Proof of Theorem. 1 The coding theorem involves constructing a common se- quence un at the legitimate terminals and using it to generate a secret key.
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).
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}.
Proof of Theorem. 5.1. 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. 6 The proof of this theorem follows the same logic as the proof of Xxxxxxx 3, and is omitted here.
Proof of Theorem. 10 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 The cost equations for the non-outsourcing case are given by (2) and (3). Consider only the costs that are dependent on the retailer’s inventory, Equations (4) and (5). A slight adaptation of Theorem 1 can be used to show that (4) and (5) are convex in y and x1, respectively. We denote the y that minimizes (4) as y¯n∗ . We assume that U VMI(x1, x2) = V¯ VMI(x1) + gn+1(x2) and we show that U VMI(x1, x2) = n+1 n+1 n V¯ VMI(x1) + gn(x2). The y that minimizes KVMI(y, xE) is y = y¯∗. Hence, x1 = y¯∗ also n n n n minimizes U VMI(x1, x2) in x1. Thus, if x2 ≥ y¯∗ (and therefore outsourcing is not required to n n reach the replenish up-to point), n n U VMI(x1, x2) = V¯ VMI(x1) + hSx2 + ∫ ∞ gn+1(x2 − ξ)dΦ(ξ). 0 If x2 < y¯n∗ , it is optimal to set y as close to y¯n∗ x2 < y¯n∗ , as possible due to convexity. Therefore, if n U VMI(x1, x2) = L¯(x2) + hSx2 + ∫ ∞ V¯ VMI(x2 − ξ)dΦ(ξ) n+1 + ∫ ∞ gn+1(x2 − ξ)dΦ(ξ). 0 Therefore, if we can show that U VMI(x1, x2) − V¯ VMI(x1) is a function of x2 alone then we are n n done. U VMI(x1, x2) − V¯ VMI(x1) = Λn(x1, x2) + ∫ ∞ gn+1(x2 − ξ)dΦ(ξ) n n where Λ (x , x ) = 0 n+1 n n L¯(x2) + hSx2 + ∫ ∞ V¯ VMI(x2 − ξ)dΦ(ξ) − V¯ VMI(x1) : x2 < y¯∗
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 Xxxxx ∩ ⊇ ⊇ ···
Proof of Theorem. 1 Based on Proposition 1, the basic idea of the proof is to find appropriate µ(t) ≤ µ(t) such that the controlled j∈Ni(σ(t)) aij to denote the interaction between the agents and weighted measurement disturbances acting on agent i, respectively, then κi defined by (8) can be rewritten as κi = υi + ωi. Proposition 2 gives an estimation on the “worst-case” divergence rate of the controlled agents.
Proof of Theorem. 1 We use backwards induction to solve for the decisions of the renewable generator. We first solve for the DA commitment Crk for each time k, taking the prior insurance contract decisions as fixed. The utility function (10) is concave in the commitment decisions. Thus, taking its the derivative with respect to Crk, we find the optimal renewable commitment. r ∂Jc rk rk Rk λp ∂Crk = λk − λpFRk (Crk − Grk) = 0 ⇒ C∗ = G∗ + F−1 .λk Σ . (A.1) Substituting (A.1) back in the expression for the expected profit, we find r k rk rk k Rk λ Jc = (λ − π ) G + λ F−1 .λk Σ p Rk Rk λ p − E ΣI .F−1 .λk Σ − RΣ λ .F−1 .λk Σ − RΣΣ p p Rk λ r r ⇒ Jc = (λk − πrk) Grk + Jb∗, (A.2) ≤ where the last term in (A.2) is the expected renewable baseline profit for the optimal baseline commitment. From this, we observe that it is individual rational for this player to purchase any available reserve through an insurance contract as long as the price is πrk λk. Otherwise, the renewable player is better off without the contract. Appendix B. Proof of Lemma 2 Let (u+, u−) and (u˜+, u˜−) be two distinct feasible storage policies such that, for every k, x x x x −u+ + u− = uk, u+u− = 0 (B.1) x x x x −u˜+ + u˜− = uk, u˜+u˜− > 0 (B.2) Following [23, Theorem 1], we can show that u+ < u˜+ and u− < u˜−. There-
