Sentence examples for follows l from inspiring English sources

If w is a weight on X (i.e. w : X → ( 0, + ∞ ) is measurable function) then the weighted Orlicz spaces, denoted by L w Φ ( X ), are defined as follows: L w Φ ( X ) : = { f | f w ∈ L Φ ( X ) }.

The operator A in (1) can be rewritten as a sum of L and N, where L and N are linear and nonlinear parts of A, respectively, as follows: L ( u ) + N ( u ) − f ( r ) = 0.

Hence, the probability density function l(t) of leaf longevity t is derived as follows l t = d L t d t = μ ⋅ e − μ ⋅ t (7).

Then, the linear space l ( p ) was defined by Maddox [1] (see also Simons [2] and Nakano [3]) as follows: l ( p ) = { x = ( x k ) ∈ ω : ∑ k | x k | p k < ∞ } ( 0 < p k ≤ H < ∞ ).

Indeed, from Theorem 1 it follows L r, 8 ?

This result may be written briefly as follows: L [ R ( 0 ) ] ⊂ S ∗ = SS ∗ ( 1 ), (1.2).

This can be checked using one of the KKT optimality conditions of the problem as follows ∂ L ( x j, y j k, λ j k, … ) ∂ y j k = μ k − λ j k = 0 → λ j k = μ k (11).

Consider the existence of solutions of nonlinear Sturm-Liouville problems in E as follows: - L x t = f t, x, t ∈ 0, 1 ; a x 0 - b x ′ 0 = 0, c x 1 + d x ′ 1 = 0 (3.1).

Following the Lagrangean method, the condition for the existence of a local extrema point at α=α0 under the unit-variance constraint is as follows: ∇ L ( α 0 ) = 0, (19).

(b) Let L X X, D ) be a class defined as follows L X X, D ) = { L : L is L  - family on X }.

In what follows (L t)) is assumed to satisfy assumption (L).