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The estimated models using data with all three bounds, with the first two bounds, and with any specific single bound mostly gave consistent parameter significances and effect directions.
In Lemma 2.2, we give an accurate bound for Green's function (G t,s)), not only an upper bound but also a lower bound, and coincidentally, the only difference between the two bounds is a function factor whose value is in ([0,1]), which leads to Lyapunov-type inverse inequalities; see Theorem 3.1 and Corollary 3.1.
The SNR below which the first of these two bounds has a higher value than the second one is the lower bound on the SNR wall.
For any variable with two bounds the data can be represented in terms of attainments from the lower bound or shortfalls from the upper bound (the maximum).
Wherever Bartoli smacked the ball, McHale would be there to hit it back - and in just two bounds - as if she had received Bartoli's game plan in advance, Bartoli said.
The two bounds derived in Section 3.3 are also shown.
The two bounds of Proposition 3.15 are also shown.
For each row in a partition, a window frame identifies two bounds in the window and only the rows between these two bounds are included for aggregate calculation.
Two bounds respectively regarding amplitude and phase can also be obtained as (10).
Recently, the two bounds for γ and (gamma (a)) have attracted the attention of many mathematicians.
A legitimate question is the following: Are these two bounds comparable?
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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