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.

A computational study of Compton profiles of water - methanol solutions

The molecular level structure of mixtures of water and alcohols is very complicated and has been under intense research in the recent past. Both experimental and computational methods have been used in the studies. One method for studying the intra- and intermolecular bindings in the mixtures is the use of the so called difference Compton profiles, which are a way to obtain information about changes in the electron wave functions. In the process of Compton scattering a photon scatters inelastically from an electron. The Compton profile that is obtained from the electron wave functions is directly proportional to the probability of photon scattering at a given energy to a given solid angle. In this work we develop a method to compute Compton profiles numerically for mixtures of liquids. In order to obtain the electronic wave functions necessary to calculate the Compton profiles we need some statistical information about atomic coordinates. Acquiring this using ab-initio molecular dynamics is beyond our computational capabilities and therefore we use classical molecular dynamics to model the movement of atoms in the mixture. We discuss the validity of the chosen method in view of the results obtained from the simulations. There are some difficulties in using classical molecular dynamics for the quantum mechanical calculations, but these can possibly be overcome by parameter tuning. According to the calculations clear differences can be seen in the Compton profiles of different mixtures. This prediction needs to be tested in experiments in order to find out whether the approximations made are valid.

Bayesian fisheries stock assessment: integrating and updating knowledge

In this thesis the use of the Bayesian approach to statistical inference in fisheries stock assessment is studied. The work was conducted in collaboration of the Finnish Game and Fisheries Research Institute by using the problem of monitoring and prediction of the juvenile salmon population in the River Tornionjoki as an example application. The River Tornionjoki is the largest salmon river flowing into the Baltic Sea. This thesis tackles the issues of model formulation and model checking as well as computational problems related to Bayesian modelling in the context of fisheries stock assessment. Each article of the thesis provides a novel method either for extracting information from data obtained via a particular type of sampling system or for integrating the information about the fish stock from multiple sources in terms of a population dynamics model. Mark-recapture and removal sampling schemes and a random catch sampling method are covered for the estimation of the population size. In addition, a method for estimating the stock composition of a salmon catch based on DNA samples is also presented. For most of the articles, Markov chain Monte Carlo (MCMC) simulation has been used as a tool to approximate the posterior distribution. Problems arising from the sampling method are also briefly discussed and potential solutions for these problems are proposed. Special emphasis in the discussion is given to the philosophical foundation of the Bayesian approach in the context of fisheries stock assessment. It is argued that the role of subjective prior knowledge needed in practically all parts of a Bayesian model should be recognized and consequently fully utilised in the process of model formulation.

Problems and Algorithms for Sequence Segmentations

The analysis of sequential data is required in many diverse areas such as telecommunications, stock market analysis, and bioinformatics. A basic problem related to the analysis of sequential data is the sequence segmentation problem. A sequence segmentation is a partition of the sequence into a number of non-overlapping segments that cover all data points, such that each segment is as homogeneous as possible. This problem can be solved optimally using a standard dynamic programming algorithm. In the first part of the thesis, we present a new approximation algorithm for the sequence segmentation problem. This algorithm has smaller running time than the optimal dynamic programming algorithm, while it has bounded approximation ratio. The basic idea is to divide the input sequence into subsequences, solve the problem optimally in each subsequence, and then appropriately combine the solutions to the subproblems into one final solution. In the second part of the thesis, we study alternative segmentation models that are devised to better fit the data. More specifically, we focus on clustered segmentations and segmentations with rearrangements. While in the standard segmentation of a multidimensional sequence all dimensions share the same segment boundaries, in a clustered segmentation the multidimensional sequence is segmented in such a way that dimensions are allowed to form clusters. Each cluster of dimensions is then segmented separately. We formally define the problem of clustered segmentations and we experimentally show that segmenting sequences using this segmentation model, leads to solutions with smaller error for the same model cost. Segmentation with rearrangements is a novel variation to the segmentation problem: in addition to partitioning the sequence we also seek to apply a limited amount of reordering, so that the overall representation error is minimized. We formulate the problem of segmentation with rearrangements and we show that it is an NP-hard problem to solve or even to approximate. We devise effective algorithms for the proposed problem, combining ideas from dynamic programming and outlier detection algorithms in sequences. In the final part of the thesis, we discuss the problem of aggregating results of segmentation algorithms on the same set of data points. In this case, we are interested in producing a partitioning of the data that agrees as much as possible with the input partitions. We show that this problem can be solved optimally in polynomial time using dynamic programming. Furthermore, we show that not all data points are candidates for segment boundaries in the optimal solution.

Architectural Solutions for Mobile RFID Services on Internet of Things

Mobile RFID services for the Internet of Things can be created by using RFID as an enabling technology in mobile devices. Humans, devices, and things are the content providers and users of these services. Mobile RFID services can be either provided on mobile devices as stand-alone services or combined with end-to-end systems. When different service solution scenarios are considered, there are more than one possible architectural solution in the network, mobile, and back-end server areas. Combining the solutions wisely by applying the software architecture and engineering principles, a combined solution can be formulated for certain application specific use cases. This thesis illustrates these ideas. It also shows how generally the solutions can be used in real world use case scenarios. A case study is used to add further evidence.

Characterization of hydrophobin proteins at interfaces and in solutions using x rays

Hydrophobins are a group of particularly surface active proteins. The surface activity is demonstrated in the ready adsorption of hydrophobins to hydrophobic/hydrophilic interfaces such as the air/water interface. Adsorbed hydrophobins self-assemble into ordered films, lower the surface tension of water, and stabilize air bubbles and foams. Hydrophobin proteins originate from filamentous fungi. In the fungi the adsorbed hydrophobin films enable the growth of fungal aerial structures, form protective coatings and mediate the attachment of fungi to solid surfaces. This thesis focuses on hydrophobins HFBI, HFBII, and HFBIII from a rot fungus Trichoderma reesei. The self-assembled hydrophobin films were studied both at the air/water interface and on a solid substrate. In particular, using grazing-incidence x-ray diffraction and reflectivity, it was possible to characterize the hydrophobin films directly at the air/water interface. The in situ experiments yielded information on the arrangement of the protein molecules in the films. All the T. reesei hydrophobins were shown to self-assemble into highly crystalline, hexagonally ordered rafts. The thicknesses of these two-dimensional protein crystals were below 30 Å. Similar films were also obtained on silicon substrates. The adsorption of the proteins is likely to be driven by the hydrophobic effect, but the self-assembly into ordered films involves also specific protein-protein interactions. The protein-protein interactions lead to differences in the arrangement of the molecules in the HFBI, HFBII, and HFBIII protein films, as seen in the grazing-incidence x-ray diffraction data. The protein-protein interactions were further probed in solution using small-angle x-ray scattering. Both HFBI and HFBII were shown to form mainly tetramers in aqueous solution. By modifying the solution conditions and thereby the interactions, it was shown that the association was due to the hydrophobic effect. The stable tetrameric assemblies could tolerate heating and changes in pH. The stability of the structure facilitates the persistence of these secreted proteins in the soil.