Isoparametric foliation and Yau conjecture on the first eigenvalue

A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.Comment: to appear in J.Diff.Geo

Strain engineering and photocatalytic application of single-layer ReS2_2

We present a theoretical study on the electronic, dynamical, and photocatalytic properties of single-layer ReS$_2$ under uniaxial and shear strains. The single-layer ReS$_2$ shows strong anisotropic responses to straining. It remains dynamically stable for a wide range of $x$-axial strain, but becomes unstable for 2\% $y$-axial compressive strain. The single-layer ReS$_2$ is calculated to be an indirect bandgap semiconductor, and there is an indirect$-$direct bandgap transition under 1$-$5\% $x$-axial tensile straining. The single-layer ReS$_2$ is predicted incapable of catalyzing the water oxidation reaction. However, 1$-$5\% $y$-axial tensile strain can enable the single-layer ReS$_2$ for overall photocatalytic water splitting. Besides, the single-layer ReS$_2$ can also catalyze the overall water splitting and be most efficient under acidic water solutions with pH=3.8

The dynamic dipole polarizabilities of the Li atom and the Be+ ion

The dynamic dipole polarizabilities for the Li atom and the Be+ ion in the 2s and 2p states are calculated using the variational method with a Hylleraas basis. The present polarizabilities represent the definitive values in the non-relativistic limit. Corrections due to relativistic effects are also estimated. Analytic representations of the polarizabilities for frequency ranges encompassing the n=3 excitations are presented. The recommended polarizabilities for ^7Li and ^9Be+ were 164.11 \pm 0.03 a.u. and 24.489 \pm 0.004 a.u.