Sentence examples for bound proposition from inspiring English sources

In Theorem 1 'if' should be replaced by 'iff' and should read as follows: For odd k, an RC complete sequence s achieves the lower bound (Proposition 1) iff there exist two edge-disjoint paths with no repeating edges, corresponding to s and RC s), that together cover all edges of the de Bruijn graph of order k − 1.

Theorem 1 For odd k, an RC complete sequence s achieves the lower bound (Proposition 1) if there exist two edge-disjoint paths with no repeating edges, corresponding to s and RC(s ), that together cover all edges of the de Bruijn graph of order k − 1. Proof Observe that the lower bound assumes one occurrence of either w or RC (w ) but not both in the sequence for each k -mer w.

For odd k, an RC complete sequence s achieves the lower bound (Proposition 1) if there exist two edge-disjoint paths with no repeating edges, corresponding to s and RC s), that together cover all edges of the de Bruijn graph of order k − 1. Observe that the lower bound assumes one occurrence of either w or RC w) but not both in the sequence for each k-mer w.

The outer bound in Proposition 3 is better than the trivial outer bound in Proposition 2 obtained by giving the channel state to the decoder.

These reasons explain why the bound in Proposition 3.13 is not very tight.

As we shall see, the bound in Proposition 1 holds with strict equality in the case of K = 2 transmitters and S = 2 channels, that is, 1 ⩽ L ⩽ 2.

The asymptotes on the PER given by a numerical computation of the coding gain and the ones using the bound of Proposition 1 are very close to one another.

In this section, we prove constructively that for odd k there exists an RC complete sequence that achieves the lower bound of Proposition 1 and is thus optimal.

This is an addition of characters, where L denotes the lower bound in Proposition 1 for an RC complete sequence of even order k.

It can be seen that the lower bound proposed in Proposition 3 gets tighter as c decreases.

Here we define the sets mathcal{X}_{m} :=bigl{ (x,y inmathcal{X}^{2}:M y,x leq m bigr} quad forall min{ 1,2,3,ldots}, where (M y,x) ) is the maximum bound defined in Proposition 4.1.