Guess & Check Codes for Deletions and Synchronization

We consider the problem of constructing codes that can correct $\delta$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions $(\delta=1)$ with an asymptotically optimal redundancy. Finding similar codes for $\delta \geq 2$ deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions $\delta$ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asymptotically optimal redundancy that is $\Theta(\delta \log n)$. We provide deterministic polynomial time encoding and decoding schemes for these codes. We also describe the applications of GC codes to file synchronization.Comment: Accepted in ISIT 201

Cold reactive and non-reactive collisions of Li and Rb with C2−_2^-: implications for hybrid trap experiments

We present a theoretical investigation of cold reactive and non-reactive collisions of Li and Rb atoms with C$_{2}^{-}$. The potential energy surfaces for the singlet and triplet states of the Li--C$_{2}^{-}$ and Rb--C$_{2}^{-}$ systems have been obtained using the CASSCF/ic-MRCI+Q approach with extended basis sets. The potential energy surfaces are then used to investigate the associative detachment reaction and to calculate rotationally inelastic cross sections by means of the close-coupling method. The effect of the core correlation on the potential energy surfaces is discussed and we estimate the error on the collisional cross sections induced by freezing the $1s$ orbitals of the carbon atoms. The results are compared to those obtained for the Rb-OH$^{-}$ system and the applications for hybrid trap experiments are explored. Furthermore, we discuss the possibility to perform Doppler thermometry on the C$_{2}^{-}$ anion and investigate the collision process involving excited states. The implications for sympathetic cooling experiments are also discussed

A stable classification of Lefschetz fibrations

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with f^0_g they become isomorphic. As a consequence, any two compact integral symplectic 4-manifolds with the same values of (c_1^2, c_2, c_1.[w], [w]^2) become symplectomorphic after blowups and symplectic sums with f^0_g.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper6.abs.htm

Reactivity of hydrated hydroxide anion cluster OH(H2_{2}O)n−_{n}^{-} with H and Rb: an ab initio study

We present a theoretical investigation of the hydrated hydroxide anion clusters OH(H$_{2}$O)$_{n}^{-}$ and of the collisional complexes H-OH(H$_{2}$O)$_{n}^{-}$ and Rb-OH(H$_{2}$O)$_{n}^{-}$ (with n$=1-4$). The MP2 and CCSD(T) methods are used to calculate interaction energies, optimized geometries and vertical detachment energies. Part of the potential energy surfaces are explored with a focus on the autodetachment region. We point out the importance of diffuse functions to correctly describe the latter. We use our results to discuss the different water loss and electronic detachment channels which are the main reaction routes at room temperature. In particular, we have considered a direct and an indirect process for the electronic detachment, depending on whether water loss follows or precedes the detachment of the excess electron. We use our results to discuss the implication for astrochemistry and hybrid trap experiments in the context of cold chemistry