Narrow Results

Because of their negligible mass and size, asteroids act as particle test in the gravitational field of the Sun (or the solar system at large); hence the knowledge of their orbit can provide local tests of general relativity. In addition to the "3D census of the Galaxy" the space-mission Gaia will enable, this ESA astrometric mission will provide highly accurate positions of a large number of solar systems objects. Given the expected nominal precision — ranging from a few milli-arcseconds for the faintest bodies down to sub-mas precision for the brightest ones — one can expect to perform the classical perihelion precession test of the GR from the motion of the asteroids. We present preliminary results of a variance analysis involving realistic simulations of a subset of asteroids including Near-Earth objects and main-belt asteroids that will be observed by Gaia. These show the formal precision achievable for the joint determination of the Solar J₂ together with the PPN parameter β, as well as the precision for and the link of the dynamical reference frame to the kinematically non-rotating conventional ICRS.

We devise an algorithm which, given a bounded automaton $A$, decides whether the group generated by $A$ is finite. The solution comes from a description of the infinite sequences having an infinite $A$-orbit using a deterministic finite-state acceptor. This acceptor can also be used to decide whether the bounded automaton acts level-transitively.

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λ_n = K ⌊x₁, …, x_n⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λ_n. It is proved that δ is a unique sum δ = δ^ev + δ^od of an even and odd skew derivation. Explicit formulae are given for δ^ev and δ^od via the elements δ (x₁), …, δ (x_n). It is proved that the set of all even skew derivations of Λ_n coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λ_n. In particular, Der_K(Λ_n) is a faithful but not simple Aut_K(Λ_n)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λ_n are found. It is proved that the set of generic normal elements of Λ_n that are not units forms a single Aut_K(Λ_n)-orbit (namely, Aut_K(Λ_n)x₁) if n is even and two orbits (namely, Aut_K(Λ_n)x₁ and Aut_K(Λ_n)(x₁ + x₂ ⋯ x_n)) if n is odd.

Let $b\ge 2$ be an integer, and write the base $b$ expansion of any non-negative integer $n$ as $n=x_0+x_{1}b+\cdots +x_d{b}^d$, with $x_d>0$ and $0\le x_i<b$ for $i=0,\dots ,d$. Let $\varphi (x)$ denote an integer polynomial such that $\varphi (n)>0$ for all $n>0$. Consider the map $S_{\varphi ,b}:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$, with $S_{\varphi ,b}(n):=\varphi (x_0)+\cdots +\varphi (x_d)$. It is known that the orbit set $\{n,S_{\varphi ,b}(n),S_{\varphi ,b}(S_{\varphi ,b}(n)),\dots ,\}$ is finite for all $n>0$. Each orbit contains a finite cycle, and for a given $b$, the union of such cycles over all orbit sets is finite.

Fix now an integer $\ell \ge 1$ and let $\varphi (x)=x^2$. We show that the set of bases $b\ge 2$ which have at least one cycle of length $\ell$ always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.