Sentence examples for using this fact and from inspiring English sources

Using this fact and molecular mechanics simulation data we derive an expression for the nonlocal parameter.

Using this fact and the assumption (2.4), applying Theorem 4 we obtain (2.5).

The integral in (27) can be upper bounded using this fact and Lemma 1: (29).

Using this fact and (2.3) in (2.1) we easily arrive at our required result.

Using this fact and a similar reason to that in the proof of (i), we see that B is a unitary non-associative subalgebra of the generalized octonions algebra (mathcal{O}_{mathbb{Q}} (alpha,beta,gamma )).

Using this fact and (3.17), we have lim_{i toinfty}{biglVert x^{{k}_{i}}-y^{{k}_{i}} bigrVert } = 0. (3.18) Recall that (x^{k} rightharpoonup p), together with (3.18), implies that (y^{k_{i}} rightharpoonup p) as (i toinfty).

(Quick: How wide is a football field? Fifty-three yards! That's right — the field is well more than half as wide as it is long. Use this fact and win a bar bet).

Then, we use this fact and arguments derived from [5] to conclude that the nonnegative critical point for I λ, obtained from the mountain pass theorem, is not the trivial one.

Using this fact, we developed and implemented a recursive filtering procedure called continuity process (CP) to eliminate these spurious windows (see Figure 10).

Also, since (M_theta ) is totally umbilical in M [5, 9], using this fact with (28) and the second condition of (27), we observe that (M_theta ) is totally umbilical in ({tilde{M}}).

Using this fact, the linearity and continuity of the operator, we get biglVert D^{n}_{varphi, u} f_{w}bigrVert _{H^{infty}_{mu}} leq Cbigl 1-|w|^{2}bigr)sum _{j=n}^{infty}|w|^{j}j^{alpha-1} biglVert D^{n}_{varphi, u} (p_{j})bigrVert _{H^{infty}_{mu}}preCbigl 1-|w|infty,quad winmathbb{D}.