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We present models that can reconcile the solar and atmospheric neutrino data with the existence of a hot dark matter component in the universe. This dark matter is a quasi-Dirac neutrino whose mass m_DM arises at the one-loop level. The solar neutrino deficit is explained via nonadiabatic conversions of ν_e to a sterile neutrino ν_s and the atmospheric neutrino data via maximal ν_μ to ν_τ oscillations generated by higher order loop diagrams. For m_{DM ∼} 30 eV the radiative neutrino decay can lead to photons that can ionize intersteller hydrogen. in one of the models one can have observable ν_e to ν_τ oscillations rates, with no appreiable ν_μ oscillations at accelerator experiments. In addition, there can be observable rates for tau number violating processes such as τ→3e and τ→e + γ. In the other model one can have sizeable ν_e to ν_μ oscillation rates, as well as sizeable rates for muon number violating processes such …