Sentence examples for an h g from inspiring English sources

From Example 2.21, M is an ( H g, F ) -closed set which satisfies the transitive property.

It is easy to see that M = { 0, 1 } 4 ⊂ X 4 is an ( H g, F ) -closed set but not an F-closed set.

Let M be a subset of X 4. We say that M is an ( H g, F ) -closed subset of X 4 if for all x, y, u, v ∈ X, ( H ( g x, g y ), H ( g y, g x ), H ( g u, g v ), H ( g v, g u ) ) ∈ M ⇒ ( F ( x, y ), F ( y, x ), F ( u, v ), F ( v, u ) ) ∈ M. Definition 2.18 Let ( X, d ) be a metric space and H X × X → → X be a given mapping.

Let ( H ( g x, g y ), H ( g y, g x ), H ( g u, g v ), H ( g v, g u ) ) ∈ M. Since F is H g -increasing with respect to ⪯, we have F ( x, y ) ⪯ F ( u, v ) and F ( y, x ) ⪰ F ( v, u ), and this implies that ( F ( x, y ), F ( y, x ), F ( u, v ), F ( v, u ) ) ∈ M. Then M is an ( H g, F ) -closed subset of X 4 which satisfies the transitive property.

The means A, H, G, and Q are stable.

The following assertions hold: (i) The means A, H, G, and Q are stable.

This break-up into two separate parts, A h - 1 - P H h A H - 1 P h H and A h G h ν, greatly helps the convergence analysis, see [26].

The symmetric Gauss Seidel method (45) satisfies the smoothing property (54) ‖ A h G h ν ‖ ⪯ η ‖ A h ‖, where the function η → 0 as ν → ∞.

Then for ν ⩾ 1 the following smoothing property holds (57) ‖ A h G h ν ‖ ⪯ 2 / ‖ A h ‖.

□ In Table 4, we list the spectral norm of A h G h ν for ν = 1, …, 4, symmetric Gauss Seidel iterations, which confirms the estimate (55).

The factor A h - 1 - P H h A H - 1 P h H is related to the approximation property and the factor A h G h ν is related to the smoothing property.