Sentence examples for seed elements from inspiring English sources

We introduce the concepts of mixed-monotone seed elements as follows.

Core networks were constructed for both direct and indirect interactions using default parameters, and the focus genes with the highest connectivity to other focus genes were selected as seed elements for network generation.

Now, we shall consider another concept of seed element.

By considering the mixed ≼ I -monotone seed element instead of mixed ⪯-monotone seed element, the assumptions for inequalities (9) and (10) can be weakened, which is shown below.

It is obvious that if x 0 is a ≼ I -monotone seed element, then it is also a mixed ≼ I -monotone seed element.

From observation (b) of Remark 2.4, it follows that if x 0 is a mixed ≼ I -monotone seed element, then it is also a mixed ⪯-monotone seed element.

Moreover, the initial element x 0 is a ≼ I -monotone seed element in X m.

We introduce the concepts of mixed monotone seed element as follows.

In the previous section, we consider the mixed ≼ I -monotone seed element.

Proof From the proof of Theorem 4.15, we see that the initial element x 0 is a mixed ≼ I -monotone seed element in X m.

Proof According to argument in the proof of Theorem 5.2, we see that the initial element x 0 is a mixed ⪯-monotone seed element in X m.