**Abstract: **The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the $O(3,2)$ de Sitter group containing two $O(3,1)$ Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the $Sp\left(4\right)$ group, which serves as the basic language for two-mode squeezed states. Since the $Sp\left(4\right)$ symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.

**Abstract: **Cosmic Dark Fluid is considered as a non-stationary medium, in which electromagnetic waves propagate, and magneto-electric field structures emerge and evolve. A medium-type representation of the Dark Fluid allows us to involve in its analysis the concepts and mathematical formalism elaborated in the framework of classical covariant electrodynamics of continua, and to distinguish dark analogs of well-known medium-effects, such as optical activity, pyro-electricity, piezo-magnetism, electro- and magneto-striction and dynamo-optical activity. The Dark Fluid is assumed to be formed by a duet of a Dark Matter (a pseudoscalar axionic constituent) and Dark Energy (a scalar element); respectively, we distinguish electrodynamic effects induced by these two constituents of the Dark Fluid. The review contains discussions of 10 models, which describe electrodynamic effects induced by Dark Matter and/or Dark Energy. The models are accompanied by examples of exact solutions to the master equations, correspondingly extended; applications are considered for cosmology and space-times with spherical and pp-wave symmetries. In these applications we focused the attention on three main electromagnetic phenomena induced by the Dark Fluid: first, emergence of Longitudinal Magneto-Electric Clusters; second, generation of anomalous electromagnetic responses; third, formation of Dark Epochs in the Universe history.

**Abstract: **In this paper, we investigated inflationary solutions for a subclass of Horndeski models where a scalar field is non-minimally coupled with the Gauss–Bonnet invariant. Examples of canonical scalar field and *k*-essence to support the early-time acceleration are considered. The formalism to calculate the perturbations in a Friedmann–Robertson–Walker (FRW) universe and to derive the spectral index and the tensor-to-scalar ratio is furnished.

**Abstract: **Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer ( $N\ge 2$ ). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods $T/p$ with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future.

**Abstract: **The drastic increase of websites is one of the causes behind the recent information overload on the internet. A recommender system (RS) has been developed for helping users filter information. However, the cold-start and sparsity problems lead to low performance of the RS. In this paper, we propose methods including the visual-clustering recommendation (VCR) method, the hybrid between the VCR and user-based methods, and the hybrid between the VCR and item-based methods. The user-item clustering is based on the genetic algorithm (GA). The recommendation performance of the proposed methods was compared with that of traditional methods. The results showed that the GA-based visual clustering could properly cluster user-item binary images. They also demonstrated that the proposed recommendation methods were more efficient than the traditional methods. The proposed VCR2 method yielded an F1 score roughly three times higher than the traditional approaches.

**Abstract: **For a given operator $D\left(t\right)$ of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant ${t}_{critical}$ of a spontaneous breakdown of the parity-time alias gain-loss symmetry should be given, in the rigorous language of mathematics, the Kato’s name of an “exceptional point”, ${t}_{critical}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{t}^{\left(EP\right)}$ . In the majority of conventional applications the exceptional point (EP) values are not real. In our paper, we pay attention to several exactly tractable toy-model evolutions for which at least some of the values of ${t}^{\left(EP\right)}$ become real. These values are interpreted as “instants of a catastrophe”, be it classical or quantum. In the classical optical setting the discrete nature of our toy models might make them amenable to simulations. In the latter context the instant of Big Bang is mentioned as an illustrative sample of possible physical meaning of such an EP catastrophe in quantum cosmology.