringkton- Xxxxxxxxxxx formalism. Denote by G an undirected weighted complete graph with n vertices. The problem consists in finding a bisection of the graph (a partition in two subsets of equal size) of minimum cost. More formally, define by gij the weight assigned to the edge between vertices i and j (gij = gji). Σ Denote by ci ∈ {−1, 1} an indicator of the suΣbset containing vertex i. We need to find c 1, 1 n such that the sum of the weights of cut edges R(c, X) = ci= cj i<j i ci = 0 (balance condition) and
Proof of Theorem. 3.2. Recall that we have knowable sets K1, K2,..., Kn that cover Σˆ(P ) and satisfy the activity property. We wish to show that they have a nonempty intersection. The first step in adapting the proof of Theorem 3.3 is to prove an analog of Lemma ∩ ⊇ ⊇ ···
5.1. Recall that for a fragment τ the set of schedules that are quasi extensions of τ is denoted Qτ , and Qˆτ = ΣˆP Qτ . The sets Qˆτ play the role of s-balls. For example, for each compressed schedule φ, and for each integer w, let φ(w) denote the maximal prefix of φ whose weight (sum of block sizes) is at most w. If we consider the sequence of sets Qˆφ(w) we see that Qˆφ(1) Qˆφ(2) and that the intersection of all of them is just φ itself. This is analogous to a sequence of balls of decreasing radius around a particular point. The analog of Lemma 5.1 is the following. Lemma 6.1. Let K be a collection of knowable sets that covers Σˆ(P ). Then there is an integer w = w(K) with the property that for each schedule φ, some member of K contains Qˆφ(w) .
Proof of Theorem. 3 | | ≤ | |
Proof of Theorem. 2. The theorem follows directly from Lemma 1, 2, 3 and Theorem
Proof of Theorem. 4.1. Let the constants γ, δ > 0 be given as well as the function ε : (0, 1] → (0, 1]. We may assume w.l.o.g. that ε(x) ≤ x and ε(x) is small enough to enable an application of Algorithm 4.3 with d0 = x. We shall not explicitly define the constant P0 = P0(γ, δ, ε), but we will describe it within the proof. Also, we shall choose the constant N0 = N0(γ, δ, ε, P0) sufficiently large whenever needed. Furthermore, let H be a 3-uniform hypergraph on the vertex set V = V (H) of size |V | = N > N0. We now describe how to construct in O(N 6) time our desired partition Π of V and V . We will show it by induction over the number of iterations.
