Sentence examples for the given hypotheses from inspiring English sources

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).

Similarly, by applying the given hypotheses one can verify that the relations G 1 ( u, v ) ( t ) : = d ( u, v ) - K ∫ t b v ( s ) d s, t ∈ [ a, b ], G 2 ( u, v ) ( t ) : = 1 p ( t ) c ( u, v r ∫ a t f ( u, v ) ), t ∈ [ a, b ] (6.8).

Similarly, by applying also the given hypotheses one can verify that the relations G 1 ( u, v ) ( t ) : = d ( u, v ) + K ∫ a t v ( s ) d s, t ∈ ( a, b ], G 2 ( u, v ) ( t ) : = 1 p ( t ) c ( u, v ) + r ∫ a t f ( u, v ), t ∈ J, (5.7).

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

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

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.

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.

To put a realist-sounding spin on it: 'one infers, from the premise that a given hypothesis would provide a "better" explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true' (Harman 1965, p. 89).