The identities of the free product of a pair of two-element monoids

Up to isomorphism, there exist two non-isomorphic two-element monoids. We show that the identities of the free product of every pair of such monoids admit no finite basis.Comment: 5 page

Energy in ghost-free massive gravity theory

The detailed calculations of the energy in the ghost-free massive gravity theory is presented. The energy is defined in the standard way within the canonical approach, but to evaluate it requires resolving the Hamiltonian constraints, which are known, in general, only implicitly. Fortunately, the constraints can be explicitly obtained and resolved in the spherically symmetric sector, which allows one to evaluate the energy. It turns out that the energy is positive for globally regular and asymptotically flat fields constituting the "physical sector" of the theory. In other cases the energy can be negative and even unbounded from below, which suggests that the theory could be still plagued with ghost instability. However, a detailed inspection reveals that the corresponding solutions of the constraints are either not globally regular or not asymptotically flat. Such solutions cannot describe initial data triggering ghost instability of the physical sector. This allows one to conjecture that the physical sector could actually be protected from the instability by a potential barrier separating it from negative energy states.Comment: 35 pages, minor improvements, an appendix adde

De Sitter vacua in ghost-free massive gravity theory

We present a simple procedure to obtain all de Sitter solutions in the ghost-free massive gravity theory by using the Gordon ansatz. For these solutions the physical metric can be conveniently viewed as describing a hyperboloid in 5D Minkowski space, while the flat reference metric depends on the Stuckelberg field $T(t,r)$ that satisfies the equation $(\partial_t{T})^2-(\partial_r T)^2=1$. This equation has infinitely many solutions, hence there are infinitely many de Sitter vacua with different physical properties. Only the simplest solution with $T=t$ has been previously studied since it is manifestly homogeneous and isotropic, but it is unstable. However, other solutions could be stable. We require the timelike isometry to be common for both metrics, and this gives physically distinguished solutions since only for them the canonical energy is time-independent. We conjecture that these solutions minimize the energy and are therefore stable. We also show that in some cases solutions can be homogeneous and isotropic in a non-manifest way such that their symmetries are not obvious. All of this suggests that the theory may admit viable cosmologies.Comment: 14 pages, 1 figure, references adde

Random environment on coloured trees

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ101 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm