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Existing oscillation data point to nonzero neutrino masses with large mixings. We analyze the generic features of the neutrino Majorana mass matrix with inverted hierarchy and construct realistic {\it minimal schemes} for the neutrino mass matrix that can explain the large (but not maximal) \nu_e - \nu_mu mixing of MSW-LAM as well as the nearly maximal \nu_mu - \nu_tau mixing and the small (or negligible) \nu_e --> \nu_tau transition. These minimal schemes are quite unique and turn out to be extremely predictive. Implications for neutrinoless double beta decay, tritium beta decay and cosmology are analyzed.

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