m_{LL}^{\prime\prime} R_{12}
\label{SRNDmLLprime}$$ which, for $m_{LL}$ as in Eq. , are explicitly given by : $$m_{LL}^\prime \sim M^{-1}_{\nu_3} \left(\matrix{
\lambda_{13}^2 & 0 & \lambda_{13} A \cr
0 & 1 & 0 \cr
\lambda_{13} A & 0 & A^2}\right)\qquad
m_{LL}^{\prime\prime} \equiv m_{LL}^{\prime\prime\prime} 
\sim M^{-1}_{\nu_3} \left(\matrix{
0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & B^2}\right)\quad
\label{SRNDmLLprimeMTR}$$

We can see from Eq.  that if $\lambda_{23} =
\lambda_{33}$ then a maximal $\theta_{23}=45^0$ angle results. Moreover, if $\lambda_{13} \ll \lambda_{23},\lambda_{33}$ then $\theta_{13}$ is small. Although SRHND, in the limiting case of Eq. , is successful in predicting a maximal atmospheric neutrino angle, it fails to account for a viable neutrino spectrum. Indeed, from Eq. , we see that the two lightest neutrinos are massless $m_{\nu_1} = m_{\nu_2} = 0$. Moreover the solar neutrino angle $\theta_{12}$ is undetermined. These two problems can be solved by allowing the right-handed muon neutrino $\nu^c_\mu$ to play a sub-dominant/perturbative role in the structure of $m_{LL}$ in Eq. .

We now turn to the more realist model in which $M_{RR}$ can be approximated by : [^8] $$M^{-1}_{RR} \sim 
{\rm diag}(M^{-1}_{\nu_1},M^{-1}_{\nu_2},M^{-1}_{\nu_3}) \sim
{\rm diag}(0,M^{-1}_{\nu_2},M^{-1}_{\nu_3})
\label{SRNDmLLMTR2}$$ Using Eq.  into Eq.  we find that : $$m_{LL} 