Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Constructing a ‘moment of crisis’ and its adequate solution

The rejection of the European Constitution marks an important crystallization point for debate about the European Union (EU) and the integration process. The European Constitution was envisaged as the founding document of a renewed and enlarged European Union and thus it was rather assumed to find wide public support. Its rejection was not anticipated. The negative referenda in France and the Netherlands therefore led to a controversial debate about the more fundamental meaning and the consequences of the rejection both for the immediate state of affairs as well as for the further integration process. The rejection of the Constitution and the controversy about its correct interpretation therefore present an intriguing puzzle for political analysis. Although the treaty rejection was taken up widely in the field of European Studies, the focus of existing analyses has predominantly been on explaining why the current situation occurred. Underlying these approaches is the premise that by establishing the reasons for the rejection it is possible to derive the ‘true’ meaning of the event for the EU integration process. In my paper I rely on an alternative, discourse theoretical approach which aims to overcome the positivist perspective dominating the existing analyses. I argue that the meaning of the event ‘treaty rejection’ is not fixed or inherent to it but discursively constructed. The critical assessment of this concrete meaning-production is of high relevance as the specific meaning attributed to the treaty rejection effectively constrains the scope for supposedly ‘reasonable’ options for action, both in the concrete situation and in the further European integration process more generally. I will argue that the overall framing suggests a fundamental technocratic approach to governance from part of the Commission. Political struggle and public deliberation is no longer foreseen as the concrete solutions to the citizens’ general concerns are designed by supposedly apolitical experts. Through the communicative diffusion and the active implementation of this particular model of governance the Commission shapes the future integration process in a more substantial way than is obvious from its seemingly limited immediate problem-solving orientation of overcoming the ‘constitutional crisis’. As the European Commission is a central actor in the discourse production my analysis focuses on the specific interpretation of the situation put forward by the Commission. In order to work out the Commission’s particular take on the event I conducted a frame analysis (according to Benford/Snow) on a body of key sources produced in the context of coping with the treaty rejection.

On Solution Sets of Information Inequalities

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.

Semi-algebraic methods for symbolic analysis of complex reaction networks

The identification of chemical mechanism that can exhibit oscillatory phenomena in reaction networks are currently of intense interest. In particular, the parametric question of the existence of Hopf bifurcations has gained increasing popularity due to its relation to the oscillatory behavior around the fixed points. However, the detection of oscillations in high-dimensional systems and systems with constraints by the available symbolic methods has proven to be difficult. The development of new efficient methods are therefore required to tackle the complexity caused by the high-dimensionality and non-linearity of these systems. In this thesis, we mainly present efficient algorithmic methods to detect Hopf bifurcation fixed points in (bio)-chemical reaction networks with symbolic rate constants, thereby yielding information about their oscillatory behavior of the networks. The methods use the representations of the systems on convex coordinates that arise from stoichiometric network analysis. One of the methods called HoCoQ reduces the problem of determining the existence of Hopf bifurcation fixed points to a first-order formula over the ordered field of the reals that can then be solved using computational-logic packages. The second method called HoCaT uses ideas from tropical geometry to formulate a more efficient method that is incomplete in theory but worked very well for the attempted high-dimensional models involving more than 20 chemical species. The instability of reaction networks may lead to the oscillatory behaviour. Therefore, we investigate some criterions for their stability using convex coordinates and quantifier elimination techniques. We also study Muldowney's extension of the classical Bendixson-Dulac criterion for excluding periodic orbits to higher dimensions for polynomial vector fields and we discuss the use of simple conservation constraints and the use of parametric constraints for describing simple convex polytopes on which periodic orbits can be excluded by Muldowney's criteria. All developed algorithms have been integrated into a common software framework called PoCaB (platform to explore bio- chemical reaction networks by algebraic methods) allowing for automated computation workflows from the problem descriptions. PoCaB also contains a database for the algebraic entities computed from the models of chemical reaction networks.

Biofuel as the solution of alternative energy production?

There have being increasing debate on the prospects of biofuel becoming the next best alternative to solving the problem of CO2 emission and the escalating fuel prices, but the question is whether this assertion is true and also if it comes without any cost to pay. This paper seeks to find out whether this much praised alternative to solving these problems is a better option or another way for the developed countries to find more areas where they could get cheap land, labour and raw materials for the production of biofuel. This will focus mainly on some effects the growing biofuel production has on food security, livelihood of people, the environment and some land conflicts developing as a result of land grabbing for biofuel production in the developing countries.