the results of Theorem \[theorem:multirelay\_capacity\]:

-   In order to maximize $R^{opt}_{P_T}(y_1,y_2,\cdots,y_N)$, we need to place the relay nodes such that $\frac{1}{g_{0,1}}+\sum_{k=2}^{N+1}\frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}$ is minimized. This quantity is viewed as the net attenuation the power $P_T$ faces.

-   When no relay is placed, the attenuation is $e^{\lambda}$. The ratio of attenuation with no relay and attenuation with relays is called the “relaying gain” $G(N,\lambda)$. $$G(N,\lambda):=\frac{e^{\lambda}}{\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1}\frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}×}
    \label{eqn:gain_definition}$$ Rate is increasing with the number of relays, and is bounded above by $C(\frac{P_T}{\sigma^2})$. Hence, $G(N,\lambda) \in [1, e^{\lambda}]$. Also, note that $G(N,\lambda)$ does not depend on $P_T$.

-   By Theorem \[theorem:multirelay\_capacity\], we have $P_{k,j} \geq P_{i,j}$ for $i<k<j$.

-   Note that we have derived Theorem \[theorem:multirelay\_capacity\] using the fact that $g_{0,k}$ is nonincreasing in $k$. If there exists some $k \geq 1$ such that $g_{0,k}=g_{0,k+1}$, i.e, if $k$-th and $(k+1)$-st nodes are placed at the same position, then $\gamma_{k+1} = 0$, i.e., the nodes $i < k$ do not direct any power specifically to relay $k+1$. However, relay $k+1$ can decode the symbols received at relay $k$, and those transmitted by relay $k$. Then relay $(k+1)$ can transmit coherently with the nodes $l \leq k$ to improve effective received power in the nodes $j > 