[ m at h . N T ] 4 M ay 2 00 4 THE CONJUGATE DIMENSION OF ALGEBRAIC NUMBERS

2 9 M ay 2 00 4 A refined version of the Siegel - Shidlovskii theorem

Using Y.André's result on differential equations staisfied by E-functions, we derive an improved version of the Siegel-Shidlovskii theorem. It gives a complete characterisation of algebraic relations over the algebraic numbers between values of E-functions at any non-zero algebraic point.

ar X iv : m at h / 04 11 14 0 v 2 [ m at h . N T ] 4 M ay 2 00 5 THE 3 x + 1 SEMIGROUP

The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...

m at h . A T ] 2 6 M ay 2 00 5 Chromatic numbers , morphism complexes , and Stiefel - Whitney characteristic classes . Dmitry N . Kozlov

ar X iv : 0 80 4 . 14 18 v 2 [ m at h . D G ] 2 7 M ay 2 00 8 SMOOTH YAMABE INVARIANT AND SURGERY

We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N) ≥ min{σ(M),Λn}.

ar X iv : m at h / 04 05 40 1 v 1 [ m at h . G N ] 2 1 M ay 2 00 4 VARIATIONS ON KURATOWSKI ’ S 14 - SET THEOREM

Kuratowski’s 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. In this article we consider the analogous questions for any possible subcollection of the operations {closure, complement, interior, intersection, union}, and any number of initially given sets. We use the al...