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Exact(48)
A sufficient condition is that the masses form a sequence in l log l.
Let l = log L be the log-likelihood.
Further, the proportionality property between the weights g and h for domestic consumers implies that Equation (A15) can be rewritten as: d log V q d log L = ϵ D q s L d log C L d log L + ϵ D q s K d log C K d log L + 1 − ϵ D q d log C X d log L + ϵ D q s L d log w d log L + ϵ D q s K d log r d log L. (A16).
Using an analogous strategy, it can be shown that the change in the effective wealth determining aggregate demand for good y is: d log V y d log L = ϵ D y s L d log C L d log L + ϵ D y s K d log C K d log L + 1 − ϵ D y d log C X d log L + ϵ D y s L d log w d log L + ϵ D y s K d log r d log L. (A17).
log (accretion power / Eddington luminosity), in units of log L / logLEdd.
One caution: Remember that luminosities add but magnitudes don't and neither does log(L).
Similar(12)
By using the Lagrange mean value theorem, there exists some (xiin (log L,log U)) such that (log U ^{k-1}- log L)^{k-1}=(k-1) xi^{k-2}(log U ^{k-1}- log16) ThU ^{k-1}- logw from (3.14)-(3.16) that the totaL ^{k-1}= k-1he grid nodes considered in the improved aL ^{k-1}= k-1ot more than frac{(k-1)xi^{k-2}c^{k-1}(log U-log L)}{varepsilon^{k-1}}.
The resulting likelihood ratio is given by: (12) − 2 log (L 0 L 1 ) = − 2 [ log (L 0 ) − log (L 1 ) ] = − 2 (L 0 − L 1 ) If the chi-square value for this test is significant, the variable is considered to be a significant predictor.
We can obtain the maximum likelihood estimator ŵ for w by maximizing the log-likelihood l(w )=log L(w ).
Since (L^{widetilde{Phi}}(Omega)=L (log L ^{delta}(loglog log L ^{frac{beta}{2}}(Omega)) is a subspace of (L log L ^{frac {1}{2}}(Omega)) if (betageq0) and (deltageqfrac{1}{2}), we can ensure (as already observed) that (1.1) has a unique finite energy solution (vin W^{1,2}_{0}(Omega)).
The minimal assumption on f that guarantees this is (fin L (log L ^{frac{1}{2}}(Omega)).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com