  Recently we have found that a three-dimensional superconducting state with
anisotropic vortices localized at the vortex-lattice points is a stable state
in zero external magnetic field for the layered high temperature
uperconductivity materials. There exists a phase transition at the temperature
$T_{c}^{v}$ from the normal phase to the vortex superconducting state which is
of the first order. The first order phase transition shows overheating and
overcooling effects. Nucleation of the superconducting phase in the normal
phase thus may occur at temperatures higher than the transition temperature
$T_{c}^{v}$. Then the onset of the vortex-like excitations above the transition
temperature $T_{c}^{v}$ occurs in our theory. The onset of the vortex-like
excitations in Nerst signal and some other experimental evidence for these
excitations above the transition temperature $T_{c}^{v}$ in LSCO, YBCO and in
other high-temperature materials may be explained thus by our theory. The
vortex-like excitations above and below the transition temperature $T_{c}^{v}$
in high temperature materials continuously evolve. This fact may be explained
within our theory.
