Sentence examples for empty such that from inspiring English sources

Then the following conditions are equivalent: (1) have the compactly local intersection property; (2)for each, there exists an open subset of (which may be empty) such that and ; (3 there exists a set-valued mapping such that for each is open or empty in, and ; (4 for each, there exists such that and ; (5) is transfer compactly open valued on.

In particular, M ≪ N leads the null space of Φ to be non-empty such that there are many different possibilities to represent g with the elements in H.

(ii) is transfer compactly open valued; (iii there exists a nonempty set and for each, there exists a compact -subspace of   containing such that is empty or compact in, where denotes the complement of.

Theorem 3.7Let be a convex and differentiable function such that ∇Wis -Lipschitz continuous and let be a convex and lower semi-continuous function such that is not empty.

And then much of the band emptied out, such that you could hear the bassists at length, alternating their roles, one holding down the root note while the other soloed melodically and percussively, then vice versa; then in a blurred crisscross.

map with non-empty values such that the restriction of Ψ to the set (Phi(X)) is approachable.

Note that if ({mathbb {A}}) = (mathfrak {R}) is a non-empty class such that (E,,Finn mathfrak {R},Ecupp Fin mathfrak {R}) and (E-Fin mathfrak {R}) and (g) satisfies the (lambda )-rule, then (tau ) satisfies the finite (lambda )-rule.

Amenability of (X, d) is equivalent to the existence of a net ({F_i}_{iin I}) of finite non-empty subsets such that begin{aligned} lim _{i} frac{|partial _R F_i|}{|F_i|} = 0, quad mathrm {for~all}quad R > 0. end{aligned}.

More precisely, a locally finite metric space (X, d) is said to be amenable if there exists a net ({F_i}_{iin I}) of finite non-empty subsets such that begin{aligned} lim _{i} frac{| N_R F_i |}{|F_i|} = 1, quad text {for any } R > 0, end{aligned}where (N_R F_i:= {xin X:d x,F_i le R}), the R-neighborhood of (F_i) (cf., Definition 2.1 and Remark 2.2).

map with non-empty closed values such that the restriction (Phi|_{operatorname{conv}{N}}) is approachable for each finite subset N of X; (Psi:Xrightrightarrows X) be a compact l.s.c.

map with non-empty closed values such that the restriction (Phi|_{operatorname{conv}{N}}) is approachable for each finite subset N of X;   (ii) (Psi:Xrightrightarrows X) be a compact l.s.c.