With DPC, the throughput maximization problem with individual power constraints (9) has been solved by converting it into a dual uplink with sum power constraint across users and uncertain noise and employing an interior-point algorithm [57].
This problem, in some sense, is the dual problem of the sum rate maximization under a fixed power constraint.
Consistently, the total FFA yield of the tested dual enzymatic systems calculated by summing up the reacted FFA (equals to the produced alkene in mole) and the remaining FFA was slightly improved compared to that of the single lipase hydrolytic system.
Moreover, by substituting γi+1 in (39), we can show that the sum of the dual variables ∑ n ∈ N v n i is equal to zero.
Alternatively, the proof can be done by following the proof of Proposition 3.7 with the help of the dual formula for left fractional sum in Lemma 3.1 after its arrangement according to our definitions.
Moreover, by substituting z nl i + 1 in (20 k, we can show that the sum of the dual variables u n, nl i + u tran ( l ), nl i is equal to zero.
where λBS is the dual variable associated with the sum power constraint for the BS and μ = ( μ 1, μ 2, …, μ | M nRT | ) is the vector of dual variables for nRT service constraints, where | M nRT | denotes the total number of nRT users.
Furthermore, the intensity for the 1.5K array is the sum of a dual color assay (cy3+cy5 channels), whereas the 24K assay is a single-color assay (cy3), the hybridization conditions and washes are different, and the readouts are different (Universal Array Matrix versus whole genome BeadChip) and therefore, the scan settings are different.
We investigate two types of dual identities for Riemann fractional sums and differences.
The dual relations for left fractional sums and differences were investigated in [12].
The higher Gd per liposome observed for Dual-Gd compared to the sum total of CE-Gd and SC-Gd liposomes is therefore most likely a result of increased encapsulated Gd fraction compared to the CE-Gd liposomes.
