Sentence examples for useful sufficient from inspiring English sources

Now we shall prove a useful sufficient condition for Cauchy sequence in cone metric spaces over a solid vector space.

Applying Theorem 9.2, we shall prove a useful sufficient condition for convergence of a sequence in a cone metric space over a solid vector space.

By introducing suitable Lyapunov functional and employing useful inequality techniques, some simple and useful sufficient conditions ensuring the GDS of considered NNs are established via designing a novel nonlinear feedback controller.

Let E and F be Banach spaces and T a bounded linear map from E into F. Using a certain perturbation function ƒ(·, T) we find a useful sufficient criterion that T maps the closed unit ball onto a closed set.

While this is often a useful and sufficient simplification, it does not allow us to learn as much as possible from historical data or laboratory experiments that investigate responses to global change.

This revealed to be useful and sufficient in all the practical cases we studied.

By the generalized Kalman Yakubovich Popov lemma and some useful lemmas, sufficient design conditions are obtained.

Although imaging of fixed cells is useful and sufficient for many applications, live cell microscopy is ideal for visualizing cellular processes, which is considerably more difficult.

This inference is supported by the evidence provided by other inferences in this chain and proved that score-based interpretations are relevant, useful, and sufficient for evaluating the adequacy of test takers' English language proficiency.

In order to provide useful and sufficient information to increase students' knowledge and attitudes, to develop a reliable and valid instrument is essential and helpful for faculty as well as the program director to pinpoint students' strengths and weaknesses.

Seven respondents reported that the training had been useful and sufficient.