Sentence examples for implications hold for from inspiring English sources

The same implications hold for the corresponding right notions.

The analysis is limited to 2D depth-averaged systems, but implications hold for 3D cases.

It is also possible that these implications hold for other biofilm forming bacteria.

This implication holds for the following reasons.

We say that a Banach space is strictly convex if the following implication holds for : (2.1).

We say that E is strictly convex if the following implication holds for x, y ∈ E : ∥ x ∥ = ∥ y ∥ = 1, x ≠ y ⇒ ∥ x + y 2 ∥ < 1. (2.1).

strictly convex if the following implication holds for all x, y ∈ X : ∥ x ∥ = ∥ y ∥ = 1 and x ≠ y ⟹ ∥ x + y 2 ∥ < 1 ; uniformly convex if for each ϵ with 0 < ϵ ≤ 2, there exists δ > 0 such that the following implication holds for all x, y ∈ X : ∥ x ∥ ≤ 1, ∥ y ∥ ≤ 1 and ∥ x − y ∥ ≥ ϵ ⟹ ∥ x + y 2 ∥ < 1 − δ.

Then f ∈ C n. Proof (i) First, we will show that this implication holds for j 0 = 2. Letting F ( z ) = z f ′ ( z ), then f ∈ C n if and only if F ∈ S n ∗.

We say that a Banach space E is strictly convex if the following implication holds for x, y ∈ E : ∥ x ∥ = ∥ y ∥ = 1, x ≠ y ⇒ ∥ x + y 2 ∥ < 1. (2.1).

A Banach space X is said to be (i) strictly convex if the following implication holds for all x, y ∈ X : ∥ x ∥ = ∥ y ∥ = 1 and x ≠ y ⟹ ∥ x + y 2 ∥ < 1 ;   (ii) uniformly convex if for each ϵ with 0 < ϵ ≤ 2, there exists δ > 0 such that the following implication holds for all x, y ∈ X : ∥ x ∥ ≤ 1, ∥ y ∥ ≤ 1 and ∥ x − y ∥ ≥ ϵ ⟹ ∥ x + y 2 ∥ < 1 − δ.  .

For the problem of total Lagrange duality, one seeks conditions ensuring that the following implication holds for x 0 ∈ dom ( f − g ) ∩ A : [ f ( x 0 ) − g ( x 0 ) = min x ∈ A { f ( x ) − g ( x ) } ] ⟹ [ ∃ λ ∈ R + ( T ), ∀ w ∗ ∈ H ∗, L ( w ∗, λ ) = f ( x 0 ) − g ( x 0 ) ]. (1.9).