The converse of the above proposition is valid when k = 1.
By uttering example 1, the speaker conveys to the hearer the information that the proposition is valid without any qualification.
CTL formula involves temporal expressions, with φ and Ψ being atomic propositions, X φ (next φ), φ U Ψ (φ atomic proposition is valid until Ψ), G φ (φ is always globally valid) and F φ (φ is valid in the future).
Their proposition is valid even when temporal synchronization remains unaffected with increasing species richness; an assertion largely supported by our results on the co-variance CV (Figures 1 and 2).
If the condition in Proposition 2.1 is valid, then from Proposition 2.3 we have begin{aligned} bigl|!bigl|!bigl|bigl{ y^,sigma^ bigr} bigr|!bigr|!bigr|_{delta} le C bigl(|f |_{L^{2}(Omega)}+bigl| u^bigr| _{L^{2}(Omega)} bigr).
Proposition 4.1 is valid.
Assume the condition in Proposition 2.1 is valid.
If the condition in Proposition 2.1 is valid, then we have begin{aligned} bigl|!bigl|!bigl|{y,sigma}bigr|!bigr|!bigr|_{delta} leq C |f|_{L^{2}(Omega)}, end{aligned} (2.12) where the constant C depends on the Poincaré constant (C_{Omega }), and the reciprocal of (c_{delta}) and (delta_{0}).
From Proposition 2.8, (4.6) is valid if and only if for any, (4.7).
It is easy to see that ∥ u ∥ H a + 1 2 ( Q T, γ 2 + σ ) 2 ≤ ∑ k = 0 2 ∥ u t k ∥ H a + 1 2 - k, 0 ( Q T, γ k + σ ) 2. Hence, Proposition 3.2 implies that the theorem is valid for m = 0. Assume that the theorem is true for m - 1, we will prove that it also holds for m.
