The phrase "and thus the sequence" is correct and usable in written English.
It can be used to indicate a conclusion or result that follows logically from the previous statements, often in a mathematical or logical context.
Example: "The initial conditions were established, and thus the sequence converges to a limit."
Alternatives: "and therefore the sequence" or "and so the sequence".
The second argument, concerning similarity in the sequence of nucleotides in the DNA (and thus the sequence of amino acids in the proteins), says that books with very similar texts cannot be of independent origin.
Then from (2.11), ({ Vert x_{k}-bar{x} Vert }) converges, and thus the sequence ({x_{k}}) globally converges to x̄.
So, lim n → ∞ p ( x n, x ∗ ) − p ( x ∗, x ∗ ) = 0, and thus the sequence { x n } n ∈ N is convergent to x ∗ with respect to τ ( p ).
Hence we have biglVert u_{k+1}-u^{infty} bigrVert leq biglVert u_{k}-u^{infty} bigrVert, and thus the sequence ({u_{k}}) converges to (u^{infty}).
Section 2 shows that operator satisfies the hypothesis of the Banach fixed point theorem and thus the sequence converges to the solution of (1.1) for any However, such a sequence cannot be determined in an explicit way.
Since (X, p) is complete, by Lemma 2, (X, d p ) is complete, and thus the sequence {x n } is convergent in X, say z ∈ X. Again by Lemma 2, (2.15).
The proposed immune procedure is built on the following fundamental idea: the bottleneck jobs existing in each scheduling instance generally constitute the key factors in the attempt to improve the quality of final schedules, and thus, the sequencing of these jobs needs more intensive optimization.
It is to be noted that only one sequence at each intron is available for SHAMO, WL, RedDB, QUAIL, and TURKEY, and thus the sequences from various introns of these individuals were uniquely concatenated.
If the number and distribution of such stems are significantly different from those that can occur in the randomly shuffled version of the alignment, then the alignment and thus the sequences are declared as ncRNAs.
It should be noted that intron sequences have not been previously reported for Apodemus paralogues and thus the sequences we report here should not be confused with the cDNA sequences we used in an earlier study [ 8].
Results of clustering sequences showed that most of the MLV variation was in the LTRs, and thus the sequences from target regions 1 and 6 (which each included part of an LTR) formed the greatest number of clusters (Fig. 1).
