Sentence examples for under the same hypotheses from inspiring English sources

Let (u overline{x},t)) be the solution of (3.1) under the same hypotheses of Theorem 3.5.

Corollary 3.11 Under the same hypotheses and notations as in Theorem  3.10, let ℱ be a P-filter of ℕ.

For example, under the same hypotheses of Theorem 4.2, we have the weighted form of (3.18) as follows: (4.14).

Under the same hypotheses of Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies (4.34).

So, Theorem 2.4, Propositions 2.1 and 2.2 are still true under the same hypotheses for them if φ ∈ S ( R d ).

Remarks 2.2 The bounds of Theorem 2.1 can be improved under the same hypotheses and computational cost in two cases as follows.

Under the same hypothesis one has that all of the "provably-Δ21" sets of reals are universally Baire.

Under the same hypothesis as Theorem 3 in [2], there exists a smooth bifurcation branch from ((d_{j}, (0,0))).

In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is Z-stable.

This correction enables the data from all five pairs to be considered under the same hypothesis.

Next, we will show that, under the same hypothesis, the orbit O = G ⋅ 0 must satisfy the order 3 condition.