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We report briefly recent experimental studies of axial-vector weak couplings for neutrino nuclear responses (squares of nuclear matrix elements NMEs) in the wide energy (0-50 MeV) and momentum (0-100 MeV/c) regions, which are relevant to neutrinoless double beta decays (DBDs) and supernova neutrinos. High energy-resolution studies of charge exchange reactions (CERs) of (³He,t) are used for studying the τ- side NMEs, while charge exchange muon reactions (ordinary muon capture reactions OMC) are for studying the τ+ side NMEs. The experimental NMEs derived from the single β/EC NMEs and the CER NMEs show that the (J± ≤4) spin isospin (axial vector) NMEs are uniformly reduced by the coefficient geffA/gfreeA≈0.5−0.6 with respect to A pnQRPA NMEs. Then the DBD NMEs M0ν may be reduced by the coefficient around 0.5, depending on the ratio of the axial-vector to vector NMEs, with respect to the pnQRPA NMEs. [ABSTRACT FROM AUTHOR]