Sentence examples for and hence the sequence from inspiring English sources

Note that y n - x * ≤ x n - x *, and hence the sequence {y n } is bounded.

Then by induction, for all and hence the sequence generated by (3.1) is well defined.

Let j → ∞, and hence the sequence { ξ k j } converges strongly to x ∗.

Since and are bounded, we have the sequence and hence the sequence is bounded.

So, we have for all and hence the sequence is well defined.

Therefore, limn→∞||x n - p|| exists, and hence, the sequence {x n } is bounded.

If p ∈ S C, A), then limn→∞||x n - p|| exists, and hence, the sequences {x n } and {Ax n } are both bounded.

Since f and the projection operator P C are continuous and bounded, we obtain that the sequence { y k } and hence the sequences { f ( x k ) } and { f ( z k ) } are bounded, and for some M > 0, ∥ α η k ( x k − y k ) + β f ( x k ) + α μ f ( z k ) ∥ ≤ M, ∀ k. (4.8).

The aforementioned different approaches for focusing the library preparation and hence the sequencing process to the virus nucleic acids are less restricted than our approach [5], [6], [7], [8] but, nevertheless, strong constraints exist: The protocol of Potgieter and co-workers [6] only works with dsRNA, excluding its use for influenza sequencing.

For all mismatched oligonucleotide combinations, the mismatch bubbles are larger by one base pair when the oligonucleotides anneal to the mutant sequence, as compared to wild-type sequence, and hence the mutant sequence is rendered more susceptible to cleavage by S1 nuclease than the wild-type sequence.

With this transformation the DBN-TE achieves full convergence and is able to perform near-optimal inference in a small number of iterations, in order to estimate the coded sequence c, and hence the uncoded sequence s.