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Exact(20)
Now, if we consider ( X, D 2 ).
Now, if we consider the large magnitude earthquakes viz.
Now, if we consider that the speed of sound can depend on t: (25).
Now, if we consider the guesswork on the key space (of k bits), we would have (5).
Now, if we consider that the adversary is able to generate a set of fake queries (lines 20 to 22).
Now if we consider this inequality and previous theorem we deduce as in Lemma 2 the following result.
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If we consider now the scenario from Figure 3, there is virtually no difference in the average sum rate between the two initialization methods.
If we consider now the space, then the assumption of existence of a -convergent subsequence for every sequence in can be removed and we can state the following result.
If we consider now the k th component h ^ x, k m of all these h ^ x m m ∈ 0, M − 1 vectors, we can introduce the h ^ x, k vector defined as follows h ^ x, k = h ^ x, k 0 ⋮ h ^ x, k M − 1 (46).
If we consider now a two-antenna spatial filter (M = 2), in this case, the optimal spatial filter given by Equation 27 becomes w k = 1 h x, k H C nn, k h x, k C zz, k h x, k * + σ n 2 h x, k * (28).
If we consider now the opposite case, when node 2 is a cooperator and node 1 is a defector, then cooperation spreads when k2>T−S+1.
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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