N1, N2 are full rank matrices and therefore have inverse matrices, as proved in Appendix.
P d,k reaches the maximum value (P_{d,k}^) only when P f,k =P f,t a r g e t, i.e., the optimization problem in Eq. (15) can be rewritten as (proved in Appendix 2) begin{array}{*{20}l} &max_{lambda_{0,k},lambda_{1,k}} P_{d,k} (lambda_{0,k},lambda_{1,k}) &s.t.t
As proved in Appendix B, a family of solutions to Eq. (3) or Eq. (4) can be expressed using pointer states [26, 27]: rho= sum_{i,j in B} rho_{i,j}(t) vert i rangle langle j vert otimes bigotimes _{k in C} biglvert alpha_{k,i}(t) bigrrangle bigllangle alpha_{k,j}(t) bigrvert.
Furthermore, as proved in Appendix 1, the additive genetic value in Equation 4 equals the breeding value, that is, twice the mean progeny value when the current population (regardless of its structure) is randomly mated.
But since both approaches are equivalent, as proven in the "Appendix", it seems more convenient to the authors to apply the first approach: Channel estimates are usually already available in communication systems and the metric derived from (9) is less complex than the metric derived from (10).
This property is proven in Appendix A.1 in Additional file1.
The (mathcal {NP} -hardness of the JLTAP is proven in Appendix 1.
It is proven in Appendix 1 that the second term in the Equation (9) is given by, E [ Δ As ( t ) s H ( t ) Δ A H ] = σ Δ a 2 T r ( R ss ) I (11).
As proved in the Appendix, h(x) has only a single root.
where the integral is a polynomial function whose degree does not exceed, as proved in the appendix.
