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The first contribution of the present work is to rigorously prove two observed properties of the FLG matrix; that it is substantially real-valued, and that its rows sum close to one.
Overall, though, driving today is substantially cheaper, in real terms, than it was about a generation ago.
Workers' real income is substantially higher than a year ago.
Overall the real picture is substantially different than the one Reagan paints for his audiences.
The real rate is substantially higher, which means that more than 1 in 10 New Yorkers who would like to work cannot find a job.
Restaurant owners estimated that the real number is substantially higher, as many chefs were embarrassed to admit the short cuts that, in effect, hoodwink their customers.
If official statistics are to be believed, only 4.6% of the world's most populous nation still live in poverty in rural areas, but many experts believe the real number is substantially higher.
Atletico Madrid net spending has been on par with Real Madrid but is substantially less than Barcelona.
In our simulations, we consider homogeneous particles, but real cometary dust is substantially inhomogeneous by chemical composition and a realistic simulation of light scattering by comets must take into account this fact.
Further, because of the problems in the questionnaire distribution, it is likely that the real response rate is substantially higher than the estimated proportion.
America now has a real gross national product that is substantially greater than it was when the Great Society was launched.
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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