Sentence examples for proved by using a from inspiring English sources

Theorem 3 is proved by using a similar method to Theorem 4.

In case of (p=1), the theorem can be proved by using a similar method.

The following lemma is well known and can be easily proved by using a standard method.

The stability analysis in this article is proved by using a well-known Lyapunov stability theorem.

The stability of the overall control systems is proved by using a Lyapunov function.

The global stability of the endemic equilibrium is also proved by using a graph-theoretic approach to the method of Lyapunov functions.

The nuclear localization of the chimeric apoaequorin in tobacco BY-2 cells was proved by using an anti-apoaequorin serum from rabbit that mainly localized the transgenic apoaequorin within the nucleus (Fig. 5) as it was reported for N. plumbaginifolia cells before (19).

The finite-time stability nature of the TSMC law is proven by using a Lyapunov-based approach.

(b) Unlike in the previous works, the stability analysis is rigorously proven by using a SPR-based Lypunov approach.

Proof We see that this result can be proven by using a similar method to Theorem 3.1.

This result is proven by using an inductive argument.