Sentence examples for a different proof for from inspiring English sources

In this paper we give a different proof for the case in which G is abelian but not necessarily discrete.

As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11].

We now give a different proof for it.

Proof Although there is a proof in [54], we give a different proof for completeness.

Soon Bassily et al. 2015 [13] gave a different proof for the concentration inequality, which enlightens our error analysis.

Then, we give a different proof for the existence and uniqueness of the solutions for linear coupled boundary value conditions for a differential system, i.e., Lemma 2.5.

For a different proof of (2.5), see [18] for details.

A different proof of Theorem 1.1 for the case was given in [7] and Bennett used essentially the same approach in [8, 9] to study (1.2) for the cases being lower triangular matrices, namely, if.

This theorem has been proved in [40] in a slightly different form and with a different proof; Theorem 12 is essential for the proof of Theorem 9.

We also utilize a sharp and convenient condition using iterated Malliavin derivatives, to arrive at similar conclusions for suprema, via a different proof, which does not require full knowledge of the covariance structure.

The case for all with a different proof is given.