Sentence examples for essential set with from inspiring English sources

Hence, (F_{s}(E)) is the essential set with respect to (rho_{s}^{u}).

(1) By Definition 3, Definition 4 and Proposition 1, an essential set with respect to (rho_{s}^{u}) is clearly an essential set with respect to (rho_{s}), an essential set with respect to (rho _{k}^{u}) is clearly an essential set with respect to (rho_{m}) (or (rho_{1})).

For every (E inmathcal{E} ), (F_{s}(E)) has at least one minimum essential set with respect to (rho_{s}^{u}).

From Theorem 3.1, the set-valued mapping (F') is upper semi-continuous on S. Then the set (F'(f,q)) itself is an essential set with respect to (C(S)).

If a connected component Ξ α ( R ) of Ξ ( R ) is an essential set with respect to M, then Ξ α ( R ) is said be an essential component of Ξ ( R ) with respect to M. Lemma 3.1 ([23.1.

(1) If (e(E)={y_{0}}) is a singleton set and an essential set with respect to (rho_{s}), then the point (y_{0}) is said to be an essential point with respect to (rho_{s}).

(2) If (mathcal{S}) is a minimum element of the family of all the essential sets with a partial order defined by the inclusion relation, then (mathcal{S}) is said to be a minimum essential set.

For each (u= f,S in P), given a triangulation of S with a grid size (frac{1}{q}), there exists a minimal essential set in (T u,q)) with respect to P. Theorems 3.1, 3.2 and Theorem 4.1 obtain the stability for approximate fixed points with simplicial methods.

A nonempty closed subset (e varphi)) of (F_{k} varphi)) is said to be an essential set of (F_{k} varphi)) with respect to (rho_{k}^{u}) (or (rho_{m})) if, given any number (epsilon>0), there exists a corresponding number (delta>0 ) such that (F_{k} varphi^{prime} cap[e varphi )+B_{epsilon} (0)]neqemptyset) for all (varphi^{prime}inmathcal{M}) such that (rho _{k}^{u} varphi^{prime},varphi

For every (Einmathcal{E} ), if the minimum essential set of (F_{s}(E)) with respect to (rho_{s}^{u}) is connected, then it is a stable set.

A nonempty closed subset (e(E)) of (F_{s}(E)) is said to be an essential set of (F_{s}(E)) with respect to (rho_{s}^{u}) if, given any number (epsilon>0), there exists a corresponding number (delta>0 ) such that (F_{s}(E^{prime } cap[e(E +B_{epsilon} (0)]neqemptyset) for all (E^{prime}inmathcal{E} ) such that (rho_{s}^{u}(E^{prime },E