on the results of the statistical analysis are also considered.

Finding CP-Violation Associated with Majorana Neutrinos
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General Constraints on the Nuclear Matrix Element Uncertainty
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### Inverted Neutrino Mass Hierarchy: ${\mbox{$ \Delta  m^2_{\odot}$}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$, $m_1 < 0.02 \ {\rm eV}$

1.0cm If ${\mbox{$ \Delta  m^2_{\odot}$}}= \Delta m^2_{32}$, one has [@BGGKP99; @BPP1]: $$\begin{aligned}
{\mbox{$\left|  < \! m  \! > \right| \ $}}=  & \Big| m_1 |U_{\mathrm{e}1}|^2  
  + 
 \sqrt{m_1^2 + {\mbox{$\Delta  m^2_{\mathrm{atm}}$}}\! \!  - {\mbox{$ \Delta  m^2_{\odot}$}}}
 \cos^2 \theta_\odot (1 - |U_{\mathrm{e}1}|^2)  e^{i\alpha_{21}}
 \nonumber \\ 
& + 
 \sqrt{ m_1^2 + {\mbox{$\Delta  m^2_{\mathrm{atm}}$}}} \sin^2 \theta_\odot 
 (1 - |U_{\mathrm{e}1}|^2) e^{i\alpha_{31} }
 \Big|
\label{eqmassinvhierarchy01} \\ 
&  \hspace{-4.7cm}  \simeq   \Big|
 \sqrt{ {\mbox{$\Delta  m^2_{\mathrm{atm}}$}}}
 \big( \cos^2 \theta_\odot   
+  \sin^2 \theta_\odot 
  e^{i\alpha_{32} }\big)
 \Big| = \sqrt{ {\mbox{$\Delta  m^2_{\mathrm{atm}}$}}} \sqrt{ 1 - 
\sin^22\theta_{\odot}~\sin^2\frac{\alpha_{32}}{2}}~,
\label{eqmassinvhierarchy02} \end{aligned}$$ where $\alpha_{32} = \alpha_{31} - \alpha_{21}$. In eq. (\[eqmassinvhierarchy02\]) we have neglected ${\mbox{$ \Delta  m^2_{\odot}$}}$ and $m_1^2$ with respect to as well as the terms proportional to which is limited by the CHOOZ data. For the best fit value of ${\mbox{$\Delta  m^2_{\mathrm{atm}}$}}\cong 3.0\times 10^{-3}~{\rm eV^2}$, $m_1 < 0.01$ eV and the best fit value of ${\mbox{$ \Delta  m^2_{\odot}$}}\cong (5.0 - 5.5)\times 10^{-5}~{\rm eV^2}$, the corrections due to the 