Sentence examples for stated in this theorem from inspiring English sources

Proof Let f ∈ g h x ( h, p, I ) and let x 1, x 2, x 3 ∈ I be the numbers stated in this theorem.

Then, by analyzing the conditions for equality for Theorem 2 for the pairs of unitary matrices U a and V, we conclude that the necessary and sufficient conditions stated in this theorem are true for the case of.

Applying Lemma 1 to analyze the condition for equality of the first inequality in Equation 10, we get the necessary and sufficient conditions for equality as stated in this theorem for the case of.

Summing up all facts, the basis property stated in the theorem finally follows according to (2.15).

Such sufficient conditions, which are the main theoretical contribution of the paper, are stated in the theorem below.

Similar running time analysis as that applied in the case of the Inside VMT algorithm can be shown, yielding the result stated in Theorem 2. Theorem 2 For every Outside VMT problem there is an algorithm whose running time over an instance s is o (|| s||).

Note that the condition (stated in Theorem 1 and Theorem 2) for the uniqueness of a fixed point is here not satisfied.

Similarly, conditions stated in Theorem 3.3(b) and Theorem 3.4 are sufficient conditions for the existence of one or two interior steady states.

As a corollary, we give the proof of the stochastic Fubini theorem stated in Theorem 1.1.

We state here the form stated in [16, Theorem 8.2(b)].

The above corollary has been stated in [7, Theorem  5.1.7], and proved by a different way.