Theorem 2. 2.3 for smaller p = p(n) We first consider the case in which n—1 p n—2/3. S Sp p
Theorem 2. 2.3 for larger p = p(n) We now consider the wider range n—1 p ≤ 2n—2/3. Proof of (2.9) in Theorem 2.2.3 We have already shown that, if n—1 p n—2/3, then F ([n]p) = (1 + o(1))np holds almost surely. Therefore, it suffices to show that (2.9) holds if, for example, n—2/3/ log n ≤ p ≤ 2n—2/3. We pro- ceed as in the proof of (2.8), given in Section 2.6.1 above. We have already observed that |[n]p| = np(1 + o(1)) almost surely as long as p n—1, and therefore F ([n]p) ≤ np(1 + o(1)) almost surely in this range of p. It now suf- fices to recall that F ([n]p) ≥ |[n]p| − X and to prove that, almost surely, we have X ≤ (2/3 +o(1))np if n—2/3/ log n ≤ p ≤ 2n—2/3. But with this assumption on p, Xxxxx 2.5.4 tells us that, w.o.p., X = 1 n3p4 + o(n3p4) = 1 n3p4 + o(np) ≤ 2 + o(1) np, (2.72) as required.
Theorem 2. .5. If 1 − ρ ≤ δ and ϕ(x, y, θ) is a special phase function, then Lm (ϕ, X) coincides with the class of pseudodifferential operators Lm (X). ρ,δ ρ,δ Proof. Let ϕ, ϕ1 be two special phase functions. Clearly, it suffices to verify the inclusion ρ,δ ρ,δ Lm (ϕ, X) ⊆ Lm (ϕ1, X). (2.10) First, we claim that for nearby x and y, the difference ϕ1 − ϕ can be written in Σ the form ϕ1 − ϕ = n bjk ∂θj ϕ ∂θk ϕ (2.11) j,k=1 where bjk is positively homogeneous of degree 1 in θ. In fact, ∂ϕ ∂θj = (xj n Σ − yj) + ajk k=1 (xk − yk), where ajk are positively homogeneous of degree 0 in θ and a(x, x, θ) = 0. This can also be written in the form ∇θϕ = (I + A)(x − y), where I is the unit matrix and A is a matrix with elements positively homogeneous of degree 0 in θ, equal to 0 for x = y. Therefore for nearby x and y the matrix (I + A)—1 exists and has elements positively homogeneous of degree 0 in θ. This means that we may write Σ n xj − yj = a˜jk ∂θk ϕ, (2.12) k=1 where a˜jk are positively homogeneous of degree 0 in θ. Now, using (2.7), we observe that ϕ1 − ϕ has a zero of order two on the diagonal D = {(x, y) ∈ X × X : x = y} and by Xxxxxx’x formula ϕ1 − ϕ = n Σ ˜bjk(xj − yj)(xk − yk), (2.13) j,k=1 where ˜bjk are positively homogeneous of degree 1 in θ. In order obtain (2.11), we just need to put together (2.12) and (2.13). Now, consider the homotopy ϕt = ϕ + t(ϕt − ϕ), 0 ≤ t ≤ 1. jk
Theorem 2. 8.1. Plane wave solutions of rotational elasticity can, up to rescaling and rotation, be explicitly written down in the form (2.8.1), (2.8.19) with arbitrary nonzero p0 and p = (p1, p2, p3) determined as follows. • If v1 > 0 and v2 > 0 and v1 v2 then we have two possibilities: v1 – p = 0, 0, ±p0 (type 1 wave), or v2 v2 – p = |p0| cos ϕ, |p0| sin ϕ, 0 where ϕ ∈ R is arbitrary (type 2 wave). • If v1 > 0 and v2 > 0 and v1 = v2 then p is an arbitrary 3-vector satisfying p = |p0| . v1 • If v1 > 0 and v2 = 0 then p = 0, 0, ±p0 . • If v = 0 and v > 0 then p = |p0| cos ϕ, |p0| sin ϕ, 0 where ϕ ∈ R is 1 2 arbitrary.
Theorem 2. 9.2. In the case of a purely axial material a nonvanishing stationary spinor field is a solution of rotational elasticity if and only if it is a solution of one of the two massless Dirac equations (2.9.1). Theorem 2.9.2 was proved in [23] and the proof is quite delicate. It involves an argument which reduces a nonlinear second order partial differential equation of a particular type to a pair of linear first order partial differential equations, which is, effectively, a form of integrability. An abstract self-contained version of this argument is given in Appendix B of [16]. It is interesting that Xxxxxxx and Xxxxxx wrote in Section 6 of [13] “we are not aiming at a model for ether”. Theorem 2.9.2 shows that one can, in fact, model world aether (ether) based on their idea of gyrocontinuum.
Theorem 2. .2.3. Suppose that S(K) contains a K-line A, and let Q0 ∈ A(K). Then, for all P ∈ S(K) we have that 2P + Q0 ∈ GjS (K). Proof. Let P ∈ S(K). By the definition of GjS (K), if P ∈ A(K) there is nothing to prove. Hence, we may assume that P ∈/ A(K). Let TP S be the
Theorem 2. 2.2. Every F ∈ H2−k can be written in the following way: F (z) = F +(z) + (4πy)1−x x − 1 c0(y) + F −(z), where F + and F− have Fourier expansions as follows, for some m0 ∈ Z: ∞ F F +(z) = nΣ=m0 c+(n)qn, and F−(τ ) = Σ c−(n)Γ(1 − k, 4πny)q−n. F n>0 In the theorem, F + is called the holomorphic part of F , and (4πy)1−k c (y) + F−(τ ) is called the nonholomorphic part of F . When the nonholomorphic part is nonzero, F + is called a mock modular form.
Theorem 2. 4. If the reference gas price is less than p¯χ,a, then in order to guarantee the competitiveness (in expectation) of the electrolysis technology compared to that of the steam- reforming technology, a CCfD can be proposed to the producers of hydrogen by electrolysis, the payment of which, noted γ¯χ,a, is a function of market carbon prices (σ) and of the selected strike : γ¯χ,a(σ) = Γ1(σ¯ Γ2 ¯ M 2 M — σ) — (σ — σ)2 = —Γ1(σ¯ — σ Γ2 σ¯ — σ)2 (2.16) m ) — 2 ( m where Γ1 is defined by (2.14) and Γ2 by (2.15). The CCfD’s payment formula depends on electricity price parameters. However, these are linked, among other things, to the electricity production fleet, which differs from one country to another. Consequently, it is preferable to have CCfDs differentiated by country even if the hydrogen production technologies are identical from one country to another. We illustrate this point in the following section devoted to an analysis of the French and German cases.
Theorem 2. .5.6. Suppose b and σ satisfy the conditions in Theorem 2.3.2 so that a unique strong solution exists to the SDE in (2.5.1). Suppose also that a smooth den- sity, ρ, exists (for example, by satisfying Theorem 2.5.3). Then ρ satisfies (2.5.2).
