Four neutrino oscillation analysis of the Superkamiokande atmospheric   neutrino data
  Superkamiokande atmospheric neutrino data (contained and upward going through $\mu$ events) for 990 days are analyzed in the framework of four neutrinos (three active and one sterile neutrinos) without imposing constraints of Big Bang Nucleosynthesis. It is shown that the wide range of the oscillation parameters is allowed at 90% confidence level ($0.1\lesssim |U_{s1}|^2+|U_{s2}|^2\le 1$), where the best fit point has some contribution of $\nu_\mu\leftrightarrow\nu_s$ and contribution of $\Delta m^2_{\rm\scriptsize LSND}$. The case of pure $\nu_\mu\leftrightarrow\nu_s$ oscillation is excluded at 99.7%CL (3.0$\sigma$) which is consistent with the recent analysis by the Superkamiokande group. Combining this result with the analysis by Giunti, Gonzarez-Garcia and Pe\~na-Garay, it is found that the Large Mixing Angle and Vacuum Oscillation solutions of the solar neutrino problem are also allowed. 
Osamu Yasuda