Sentence examples for as proved in appendix from inspiring English sources

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.

As proven in Appendix 1, this is not true.

The proof relies on the optimality of an opportunistic scheduling structure as first proved in Appendix 1: opportunistic scheduling structure of the optimal power policy (see Lemmas 1 and 2).

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.

which are proved in Appendix C.

The above results are proved in Appendix 1.

This conclusion has been proved in Appendix 3 and verified by simulation examples in Section 8.