Sentence examples for by direct computations using from inspiring English sources

Above equalities can be obtained by direct computations using definition of operator and (2.3).

By direct computation, using Matlab software package, we get (R_{0}=1.3281).

Furthermore by direct computations we have (2.4).

By direct computations, we get (2.35).

By direct computations, one has that solves (3.5).

By direct computations, we get the following results.

The results estimated by the software show good correspondences with the ones achieved by direct computations.

Now, we verify the effectiveness of the conclusion by direct computations.

Finally, it is easy to show by direct computations that ((x_{6}, y_{6}) = (2b + d, b)).

Proof From Lemma 2.9, we obtain T ( P ) ⊆ P. By direct computations, we have (3.3) (3.3).

Proof This can be verified by direct computations, so we omit it.