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An extra factor of g multiplies the second ODE because g acts as a multiplier and must remain nonnegative.
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Multiply the second equation of (2.10) by and integrate over.
Multiplying the second inequality of (2.5) by, one gets (2.6).
Multiplying the second equation of (1.4) by and integrating it over, we obtain (4.11).
Multiply the second equation of (1.4) by and integrate it over to obtain (4.15).
Now, multiplying the second equation of (1.1) by and integrating over, we have (4.3).
Multiplying the second equation of (1.4) by and integrating it over, we have (4.31).
Therefore, to hold on the uniformity of vectors z and x, we need to multiply the second term by s.
Multiplying the second equation of (1.4) by, adding up the first equation of (1.4), and integrating the result over, we obtain (4.3).
We multiply the second equation of (1.1) by | w | 2 w, then integrate the resulting equation with respect to x over R 3 and use integrating by parts.
For nonzero frequencies we may multiply the second block row by ω − 1 and introduce the new variable v ˆ = ω − 1 p ˆ.
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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