Sentence examples for proposition is due from inspiring English sources

The case i = 0 of the following proposition is due to Lutwak [19].

The approximation in this proposition is due to neglecting the spatial interference correlation resulting from the fact that multiple interfering streams are coming from the same location (more details can be found in Appendix 1: Proof of Proposition 1).

Now we consider the functional Φ λ defined on H G λ by (2.2), the following Proposition 3.3 is due to [9].

First of all, we notice that the proof is vastly more complicated than that of Proposition 2.1, and this is due precisely to the presence of the nonlocal elements in the definition of the fractional difference.

The following assertion is due to Proposition 2.1 and its proof can be omitted.

For (u le r), optimal fixed compensation exactly resembles Proposition 4. This observation is due to the fixed compensation (hat{beta }_1 = hat{b}_1 = 0) in both cases.

Combining the latter with the hypothesis (5.10) we obtain 1 ( 2 π i ) n ∫ w ∈ D f ( w ) ( ∂ ¯ w η ˜ ) n ( w, z ) = ∫ w ∈ b D f ( w ) j ∗ Ω 0 ( w, z ) = f ( z ) where the last identity is due to Proposition 2.

However, Gilbert Harman has said that the best explanation of our failure to find a counterexample to some proposition may be due to our limited imaginations (Harman 2001).

Lastly, a recent proposition has been made that gene clustering is due to the relative difficulty of protein movement through the cellular matrix [ 18].

Around $9 billion is due later this year, and refinancing will be a tricky proposition given the parlous state of debt markets.

This is due to the fact that these theories contain general propositions that have the logical form of universal axioms.