Long-term evolution of anomalous X-ray pulsars and soft gamma repeaters

We have investigated the long-term evolution of individual anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs) with relatively well constrained X-ray luminosity and rotational properties. In the frame of the fallback disc model, we have obtained the ranges of disc mass and dipole field strength that can produce the observed source properties. We have compared our results with those obtained earlier for dim isolated neutron stars (XDINs). Our results show that (1) the X-ray luminosity, period and period derivative of the individual AXP/SGR sources can be produced self-consistently in the fallback disc model with very similar basic disc parameters to those used earlier in the same model to explain the long-term evolution of XDINs, (2) except two sources, AXP/SGRs are evolving in the accretion phase; these two exceptional sources, like XDINs, completed their accretion phase in the past and are now evolving in the final propeller phase and still slowing down with the disc torques, (3) the dipole field strength (at the poles) of XDINs are in the $10^{11}-10^{12}$ G range, while AXP/SGRs have relatively strong dipole fields between $1-6 \times 10^{12}$ G, and (4) the source properties can be obtained with large ranges of disc masses which do not allow a clear test of correlation between disc masses and the magnetic dipole fields for the whole AXP/SGRs and XDIN population.Comment: 10 pages, 6 figures, Accepted for publication in MNRA

Uncountably many 22-generated just-infinite branch pro-22 groups

The aim of this note is to prove that there are $2^{\aleph_0}$ non-isomoprhic 2-generated just-infinite branch pro-$2$ groups

On the condensation property of the Lamplighter groups and groups of Intermediate growth

The aim of this short note is to revisit some old results about groups of intermediate growth and groups of the lamplighter type and to show that the Lamplighter group $L=\mathbb{Z}_2\wr \mathbb{Z}$ is a condensation group and has a minimal presentation by generators and relators. The condensation property is achieved by showing that $L$ belongs to a Cantor subset of the space $\mathcal{M}_2$ of marked 2-generated groups consisting mostly of groups of intermediate growth

Central Compact Objects: some of them could be spinning up?

Among confirmed central compact objects (CCOs), only three sources have measured period and period derivatives. We have investigated possible evolutionary paths of these three CCOs in the fallback disc model. The model can account for the individual X-ray luminosities and rotational properties of the sources consistently with their estimated supernova ages. For these sources, reasonable model curves can be obtained with dipole field strengths $\sim$ a few $\times 10^9$ G on the surface of the star. The model curves indicate that these CCOs were in the spin-up state in the early phase of evolution. The spin-down starts, while accretion is going on, at a time $t \sim 10^3 - 10^4$ yr depending on the current accretion rate, period and the magnetic dipole moment of the star. This implies that some of the CCOs with relatively long periods, weak dipole fields and high X-ray luminosities could be strong candidates to show spin-up behavior if they indeed evolve with fallback discs.Comment: 5 pages, 3 figures, accepted for publication in MNRA

Universal groups of intermediate growth and their invariant random subgroups

We exhibit examples of groups of intermediate growth with $2^{\aleph_0}$ ergodic, continuous, invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.Comment: Functional Analysis and its Applications, 201