Sentence examples for by using the assumptions from inspiring English sources
By using the assumptions of the theorem, we conclude the proof.
Proof By using the assumptions of this lemma, we have G 1 ( t, s ) ≥ 0 for all ( t, s ) ∈ [ 0, 1 ] × [ 0, 1 ], and so u ( t ) ≥ 0 for all t ∈ [ 0, 1 ].
On the other hand, by using the assumptions of the theorem we easily get vert x_{n}vert lefrac{Vert fVert _{infty}}{sqrt{vert qvert } (sqrt{vert qvert }-1)}< infty, quad nin mathbb{N}_{0}, from which the boundedness of ((x_{n})_{ninmathbb{N}_{0}}), follows.
Both Bürgi and Kepler were astronomical observers, and Kepler included logarithmic tables in his famous Tabulae Rudolphinae (1627; "Rudolphine Tables"), astronomical tabulations of planetary motion derived by using the assumption of elliptical orbits about the Sun.
By using the assumption that Δ ⊆ E ( G ), we immediately obtain X f ≠ ∅.
By using the assumption of, we can get following equality: (3.5).
By using the assumption, we have α ( x n, x ∗ ) ≥ 1 for all n.
By using the assumption of (xg(x le0) for (xin -infty,0)cup(0,+infty)), we have (u_{0}(t^)le0), which contradicts (3.33).
These higher response moments should enable better reliability predictions than those obtained by using the assumption of Gaussian responses.
Both foreshocks and aftershocks, which are meaningless in seismotectonic investigations, were identified and removed by using the assumption reported by Gardner and Knopoff (1974).
