Additive combinatorics methods in associative algebras

We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras. Finally, we give lower bounds for the Minkowski product of two subsets in finite monoids by using their associated monoid algebras.Comment: In this second version, we clarify and extend the domain of validity of Diderrich-Kneser's theorem for associative algebras. We simplify the proofs and we also add a section on Kneser's and Hamidoune's theorem in monoi

Phase synchronization in tilted deterministic ratchets

We study phase synchronization for a ratchet system. We consider the deterministic dynamics of a particle in a tilted ratchet potential with an external periodic forcing, in the overdamped case. The ratchet potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of external periodic forcing. This oscillator has an intrinsic frequency that can be entrained with the frequency of the external driving. We introduced a linear phase through a set of discrete time events and the associated average frequency, and show that this frequency can be synchronized with the frequency of the external driving. In this way, we can properly characterize the phenomenon of synchronization through Arnold tongues, which represent regions of synchronization in parameter space, and discuss their implications for transport in ratchets.Comment: 14 pages, 6 figures, Physica A, in pres

Circumbinary Ring, Circumstellar disks and accretion in the binary system UY Aurigae

Recent exo-planetary surveys reveal that planets can orbit and survive around binary stars. This suggests that some fraction of young binary systems which possess massive circumbinary disks (CB) may be in the midst of planet formation. However, there are very few CB disks detected. We revisit one of the known CB disks, the UY Aurigae system, and probe 13CO 2-1, C18O 2-1, SO 5(6)-4(5) and 12CO 3-2 line emission and the thermal dust continuum. Our new results confirm the existence of the CB disk. In addition, the circumstellar (CS) disks are clearly resolved in dust continuum at 1.4 mm. The spectral indices between the wavelengths of 0.85 mm and 6 cm are found to be surprisingly low, being 1.6 for both CS disks. The deprojected separation of the binary is 1.26" based on our 1.4 mm continuum data. This is 0.07" (10 AU) larger than in earlier studies. Combining the fact of the variation of UY Aur B in $R$ band, we propose that the CS disk of an undetected companion UY Aur Bb obscures UY Aur Ba. A very complex kinematical pattern inside the CB disk is observed due to a mixing of Keplerian rotation of the CB disk, the infall and outflow gas. The streaming gas accreting from the CB ring toward the CS disks and possible outflows are also identified and resolved. The SO emission is found to be at the bases of the streaming shocks. Our results suggest that the UY Aur system is undergoing an active accretion phase from the CB disk to the CS disks. The UY Aur B might also be a binary system, making the UY Aur a triple system.Comment: 14 pages, 11 figures; accepted for publication in Ap