Sentence examples for proven by using the from inspiring English sources

This Lemma can be easily proven by using the properties of the positive (semi) definite matrices.

Theorem 2.3 can be proven by using the ideas in the proof of [30], Theorem 2.

Theorem 2.2 can be proven by using the ideas presented in the proof of [28], Theorem 3.3.

The global stability and error convergence of the closed-loop system are proven by using the Lyapunov stability theorem.

The feasibility of this approach has been experimentally proven by using the commercially available biomedical polymer poly d,l-lactic acid) (PDLLA) as a model system.

However, the third property of the original objective function still holds, i.e., the objective function in (15) provides shortest paths in terms of hop distances satisfying both the transmission rate and QoS requirement of each sublayer, which can be proven by using the similar approach to the third property of Theorem 1.

This result is proved by using the implicit function theorem.

The asymptotic stability of the closed-loop MRR system is proved by using the Lyapunov theory.

The convergence of the identification error is proved by using the Lyapunov theory.

This can be proved by using the techniques similar to those used in Theorem 2.2.

Furthermore, the uniform validity of the solutions is proved by using the maximum principle.