ar X iv : m at h / 06 08 17 8 v 1 [ m at h . O C ] 7 A ug 2 00 6 A regularity theory for multiple - valued Dirichlet minimizing maps ∗

ar X iv : m at h / 06 08 16 3 v 1 [ m at h . L O ] 7 A ug 2 00 6 IND - AND PRO - DEFINABLE SETS

We describe the indand procategories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

ar X iv : m at h / 06 04 63 5 v 1 [ m at h . A P ] 2 8 A pr 2 00 6 PARTIAL REGULARITY FOR HARMONIC MAPS , AND RELATED PROBLEMS

Via gauge theory, we give a new proof of partial regularity for harmonic maps in dimensions m ≥ 3 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of ”minimal” C regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.

ar X iv : m at h / 06 08 37 7 v 1 [ m at h . R T ] 1 5 A ug 2 00 6 ALGEBRAS OF DERIVED DIMENSION ZERO

ar X iv : h ep - p h / 06 08 05 0 v 1 4 A ug 2 00 6 Percolation and high energy cosmic rays above 10 17 eV

ar X iv : m at h / 06 08 73 5 v 1 [ m at h . C O ] 2 9 A ug 2 00 6 SUFFICIENT CONDITIONS FOR LABELLED 0 – 1 LAWS

Exponentiating a power series can have the effect of smoothing out the behavior of the coefficients. In this paper we look at conditions on the growth of the coefficients of G(x) = ∑ g(n)x, where g(n) ≥ 0, which ensure that f(n−1)/f(n)→ ∞, where F(x) = e. Useful notation will be f(n) ≺ g(n) for f(n) eventually less than g(n) and f(n) ∈ RT∞ for f(n− 1)/f(n) → ∞; the notation RT stands for the ra...