Distortion of Dimension under Quasiconformal Mappings

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.

Generation and Characterization of the Cation-Chloride Cotransporter KCC2 Hypomorphic Mouse

The cation-Cl- cotransporter (CCC) family comprises of Na+-Cl- cotransporter (NCC), Na+-K+-2Cl- cotransporters (NKCC1-2), and four K+-Cl- cotransporters (KCC1-4). These proteins are involved in several physiological activities, such as cell volume regulation. In neuronal tissues, NKCC1 and KCC2 are important in determining the intracellular Cl- levels and hence the neuronal responses to inhibitory neurotransmitters GABA and glycine. One aim of the work was to elucidate the roles for CCC isoforms in the control of nervous system development. KCC2 mRNA was shown to be developmentally up-regulated and follow neuronal maturation, whereas NKCC1 and KCC4 transcripts were highly expressed in the proliferative zones of subcortical regions. KCC1 and KCC3 mRNA displayed low expression throughout the embryogenesis. These expression profiles suggest a role for CCC isoforms in maturation of synaptic responses and in the regulation of neuronal proliferation during embryogenesis. The major aim of this work was to study the biological consequences of KCC2-deficiency in the adult CNS, by generating transgenic mice retaining 15-20% of normal KCC2 levels. In addition, by using these mice as a tool for in vivo pharmacological analysis, we investigated the requirements for KCC2 in tonic versus phasic GABAA receptor-mediated inhibition. KCC2-deficient mice displayed normal reproduction and life span, but showed several behavioral abnormalities, including increased anxiety-like behavior, impaired performance in water maze, alterations in nociceptive processing, and increased seizure susceptibility. In contrast, the mice displayed apparently normal spontaneous locomotor activity and motor coordination. Pharmacological analysis of KCC2-deficient mice revealed reduced sensititivity to diazepam, but normal gaboxadol-induced sedation, neurosteroid hypnosis and alcohol-induced motor impairment. Electrophysiological recordings from CA1-CA3 subregions of the hippocampus showed that KCC2 deficiency affected the reversal potentials of both the phasic and tonic GABA currents, and that the tonic conductance was not affected. The results suggest that requirement for KCC2 in GABAergic neurotransmission may differ among several functional systems in the CNS, which is possibly due to the more critical role of KCC2 activity in phasic compared to tonic GABAergic inhibition.

Observations of the solar wind-magnetosphere-ionosphere coupling

In this thesis, the solar wind-magnetosphere-ionosphere coupling is studied observationally, with the main focus on the ionospheric currents in the auroral region. The thesis consists of five research articles and an introductory part that summarises the most important results reached in the articles and places them in a wider context within the field of space physics. Ionospheric measurements are provided by the International Monitor for Auroral Geomagnetic Effects (IMAGE) magnetometer network, by the low-orbit CHAllenging Minisatellite Payload (CHAMP) satellite, by the European Incoherent SCATter (EISCAT) radar, and by the Imager for Magnetopause-to-Aurora Global Exploration (IMAGE) satellite. Magnetospheric observations, on the other hand, are acquired from the four spacecraft of the Cluster mission, and solar wind observations from the Advanced Composition Explorer (ACE) and Wind spacecraft. Within the framework of this study, a new method for determining the ionospheric currents from low-orbit satellite-based magnetic field data is developed. In contrast to previous techniques, all three current density components can be determined on a matching spatial scale, and the validity of the necessary one-dimensionality approximation, and thus, the quality of the results, can be estimated directly from the data. The new method is applied to derive an empirical model for estimating the Hall-to-Pedersen conductance ratio from ground-based magnetic field data, and to investigate the statistical dependence of the large-scale ionospheric currents on solar wind and geomagnetic parameters. Equations describing the amount of field-aligned current in the auroral region, as well as the location of the auroral electrojets, as a function of these parameters are derived. Moreover, the mesoscale (10-1000 km) ionospheric equivalent currents related to two magnetotail plasma sheet phenomena, bursty bulk flows and flux ropes, are studied. Based on the analysis of 22 events, the typical equivalent current pattern related to bursty bulk flows is established. For the flux ropes, on the other hand, only two conjugate events are found. As the equivalent current patterns during these two events are not similar, it is suggested that the ionospheric signatures of a flux rope depend on the orientation and the length of the structure, but analysis of additional events is required to determine the possible ionospheric connection of flux ropes.

Late Pleistocene and Holocene environmental evolution of the Murchisonfjorden area, Nordaustlandet, Svalbard

The purpose of this study was to establish the palaeoenvironmental conditions during the late Quaternary in Murchisonfjorden, Nordaustlandet, based on foraminiferal assemblage compositions, and to determine the onset and termination of the Weichselian glaciations. The foraminiferal assemblage compositions were studied in marine sediments from three different archives, from sections next to the present shoreline in the Bay of Isvika, from a core in the Bay of Isvika and from a core from Lake Einstaken. OSL and AMS 14C age determinations were performed on samples from the three archives, and the results show deposition of marine sediments during ice-free periods of the Early Weichselian, the Middle Weichselian and the Late Weichselian, as well as during the Holocene in the investigated area. Marine sediments from the Early and Middle Weichselian were sampled from isostatically uplifted sections along the present shoreline.Sediments from the transition from the Late Weichselian to early Holocene time intervals were found in the bottom of the core from Lake Einstaken. Holocene sediments were investigated in the sections and in the core from the Bay of Isvika. The marine sediments from the sections are comprised of five benthic foraminiferal assemblages. The Early Weichselian is represented by two foraminiferal assemblages, the Middle Weichselian, the early and the late Holocene each by one. All five foraminiferal assemblages were deposited in glacier-distal shallow-water environments, which had a connection to the open ocean. Changes in the composition of the assemblages can be ascribed to differences in the bottom-water currents and changes in the salinity. The Middle Weichselian assemblage is of special importance, because it is the first foraminiferal assemblage to be described from this time interval from Svalbard. Four benthic foraminiferal assemblages were deposited shortly before the marine to lacustrine transition at the boundary between the Late Weichselian and Holocene in Lake Einstaken. The foraminiferal assemblages show a change from a high-arctic, normal marine shallow-water environment to an even shallower environment with highly fluctuating salinity. The analyses of the core from 100 m water depth in the Bay of Isvika resulted in the determination of four foraminiferal assemblages. These indicated changes from a glacier-proximal environment during deglaciation, to a more glacier-distal environment during the Early Holocene. This was followed by a period with a marked change to a considerably cooler environment and finally to a closed fjord environment in the middle and late Holocene times. Additional sedimentological analyses of the marine and glacially derived sediments from the uplifted sections, as well as observations of multiple striae on the bedrock, observations of deeply weathered bedrock and findings of tills interlayered with marine sediments complete the investigations in the study area. They indicate weak glacial erosion in the study area. It can be concluded that marine deposition occurred in the investigated area during three time intervals in the Weichselian and during most of the Holocene. The foraminiferal assemblages in the Holocene are characterized by a transition from glacier-proximal to glacier-distal faunas. The palaeogeographical change from an open fjord to a closed fjord environment is a result of the isostatic uplift of the area after the LGM and is clearly reflected in the foraminiferal assemblages. Another influencing factor on the foraminiferal assemblage composition are changes in the inflow of warmer Atlantic waters to the study area.