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.

Let n be any positive integer and ℤ_n = {0, 1, …, n - 1} be the ring of integers modulo n. Let X_n be the set of all nonzero, nonunits of ℤ_n, and G_n the group of all units of ℤ_n. In this paper, by considering the regular representation π : G_n → Sym(X_n), the following are investigated as follows: (1) G_n is not fixed-point free; (2) If Fix(g) = {x ∈ X_n : gx = x }≠ ∅ for some g ∈ G_n, then Fix(g) is a union of orbits under the regular action of G_n on X_n; (3) B(⊆ o(x₁) ∪ ⋯ ∪ o(x_ℓ)) is a set of imprimitivity under π for some orbits o(x₁), …, o(x_ℓ) if and only if B = g₁H₁x₁∪⋯∪g_ℓH_ℓx_ℓ for some subgroups H₁, …, H_ℓ of G_n and some elements g₁, …, g_ℓ ∈ G_n satisfying that if (gB)∩B ≠ ∅ for some g ∈ G, then g ∈ stab(x_i)H_i for each i = 1, …, ℓ.

We prove that each residuated commutative ℓ-monoid is a monoid of all residuated endomorphisms of some universal algebra. This result demonstrates that similarly to groups and semigroups, a monoidal operation of a residuated commutative ℓ-monoid is represented by a composition of endomorphisms that are moreover, residuated.