where, with, denotes the -Laplacian operator; and are continuous functions not identically zero and is a real parameter.
for some with denotes the input covariance satisfying and, denotes the channel matrix of receiver 1, 2, respectively.
where, by definition, with, denotes Brownian Motion with unit variance parameter, and, are constants reflecting the (spatial) variability of the field and its noisy behaviour, respectively.
where with, denotes the power allocated to the user on subchannel, represents the statistical expectation with respect to, is user 's average data rate so far at the allocation time, and is a tunable fairness parameter.
with, denotes the all-ones -dimensional vector, is the set of all -dimensional vectors with entries taking values 0 of 1, that is, contains the binary words of length and is the th component of.
where, are the (complex conjugated) channel vectors of users through, with ( denotes the -dimensional complex space), denotes the -dimensional signal transmitted by the BS, and are independent complex Gaussian additive noise terms with zero mean and unit variances.
