Sentence examples for by direct calculations using from inspiring English sources

The following lemma follows by direct calculations using Dirichlet formula, [36].

Properties (14) and (15) can be verified by direct calculation using Eq. (16) and Eq. (17).

For the remaining 2 ≤ n ≤ 6, one can verify (10) by a direct calculation using the values reported in [22].

This result corresponds to [1], Theorem 1] by a direct calculation using identity (7) and reduces to [5], Lemma 2] in the special case T = R. (v) As mentioned in the introduction, Theorem 3 is new even in the discrete time case, i.e., for T = Z.

A direct calculation using (2) is sometimes difficult to be done, especially for complex models.

Moreover, for quantitative sensitivity analyses, the MCA [ 36, 37] were carried out by direct calculations of the concentration control coefficient from the calculations using ODE simulations.

A two-dimensional approach is adopted for the direct aeroacoustic calculations using a direct numerical simulation (DNS) technique.

By direct calculations we obtain (e).

By direct calculations, we can obtain that.

that can be verified by direct calculations (see, e.g., [27]).

By direct calculations from the first quadratic equation (2.30).