Sentence examples for sufficient in order that from inspiring English sources

We establish conditions on the parameters which are both necessary and sufficient in order that Besov and Triebel Lizorkin spaces of generalized smoothness contain only regular distributions.

for j = 0, 1, is necessary and sufficient in order that the Neumann problem might have a solution in R E 1, p ( D a, φ ) Open image in new window.

The classical notion of uniform integrability of a sequence of integrable random variables { X n, n ≥ 1 } is defined through the condition lim a → ∞ sup n ≥ 1 E | X n | I ( | X n | > a ) = 0. Landers and Rogge [7] proved that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak laws of large numbers.

We give sufficient conditions in order that this set contains at least two different elements.

In this paper we give necessary and sufficient conditions in order that the crossed product G ×xA be simple.

For other values of p, we point out some necessary and some sufficient conditions in order that ddx∈Sp.

Necessary and sufficient conditions in order that two Krein space isometries whose defect subspaces are Hilbert spaces admit commuting unitary Hilbert space extensions are discussed.

As an application we give necessary and sufficient conditions in order that (A, P) is isomorphic to (Oλ(G), Σ) with λ the regular representation of a compact and metrizable group G and Σ the set of endomorphisms induced by an irreducible decomposition of λ.

The next theorem states a necessary and sufficient condition in order that χ is such a function.

This paper gives necessary and sufficient conditions in order that a series ∑ a n λ n should be summable | B | k, k ≥ 1, whenever ∑ a n is summable | A |. Some new results have also been obtained.

Motivated by this observation, in this paper we consider the cases p, q ≥ 0 and p, q changes sign for all large t, to give sufficient conditions in order that every solution of equation (1.1) is either oscillatory or weakly oscillatory and to study the asymptotic nature of non-oscillatory solutions of equation (1.1) with respect to their asymptotic behavior.