ar X iv : 0 71 1 . 39 28 v 1 [ m at h . N A ] 2 5 N ov 2 00 7 A POSTERIORI ERROR ESTIMATES IN THE MAXIMUM NORM FOR PARABOLIC PROBLEMS ∗

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 . 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.

ar X iv : 0 71 1 . 27 39 v 1 [ m at h . N T ] 1 9 N ov 2 00 7 ASYMPTOTIC COHOMOLOGY OF CIRCULAR UNITS

— Let F be a number field, abelian over Q, and fix a prime p 6= 2. Consider the cyclotomic Zp-extension F∞/F and denote Fn the n th finite subfield and Cn its group of circular units. Then the Galois groups Gm,n = Gal(Fm/Fn) act naturally on the Cm’s (for any m ≥ n >> 0). We compute the Tate cohomology groups Ĥ(Gm,n, Cm) for i = −1, 0 without assuming anything else neither on F nor on p.

ar X iv : 1 71 1 . 04 08 1 v 1 [ m at h . A P ] 1 1 N ov 2 01 7 ON THE SECOND ORDER DERIVATIVE ESTIMATES FOR DEGENERATE PARABOLIC EQUATIONS

δ(t) holds near t = 0 (see (1.3)). Here B 2−2/(βp) p is the Besov space of order 2 − 2/(βp) and β > 0 is the constant related to the asymptotic behavior in (1.3). For instance, if d = 1 and a11(t) = δ(t) = 1 + sin(1/t), then (0.4) holds with β = 1, which actually equals the maximal regularity of the heat equation ut = ∆u. Combining above two results we obtain the second order derivative estimat...