Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow

In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=\Delta u+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=\Delta u+\lambda u^{\alpha}$$ with $\lambda,\alpha$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang type gradient estimates.Comment: All comments are welcom

Constraints on absolute neutrino Majorana mass from current data

We present new constraints on the neutrino Majorana masses from the current data of neutrinoless double beta decay and neutrino flavour mixing. With the latest results of $0\nu\beta\beta$ progresses from various isotopes, including the recent calculations of the nuclear matrix elements, we find that the strongest constraint of the effective Majorana neutrino mass is from the $^{136}\rm{Xe}$ data of the EXO-200 and KamLAND-Zen collaborations. Further more, by combining the $0\nu\beta\beta$ experimental data with the neutrino mixing parameters from new analyses, we get the mass upper limits of neutrino mass eigenstates and flavour eigenstates and suggest several relations among these neutrino masses.Comment: 6 latex pages, 2 figures. Final version for publication in "The Universe