Sentence examples for completeness of the paper from inspiring English sources

For the completeness of the paper, we give proofs below.

For the sake of completeness of the paper, we recall several pertinent notions and fundamental results.

The rest of the proof follows from the proof of proposition 1 in [18] and we provide the details for the completeness of the paper.

For convenience of the readers and completeness of the paper, we recall the following basic concepts and results for the calculus on time scales.

Two case studies are discussed for the completeness of the paper, showing how the method and the quality indexes can be used in practice.

For the completeness of this paper we describe the approach with some improvements in the following.

holds for every y ∈ ( 2 B ) c and f ∈ M p, λ ( w ), where ( 2 B ) c = R n ∖ 2 B. Proof The proof of Lemma 2.4 has a root in [30], which we adopted here for the completeness of this paper.

For reasons of completeness, the paper provides an overall description of those income distribution functions for which a majority winning tax exists (or does not exist), when the quadratic taxation model is not purely redistributive.

For the self-completeness of this paper, we also present our MapReduce algorithms to compute the frequencies of word-pair.

Furthermore, it was not possible to verify consecutive enrollment or the completeness of the information on the paper CRF by source data monitoring.

For the completeness of the structure of the paper, we write down the proofs.