Almost maximally broken permutation symmetry for neutrino mass matrix

Assuming three light neutrinos are Majorana particles, we propose mass matrix ansatz for the charged leptons and Majorana neutrinos with family symmetry $S_{3}$ broken into $S_{1}$ and $S_{2}$, respectively. Each matrix has three parameters, which are fixed by measured charged lepton masses, differences of squared neutrino masses relevant to the solar and the atmospheric neutrino puzzles, and the masses of three light Majorana neutrinos as a candidate for hot dark matter with $\sum |m_{\nu}| \sim 6 eV$. The resulting neutrino mixing is compatible with the data for the current upper limit, $_{th} < 0.8 eV$, of neutrino-less double beta decay experiments, and the current data for various types of neutrino oscillation experiments. One solution of our model predicts that $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation probability is about $< 0.008$ with $\Delta m^{2} \sim 10^{-2} eV^2$, which may not be accessible at CHORUS and other ongoing experiments.Comment: 9 pages, RevTex, no figure

The Riemann mapping theorem from Riemann's viewpoint

This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.Comment: 14 pages, 2 figure

A simple modification of the maximal mixing scenario for three light neutrinos

We suggest a simple modification of the maximal mixing scenario (with $S_3$ permutation symmetry) for three light neutrinos. Our neutrino mass matrix has smaller permutation symmetry $S_{2}$ ($\nu_{\mu} \leftrightarrow \nu_{e}$), and is consistent with all neutrino experiments except the $^{37}$Cl experiment. The resulting mass eigenvalues for three neutrinos are $m_{1} \approx (2.55-1.27) \times 10^{-3} eV, m_{2,3} \approx (0.71-1.43) eV$ for $\Delta m_{LSND}^{2} = 0.5 - 2.0 eV^2$. Then these light neutrinos can account for $\sim (2.4-4.8)$ of the dark matter for $h = 0.8 (0.5)$. Our model predicts the $\nu_{\mu} \rightarrow \nu_{\tau}$ oscillation probability in the range sensitive to the future experiments such as CHORUS and NOMAD.Comment: The title has been changed, to appear in Z. Phys.