Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Two bracketing schemes for the Penn Treebank

The trees in the Penn Treebank have a standard representation that involves complete balanced bracketing. In this article, an alternative for this standard representation of the tree bank is proposed. The proposed representation for the trees is loss-less, but it reduces the total number of brackets by 28%. This is possible by omitting the redundant pairs of special brackets that encode initial and final embedding, using a technique proposed by Krauwer and des Tombe (1981). In terms of the paired brackets, the maximum nesting depth in sentences decreases by 78%. The 99.9% coverage is achieved with only five non-top levels of paired brackets. The observed shallowness of the reduced bracketing suggests that finite-state based methods for parsing and searching could be a feasible option for tree bank processing.

Two bracketing schemes for the Penn Treebank

The trees in the Penn Treebank have a standard representation that involves complete balanced bracketing. In this article, an alternative for this standard representation of the tree bank is proposed. The proposed representation for the trees is loss-less, but it reduces the total number of brackets by 28%. This is possible by omitting the redundant pairs of special brackets that encode initial and final embedding, using a technique proposed by Krauwer and des Tombe (1981). In terms of the paired brackets, the maximum nesting depth in sentences decreases by 78%. The 99.9% coverage is achieved with only five non-top levels of paired brackets. The observed shallowness of the reduced bracketing suggests that finite-state based methods for parsing and searching could be a feasible option for tree bank processing.

Actuarial report 2011: Social security schemes administered by Kela 2010-2060

Projections concerning the long-term outlook of the social security schemes administered by the Social Insurance Institution of Finland (Kela) are made regularly by the Institution’s Actuarial Section. The report at hand was compiled with the help of an aggregate model devised by the Actuarial Section. In the model, various universal factors influencing benefit trends are consolidated and the interactions between individual benefits are taken into account. The demographic forecasts underlying the report have also been made by the Actuarial Section. Estimates of income and administrative costs have been made in cooperation with the Financial Planning Section.