Sentence examples for underlying spaces from inspiring English sources

Some assumptions on the underlying spaces and involved operators are listed below.

Now we give the assumptions on the underlying spaces, involved operators and additional parameters, throughout.

The underlying spaces may not be equipped with a topological structure or an algebraic structure.

Moreover, we will investigate some conditions for the considered mappings to substitute the chain-completeness of the underlying spaces.

By using the theorems, we also obtain several interesting corollaries and fixed point theorems in the underlying spaces.

As applications of our results we obtain several interesting corollaries and fixed point theorems in the underlying spaces.

We fix as an underlying space a (non-compact) Riemannian manifold X and an intensity measure σ on it.

In fact, we have that If, the notion of -boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space [5, Proposition 1.17].

The following result does not require the conditions on the completeness (or the closedness) of the underlying space together with the conditions on continuity and Hadžić's condition of t.

Identification of least-cost paths, rooted in ecological and behavioral mechanisms underlying space use, can identify realistic putative corridors for focused implementation of control measures for introduced species in ecologically sensitive areas.

Realistic physiological simulations present a grand challenge, because of the wide range of underlying space and time scales, as well as the widely disparate organization and properties of different cells.