Proposition 2. 4.1. A, h, ρ |=SSL ϕ if and only if A, h, ρ |=FSL finh ∧ T (ϕ).
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. 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. For all c ≥ 0 and n ≥ 1, we have cn log2(n) ≤ fc(n) ≤ (c + 1)n[log2(n)|.
Proposition 2. 3.3. Let X be a k-variety, L/k a cyclic extension, and f an element of k(X). The class of the cyclic algebra (L/k, f ) e Br k(X) is in the image of Br X ‹→ Br k(X) if and only if ( f ) = NormL/k(D) for a D e Div(XL). Moreover, if Pic XL = (Pic X)Gal(L/k) then (L/k, f ) e Br k if and only if we can take D to be principal.
Proposition 2. If in a T1 triopoly, the branded 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,
Proposition 2. .1.5. For n = 1, 2 we have B1 ∞,1 (Rn) C OL(Rn), and ≤ B f OL(Rn) C f 1 ∞,1 (Rn), ∀f ∈ B1 (Rn). ∞,1
Proposition 2. .2.4. For i = 0, . . . , r − 1, the Dedekind sums σi . 1 (a1, . . . , an)Σ are the unique solution to the r × r system of equations Σ r−1 i=0 σisi = 0 if s ∈ µaj for some j, j )
Proposition 2. .27. The real moment-angle manifold RP corresponding to a 3- polytope in the Pogorelov class P has the structure of a hyperbolic 3-manifold. The fundamental group of RP is isomorphic to the commutator subgroup G(P )′ of the corresponding right-angled Coxeter group.
