Sentence examples for by using these propositions from inspiring English sources

By using these propositions, we give the proof of our main result in Section 5.

We use the propositions 1 to 4 in Section 3 to reduce the searching range effectively, since all the QPBTOCAS and the APBTOCAS seem only one if they can get from one sequence by using the propositions 1 to 4. Therefore, by means of an exhaustive computer search for the possible class of PBTOCAS, we get some QPBTOCAS within length 25 in Table 1 and APBTOCAS within length 26 listed in Table 2.

Proposition 2.2 can be easily shown by using the Proposition 2.1 (see [6]).

By using the Proposition 2 we obtain the following result: (overline{x}_{2}) is a saddle point if (q-p<1), (overline{x}_{2}) is a repeller if (q-p>1).

Concerning (3.49), it can be obtained in a similar way by using Propositions 2.1(f) and 2.2, (2.14), Propositions 3.1 and 3.3(b).

However, we are more interested in how to establish new inequalities for two-parameter trigonometric means from these ones derived by using Propositions 3.1-5.2 3.1-5.2asning an inequality fobtainingane mans from the correspondinequalityr hyperbolic forctions (see [22–24, 30]).

Finally, by using Propositions 3-5, we prove Theorem 1.

(26) Again, by using Propositions 1.6 and 1.7, we have (p x,Tx)=0), which gives (xin Tx).

(25) Now by using Propositions 1.6 and 1.7, we have (p x,Sx =0), and thus (xin Sx).

And we remark that under the assumptions of Theorem 1.3, can be completely computed by using Propositions 2.4 and 2.5.

As shown in the previous sections-, by using Propositions 3.1-5.2 3.1-5.2establish a series of neweinequalities for trigonometricanunctions and restablishme known ones.