Proof The assumption (1.2) says that the domain p ( U r ) lies in a sector between two rays arg { w } = − π β / 2 and arg { w } = π α / 2, and it contacts with the rays at p ( z 1 ) and at p ( z 2 ).
To proof the assumption that the BLAST analysis gives the same result irrespective which species is used as query and which as reference in a species pair, we conducted some selected reciprocal runs for the species pairs Dmel Dsim, Dmel Dpse, Dmel Dvir, Amel Soli, and Acep Soli (for abbreviations see fig. 1).
Proof Let the assumption conditions in Theorem 3.2 hold.
Proof Let the assumption of corollary hold and denote λ = max { α, 2 β, 2 γ }.
Proof From the assumption, there exist x 0, y 0 ∈ X such that x 0 ⪯ F ( x 0, y 0 ) and y 0 ⪰ F ( y 0, x 0 ).
Proof By the assumption that T : K → C B ( K ) is a multi-valued mapping with convex-values, hence Tp is a nonempty closed and convex subset of K.
Proof Obviously, the assumption (3.5) implies that there exists α > 0 such that f ( k, α ) ≤ r k α for all k ∈ [ 2, N − 1 ] Z.
end{aligned} (3.17) Taking into account (2.12), (2.17), (3.12), the convergence results (3.2) obtained in the first item of the proof, and the assumption that (alpha_{k} geq1), we can pass to the limit in (3.17) as (varepsilonto0).
Several authors supposed the generation of p-benzoquinone (PBQ; Table 2) from PPD under oxidation conditions relevant for the formation of azo (hair) dyes, but a reliable proof of the assumption is missing [3, 16, 49, 50].
Proof By the assumption, there exists x 0 ∈ X such that x 0 ⪯ f x 0. We construct a sequence { x n } in the following way: f x n = x n + 1 for all n ∈ N ∪ { 0 }.
As a result, is Fredholm with the same index as Since the coefficients of and and of and have the same smoothness, respectively, we may, upon replacing by and by continue the proof under the assumption that is a submanifold of (but the are still and the still ).
