The consumer surplus that is derived from these optimal consumption plans is given by substituting (10) back into (7) and (8), respectively: u_{rm PU}(n^*_{rm PU})= frac{left( beta_1 - p_{rm PU}right) ^2}{4 beta_2},quad u_{rm FR}(n^*_{rm FR})= frac{beta_1^2}{4 beta_2} - b_{rm FR}.
While, for a targeted false alarm probability p f, i T, the probability of the detection is given by substituting (6) into (5), P d, i τ i = Q Q − 1 p f, i T − γ i h i 2 τ i f s 2 γ i h i 2 + 1. (10).
The force vectors fTM and fRL caused by the shear motion between the TM and the RL are given by substituting the relative velocity vector Vrelative of the TM in relation to the RL into (15) and (16), respectively.
Substituting Eq. (49) into Eq. (39), the power allocated to flow g in the linear case can be given by.
The probability of bit error at a reliable relay is given by [11] (6). Substituting (5) into (4), we obtain the e2e BER conditioned on as (7).
Throughput for any recipient from set is given by (17), where we substitute for.
Since the value of initial condition corresponding to the start point z s is given by (2.6) and (2.7), and substituting already calculated coefficients a 0 and a 1 given by (2.13), we get (2.15).
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As can be easily seen from Equation 1 in the main text, if the best linear combination of regulators provides a correlation R with the promoter's desired levels, the optimal value S* of the signal-to-noise of this composite regulator is given by S* = RY/ X. Substituting this back into Equation 1, we find for the optimal coupling strength X * 2 = max [ 0, (1 − R 2 ) Y 2 − 1 ].
The conservation law gives: (16) cm 00 + cm 10 + cm 01 + cm 11 = 1 In steady-state, the matrix for the system (15) with the last equation substituted by (16) is given by: (17) N = (− (k 1 + k 2 ) ⋅ u d 1 d 2 0 k 1 ⋅ u − d 1 − k c 2 ⋅ u 0 d c 2 k 2 ⋅ u 0 − d 2 − k c 1 ⋅ u d c 1 1 1 1 1 ) The matrix (17) can be solved using Cramer's method.
