Average Symmetry and Complexity of Binary Sequences

The concept of complexity as average symmetry is here formalised by introducing a general expression dependent on the relevant symmetry and a related discrete set of transformations. This complexity has hybrid features of both statistical complexities and of those related to algorithmic complexity. Like the former, random objects are not the most complex while they still are more complex than the more symmetric ones (as in the latter). By applying this definition to the particular case of rotations of binary sequences, we are able to find a precise expression for it. In particular, we then analyse the behaviour of this measure in different well-known automatic sequences, where we find interesting new properties. A generalisation of the measure to statistical ensembles is also presented and applied to the case of i.i.d. random sequences and to the equilibrium configurations of the one-dimensional Ising model. In both cases, we find that the complexity is continuous and differentiable as a function of the relevant parameters and agrees with the intuitive requirements we were looking for.Comment: 9 pages, 5 figure

Interactive mixture of inhomogeneous dark fluids driven by dark energy: a dynamical systems analysis

We examine the evolution of an inhomogeneous mixture of non-relativistic pressureless cold dark matter (CDM), coupled to dark energy (DE) characterised by the equation of state parameter $w<-1/3$, with the interaction term proportional to the DE density. This coupled mixture is the source of a spherically symmetric Lema\^\ itre-Tolman-Bondi (LTB) metric admitting an asymptotic Friedman-Lema\^\ itre-Robertson-Walker (FLRW) background. Einstein's equations reduce to a 5-dimensional autonomous dynamical system involving quasi--local variables related to suitable averages of covariant scalars and their fluctuations. The phase space evolution around the critical points (past/future attractors and five saddles) is examined in detail. For all parameter values and both directions of energy flow (CDM to DE and DE to CDM) the phase space trajectories are compatible with a physically plausible early cosmic times behaviour near the past attractor. This result compares favourably with mixtures with the interaction driven by the CDM density in which conditions for a physically plausible past evolution are more restrictive. Numerical examples are provided describing the evolution of an initial profile that can be associated with idealised structure formation scenariosComment: 23 pages, IOP format, 8 figure

On the weakness of disc models in bright ULXs

It is sometimes suggested that phenomenological power-law plus cool disc-blackbody models represent the simplest, most robust interpretation of the X-ray spectra of bright ultraluminous X-ray sources (ULXs); this has been taken as evidence for the presence of intermediate-mass black holes (BHs) (M ~ 10^3 Msun) in those sources. Here, we assess this claim by comparing the cool disc-blackbody model with a range of other models. For example, we show that the same ULX spectra can be fitted equally well by subtracting a disc-blackbody component from a dominant power-law component, thus turning a soft excess into a soft deficit. Then, we propose a more complex physical model, based on a power-law component slightly modified at various energies by smeared emission and absorption lines from highly-ionized, fast-moving gas. We use the XMM-Newton/EPIC spectra of two ULXs in Holmberg II and NGC 4559 as examples. Our main conclusion is that the presence of a soft excess or a soft deficit depends on the energy range over which we choose to fit the ``true'' power-law continuum; those small deviations from the power-law spectrum are well modelled by disc-blackbody components (either in emission or absorption) simply because they are a versatile fitting tool for most kinds of smooth, broad bumps. Hence, we argue that those components should not be taken as evidence for accretion disc emission, nor used to infer BH masses. Finally, we speculate that bright ULXs could be in a spectral state similar to (or an extension of) the steep-power-law state of Galactic BH candidates, in which the disc is now completely comptonized and not directly detectable, and the power-law emission may be modified by the surrounding, fast-moving, ionized gas.Comment: 12 pages, accepted by MNRA