Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

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.

Lightcurve inversion for asteroid spins and shapes

Knowledge of the physical properties of asteroids is crucial in many branches of solar-system research. Knowledge of the spin states and shapes is needed, e.g., for accurate orbit determination and to study the history and evolution of the asteroids. In my thesis, I present new methods for using photometric lightcurves of asteroids in the determination of their spin states and shapes. The convex inversion method makes use of a general polyhedron shape model and provides us at best with an unambiguous spin solution and a convex shape solution that reproduces the main features of the original shape. Deriving information about the non-convex shape features is, in principle, also possible, but usually requires a priori information about the object. Alternatively, a distribution of non-convex solutions, describing the scale of the non-convexities, is also possible to be obtained. Due to insufficient number of absolute observations and inaccurately defined asteroid phase curves, the $c/b$-ratio, i.e., the flatness of the shape model is often somewhat ill-defined. However, especially in the case of elongated objects, the flatness seems to be quite well constrained, even in the case when only relative lightcurves are available. The results prove that it is, contrary to the earlier misbelief, possible to derive shape information from the lightcurve data if a sufficiently wide range of observing geometries is covered by the observations. Along with the more accurate shape models, also the rotational states, i.e., spin vectors and rotation periods, are defined with improved accuracy. The shape solutions obtained so far reveal a population of irregular objects whose most descriptive shape characteristics, however, can be expressed with only a few parameters. Preliminary statistical analyses for the shapes suggests that there are correlations between shape and other physical properties, such as the size, rotation period and taxonomic type of the asteroids. More shape data of, especially, the smallest and largest asteroids, as well as the fast and slow rotators is called for in order to be able to study the statistics more thoroughly.