Sentence examples for using the above inequality from inspiring English sources

and consequently, using the above inequality, we obtain that (3.35).

Using the above inequality and the definition of we have (3.63).

Using the above inequality, Minkowski's inequality and Fubini's theorem, we get (3.20).

The required inequality can be proved by using the above inequality and following the steps of Theorem 2.1.

Using the above inequality, using (24) and the property of ψ, we get ϕ(d(gx, F x, y))) = 0, thus d(gx, F x, y)) = 0. Hence gx = F x, y).

By using the above inequality, vert J n,l) vert leq frac{n^{1/ beta }}{ 2 pi} int_{r leq vert theta vert leq pi } vert phi (theta)vert^{n} d theta leq n^{1/ beta } rho^{n}.

Using the above inequalities, and applying Gronwall's inequality to (3.9), one can easily get that ∥ w ( t ) ∥ B p, r s ≤ ∥ w 0 ∥ B p, r s exp { c ∫ 0 t ( ∥ u ∥ B p, r s + 1 2 + ∥ u ∥ B p, r s + 1 ∥ w ∥ B p, r s + ∥ w ∥ B p, r s 2 ) d τ }. (3.10).

By using the above inequalities combined with (4.10), (4.9), and Lemma 3.5, and choosing a sufficiently small (varepsilon >0), we obtain that begin{aligned} Vert (delta _{j,h} p) eta Vert _{L_2(B_{R_1}^+)} le N d,delta ) (R-r)^{-2} Vert DuVert _{L_2(B_R^+)}.

After conducting a different error decomposition using the above concentration inequality, we provide an upper bound or learning rate of the expected generalization error.

Thus, using the above recursive inequalities repeatedly, we have (3.17).

Therefore, by using the above two inequalities, we have f z = z.