Sentence examples for extending the proof from inspiring English sources

Proofs of parts (i) and (ii) come from extending the proof of Kim et al. (2006), which shows (ii) is true in the special case where ; that of part (iii) comes from applying Theorem 2 of Robins and Wang (2000).

To help reasoning about the correctness of quantum programs, we extend the proof rules presented by Morgan for classical probabilistic loops to quantum loops.

The idea of the proof follows to a certain extend the proof of Theorem  5.1 in [2].

In particular, Kuroda [398, 399] was able to extend the proof of Theorem 15.1 to (alpha > 1).

This result was originally obtained by Broyden in [9] in a different way for Algorithm 3, but we extended the proof of Galantai [12] for Algorithm 1.

That being done, it becomes straightforward to transfer results from the scalar case [8, 9] and thus extend the proof, to the MIMO case, that a relay-assisted communication achieves secrecy when the conventional scheme fails.

Suzuki extended the proof to all primes m ≤ 113.

Proof Directly extend the proof of Theorem 2 and Corollary 1. □ Directly extend the proof of Theorem 2 and Corollary 1. □ The solution to the identification problem in Figure 2 when q ≥ 2, G2 → Y2(t ), t=0,…, q, and dim(G2) ≥ q involves an unusual use of IV.

Phase IV: Certification and recertification of other breast centres in Germany was to extend the proof of concept to the multicentre level.

Phase IV finally extended the proof of concept to the multicentre level by achieving certification and recertification of numerous other breast centres in Germany, the total number of DKG/DGS-certified breast centres reaching 175 by mid 2008.

The proof follows directly by extending the TWRC scheme in [13], using Tuncel's "virtual binning" [31] at the decoder.