ar X iv : 0 90 1 . 35 09 v 1 [ m at h . C O ] 2 2 Ja n 20 09 Catalan numbers and relations ∗

ar X iv : 0 90 6 . 48 35 v 1 [ m at h . O C ] 2 6 Ju n 20 09 The Complex Gradient Operator and the CR - Calculus

ar X iv : 0 90 1 . 42 99 v 1 [ m at h . C O ] 2 7 Ja n 20 09 TRIANGLE - FREE TRIANGULATIONS

The flip operation on colored inner-triangle-free triangulations of a convex polygon is studied. It is shown that the affine Weyl group e Cn acts transitively on these triangulations by colored flips, and that the resulting colored flip graph is closely related to a lower interval in the weak order on e Cn. Lattice properties of this order are then applied to compute the diameter.

ar X iv : 0 90 1 . 38 34 v 1 [ m at h . N T ] 2 4 Ja n 20 09 On the Inverse Problem Relative to Dynamics of the w Function

In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.

ar X iv : 0 90 1 . 46 64 v 1 [ cs . L O ] 2 9 Ja n 20 09 Square root meadows ∗

Let Q0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0 = 0. Q0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q0(s, √ ) of the signed rationals in which every number has a unique square root.

ar X iv : 0 90 1 . 18 06 v 1 [ m at h . A G ] 1 3 Ja n 20 09 GREENBERG APPROXIMATION AND THE GEOMETRY OF ARC SPACES

We study the differential properties of generalized arc schemes and geometric versions of Kolchin’s Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.