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.

Confronting Confucian Understandings of the Christian Doctrine of Salvation : A systematic theological analysis of the basic problems in the Confucian-Christian dialogue

The aim of the present study is to analyze Confucian understandings of the Christian doctrine of salvation in order to find the basic problems in the Confucian-Christian dialogue. I will approach the task via a systematic theological analysis of four issues in order to limit the thesis to an appropriate size. They are analyzed in three chapters as follows: 1. The Confucian concept concerning the existence of God. Here I discuss mainly the issue of assimilation of the Christian concept of God to the concepts of Sovereign on High (Shangdi) and Heaven (Tian) in Confucianism. 2. The Confucian understanding of the object of salvation and its status in Christianity. 3. The Confucian understanding of the means of salvation in Christianity. Before beginning this analysis it is necessary to clarify the vast variety of controversies, arguments, ideas, opinions and comments expressed in the name of Confucianism; thus, clear distinctions among different schools of Confucianism are given in chapter 2. In the last chapter I will discuss the results of my research in this study by pointing out the basic problems that will appear in the analysis. The results of the present study provide conclusions in three related areas: the tacit differences in the ways of thinking between Confucians and Christians, the basic problems of the Confucian-Christian dialogue, and the affirmative elements in the dialogue. In addition to a summary, a bibliography and an index, there are also eight appendices, where I have introduced important background information for readers to understand the present study.

On the Benefits and Problems of the Object-Oriented Paradigm including a Finnish Study

Several researchers are of the opinion that there are many benefits in using the object-oriented paradigm in information systems development. If the object-oriented paradigm is used, the development of information systems may, for example, be faster and more efficient. On the other hand, there are also several problems with the paradigm. For example, it is often considered complex, it is often difficult to make use of the reuse concept and it is still immature in some areas. Although there are several interesting features in the object-oriented paradigm, there is still little comprehensive knowledge of the benefits and problems associated with it. The objective of the following study was to investigate and to gain more understanding of the benefits and problems of the object-oriented paradigm. A review of previous studies was made and twelve benefits and twelve problems were established. These benefits and problems were then analysed, studied and discussed. Further a survey and some case studies were made in order to get some knowledge on what benefits and problems with the object-oriented paradigm Finnish software companies had experienced. One hundred and four companies answered the survey that was sent to all Finnish software companies with five or more employees. The case studies were made with six large Finnish software companies. The major finding was that Finnish software companies were exceptionally positive towards the object-oriented information systems development and had experienced very few of the proposed problems. Finally two models for further research were developed. The first model presents connections between benefits and the second between problems.