Sentence examples for been proved using these from inspiring English sources

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The notion of invariant set was introduced in order to avoid the condition of mixed monotone property, and many statements have been proved using these hypotheses.

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The stability and robustness of the proposed controller have been proved using the classic Lyapunov criterion.

The finite time stability has been proved using the Lyapunov theory.

The stability of the EASFSC has been proven using the Lyapunov stability theorem.

The positive effect of folic acid on dermal fibroblast has already been proven using the Host Cell Reactivation Assay.

It should be noted that these inequalities were proved using the variation of some functions and their derivatives, meanwhile, some of the above inequalities were improved using Taylor's expansions of inverse trigonometric functions.

Later some of these identities have been proved using only elementary techniques.

The strong bounding theorem of micromixing has been proved using Bellman's principle of optimality.

Remark 4.2 Equality (28) has been proved, using different method, by Luo [[11], Theorem 2.1].

The involvement of caspases 1 and 3 in apoptotic cell death has been proved using PD animal models [ 58].

The next lemma is proved using the phase spaces axioms.

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