Sentence examples for manifold provided from inspiring English sources

We conclude that (64) has a stable invariant manifold provided that (H1 - H6) and (16 - 19 16 - 19

Upper bounds are obtained for the heat content of an open set D in a complete Riemannian manifold, provided the Dirichlet Laplace Beltrami operator satisfies a strong Hardy inequality, and the distance function on D satisfies an integrability condition.

We note that for each manifold provided by Theorem 5.2, determining the (N^2 -dimensional coN^2 -dimensional connected of identitymorphism group explicomponentnof hard (see, e.g., [41], p. 31).

for any vector field X on M ¯, where the indices are taken from { 1, 2, 3 } modulo 3. We remark that any quaternionic Kähler manifold is an Einstein manifold, provided that dim M > 4 (see [32 34]).

We note that for pseudo-Riemannian structures on Lie N-algebroids, the d-torsions can be induced by the N-connection coefficients and reflect the nonholonomic character of the corresponding manifold provided with a nonintegrable distribution.

Upper bounds are obtained for the heat content of an open set D with singular initial condition f on a complete Riemannian manifold, provided (i) the Dirichlet Laplace Beltrami operator satisfies a strong Hardy inequality, and (ii) f satisfies an integrability condition.

The cubic shape of the critical manifold provides a mechanism for spike generation and relaxation oscillations.

The Grassmann manifold provides a robust framework for measuring data similarity in a subspace context.

Based on this model we design a novel manifold providing uniform fluid distribution.

This conceptual manifold provides a unified and intuitive representation of the view space and supports efficient dynamic view estimation.

Condition combines flux functions and curved boundary manifold, providing necessary condition for the convex property of the new flux function which will be constructed in formula (4.5).