ar X iv : 0 80 2 . 42 23 v 1 [ m at h . Q A ] 2 8 Fe b 20 08 Local analytic classification of q - difference equations

ar X iv : 0 80 1 . 07 10 v 2 [ m at h . C V ] 1 1 Fe b 20 08 KOPPELMAN FORMULAS AND THE ∂̄ - EQUATION ON ANALYTIC VARIETIES

Let Z be an analytic subvariety of pure codimension p of a pseudoconvex set X in C n. We introduce a formalism to generate weighted Koppelman formulas on Z that provide solutions to the ¯ ∂-equation. We prove that if φ is a smooth (0, q + 1)-form on Z with ¯ ∂φ = 0, then there is a smooth (0, q)-form ψ on Z reg with at most polynomial growth at Z sing such that ¯ ∂ψ = φ. The integral formulas a...

ar X iv : 0 80 2 . 30 10 v 1 [ m at h . Q A ] 2 1 Fe b 20 08 THE OPERAD LIE IS FREE

We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e.

ar X iv : 0 80 2 . 38 39 v 1 [ m at h . G R ] 2 6 Fe b 20 08 Quadratic equations over free groups are NP - complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

ar X iv : 0 80 4 . 22 27 v 1 [ m at h . A G ] 1 4 A pr 2 00 8 On equations of double planes with p g = q = 1

This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with pg = q = 1 and K = 2, . . . , 8. A double plane with K = 8 having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma. 2000 Mathematics Classification: 14J29, 14Q05.

ar X iv : 0 80 1 . 48 21 v 2 [ he p - ph ] 2 8 Fe b 20 08 X ( 1835 ) and η ( 1760 ) observed by BES Collaboration