We explore singularity-free and geodesically-complete cosmologies based on manifolds that are not quite Lorentzian. The metric can be either smooth everywhere or non-degenerate everywhere, but both, depending the coordinate system. gives an Einstein tensor is first order in derivatives while has a piecewise FLRW form. On such manifold universe transition from expanding to contracting, vice vers...

in this paper, the asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated. furthermore, we obtain the zeros of eigenfunctions.

This paper presents a new method for predicting turning points. The paper formally defines a turning point; develops a probit model for estimating the probability of a turning point; and then examines both the in-sample and out-of-sample forecasting performance of the model. The model performs better than some other methods for predicting turning points. *While working on this paper, all three ...