Acceleration and Deceleration in Curvature Induced Phantom Model of the Late and Future Universe, Cosmic Collapse as Well as its Quantum Escape

Here, cosmology of the late and future universe is obtained from $f(R)$-gravity with non-linear curvature terms $R^2$ and $R^3$ ($R$ being the Ricci scalar curvature). It is different from $f(R)$-dark enrgy models, where non-linear curvature terms are taken as gravitational alternative of dark energy. In the present model, neither linear nor no-linear curvature terms are taken as dark energy. Rather, dark energy terms are induced by curvature terms in the Friedmann equation derived from $f(R)$-gravitational equations. It has advantage over $f(R)$- dark energy models in the sense that the present model satisfies WMAP results and expands as $\sim t^{2/3}$ during matter-dominance. So, it does not have problems due to which $f(R)$-dark energy models are criticized. Curvature-induced dark energy, obtained here, mimics phantom. Different phases of this model, including acceleration and deceleration during phantom phase, are investigated here.It is found that expansion of the universe will stop at the age $(3.87 t_0 + 694.4 {\rm kyr})$ ($t_0$ being the present age of the universe) and after this epoch, it will contract and collapse by the time $(336.87 t_0 + 694.4 {\rm kyr})$. Further,it is shown that universe will escape predicted collapse (obtained using classical mechanics) on making quantum gravity corrections relevant near collapse time due to extremely high energy density and large curvature analogous to the state of very early universe. Interestingly, cosmological constant is also induced here, which is very small in classical domain, but very high in quantum domain.Comment: 33 page

Online codes for analog signals

This paper revisits a classical scenario in communication theory: a waveform sampled at regular intervals is to be encoded so as to minimize distortion in its reconstruction, despite noise. This transformation must be online (causal), to enable real-time signaling; and should use no more power than the original signal. The noise model we consider is an "atomic norm" convex relaxation of the standard (discrete alphabet) Hamming-weight-bounded model: namely, adversarial $\ell_1$-bounded. In the "block coding" (noncausal) setting, such encoding is possible due to the existence of large almost-Euclidean sections in $\ell_1$ spaces, a notion first studied in the work of Dvoretzky in 1961. Our main result is that an analogous result is achievable even causally. Equivalently, our work may be seen as a "lower triangular" version of $\ell_1$ Dvoretzky theorems. In terms of communication, the guarantees are expressed in terms of certain time-weighted norms: the time-weighted $\ell_2$ norm imposed on the decoder forces increasingly accurate reconstruction of the distant past signal, while the time-weighted $\ell_1$ norm on the noise ensures vanishing interference from distant past noise. Encoding is linear (hence easy to implement in analog hardware). Decoding is performed by an LP analogous to those used in compressed sensing