The phrase "and established a sharp" is grammatically correct and can be used in written English.
It can be used in contexts where you are describing the creation or implementation of a clear distinction or boundary in a discussion or analysis.
Example: "The researchers conducted a thorough study and established a sharp contrast between the two methodologies."
Alternatives: "and created a clear" or "and defined a distinct".
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).
In [4], Chen extended the notion of Ricci curvature to k-Ricci curvature for a Riemannian manifold and established a sharp relationship between k-Ricci curvatures and the shape operator and also a sharp relationship between k-Ricci curvatures and the squared mean curvature for an n-dimensional Riemannian submanifold in a real space form M ¯ with constant sectional curvature c.
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.
But the operative word here is risk, and the man has established a sharp eye for the untested, which extends beyond performance and tech (he gave the digital-art visionary Ryan Trecartin his first New York museum show) to conventional mediums.
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).
Also, Chen established a sharp inequality between the main intrinsic curvatures (the sectional curvature and the scalar curvature) and the main extrinsic curvatures (the squared mean curvature) for a submanifold in a real space form R m ( c ¯ ) in [5] as follows: For each unit tangent vector X ∈ T p M n, H 2 ( p ) ≥ 4 n 2 { Ric ( X ) − ( n − 1 ) c ¯ }, (1.2).
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).
Directional transport of cargos is powered by the small GTPase Ran and a system of compartment-specific GTP hydrolysis and GDP-to-GTP exchange, which establishes a sharp concentration gradient of RanGTP across the nuclear envelope (Izaurralde et al., 1997; Kalab et al., 2002).
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.
