Given, in view of Lemma 2.11, we have that is a pseudo-almost periodic function, and hence bounded in.
The use of identical smoothing operators indicates that T-F coherence satisfies the Cauchy-Schwarz inequality and is hence bounded within [0,1] (see also [20, 21]).
Indeed the image (f([0,T]) subset L^p({{mathbb R}^{N}})) is weakly compact, hence bounded and so, the claimed Lipschitz continuity results from the detailed reformulation (6) of weak solutions.
Second, we suppose additionally that A ℓ (L 2 ⊆ H 1. Then, the H -stability (78) yields that A ℓ ∈ L 2 (H 1 ; H 1 are uniformly continuous operators. For v ∈ H 1, the sequence (A ℓ v ) ℓ is hence bounded in H 1 and thus admits a weakly convergent subsequence A ℓ k v → w weakly in H 1 as k → ∞.
Hence, confidence bounds on α can be determined from the quantiles of the bootstrap distribution of, e.g. two-sided confidence bounds with nominal error probability β will be: Here, is the estimated IS incidence computed from the original data and.
Hence, verified bounds are a key step in the optimisation algorithm.
Hence, the bounds in Theorem 2.2 are tight and sharper than those in Theorem 1.1 in [15].
Hence, is bounded, that is, is bounded.
Hence is bounded and also, and are bounded.
Hence is bounded and therefore, and are also bounded.
