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

ar X iv : h ep - p h / 04 05 11 9 v 1 1 3 M ay 2 00 4 The Lorentz and CPT violating effects on the charged

ar X iv : m at h / 04 05 52 2 v 5 [ m at h . D S ] 2 5 M ay 2 00 5 Dimensions of Julia sets of expanding rational semigroups

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.

ar X iv : h ep - p h / 04 02 19 0 v 2 2 6 M ay 2 00 4 1 Pentaquarks in the 10 F - and 10 F - plets

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...

ar X iv : h ep - t h / 04 12 11 9 v 3 4 M ay 2 00 5 Some remarks on Gopakumar - Vafa invariants

We show that Gopakumar-Vafa (GV) invariants can be expressed in terms of the cohomology ring of moduli space of D-branes without reference to the (sl2)L ⊕ (sl2)R action. We also give a simple construction of this action.