Sentence examples for be given explicitly by from inspiring English sources

Let be equipped with the -norm, then the volume of the -dimensional parallelepiped spanned by the vectors, which may be given explicitly by the formula (1.1).

With these formulations, highpass filters {Gk} and {˜Gk} of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters {Hk} and {˜Hk} of their corresponding biorthogonal scaling functions.

A trivial example of an 2-normed space is X=R2, equipped with the Euclidean 2-norm ||x1,x2|| E = the volume of the parallellogram spanned by the vectors x1,x2 which may be given explicitly by the formula | | x 1, x 2 | | E = | det ( x ij ) | = abs det ( < x i, x j > ) Open image in new window.

Example 2.1 Take X = R 2 being equipped with the 2-norm ∥ x, y ∥ = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula ∥ x, y ∥ = | x 1 y 2 − x 2 y 1 |, where  x = ( x 1, x 2 ), y = ( y 1, y 2 ).

Take X = ℝ2 being equipped with the 2-norm ∥x, y∥ = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula x, y = x 1 y 2 - x 2 y 1, where x = ( x 1, x 2 ), y = ( y 1, y 2 ).

For example, we may take X = R n being equipped with the Euclidean n-norm ∥ x 1, x 2, …, x n ∥ E, the volume of the n-dimensional parallelepiped spanned by the vectors x 1, x 2, …, x n which may be given explicitly by the formula ∥ x 1, x 2, …, x n ∥ E = | det ( x i j ) |, where x i = ( x i 1, x i 2, …, x i n ) ∈ R n for each i = 1, 2, …, n.

It is given explicitly by (2.8).

The minimizer is given explicitly by u^{k+1}=mathcal{P}_{Omega}left[f^{k} + frac{{lambda_{3}^{k}}}{beta_{2}}right].

Then, the elements,, and for, are given explicitly by using Corollary 3.5, as follows: (4.18).

They are given explicitly by cubic expressions in the usual triangle logical (areal) coordinates.

In each case trajectories are given explicitly by (23) with (G xi,xi')=G xi-xi')) and (G=G xi-xi'where (G_{mu}(xi) = exp(andmu}xi)).