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We use actual data covariance matrix R in the theoretical analysis; the affect of finite sample effect with different snapshots is simulated in the sixth example, SNR=25 dB.
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The coupleof in the second example merges into couplea (which I would spell coupluh).
In the first example above, the doctor cited the relative risk reduction — an impressive-sounding 50 percent.
As in the first example, we should have a comma after the introductory phrase.
The more important logical relations are implication (as in the first example above) and equivalence; and, as displayed in the square of opposition (see opposition, square of), contrariety (as in the second example), contradiction, subcontrariety, and subimplication.
In the first example, we take (varepsilon=gamma=1).
This is done in the third example.
In the first example, you have 0 debt.
These figures illustrate the result proved in the first example.
And a feedback controller is designed in the third example.
That ecological process was identified as natural selection in the first example; therefore it must be natural selection in the second example as well.
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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