Sentence examples for applying this lemma from inspiring English sources

Applying this lemma yields the following new integral inequality.

Applying this lemma and the results in the previous section to Theorem 3.2, we obtain the following convergence theorem of an iterative scheme approximating a fixed point of a nonexpansive mapping.

Finally, we will see that by applying this lemma, we can easily derive entropy bounds for the graph classes under consideration.

Thus, if ϕ ∈ F 1 Hom M o d ∞ (A, M ) is a homomorphism, then ∂ (b ϕ h A ) = b ϕ ∂ h A = b ϕ implying that it is a boundary and therefore F 1 Hom (U A, M ) = 0.  □ Applying this lemma yields the following corollary.

We apply this lemma in Sect.

We will revisit that and apply this lemma to a great effect in the next section.

In that lemma we denote (u^a=a^{-1}ua) and (u^{a+b}=u^au^b) (note that although (u^{a+b}) is not necessarily equal to (u^{b+a}), the equality will hold if the normal subgroup generated by u is Abelian, which is going to be the case every time we apply this lemma).

Assume that (A1 - A5) hold, and applying this in Lemma 3.4, we can obtain that sup_{y} biglvert {mathrm{P}}bigl(U_{1n}^{prime } leq ybigr -mathrm{P}(H_{n}leq ybigr -mathrm{P}C bigl( xi_{4n} + V^{1/3}(q)bigr).

Assume that (A1 - A6) hold, and applying this in Lemma  3.4, we can get sup_{u}biglvert mathrm{P} (T_{n}/s_{n} leq u )-Phi u bigrvert leq Czeta _{2n}^{delta/2}.

Assume that (A1 - A6) hold, and applying this in Lemma  3.4, we can get sup_{u}biglvert mathrm{P}bigl(S_{1n}^{prime} leq ubigr -mathrm{P}(T_{n}leq ubigr -mathrm{P} bigl{ zeT_{n}leq{delta/2}+ zeta_{4n}^{1/2} bigr}.

Applying this bound and Lemma 2 to the second term yields the following: <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0000830.e011.PNG" class= inline-graphic"/> This completes the proof of Claim 2. The Province of Quebec has a unique history in North America.