Sentence examples for established a sharp from inspiring English sources

He has established a sharp relationship between the squared norm of the second fundamental form and the warping function.

Chen established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, and the squared mean curvature.

This theory was initiated in [2]: Chen established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature (both being intrinsic invariants) and squared mean curvature (the main extrinsic invariant).

In [3], Schuster and Weberndorfer established a sharp bound for the volume of the Wulff shape W ν, f determined by an f-centered isotropic measure ν as follows.

In [4], Chen established a sharp relationship involving the Ricci curvature and the squared mean curvature for an arbitrary n-dimensional Riemannian submanifold of a real space form of constant sectional curvature c, Ric ( X ) ≤ ( n − 1 ) c + n 2 4 ∥ H ∥ 2, (23).

Therefore, in [1], Chen obtained a necessary condition for the existence of minimal isometric immersion from a given Riemannian manifold M into Euclidean space and established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature (both being intrinsic invariants) and squared mean curvature (the main extrinsic invariant).

I rarely write profiles, and in the few instances when I have, I've tended to establish a sharp critical distance between myself and my subject.

Though he is clearly trying to establish a sharp contrast with his rival, former President Carlos Menem, the few outside analysts who have examined Mr. Kirchner's record here see a more complicated picture.

Though every woman in the film is either shrewish or sluttish, the sexual politics of the film are driven not so much by wilful misogyny as they are by a need to establish a sharp opposition between opposing political coteries bent on swaying the queen, and failing that, the people.

In this article we establish a sharp two-sided inequality for bounding the Wallis ratio.

This study has shown, however, that the critical size does not establish a sharp cutoff point.