Gleichmäßig dünne Basen zweiter Ordnung

Eine Menge B nicht negativer ganzer Zahlen heißt Basis h-ter Ordnung, wenn jede nicht negative ganze Zahl Summe von höchstens h Elementen von B ist. Eine der großen Fragen der additiven Zahlentheorie ist die nach der effektivsten Basis h-ter Ordnung für ein gegebenes h>=2. Im Fokus des Interesses steht dabei der immer noch offene Fall h=2. Bezeichnet B(x) die Anzahl der Elemente b aus B mit 0= af(x), wobei f die Wurzelfunktion bezeichne. Andererseits gibt es Basen B zweiter Ordnung mit B(x) <= cf(x). Daher kann man den Limes superior S(B), den Limes inferior s(B) sowie im Falle der Existenz den Limes d(B) des Quotienten B(x) / f(x) als Dichtefunktionen von Basen zweiter Ordnung betrachten. J. W. S. Cassels konstruierte 1957 eine Basis C zweiter Ordnung mit d(C)=5,196…. G. Hofmeister gab 2001 eine Basis H zweiter Ordnung mit asymptotischer Wurzeldichte d(H)=4,638… an. In der vorliegenden Arbeit wird eine Basis S zweiter Ordnung mit asymptotischer Wurzeldichte d(S)=3,464… konstruiert. Darüber hinaus wird für die von J. W. S. Cassels, für die von G. Hofmeister und für die in dieser Arbeit verwendete Klasse von Basen zweiter Ordnung gezeigt, dass die asymptotische Wurzeldichte innerhalb der jeweiligen Klasse nicht mehr zu verbessern ist. Bisher war die Frage nach möglichen Verbesserungen innerhalb der jeweiligen Konstruktionsprinzipien offen geblieben.

Sensitivity analysis of a parabolic-elliptic problem

We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative L-infinity-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems.

Polycyclic aromatic hydrocarbons (PAHs) in river bank soils

Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous and found in the atmosphere, aquatic environment, sediments and soils. For environmental risk assessments and the allocation of the polluter it is important to know the PAH sources. PAH contamination sites are usually the result of anthropogenic processes. Three major sources are known: i) petroleum, including crude oil and its refined products and coals (petrogenic PAHs), ii) burning of organic matter (pyrogenic PAHs) and iii) transformation products of natural organic precursors present in the environment (diagenetic processes). In one case elevated PAH concentrations were found in river bank soils when building a retention area along the Mosel River. The source of the PAHs in this area was unclear and required the investigation of possible sources. To evaluate the PAH distribution along the Mosel River, a section of ~ 160 km along the river and a short section along the Saar River were investigated within this study. Concentrations of the Σ16 EPA PAHs were as high as 81 mg kg-1 dry weight (dw). Additionally, coal particles were identified in some soils, which originated from mining activities in the Saarland region. PAH distribution patterns of the 16 EPA PAHs suggest a mainly pyrogenic origin and in some cases a mixture of pyrogenic and petrogenic origin. For a comprehensive investigation five sampling sites were selected. Two sites were located before the confluence of the Mosel and Saar River, one site at the confluence and two sites after the confluence. The examination included typical forensic methods such as PAH distribution patterns of 45 PAHs (including alkylated PAHs), calculation of PAH ratios, determination of PAH alkyl homologues, n-alkanes, principal component analysis (PCA) and coal petrography. The results revealed a mainly pyrogenic source at sampling sites before the confluence of the two rivers. At and after the confluence, a mixture of pyrogenic and petrogenic inputs were present. With the help of coal petrography, coal derived particles could be identified in these soils. Therefore, coal was suggested to be the petrogenic source. It could be shown that sites with diffuse sources of contaminants, like the bank soils of the Mosel River, are difficult to characterize. As previously mentioned for detailed source identifications, the use of various forensic methods is essential. Determination of PAH alkyl homologue series, biomarkers and isotopes are often recommended. Source identification was evaluated using three different methods (i.e. PAH distribution patterns of an extended PAH spectrum, PAH ratios and analyses of n-alkanes). It was assessed if these methods were sufficient for the initial steps in identifying sources of PAHs in selected samples, and if they could be used for decision-making purposes. Point- and non-point sources were identified by applying the three methods and it could be shown that these relatively simple methods are sufficient in determining the primary source. In a last step of this study two soils (one before the confluence of the Mosel and Saar rivers and one after the confluence), and one sediment of the Mosel River were evaluated by investigating the mutagenic potential of the soils and the sediment with a fluctuation version of the Ames-test. The study showed that coal bearing soils at the Mosel River do not exhibit a greater mutagenic potential than other soils or sediments without coal particles.