[en] The spectrum of gravitational waves emitted after the formation of a black hole is dominated by damped oscillations, or ringing tones, represented by quasinormal modes. An analogy with the natural frequencies of a bell is quite fitting, since any system with a continuous spectrum permits complex frequency resonances. Some applications of spectral techniques are considered here and the following question is posed: to what extent is the geometry determined by the quasinormal mode spectrum?

[en] We study limits of vacuum, isotropic universes in the full, effective, four-dimensional theory with higher derivatives. We show that all flat vacua as well as general curved ones are globally attracted by the standard, square root scaling solution at early times. Open vacua asymptote to horizon-free, Milne states in both directions while closed universes exhibit more complex logarithmic singularities, starting from initial data sets of a possibly smaller dimension. We also discuss the relation of our results to the asymptotic stability of the passage through the singularity in ekpyrotic and cyclic cosmologies.

[en] We consider a statistical system in a planar wedge, for values of the bulk parameters corresponding to a first order phase transition and with boundary conditions inducing phase separation. Our previous exact field theoretical solution for the case of a single interface is extended to a class of systems, including the Blume–Capel model as the simplest representative, allowing for the appearance of an intermediate layer of a third phase. We show that the interfaces separating the different phases behave as trajectories of vicious walkers, and determine their passage probabilities. We also show how the theory leads to a remarkable form of wedge covariance, i.e. a relation between properties in the wedge and in the half plane, which involves the appearance of self-Fourier functions.

[en] We present the class of regular homogeneous T-models with vacuum dark fluid, associated with a variable cosmological term. The vacuum fluid is defined by the symmetry of its stress-energy tensor, i.e., its invariance under Lorentz boosts in a distinguished spatial direction (p_j = -ρ), which makes this fluid essentially anisotropic and allows its density to evolve. Typical features of homogeneous regular T-models are: the existence of a Killing horizon; beginning of the cosmological evolution from a null bang at the horizon; the existence of a regular static pre-bang region visible to cosmological observers; creation of matter from anisotropic vacuum, accompanied by very rapid isotropization. We study in detail the spherically symmetric regular T-models on the basis of a general exact solution for a mixture of the vacuum fluid and dust-like matter and apply it to give numerical estimates for a particular model which illustrates the ability of cosmological T-models to satisfy the observational constraints

[en] Homotopy perturbation method (HPM) proposed by Ji-Huan He is very effective and convenient for single-degree-of-freedom systems. In this paper a coupling technique of He's method and precise integration method (PIM) is suggested to solve multi-degree-of-freedom nonlinear dynamic systems. The new technique keeps the merits of the two methods. Some examples are given to illustrate its effectiveness and convenience. Furthermore the obtained solution is of high accuracy

[en] In this work we extend the range of applicability of a method recently introduced where coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. Some general consequences of the present extension are then commented

[en] We prove that Ricci-flat vacuum exact solutions are stable under linear perturbations in a new class of weakly non-local gravitational theories finite at the quantum level.

[en] We solve analytically the renormalization-group equation for the potential of the $O(N)$-symmetric scalar theory in the large-N limit and in dimensions $2<d<4$, in order to look for nonperturbative fixed points that were found numerically in a recent study. We find new real solutions with singularities in the higher derivatives of the potential at its minimum, and complex solutions with branch cuts along the negative real axis.

[en] We study synchronization properties of networks of coupled dynamical systems with interaction akin to diffusion. We assume that the isolated node dynamics possesses a forward invariant set on which it has a bounded Jacobian, then we characterize a class of coupling functions that allows for uniformly stable synchronization in connected complex networks—in the sense that there is an open neighbourhood of the initial conditions that is uniformly attracted towards synchronization. Moreover, this stable synchronization persists under perturbations to non-identical node dynamics. We illustrate the theory with numerical examples and conclude with a discussion on embedding these results in a more general framework of spectral dichotomies. (paper)

[en] The false vacuum states are unstable and they decay by tunneling. Some of them may survive up to times when their survival probability has a non-exponential form. At times much latter than the transition time, when contributions to the survival probability of its exponential and non-exponential parts are comparable, the survival probability as a function of time t has an inverse power-like form. We show that at this time region the instantaneous energy of the false vacuum states tends to the energy of the true vacuum state as 1/t² for t → ∞. (author)