\begin{align*}
  R_{ij} =& \left\{\frac{\rho(x_i)\rho(x_j)}{(d(x_i,x_j)^2+\delta)^{1/2}}\right\}^{k} \\ 
  =& \exp\bigg\{-k\bigg(\frac{1}{2p}\log\pi(x_{i})+\frac{1}{2p}\log\pi(x_{j}) + \frac{1}{2}\log(d^{2}(x_i,x_j)+\delta)\bigg)\bigg\}, \\
  =& C\exp\bigg\{-k\bigg(\frac{1}{2p}\log\gamma(x_{i})+\frac{1}{2p}\log\gamma(x_{j}) + \frac{1}{2}\log(d^{2}(x_i,x_j)+\delta)\bigg)\bigg\},
\end{align*}
