Sentence examples for sharp criterion from inspiring English sources

Moreover, a sharp criterion of weak Poincaré inequalities is presented for Poisson measures on configuration spaces.

Using this result, we derive a sharp criterion for no blow-up.

Moreover, we address a borderline problem in the regularity theory and give a sharp criterion for continuity of the gradient of a solution.

We provide a sharp criterion in term of generalized principal eigenvalues for existence/nonexistence of positive solution of (Pβ) in suitable classes of functions.

In order to show the results of the spreading-vanishing dichotomy, the eigenvalue problems associated with model (1.3) are given in Section 3, and a sharp criterion for spreading and vanishing is established in Section 4.

A sharp criterion for the transition from a dimerized state to a long-range ordered state can be inferred from Figure 2(b), where the energy of the lowest singlet and the lowest triplet excitation is plotted.

There were no sharp criteria for distinguishing the emotions.

(Sharp criteria).

Combining Theorems 4.5-4.7 4.5-4.7 derive the sharp criteria for spreading-vanishing for the invasion species.

The following theorem reveals the sharp criteria for spreading and vanishing, which provides some sufficient conditions for vanishing.

This section is devoted to the proofs of the spreading-vanishing dichotomy, and the sharp criteria for spreading and vanishing.