Multiplicity of closed characteristics on symmetric convex hypersurfaces in R2n\R^{2n}

Let $\Sigma$ be a compact $C^2$ hypersurface in $\R^{2n}$ bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least $n$ geometrically distinct closed characteristics on $\Sigma$ if $\Sigma$ is symmetric with respect to the origin.Comment: 16 page

Pulsar timing array observations as possible hints for nonsingular cosmology

Recent pulsar timing array (PTA) experiments have reported strong evidence of the stochastic gravitational wave background (SGWB). If interpreted as primordial gravitational waves (GWs), the signal favors a strongly blue-tilted spectrum. Consequently, the nonsingular cosmology, which is able to predict a strongly blue-tilted GW spectrum with $n_T \simeq 2$ on certain scales, offers a potential explanation for the observed SGWB signal. In this paper, we present a Genesis-inflation model capable of explaining the SGWB signal observed by the PTA collaborations while also overcoming the initial singularity problem associated with the inflationary cosmology. Furthermore, our model predicts distinctive features in the SGWB spectrum, which might be examined by forthcoming space-based gravitational wave experiments.Comment: 20 pages, 8 figures; references added, published in EPJ

Solving and Completing the Rabi-Stark Model in the Ultrastrong Coupling Regime

In this work,we employ a unitary transformation with a suitable parameter to convert the quantum Rabi-Stark model into a Jaynes-Cummings-like model. Subsequently, we derive the analytical energy spectra in the ultrastrong coupling regime. The energy spectra exhibit a phenomenon known as spectral collapse, indicating the instability of the model due to the unboundedness of its energy from below at higher coupling parameters. To stabilize the Rabi-Stark model, we introduce a nonlinear photon-photon interaction term. We then compare the modified model with the original model in the classical oscillator (CO) limit. Interestingly, we observe a regular "staircase" pattern in the mean photon number of the ground state. This pattern exhibits a fixed slope and equal step width, which we determine analytically. Moreover, we analytically determine the phase boundary, which slightly differs from that in the original Rabi-Stark model. These findings offer insights into the investigation of those superradiant phase transitions that are unbounded from below due to the phenomenon of spectral collapse.Comment: 11 pages, 7 figure