Sentence examples for with respect to the ordering from inspiring English sources

In an abstract setting, these mean that is a strictly positive operator with respect to the ordering defined by a.e.

The set of all weak minimal points of K with respect to the ordering cone C is defined as operatorname{wMin}_{C}(K)= bigl{ xin K: (x-K cap operatorname{int} C=emptyset bigr}.

An element u ∈ K is called a best ordered approximation of f on K, if it satisfies d X ( u, f ( u ) ) = min { d X ( s, f ( u ) ) : s ∈ K }, where min { d X ( s, f ( u ) ) : s ∈ K } is the smallest (minimum) element of the set { d X ( s, f ( u ) ) : s ∈ K } with respect to the ordering ≽ X on X.

An element u ∈ K is called an extended best ordered approximation of f on K, if it satisfies d X ( u, f ( u ) ) ∈ Min { d X ( s, f ( u ) ) : s ∈ K }, where Min { d X ( s, f ( u ) ) : s ∈ K } is the set of minimal elements of the set { d X ( s, f ( u ) ) : s ∈ K } with respect to the ordering ≽ X on X.

is called upper (lower) bounded with respect to the ordering if its upper (lower) bounds exist.

First, let ((g(x),g y))) be comparable to ((g(x^{ast}),g y^{ast}))) with respect to the ordering in (A times A).

Definition 3.10 Let ( S, d X ) and ( T, d Y ) be ordered metric spaces, with respect to the ordered vector spaces ( X, ≽ X ) and ( Y, ≽ Y ), respectively.

We now consider the matrix, N, that represents the homomorphism on 1-chains with respect to the ordered basis.

A mapping f : S → T is said to be sequentially or σ-continuous, whenever, for any sequence { s n } ⊂ S, s n o → s, with respect to the ordered metric d X on S, implies f ( s n ) o → f ( s ), with respect to the ordered metric d Y on T.

Then ( S, d X ) is called an ordered metric space, and d X ( u, v ) is called the ordered distance between u and v, with respect to the ordered vector space ( X, ≽ X ).

Thus, T is a nondecreasing mapping with respect to the order ≺ on X̃.