Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Burnt area mapping in insular Southeast Asia using medium resolution satellite imagery

Burnt area mapping in humid tropical insular Southeast Asia using medium resolution (250-500m) satellite imagery is characterized by persisting cloud cover, wide range of land cover types, vast amount of wetland areas and highly varying fire regimes. The objective of this study was to deepen understanding of three major aspects affecting the implementation and limits of medium resolution burnt area mapping in insular Southeast Asia: 1) fire-induced spectral changes, 2) most suitable multitemporal compositing methods and 3) burn scars patterns and size distribution. The results revealed a high variation in fire-induced spectral changes depending on the pre-fire greenness of burnt area. It was concluded that this variation needs to be taken into account in change detection based burnt area mapping algorithms in order to maximize the potential of medium resolution satellite data. Minimum near infrared (MODIS band 2, 0.86μm) compositing method was found to be the most suitable for burnt area mapping purposes using Moderate Resolution Imaging Spectroradiometer (MODIS) data. In general, medium resolution burnt area mapping was found to be usable in the wetlands of insular Southeast Asia, whereas in other areas the usability was seriously jeopardized by the small size of burn scars. The suitability of medium resolution data for burnt area mapping in wetlands is important since recently Southeast Asian wetlands have become a major point of interest in many fields of science due to yearly occurring wild fires that not only degrade these unique ecosystems but also create regional haze problem and release globally significant amounts of carbon into the atmosphere due to burning peat. Finally, super-resolution MODIS images were tested but the test failed to improve the detection of small scars. Therefore, super-resolution technique was not considered to be applicable to regional level burnt area mapping in insular Southeast Asia.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

Coastal environmental gradients – Key to reproduction habitat mapping of freshwater fish in the Baltic Sea

Habitat requirements of fish are most strict during the early life stages, and the quality and quantity of reproduction habitats lays the basis for fish production. A considerable number of fish species in the northern Baltic Sea reproduce in the shallow coastal areas, which are also the most heavily exploited parts of the brackish marine area. However, the coastal fish reproduction habitats in the northern Baltic Sea are poorly known. The studies presented in this thesis focused on the influence of environmental conditions on the distribution of coastal reproduction habitats of freshwater fish. They were conducted in vegetated littoral zone along an exposure and salinity gradient extending from the innermost bays to the outer archipelago on the south-western and southern coasts of Finland, in the northern Baltic Sea. Special emphasis was placed on reed-covered Phragmites australis shores, which form a dominant vegetation type in several coastal archipelago areas. The main aims of this research were to (1) develop and test new survey and mapping methods, (2) investigate the environmental requirements that govern the reproduction of freshwater fish in the coastal area and (3) survey, map and model the distribution of the reproduction habitats of pike (Esox lucius) and roach (Rutilus rutilus). The white plate and scoop method with a standardized sampling time and effort was demonstrated to be a functional method for sampling the early life stages of fish in dense vegetation and shallow water. Reed-covered shores were shown to form especially important reproduction habitats for several freshwater fish species, such as pike, roach, other cyprinids and burbot, in the northern Baltic Sea. The reproduction habitats of pike were limited to sheltered reed- and moss-covered shores of the inner and middle archipelago, where suitable zooplankton prey were available and the influence of the open sea was low. The reproduction habitats of roach were even more limited and roach reproduction was successful only in the very sheltered reed-covered shores of the innermost bay areas, where salinity remained low (< 4‰) during the spawning season due to freshwater inflow. After identifying the critical factors restricting the reproduction of pike and roach, the spatial distribution of their reproduction habitats was successfully mapped and modelled along the environmental gradients using only a few environmental predictor variables. Reproduction habitat maps are a valuable tool promoting the sustainable use and management of exploited coastal areas and helping to maintain the sustainability of fish populations. However, the large environmental gradients and the extensiveness of the archipelago zone in the northern Baltic Sea demand an especially high spatial resolution of the coastal predictor variables. Therefore, the current lack of accurate large-scale, high-resolution spatial data gathered at exactly the right time is a considerable limitation for predictive modelling of shallow coastal waters.