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(1) and are monotonously decreasing in interval ; (2) (1.5) .
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This leads to the fact that ȷ ′ ( t ) has a unique positive real zero in the interval ( 0, 1 ) around t ≃ 1 2. A computation yields ȷ ( t ) is decreasing in the interval t ∈ ( 1 2, 1 ) and assuming its maximums at t = 0.5 and r = 0.5.
Case 2: Similarly to Case 1, one shows that z(x) is decreasing in the interval (b + c+r13)/a ≤ x ≤ 1 (UPC), respectively z(x) is increasing in the interval − 1 ≤ x ≤ (c−r13 − b /a (LPC).
The free boundary is decreasing in the interval.
Then, which implies that is strictly decreasing in that interval.
f + is symmetric about t = 1 / 2. f + is decreasing in the interval [ 0, 1 / 2 ].
Using Lemma 3.3 (vi), we obtain for, from which it follows that the sequence is strictly decreasing in the interval.
The ozone density increases in the height interval 15.5 18.5 km and decreases in the interval 24.5 35 km.
The ranking of Trade increases when the number of patent applications is in the interval of 60%to70%0%, and then decreases in the interval of 70%to90%0%.
monotonically decreases in the interval.
The range 0 < H < 0.5 (1 < β < 2) indicates antipersistency, which means that if the fluctuations increase in a period, it is likely to decrease in the interval immediately following and vice versa.
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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