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.

Individuals on the Move : Body Condition Dependent Dispersal and Quasi-Local Competition in Metapopulations

Individual movement is very versatile and inevitable in ecology. In this thesis, I investigate two kinds of movement body condition dependent dispersal and small-range foraging movements resulting in quasi-local competition and their causes and consequences on the individual, population and metapopulation level. Body condition dependent dispersal is a widely evident but barely understood phenomenon. In nature, diverse relationships between body condition and dispersal are observed. I develop the first models that study the evolution of dispersal strategies that depend on individual body condition. In a patchy environment where patches differ in environmental conditions, individuals born in rich (e.g. nutritious) patches are on average stronger than their conspecifics that are born in poorer patches. Body condition (strength) determines competitive ability such that stronger individuals win competition with higher probability than weak individuals. Individuals compete for patches such that kin competition selects for dispersal. I determine the evolutionarily stable strategy (ESS) for different ecological scenarios. My models offer explanations for both dispersal of strong individuals and dispersal of weak individuals. Moreover, I find that within-family dispersal behaviour is not always reflected on the population level. This supports the fact that no consistent pattern is detected in data on body condition dependent dispersal. It also encourages the refining of empirical investigations. Quasi-local competition defines interactions between adjacent populations where one population negatively affects the growth of the other population. I model a metapopulation in a homogeneous environment where adults of different subpopulations compete for resources by spending part of their foraging time in the neighbouring patches, while their juveniles only feed on the resource in their natal patch. I show that spatial patterns (different population densities in the patches) are stable only if one age class depletes the resource very much but mainly the other age group depends on it.

Tumour necrosis factor receptor-associated periodic syndrome (TRAPS) : Identification of novel TNFRSF1A mutations and intracellular signalling defects

The systemic autoinflammatory disorders are a group of rare diseases characterized by periodically recurring episodes of acute inflammation and a rise in serum acute phase proteins, but with no signs of autoimmunity. At present eight hereditary syndromes are categorized as autoinflammatory, although the definition has also occasionally been extended to other inflammatory disorders, such as Crohn s disease. One of the autoinflammatory disorders is the autosomally dominantly inherited tumour necrosis factor receptor-associated periodic syndrome (TRAPS), which is caused by mutations in the gene encoding the tumour necrosis factor type 1 receptor (TNFRSF1A). In patients of Nordic descent, cases of TRAPS and of three other hereditary fevers, hyperimmunoglobulinemia D with periodic fever syndrome (HIDS), chronic infantile neurologic, cutaneous and articular syndrome (CINCA) and familial cold autoinflammatory syndrome (FCAS), have been reported, TRAPS being the most common of the four. Clinical characteristics of TRAPS are recurrent attacks of high spiking fever, associated with inflammation of serosal membranes and joints, myalgia, migratory rash and conjunctivitis or periorbital cellulitis. Systemic AA amyloidosis may occur as a sequel of the systemic inflammation. The aim of this study was to investigate the genetic background of hereditary periodically occurring fever syndromes in Finnish patients, to explore the reliability of determining serum concentrations of soluble TNFRSF1A and metalloproteinase-induced TNFRSF1A shedding as helpful tools in differential diagnostics, as well as to study intracellular NF-κB signalling in an attempt to widen the knowledge of the pathomechanisms underlying TRAPS. Genomic sequencing revealed two novel TNFRSF1A mutations, F112I and C73R, in two Finnish families. F112I was the first TNFRSF1A mutation to be reported in the third extracellular cysteine-rich domain of the gene and C73R was the third novel mutation to be reported in a Finnish family, with only one other TNFRSF1A mutation having been reported in the Nordic countries. We also presented a differential diagnostic problem in a TRAPS patient, emphasizing for the clinician the importance of differential diagnostic vigiliance in dealing with rare hereditary disorders. The underlying genetic disease of the patient both served as a misleading factor, which possibly postponed arrival at the correct diagnosis, but may also have predisposed to the pathologic condition, which led to a critical state of the patient. Using a method of flow cytometric analysis modified for the use on fresh whole blood, we studied intracellular signalling pathways in three Finnish TRAPS families with the F112I, C73R and the previously reported C88Y mutations. Evaluation of TNF-induced phosphorylation of NF-κB and p38, revealed low phosphorylation profiles in nine out of ten TRAPS patients in comparison to healthy control subjects. This study shows that TRAPS is a diagnostic possibility in patients of Nordic descent, with symptoms of periodically recurring fever and inflammation of the serosa and joints. In particular in the case of a family history of febrile episodes, the possibility of TRAPS should be considered, if an etiology of autoimmune or infectious nature is excluded. The discovery of three different mutations in a population as small as the Finnish, reinforces the notion that the extracellular domain of TNFRSF1A is prone to be mutated at the entire stretch of its cysteine-rich domains and not only at a limited number of sites, suggesting the absence of a founder effect in TRAPS. This study also demonstrates the challenges of clinical work in differentiating the symptoms of rare genetic disorders from those of other pathologic conditions and presents the possibility of an autoinflammatory disorder as being the underlying cause of severe clinical complications. Furthermore, functional studies of fresh blood leukocytes show that TRAPS is often associated with a low NF-κB and p38 phosphorylation profile, although low phosphorylation levels are not a requirement for the development of TRAPS. The aberrant signalling would suggest that the hyperinflammatory phenotype of TRAPS is the result of compensatory NF-κB-mediated regulatory mechanisms triggered by a deficiency of the innate immune response.

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.