Weimar, BauhausUniversiẗat, Diss., 2000.

An m charge in the n dimensional Euclidean space is a linear functional acting on m dimensional polyhedral chains and satisfying the following continuity condition. The value of the linear functional approaches zero on chains whose normal masses are bounded and whose flat norms asymptotically vanish. Our main theorem relates m charges to pairs of continuous differential forms. Luzin's theorem states that every measurable function on the line is the derivative of a continuous, almost everywhere differentiable function. We show this can be improved in several dimensions. Finally we prove that a compact subset C of the n dimensional Euclidean space does not support the distributional divergence of a bounded measurable vector field if and only if C has vanishing (n-1) dimensional Hausdorff measure.

In calculus the concept of an infinite derivative – i.e. DF(x) = ±∞ – is seldom studied due to a plethora of complications that arise from this definition. For instance, in this extended sense, algebraic expressions involving derivatives are generally undefined; and two continuous functions possessing identical derivatives at every point of an interval generally differ by a non-constant function. These problems are fundamentally irremediable insofar as calculus is concerned and must therefore be addressed in a more general setting. This is quite difficult since the literature on infinite derivatives is rather sparse and seldom accessible to non-specialists. Therefore we supply a self-contained thesis on continuous functions with infinite derivatives aimed at graduate students with a background in real analysis and measure theory.  Predominately we study continuous primitives which satisfy the Luzin condition (N) by establishing a deep connection with the strong Luzin condition – a weak form of absolute continuity which has its origins in the Henstock–Kurzweil theory of integration. The main result states that a function satisfies the strong Luzin condition if and only if it can be expressed as a sum of two such primitives. Furthermore, we establish some pathological properties of continuous primitives which fail to satisfy the Luzin condition (N).

Gegenstand der Arbeit war die Untersuchung der Rolle der Schwebegarnele Mysis relicta im pelagischen Nahrungsnetz des Breiten Luzins unter besonderer Berücksichtigung der trophischen Interaktionen zur Kleinen Maräne Coregonus albula. Schwerpunkte waren die Abundanz und räumlich-zeitliche Verteilung von M. relicta, C. albula und dem Zooplankton sowie die trophischen Beziehungen im Pelagial. Im Vergleich zu früheren Studien war M. relicta weiträumiger und mit höherer Dichte im Untersuchungsgebiet verbreitet. Dies war auf die Verbesserung der Wasserqualität zurückzuführen. Ähnliche Verteilungsmuster führten zu starken räumlichen Überlappungen von M. relicta und C. albula. M. relicta konsumierte vorwiegend abundantes Plankton. Es zeigte sich aber auch eine Präferenz für Cladocera. Aus den in vitro ermittelten Ingestionsraten wurde für die Situation im Freiland ein starker Fraßdruck von M. relicta auf das Zooplankton abgeleitet. In Kombination mit einer Überschneidung der Beutespektren führte das zur Nahrungskonkurrenz zwischen M. relicta und C. albula. Andererseits war M. relicta eine wichtige Beute für C. albula und wurde stark positiv selektiert. Die Fettsäuremuster der untersuchten Organismen waren überwiegend durch ihre Nahrung beeinflusst. Die Aufnahme von M. relicta wurde nicht durch das Auftreten spezifischer Fettsäuren bei C. albula nachgewiesen. Grundsätzlich entsprach der Energietransfer im Nahrungsnetz des Breiten Luzins allgemeinen Modellen mit 10 % Transfereffizienz zwischen benachbarten trophischen Ebenen. Im Einzelnen zeigten sich aber auch Abweichungen, die größtenteils auf M. relicta zurückzuführen waren. Durch ihre omnivore Ernährung und als Beute für C. albula schafft M. relicta zusätzliche Verbindungen zwischen den trophischen Ebenen. Insgesamt hat M. relicta einen weitreichenden Einfluss auf die Struktur und den Energietransfer des Nahrungsnetzes im Breiten Luzin, wodurch ihr Vorkommen für das Gewässer von besonderer Bedeutung ist. / The aim of the study was to investigate the role of the opossum shrimp Mysis relicta in the pelagic food web of Lake Breiter Luzin (Germany) with special focus on trophic interactions with European cisco Coregonus albula. Main aspects were the abundance and temporal and spatial distribution of M. relicta, C. albula and zooplankton as well as trophic relationships in the food web. As a result of improved water quality M. relicta showed a wider distribution and enhanced abundances in the study area than earlier investigations suggested. Due to similar diurnal vertical distributions a great spatial overlap was evident for M. relicta and C. albula. M. relicta fed mainly on highly abundant plankton. Besides, an overall preference for cladocerans was detected. By applying estimated in vitro feeding rates to field situations it could be shown that zooplankton was strongly exploited by M. relicta. This high demand combined with similar prey preferences of C. albula led to a strong food competition between M. relicta and C. albula. On the other hand M. relicta formed high amounts in the diet of C. albula and was always positively selected. Fatty acid composition of examined organisms resembled the composition found in their dietary resources. The intake of M. relicta could not be detected by a specific fatty acid composition in C. albula. Considering highest and lowest trophic levels, transfer efficiency in Lake Breiter Luzin fit into general models assuming a transfer of 10 % between neighbouring trophic levels. Transfer between intermediate trophic levels partly differed from this general pattern. This was connected to the existence of M. relicta. Being omnivorous and by providing an important food source for C. albula the mysid established additional links between trophic levels in the food web. In summary, M. relicta is a significant factor in structuring the food web of Lake Breiter Luzin and has profound impact on the entire lake ecosystem.

We study the optimal conditions on a homeomorphism f : Ω → Rn which guarantee that the composition u◦f is weakly differentiable and its weak derivative belongs to the some function space. We show that if f has finite distortion and q-distortion Kq = |Df|q /Jf is integrable enough, then the composition operator Tf (u) = u ◦ f maps functions from W1,q loc into space W1,p loc and the well-known chain rule holds. To prove it we characterize when the inverse mapping f−1 maps sets of measure zero onto sets of measure zero (satisfies the Luzin (N−1 ) con- dition). We also fully characterize conditions for Sobolev-Lorentz space WLn,q for arbitrary q and for Sobolev Orlicz space WLq log L for q ≥ n and α > 0 or 1 < q ≤ n and α < 0. We find a necessary condition on f for Sobolev rearrangement invariant function space WX close to WLq , i.e. X has q-scaling property. 1