Sentence examples for as noted in the proof from inspiring English sources

As noted in the proof of Proposition 2.1, according to [4, Theorem ], for some.

As noted in the proof research reviewed earlier, gestures and speech each make independent and significant contributions to predicting performance (Pier et al., 2014).

As noted in the proof of Theorem 1.2, is a solution of (2.1) with the asymptotically constant coefficients given by (3.2) and the limiting equation of (2.1) is the linearized equation (1.9).

As noted in the proof of Theorem 3.6(b), σ w ( T + R ) = σ w ( T ), T = M 0 or M C. Furthermore, since M 0 + R and M C + R are quasi-nilpotent equivalent, σ x ( M 0 + R ) = σ x ( M C + R ), σ x = σ  or  σ w (Theorem 3.4).

As noted in the Introduction, Dehua et al. [10] have proved Theorem 2.4, but their proof is not correct.

As was already noted in the proof of Lemma 2, the same proof implies that ((u_{n},v_{n})) converges in H. □.

Furthermore, the proofs of the results in [13] and [17] have the same errors as noted in [18].

As we noted in our proof-of-concept study [44], there appears to be an inconsistent application of the molecular docking methodology when studying HLA systems.

(As noted earlier, the mere assertion that one's opponent bears a burden of proof is cheap; we're interested in the situation where one side in the debate willingly shoulders the burden).

It should be noted that, as shown in the proof of Theorem 1, we separately discuss the existence and the number of the equilibria in three cases based on the sign of the discriminant (Delta_{P'(x)}).

As noted by one referee, no general proof is offered that the GroupFix algorithm accesses the full legal solution space.