On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Analysis of the Market Potential of Syrian Organic Fruit and Vegetables for Exports to Germany

Syria has been a major producer and exporter of fresh fruit and vegetables (FFV) in the Arabic region. Prior to 2011, Syrian FFV were mainly exported to the neighbouring countries, the Gulf States and Northern Africa as well as to Eastern European countries. Although the EU is potentially one of the most profitable markets of high quality FFV (such as organic ones) in the world, Syrian exports of FFV to Western European countries like Germany have been small. It could be a lucrative opportunity for Syrian growers and exporters of FFV to export organic products to markets such as Germany, where national production is limited to a few months due to climatic conditions. Yet, the organic sector in Syria is comparatively young and only a very small area of FFV is certified according to EU organic regulations. Up to the author’s knowledge, little was known about Syrian farmers’ attitudes towards organic FFV production. There was also no study so far that explored and analysed the determining factors for organic FFV adoption among Syrian farmers as well as the exports of these products to the EU markets. The overarching aim of the present dissertation focused on exploring and identifying the market potential of Syrian exports of organic FFV to Germany. The dissertation was therefore concerned with three main objectives: (i) to explore if German importers and wholesalers of organic FFV see market opportunities for Syrian organic products and what requirements in terms of quality and quantity they have, (ii) to determine the obstacles Syrian producers and exporters face when exporting agricultural products to Germany, and (iii) to investigate whether Syrian farmers of FFV can imagine converting their farms to organic production as well as the underlying reasons why they do so or not. A twofold methodological approach with expert interviews and a farmer survey were used in this dissertation to address the abovementioned objectives. While expert interviews were conducted with German and Syrian wholesalers of (organic) FFV in 2011 (9 interviews each), the farmer survey was administrated with 266 Syrian farmers of FFV in the main region for the production of FFV (i.e. the coastal region) from November 2012 till May 2013. For modelling farmers’ decisions to adopt organic farming, the Theory of Planned Behaviour as theoretical framework and Partial Least Squares Structural Equation Modelling as the main method for data analysis were used in this study. The findings of this dissertation yield implications for the different stakeholders (governmental institutions and NGOs, farmers, exporters, wholesalers, etc.) who are interested in prompting the Syrian export of organic products. Based on the empirical results and a literature review, an action plan to promote Syrian production and export of organic products was developed which can help in the post-war period in Syria at improving the organic sector.