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The phrase "allocation are given" is not correct in written English.
It should be "allocations are given" or "allocation is given" depending on the intended meaning.
Example: "Allocations are given to each department based on their budget needs."
Alternatives: "resources are allocated" or "funds are distributed."
Exact(3)
The parameters for the PSO utilized for the power allocation are given in Table 2.
Theorem 3 The expected mutual information upper bound C ̃ u is concave with respect to λ, and the necessary and sufficient conditions for the optimal power allocation are given by λ j ( i ) = ν ̃ i - p λ i ( j ) q λ i ( j ) +, (31) 1 T λ i = P i ′, f o r j = 1,..., N t ( i ), a n d i = 1,..., m, (32).
Definition 1: Given the fixed ( {S}_{-n}^F ) and ( {P}_{-n}^F ), the best response dynamics of the RB and power allocation are given by, left({S}_n^F,{P}_n^Fright)= arg underset{S_n^F,{P}_n^F}{ max }{U}_nleft({S}_n^F,{P}_n^FBig|{S}_{-n}^F,{P}_{-n}^Fright) (9 where ( {S}_{-n}^F ) and ( {P}_{-n}^F ) are the action profile of all the players except the nth one.
Similar(57)
Furthermore, a joint algorithm in subcarrier pairing, relay selection and power allocation is given.
Finally, an example about function allocation is given, that is, fault diagnosis in the cockpit of civil aircraft.
The power allocation is given by the positive root (17).
Realization of the resource allocation is given in Algorithm 1.
and the rate allocation is given by (2).
However, no method for optimizing the power allocation is given.
Then, the optimal energy allocation is given by.
The corresponding power allocation is given by (45).
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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