Sentence examples for refinements below from inspiring English sources

Since the proofs are trivial by breaking the corresponding sums in the following form x 1 j - ∑ i = 2 n x i j = x 1 j - ∑ i = 2 l x i j - ∑ i = l + 1 n x i j, and reusing the corresponding theorems, we present these refinements below without proofs.

Nationalist and pro-nationalist views mostly rely on the attachment that members of a nation have to national territory and to the formative value of territory for a nation to justify territorial claims (see Miller 2000 and Meissels 2009, with some refinements discussed below).

The lowest-energy models and the structures at the center of the clusters enter atomic-detail refinement (read below).

For the remaining sequences recognized only by one HMM, we first looked at the existence of strong homology with positive hits identified in the previous steps in order to include putative 'truncated' proteins (see Inventory refinement, below).

During subsequent refinement (see below), we also estimated particle rotations by aligning 3-frame sums (see below).

This initial map suffices for further local refinement described below.

We observed that there is no significant gain in reducing the refinement interval below 5000 wall-clock ticks for both the LP graph generation models.

What isn't seen is the constant process of character refinement just below the surface that allows them to be both flexible and strong while bending with the wind, like a stalk of bamboo.

Although the proposed inference framework is validated with the real-world profile datasets, there are still several directions needing further refinement, depicted within the below section.

First, the number of independent parameters (Nind) was calculated according to the Nyquist theorem Nind = 2/π*ΔR* Δk + 2. In all cases the number of free running parameters in the refinements was well below Nind.

The image below is before refinement.