Algebraic Equations With Continuous Coefficients and Some Problems of the Algebraic Theory of Braids

Abstract

The asymptotic-energy limit of the density P ( S ) of spacings between adjacent levels of the two-dimensional harmonic oscillator ( T D H O ) spectrum is studied. It is shown that in any integer segment [ M , M + 11, containing -h f / a levels, of the TDHO spectrum m + an, P ( S ) has the form 1 w , S ( S -S,) where i takes on at most three values. For large M , P ( S ) displays strong level repulsion for irrational a, but it does not settle on a stationary form nor does its average over M . This is in marked contrast with the behaviour of generic integrable systems for which the Poisson statistics, P ( S ) = exp(-S), is known to apply.

The homology of the Milnor fiber for classical braid groups

“…‫ޚ‬ ! 1; tells us that the homology H .Br.n/I A/ is H .Br.n/ 0 I K/ as an A-module (see for example [22], [11] or [7]); since for n ¤ 3; 4, Br.n/ 00 D Br.n/ 0 (see [18] for a proof of this), we have that H 0 .Br.n/ 0 I K/ D K , H 1 .Br.n/ 0 I K/ D 0. Moreover the A-action on H 0 is trivial and so H 0 .Br.n/I A/ D A=.q C 1/ as an A-module.…”

Untitled

“…The case n = 4 is particularly interesting: Γ 2 (B 4 (S 2 )) is a semi-direct product of the quaternion group Q 8 of order 8 by the free group of rank 2. This may be compared with Gorin and Lin's result for Γ 2 (B 4 ) [32]. Thus B 4 (S 2 ) contains an isomorphic copy of Q 8 .…”

“…In Section 3, we study the derived series of B n (S 2 ). As in the case of B n [32], (B n (S 2 )) (1) is perfect if n ≥ 5; in other words, the derived series of B n (S 2 ) is constant from (B n (S 2 )) (1) onwards. The cases n = 1, 2, 3 are straightforward, and the groups B n (S 2 ) are finite and soluble.…”

“…Using the Reidemeister-Schreier rewriting process, Gorin and Lin obtained a presentation of the commutator subgroup of B n for n ≥ 3 [32]. For n ≥ 5, they were able to infer that (B n )…”

Book Review: Braid groups (With the graphical assistance of Olivier Dodane); Ordering braids

“…• Gorin and Lin [23] regarded braids as the start-point for a way to generalize Galois theory. We single out, from their very many contributions, an open problem about braids which they solved in the context of complex function theory: the structure of the commutator subgroup of B n was first uncovered in [23].…”