Sentence examples for by using the given from inspiring English sources

Hence, we can derive these results in matrix representation and prove these results simply by using the given matrix forms.

Moreover, for butt joint with single-edge crack under fatigue load, its remaining life can be predicted by using the given expressions.

Moreover, it is seen from Theorems 5 and 6 that a solution in (mathbf{L}^{2}[a,b]) is only determined by using the given conditions.

Yang provided an algorithm to determine the coefficients of the Sturm Liouville problem by using the given nodal points in [3].

First, we reduced the problems to some systems of functional equations by using the given data and then we obtained the solutions by the tools in [1].

We reduce the problem to a system of functional equations by using the given data and then we obtain the solution by the method of Kuczma (Functional Equations in a Single Variable, 1968).

As discussed before, in this trial, learners used strategy to pose the problem by directly using the given numerical expression, "8 3".

The AE, as subjectively assessed by the patients using the given categories, "clearly improved (−2)", "slightly improved (−1)", "unchanged (0)", "slightly deteriorated (+1)", and "clearly deteriorated (+2)".

Also we will give some matrix transformations from l ( p, λ 2, p ) into l ( q ) by using the matrix given in [4].

The value of Q is obtained as.4,468 by using the formula given by Equation (32).

One can easily obtain conclusion by using the technique given in the proof of Theorem 2.8 [8].