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We apply the shell-model wave-functions of $^{76}Ge$ to calculate the nuclear matrix elements and the half-life of the $2\nu\beta\beta$ decay in $^{76}Ge$. Our result is comparable to the recent observed $2\nu\beta\beta$-decay half-life of $^{76}Ge$. Furthemore it provides the upper limit of the effective neutrino mass is about 0.4 eV from the recent experimental data of the $0\nu\beta\beta$-decay half-life., Comment: 6 pages

We apply the shell-model wave-functions of $^{76}Ge$ to calculate the nuclear matrix elements and the half-life of the $2������$ decay in $^{76}Ge$. Our result is comparable to the recent observed $2������$-decay half-life of $^{76}Ge$. Furthemore it provides the upper limit of the effective neutrino mass is about 0.4 eV from the recent experimental data of the $0������$-decay half-life., 6 pages

Neutrinoless double beta decay of 48Ca is discussed in the double charge exchange of pions in flight between two nucleons by using a diagrammatic effective operator approach. The virtual intermediate neutrino is assumed to be of the heavy Majorana type.

We discussed the influence of different mechanism for neutrinoless double decay with the heavy Majorana neutrino, which is the virtual intermediate states.