We introduce a concept of "a-reducible configuration" and show that if every permutation of [n] lacking at least r adjacencies contains an a-reducible configuration, then f(n) £ an+c_r for some constant c_r. Here an ädjacency" is a contiguous appearance of successive values in the permutation.

We present some configurations that are 13/8-reducible, seeking an asymptotic improvement in the upper bound. Say that two blocks (strings of adjacent elements) are ßuccessive" if the largest element of one block and the smallest of the other are successive in the size order. We show that every configuration having eight successive blocks, two sets of seven successive blocks, or four sets of five successive blocks is 13/8-reducible. We also put severe restrictions on permutations that have no strictly 5/3-reducible configuration.

The discovery of ground states solutions for NLS in generic dimension is due to W. Strauss. The issue of stability was settled independently by J. Shatah and M. I. Weinstein. These early works do not touch the issue of the asymptotic behaviour of solutions of NLS close to a ground state. In particular, they do not discuss whether these solutions look more and more like a fixed ground state, with the difference given by radiation dispersing in space. This issue remains largely open. The talk focuses on our positive solution to this problem, if some special hypotheses are satisfied. We sketch the framework, some of the difficulties, some of the facts that we try to exploit. Technically the core of our proof involves ideas from scattering theory for Schrödinger operators by Kato, Kato Jensen, Jensen, Yajima. However we will NOT discuss this in detail.

I also present a new result by exhibiting an uncountable set of points (of measure zero) on the unit circle where K(x) diverges.