Proposition 2. 3.7 (Major Arc Integral). Let A > 3 be fixed and let ε > 0 be fixed sufficiently small. Let (log X)19+ε ≤ H ≤ X log−A X. With M defined as in (2.3.4) and δ > 0 sufficiently small, there exists some η = η(ε) > 0 such that for all but at most O(HQ−1/3) values of 0 < |h| ≤ H we have that ∫ 2 , Σ 1 X M |S(α)| e(−hα)dα = S(h)X p + O P<p≤P 1+δ logη X , where S(h) is the singular series given in (1.0.3). Assuming Proposition 2.3.6 and Proposition 2.3.7, we can now prove The- orem 2.3.3. Proof of Theorem 2.3.3. We follow the arguments in [27, Pages 32-34]. By (2.3.3), we have that Σ Σ ϖ2(n)ϖ2(n + h) − ∫ |S(α)|2e(−hα)dα 0<|h|≤H X<n≤2X−h M ≪ 0<Σ|h|≤H ∫m |S(α)|2e(−hα)dα . We now apply a smoothing; we multiply the above by an even non-negative
Proposition 2. 3.10. Let ε > 0 be fixed sufficiently small, let A > 3 be fixed. Let Q0 be defined as in (2.3.5) 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 have that ∫ 2 , Σ 1 X |a(α)| e(−hα)dα = S(h)X + O P<p≤P 1+δ logη X , for some η = η(ε) > 0, where we define the singular series S(h) as in (1.0.3). ! .
Proposition 2. If in a T1 triopoly the brander firm’s profit are nearly constant in first mover advantage (i.e., ∂ΠT1/∂κ 0), and there exists a κ∗ [0, 1] such that the net surplus from lunching AG at κ∗ is zero, then under take-it or leave-it offer for the licensing fee, the threat to launch an AG is credible for all κ ≥ κ∗. (Proof in Appendix (A.2)). 0 The condition ∂ΠT1/∂κ ≈ 0, that the equilibrium profit for the branded firm in T1 is nearly constant, is stronger than needed. What we need for net surplus to be increasing in κ is the condition |∂ΠT1/∂κ| < |∂ΠT1/∂κ|, i.e., the branded firm’s equilibrium profit is decreasing in first mover advantage at a slower rate than the increase in the equilibrium profit of the first generic entrant so that the overall net surplus still keeps on increasing in κ (recall that ΠT0 does not change with κ, but ΠT1 can decrease in κ due to price coordination between the brand and the AG, see lower-left panel in Figure A-3 for the shape of ΠT0). Next, we can provide conditions – or values of cost θ – under which a branded firm would prefer to launch an in-house AG. To do so, we fist define two threshold values. Let θ∗(κ) = (ΠT1 +ΠT1 − ΠT0) 0 and θ∗∗(κ) = (ΠT1 + ΠT0 − ΠD0) + δ · θ∗(κ). ≤
Proposition 2. For all a > 0, each collaborator’s welfare is higher than the stand-alone nation’s welfare. uP uP uP since qP qP for a 0, i i where uP I v GP c qP , i 1, 2,
Proposition 2. 2. Let’s denote p¯χ,a the gas price that cancels the discriminant of the poly- nomial 2.9, therefore pχ,a e
Proposition 2. If the CDH problem in G is hard, the protocol in Figure 1 is a secure authenticated key establishment protocol.
Proposition 2. .3.1. Suppose ν is a non-null finite Borel measure on Rd with exact dimension α. Let µ be any non-null finite Borel measure µ on Rd with µ ν. Then µ is exact dimensional with exact dimension α. The following results about derivatives of measures will also come in useful, see for instance [Ma, Theorem 2.12].
Proposition 2. 3.2. Let µ, λ be inner regular probability measures on Rd (by inner regular we mean that for each m = µ, λ, m(B) is the supremum of m(K) over all compact subsets K of B, for any Borel set B). Then:
Proposition 2. .5.3. Let f : Σ → R be a H¨older continuous potential. Then there exists a unique invariant Xxxxx measure µf for f and the constant P from (2.1) is given by P = P (f ). Moreoever, this is the unique equilibrium state for f. Given a continuous function u : Σ → Σ we say that u − u ◦ σ is a coboundary. We say that two functions f, g : Σ → R are cohomologous (writing f ∼ g) if there exists some continuous function u : Σ → R for which g = f + u − u ◦ σ. (2.6) Note that two functions being cohomologous is an equivalence relation. Also observe that if two functions f and g are cohomologous then their Birkhoff sums coincide on periodic orbits, that is Snf (i) = Sng(i) for any i such that σni = i. Coboundaries are useful since adding a coboundary to a function preserves thermodynamic quantities, as demonstrated by the following result, see for instance [PP, Proposition 3.6].
Proposition 2. 5.4. Two H¨older continuous functions f and g have the same equi- librium state if and only if f ∼ g + c, where c = P (f ) − P (g).
