N}, where n is the length of the sequence, Reconstruct the d-dimension and (d + l) dimension phase-space.
This paper establishes forecasting model by combining phase-space reconstruction technology with neural networks of Elman, BP and RBF.
Use the best delay time and the embedding dimension to carry out the phase-space reconstruction of the original time series to obtain the new phase-space vector:
0] is a real number, we get oscillating WKB modes and the Hartle criterion is recovered since |[PSI]> is peaked on a phase-space region defined by
In other words, the issue is searching for some general method by which selecting such constants of motion related to the emergence of classical trajectories without arbitrarily choosing regions of the phase-space where momenta are conserved.
In this paper, we have shown that the reduction procedure of dynamics, related to conservation laws, can give rise to a splitting of the phase-space of a physical system, by which it is possible to achieve the complete solution of dynamics.
The relations (151, 152, 153, 154, 155) are required in order to prove the group composition law of the transformations of (137)-(140) and, consequently, in order to have a truly Maximal-Acceleration Phase Space Relativity theory resulting from a phase-space change of coordinates in the cotangent bundle of spacetime.
Also we notice that the Phase-space areas, or cells, in units of h, are also invariant
In section 6 we review the Maximal-Acceleration Relativity in Phase-Spaces , starting with the construction of the submaximally-accelerated particle action of  using Clifford algebras in phase-spaces; the U(1, 3) invariance transformations  associated with an 8-dimensional phase space, and show why the minimal Planck-Scale areas are invariant under pure acceleration boosts which suggests that there could be a principle of maximal-tension (maximal acceleration) operating in string theory .
where the analog of the Lorentz time-dilation factor in Phase-space is now given by
In  we proposed a plausible explanation of the variable fine structure constant phenomenon based on the maximal-acceleration relativity principle in phase-space by modifying the Robertson-Friedmann-Walker metric by a similar (acceleration-dependent) conformal factor as in eqs-(3.
In contrast, the results of this work are based on Born's Dual Phase-Space Relativity principle.