Before stating our main result, we first deduce certain estimates on (ϕ t, x), ψ t, x)) based on the assumption (1.16).
To this end, we need to deduce certain estimates on composite functions and to do so, we need the following result whose proof can be found in the proof of Theorem 4.3 of [5].
To prove Theorem 1.1, as in [1], we only need to deduce certain estimates on (ϕ t, x), ψ t, x)) = (u t, x) - w t, x), q t, x) + ∂ x w t, x)) since the estimates on w ( t, x ) - w R x t, ∂ x w ( t, x ) - ∂ x w R x t are well-established in [1].
Proof The first estimate follows by the best approximation property of f E. The proof of the second estimate is found e.g. in [33, Lemma 4] and relies on certain inverse-type estimates as well as on the fact that newest vertex bisection only leads to finitely many shapes of triangles.
However, kappa values estimated on certain combinations of ecological axes and phylogenetic trees are not significantly different from 0, and the proportion of non-zero kappa values increases from climatic to habitat and then trophic niches (2, 46 and 77%, respectively).
Avoidance or attraction to aerial or shipboard platforms may also influence density estimates, depending on certain species [18], [19].
It means that the optimized waveforms based on a certain parameter estimate can give a very poor performance for another reasonable estimate.
Moreover, we take advantage of the stability estimates of the true solution and we prove the constants in these estimates depend only on certain Sobolev norms of the initial and the right-hand side data.
By contrast, the WHO figure is an estimate based on certain assumptions regarding Pakistan as a whole, which may not necessarily hold in the overcrowded and impoverished peri-urban settings of this study.
Elisabeth Poole, head of Christie's rug department in New York, estimated the saving on certain rugs at 10 to 20percentt.
