We prove for small e and ~ satisfying a certain Diophantine condition the operatorhas pure point spectrum for almost all 0. A similar result is established at low d 2 energy for H = -dx--5 -K 2 (cos 2nx + cos 2n (~x + 0)) provided K is sufficiently large.

No abstract

Abstract. In this paper we present a computer assisted proof of the existence of a solution for the Feigenbaum equation 'x = 1 ''x: There exist by now various such proofs in the literature. Although the one presented here is new, the main purpose of this paper is not to provide yet another version, but to give an easy to read and self contained introduction to the technique of computer assisted proofs in analysis. Our proof is written in Prolog Programming in logic, a programming language which we found to be well suited for this purpose. In this paper we also give a n introduction to Prolog, so that even a reader without prior exposure to programming should beable to verify the correctness of the proof.

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. This corresponds to studying the stationary fluid flow past a body. The necessity to truncate for numerical purposes the infinite exterior domain to a finite domain leads to the problem of finding appropriate boundary conditions on the surface of the truncated domain. We solve this problem by providing a vector field describing the leading asymptotic behavior of the solution. This vector field is given in the form of an explicit expression depending on a real parameter. We show that this parameter can be determined from the total drag exerted on the body. Using this fact we set up a self-consistent numerical scheme that determines the parameter, and hence the boundary conditions and the drag, as part of the solution process. We compare the values of the drag obtained with our adaptive scheme with the results from using traditional constant boundary conditions. Computational times are typically reduced by several orders of magnitude

We analyse iterations of maps on an interval with an added noise term, in the neighbourhood of an intermittency threshold. We rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise.