ar X iv : 0 71 1 . 26 23 v 1 [ m at h . O C ] 16 N ov 2 00 7 THE VIRTUAL PRIVATE NETWORK DESIGN TREE ROUTING CONJECTURE FOR OUTERPLANAR NETWORKS

ar X iv : 0 71 1 . 26 23 v 3 [ m at h . O C ] 2 4 N ov 2 00 8 THE VPN TREE ROUTING CONJECTURE FOR OUTERPLANAR NETWORKS

The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of t...

ar X iv : m at h . C O / 0 41 16 10 v 1 2 7 N ov 2 00 4 CHAIN POLYNOMIALS OF DISTRIBUTIVE LATTICES ARE 75 %

It is shown that the numbers ci of chains of length i in the proper part L \ {0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c0 < . . . < c⌊l/2⌋ and c⌊3l/4⌋ > . . . > cl. This proves 75 % of the inequalities implied by the Neggers unimodality conjecture.

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 : 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 58 v 1 [ m at h . A G ] 1 7 N ov 2 00 7 Rational curves of degree 11 on a general quintic threefold ∗

We prove the “strong form” of the Clemens conjecture in degree 11. Namely, on a general quintic threefold F in P, there are only finitely many smooth rational curves of degree 11, and each curve C is embedded in F with normal bundle O(−1) ⊕ O(−1). Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with ration...