Sentence examples for manifold exists from inspiring English sources

Meanwhile, Uddin in [21] and Srivastava in [22] proved that the warped product semi-slant submanifold of a Kenmotsu manifold exists in the forms (M=M_{T}times_{f}M_{theta}) and (M=M_{theta}times_{f}M_{T}), except in the case when the structure vector field ξ is tangent to (M_{T}) and (M_{theta}), respectively.

If dynamics of fast components under given values of slow concentrations is stable, then the slow attractive manifold exists, and its zero approximation is given by the system of equations Wf(cs, cf) = 0.

Two spin-orbit manifolds exist with Ω=1/2 and Ω=3/2.

At these energies many ligand field manifolds exist that all mix and form the peak at approximately 2.5 eV (see Figure  3 b).

where is a Riemannian volume of the manifold, and is the Laplace-Beltrami operator on a manifold; (2 there exists a constant such that for any, (3.9).

The definition says that at any arbitrary point of the manifold there exists a geodesic coordinate system such that the components of any vector at that point are not altered by an infinitesimal parallel displacement with respect to it.

(2.1) On an almost contact manifold there exists a Riemannian metric g which satisfies the following: g varphi U, varphi V)=g(U, V -eta(U)eta(V -etaU etaa(U)=g(U, xi ), (2.2) for any U, V tangent to M̃.

However, for asymptotic stabilization of systems defined on noncontractible manifolds, there exists no differentiable CLF; the semiconcave CLF design problem based on the backstepping has not been discussed.

It may ultimately turn out, that manifold mechanisms exist and only some mechanisms will be operative in a single cell system [ 68].

Diverse lineages that exhibit MTOM may be predisposed to evolutionary novelty, because manifold solutions exist among species for new ecological or functional demands [ 25].

there is an isomorphism of fundamental groups (pi _1 (W) = pi _1 (hat{X})= Gamma ), where ( hat{X}) is a smooth ample divisor in a (K(Gamma, 1)) complex projective manifold Z; there exists a holomorphic map ( f : W rightarrow Z'), where (Z') is another (K(Gamma, 1)) complex manifold, such that (f_* : pi _1 (W) rightarrow pi _1 (Z') = Gamma ) is an isomorphism, and moreover.