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.

Enhanced processing of SPOT multispectral satellite imagery for environmental monitoring and modelling

The Taita Hills in southeastern Kenya form the northernmost part of Africa’s Eastern Arc Mountains, which have been identified by Conservation International as one of the top ten biodiversity hotspots on Earth. As with many areas of the developing world, over recent decades the Taita Hills have experienced significant population growth leading to associated major changes in land use and land cover (LULC), as well as escalating land degradation, particularly soil erosion. Multi-temporal medium resolution multispectral optical satellite data, such as imagery from the SPOT HRV, HRVIR, and HRG sensors, provides a valuable source of information for environmental monitoring and modelling at a landscape level at local and regional scales. However, utilization of multi-temporal SPOT data in quantitative remote sensing studies requires the removal of atmospheric effects and the derivation of surface reflectance factor. Furthermore, for areas of rugged terrain, such as the Taita Hills, topographic correction is necessary to derive comparable reflectance throughout a SPOT scene. Reliable monitoring of LULC change over time and modelling of land degradation and human population distribution and abundance are of crucial importance to sustainable development, natural resource management, biodiversity conservation, and understanding and mitigating climate change and its impacts. The main purpose of this thesis was to develop and validate enhanced processing of SPOT satellite imagery for use in environmental monitoring and modelling at a landscape level, in regions of the developing world with limited ancillary data availability. The Taita Hills formed the application study site, whilst the Helsinki metropolitan region was used as a control site for validation and assessment of the applied atmospheric correction techniques, where multiangular reflectance field measurements were taken and where horizontal visibility meteorological data concurrent with image acquisition were available. The proposed historical empirical line method (HELM) for absolute atmospheric correction was found to be the only applied technique that could derive surface reflectance factor within an RMSE of < 0.02 ps in the SPOT visible and near-infrared bands; an accuracy level identified as a benchmark for successful atmospheric correction. A multi-scale segmentation/object relationship modelling (MSS/ORM) approach was applied to map LULC in the Taita Hills from the multi-temporal SPOT imagery. This object-based procedure was shown to derive significant improvements over a uni-scale maximum-likelihood technique. The derived LULC data was used in combination with low cost GIS geospatial layers describing elevation, rainfall and soil type, to model degradation in the Taita Hills in the form of potential soil loss, utilizing the simple universal soil loss equation (USLE). Furthermore, human population distribution and abundance were modelled with satisfactory results using only SPOT and GIS derived data and non-Gaussian predictive modelling techniques. The SPOT derived LULC data was found to be unnecessary as a predictor because the first and second order image texture measurements had greater power to explain variation in dwelling unit occurrence and abundance. The ability of the procedures to be implemented locally in the developing world using low-cost or freely available data and software was considered. The techniques discussed in this thesis are considered equally applicable to other medium- and high-resolution optical satellite imagery, as well the utilized SPOT data.

Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams

This thesis is a study of a rather new logic called dependence logic and its closure under classical negation, team logic. In this thesis, dependence logic is investigated from several aspects. Some rules are presented for quantifier swapping in dependence logic and team logic. Such rules are among the basic tools one must be familiar with in order to gain the required intuition for using the logic for practical purposes. The thesis compares Ehrenfeucht-Fraïssé (EF) games of first order logic and dependence logic and defines a third EF game that characterises a mixed case where first order formulas are measured in the formula rank of dependence logic. The thesis contains detailed proofs of several translations between dependence logic, team logic, second order logic and its existential fragment. Translations are useful for showing relationships between the expressive powers of logics. Also, by inspecting the form of the translated formulas, one can see how an aspect of one logic can be expressed in the other logic. The thesis makes preliminary investigations into proof theory of dependence logic. Attempts focus on finding a complete proof system for a modest yet nontrivial fragment of dependence logic. A key problem is identified and addressed in adapting a known proof system of classical propositional logic to become a proof system for the fragment, namely that the rule of contraction is needed but is unsound in its unrestricted form. A proof system is suggested for the fragment and its completeness conjectured. Finally, the thesis investigates the very foundation of dependence logic. An alternative semantics called 1-semantics is suggested for the syntax of dependence logic. There are several key differences between 1-semantics and other semantics of dependence logic. 1-semantics is derived from first order semantics by a natural type shift. Therefore 1-semantics reflects an established semantics in a coherent manner. Negation in 1-semantics is a semantic operation and satisfies the law of excluded middle. A translation is provided from unrestricted formulas of existential second order logic into 1-semantics. Also game theoretic semantics are considerd in the light of 1-semantics.

Stochastic Filtering

The stochastic filtering has been in general an estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values as being one of the most accurate estimation, given the observations in the context of probability space. In my thesis, I have presented the theory of filtering using two different kind of observation process: the first one is a diffusion process which is discussed in the first chapter, while the third chapter introduces the latter which is a counting process. The majority of the fundamental results of the stochastic filtering is stated in form of interesting equations, such the unnormalized Zakai equation that leads to the Kushner-Stratonovich equation. The latter one which is known also by the normalized Zakai equation or equally by Fujisaki-Kallianpur-Kunita (FKK) equation, shows the divergence between the estimate using a diffusion process and a counting process. I have also introduced an example for the linear gaussian case, which is mainly the concept to build the so-called Kalman-Bucy filter. As the unnormalized and the normalized Zakai equations are in terms of the conditional distribution, a density of these distributions will be developed through these equations and stated by Kushner Theorem. However, Kushner Theorem has a form of a stochastic partial differential equation that needs to be verify in the sense of the existence and uniqueness of its solution, which is covered in the second chapter.