Sentence examples for the given hypothesis from inspiring English sources

Finally, in Section 5, we present an application that illustrates the feasibility of the given hypothesis in the abstract results.

Under the given hypothesis, every graded prime submodule which contains N2(resp. K2) contains N1 resp. K1).

Note also that the cardinal (or discrete measure) of is (i.e., infinity numerable), since otherwise, for some (contrarily to one of the given hypothesis) and.

By the given hypothesis, ( x 2 n, u ) ∈ E ( G ) for all n ∈ N. We claim that the weight assigned to the edge ( u, g u ) is zero.

We note in the next proposition that under the given hypothesis on A, it is possible to construct a strongly continuous family of bounded and linear operators for (4.2) that commutes with A and satisfies a certain 'resolvent equation'.

By the given hypothesis and the last inequality, one has ∥ 1 k g ( k x ) − 1 k h ( k x ) ∥ β ≤ C L φ ( x, 0 ), ∀ x ∈ X. (3.9).

The given hypotheses imply that both and exist.

Under the given hypotheses, it is easy to obtain the solutions of (2.5) and (2.6): (3.2.6.

Note that these works established the existence and uniqueness of the solution under the non-Lipschitz conditions, and their methods depend heavily on the given hypotheses.

Under the given hypotheses (H1) and (H2), it is easy to check that x = ( 0, 0, 0 ) T is an equilibrium point of system (2.1).

Using the one-sided binomial test with the given hypotheses, the study required 37 patients to decide whether the PFS-3yR is less than or equal to 0.5 or greater than or equal to 0.7.