The degree of validity [Grad der Gültigkeit] of a proposition relative to chosen variables is the ratio of the number of true propositions obtained by variation to the number of all propositions obtained.
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.
For the clearest example the Mohists give of it is not knowledge of an a priori proposition, but knowledge obtained by analogical inference from information provided by an informant (B70).[13] So "knowledge by explanation" seems a catch-all term for knowledge obtained by inference, whether deductive, inductive, abductive, or analogical.
Note that if, the result from Proposition 3.3(1) was first obtained by De Donder and Hindriks [3] (see Proposition 1(a) in that paper).
For bodies with ith continuous curvature functions, the equality conditions for the inequality of Proposition 3.14 are easily obtained by combining Propositions 3.10 and 3.21.
The spectral properties of a UNTF are obtained by Proposition 1 by setting α=N/m.
Finally, the estimate (6) can easily be obtained by Proposition 4.4 in [9].
For this purpose, we need the strong semismoothness of transformation H which can be obtained by Proposition 2.3 (ii).
In the last equality, we use the fact P 0 F 0 | V | 1 / 2 ϕ = | V | 1 / 2 ϕ, which can be obtained by Proposition 3.2.
Applying Lemma 1 in (19) and (21), the analytical end-to-end SEP in Proposition 2 can be obtained by the integral of [22, 2.33.10, 8.350.2].
Since SigShrink functions satisfy assumptions of [16, Proposition 3.2], the shrinkage obtained by using a function can be seen as a regularization approximation [7] by seeking the vector that minimizes the penalized least squares (12).
