Sentence examples for applying propositions from inspiring English sources

By applying Propositions 3.1 and 3.4, we deduce that if p > 1 − m g M ˜, then problem (1.1) admits solutions, whereas if p ≤ 1 − M g m ˜, then (1.1) does not admit solutions.

Applying Propositions 4.1 and 4.2 with the condition that v ∈ D C ( G 1 × G 2 ) we find that the operator S α 1, α 2 is bounded from L dec, r p ( w, G 1 × G 2 ) to L q ( v, G 1 × G 2 ) if and only if condition (v) is satisfied.

There is growing sentiment that applying Proposition 13 to business property, and maintaining the crazy-quilt of money flows among government entities that Proposition 13 has created, should be reconsidered.

Applying Proposition in [1] and (3.7) results in (4.56).

Hence, applying Proposition 1.18 in Resnick [13], we obtain (F_{k}in D(Lambda)).

By applying Proposition 2.3 we start from formula (4.12) and we put (4.29).

Letting n → ∞ and applying Proposition 3, we see that d ( x, y ) ≤ d ( x, z ) + d ( z, y ).

Applying Proposition 26, we can present upper bounds for the system algebras with restricted interaction length.

Applying Proposition 1 to the MARC leads to three possibilities (Figure 3).

From (62)–(63), applying Proposition 4.1 and Gronwall's inequality, the assertion (64) holds.

Applying Proposition 4.2 yields that is also a local -minimizer of.