Sentence examples for by using the arguments from inspiring English sources

Moreover, by using the arguments in the proof of Lemma 2.3, we have I ′ ( v 0 ) = 0 and I ( v 0 ) ∈ [ c − δ, c + δ ].

By using the arguments similar to that of the proof of Case (i) of Theorem 1.2 we can show that any solution u ( x, t ) of Problem (1.1) vanishes in finite time.

Since f ∈ DR ( G 1 × G 2 ) by using the arguments of the proof of Proposition 3.1 it can be checked that ( J α 1, α 2 f ) ( x, y ) ≈ r 1 α 1 − Q 1 ( x ) r 2 α 2 − Q 2 ( y ) ∬ B ( e 1, r 1 ( x ) ) × B ( e 2, r 2 ( y ) ) f ( t, τ ) d t d τ. (see also [4] for a similar estimate in the case of the multiple one-sided potentials on R + 2 ).

by using the argument used to prove (6.26).

Proof The proof can be easily obtained by using the argument in the proof of ([36], Theorem 3.1).

Continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper.

By using the argument as above, we obtain x 3 ∈ T x 2 such that d ( x 2, x 3 ) < ( 1 + ϵ ( x 2 ) ) d ( x 2, T x 2 ).

By using the argument in the proof of Theorem 3.4, we can also obtain the following result that is an extension of [[4], Theorem 3.8].

By using the argument in Section 3 together with the results in [25, 36, 40, 41], we can also obtain the analogous results for CAT 0) spaces.

Determination of an optimal bandwidth can present a problem, but this can be overcome by using the argument that minimizes the asymptotic mean integrated squared error (or other minimization techniques).

He is already accusing his opponent of having encouraged the Front National by using the same arguments, and will argue he is the only shield against extremism.