Sentence examples for improved the bound from inspiring English sources

Kolpakov and Kucherov (2003) improved the bound for the Hamming distance to O nklog k + s) where s is the number of TRs found.

This bound in (1.3) improved the bound in (1.1) in some cases.

The block RIC can be improved, for example, Lin and Li [1] improved the bound to (delta_{2s|mathcal{I}}<0.4931), and established another sufficient condition (delta_{s|mathcal {I}}<0.307) for exact recovery.

Subsequently, Chen [5] improved the bound in (1.1) and obtained the following result: q ( A ∘ B − 1 ) ≥ q ( A ) q ( B ) min 1 ≤ i ≤ n { ( a i i q ( A ) + b i i q ( B ) − 1 ) β i i b i i }. (1.2).

which only depends on the entries of A = ( a i j ), where R i = ∑ k ≠ i | a i k | ; d i = R i | a i i |, i ∈ N ; t j i = | a j i | + ∑ k ≠ j, i | a j k | d k | a j j |, j ≠ i, j ∈ N ; t i = max j ≠ i { t i j }, i ∈ N. Li [[9], Theorem 3.2] improved the bound (1.6) and obtained the following result: τ ( A ∘ A − 1 ) ≥ min i { a i i − m i R i 1 + ∑ j ≠ i m j i }, (1.7).

Then, it is natural to improve the bound periodically.

The result improves the bound on the block RIC (delta_{2s|mathcal{I}}) in [1].

In this section we improve the bound (1.2) based on the idea of preconditioning.

That helps to improve the bound of throughput from node 1 to node 8 to 5.14 packets/TS.

The result improves the bound on the block restricted isometry constant (delta_{2s|mathcal {I}}) of Lin and Li (Acta Math. Sin. Engl. Ser. 29 7):1401-1412013013).

Next it is proved that the bound (9) given in Theorem 3 can improve the bound (5) in Theorem 1 (Theorem 2.2 in [2]) in some cases.