ar X iv : 0 71 1 . 03 12 v 1 [ m at h . C O ] 2 N ov 2 00 7 Period Lengths for Iterated Functions . ( Preliminary Version )

ar X iv : 0 71 1 . 03 12 v 1 [ m at h . C O ] 2 N ov 2 00 7 Period Lengths for Iterated Functions . ( Preliminary Version ) Eric

Let Ωn be the n -element set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f , f , f (4) . . . must eventually repeat. Let T(f) be the period of this eventually periodic sequence of functions, i.e. the least positive integer T such that, for all m ≥ n,...

ar X iv : 0 71 1 . 23 72 v 1 [ m at h . G R ] 1 5 N ov 2 00 7 Braid groups and Artin groups

ar X iv : m at h / 03 11 30 1 v 1 [ m at h . N T ] 1 8 N ov 2 00 3 ON THE ESTIMATION OF Z 2 ( s ) Aleksandar

Estimates for Z2(s) = ∫∞ 1 |ζ(12 + ix)|4x−s dx (Re s > 1) are discussed, both pointwise and in the mean square. It is shown how these estimates can be used to bound E2(T ), the error term in the asymptotic formula for ∫ T 0 |ζ(12 + it)|4 dt.

ar X iv : 0 71 1 . 32 82 v 1 [ he p - ph ] 2 1 N ov 2 00 7 Study of Pure Annihilation Decays B d , s → D 0 D 0

With heavy quark limit and hierarchy approximation λ QCD ≪ m D ≪ m B , we analyze the

ar X iv : 0 71 1 . 26 95 v 1 [ m at h . SP ] 1 6 N ov 2 00 7 REGULARITY AND THE CESÀRO – NEVAI CLASS

We consider OPRL and OPUC with measures regular in the sense of Ullman–Stahl–Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on [−2, 2] implies lim N →∞ N −1 [ N n=1 (a n −1) 2 +b 2 n ] = 0.