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.

Effects of Organic and Inorganic Fertilizers on Growth and Yield of Banana (Musa AAA cv. Malindi) in Oman

The Sultanate of Oman is located on the south-eastern coast of the Arabian Peninsula, which lies on the south-western tip of the Asian continent. The strategic geographical locations of the Sultanate with its many maritime ports distributed on the Indian Ocean have historically made it one of the Arabian Peninsula leaders in the international maritime trade sector. Intensive trading relationships over long time periods have contributed to the high plant diversity seen in Oman where agricultural production depends entirely on irrigation from groundwater sources. As a consequence of the expansion of the irrigated area, groundwater depletion has increased, leading to the intrusion of seawater into freshwater aquifers. This phenomenon has caused water and soil salinity problems in large parts of the Al-Batinah governorate of Oman and threatens cultivated crops, including banana (Musa spp.). According to the Ministry of Agriculture and Fisheries, the majority of South Al-Batinah farms are affected by salinity (ECe > 4 dS m-1). As no alternative farmland is available, the reclamation of salt-affected soils using simple cultural practices is of paramount importance, but in Oman little scientific research has been conducted to develop such methods of reclamation. This doctoral study was initiated to help filling this research gap, particularly for bananas. A literature review of the banana cultivation history revealed that the banana germplasm on the Arabian Peninsula is probably introduced from Indonesia and India via maritime routes across the Indian Ocean and the Red Sea. In a second part of this dissertation, two experiments are described. A laboratory trial conducted at the University of Kassel, in Witzenhausen, Germany from June to July 2010. This incubation experiment was done to explore how C and N mineralization of composted dairy manure and date palm straw differed in alkaline non-saline and saline soils. Each soil was amended with four organic fertilizers: 1) composted dairy manure, 2) manure + 10% date palm straw, 3) manure + 30% date palm straw or 4) date palm straw alone, in addition to un-amended soils as control. The results showed that the saline soil had a lower soil organic C content and microbial biomass C than the non-saline soil. This led to lower mineralization rates of manure and date palm straw in the saline soil. In the non-saline soil, the application of manure and straw resulted in significant increases of CO2 emissions, equivalent to 2.5 and 30% of the added C, respectively. In the non-amended control treatment of the saline soil, the sum of CO2-C reached only 55% of the soil organic C in comparison with the non-saline soil. In which 66% of the added manure and 75% of the added straw were emitted, assuming that no interactions occurred between soil organic C, manure C and straw C during microbial decomposition. The application of straw always led to a net N immobilization compared to the control. Salinity had no specific effect on N mineralization as indicated by the CO2-C to Nmin ratio of soil organic matter and manure. However, N immobilization was markedly stronger in the saline soil. Date palm straw strongly promoted saprotrophic fungi in contrast to manure and the combined application of manure and date palm straw had synergistic positive effects on soil microorganisms. In the last week of incubation, net-N mineralization was observed in nearly all treatments. The strongest increase in microbial biomass C was observed in the manure + straw treatment. In both soils, manure had no effect on the fungi-specific membrane component ergosterol. In contrast, the application of straw resulted in strong increases of the ergosterol content. A field experiment was conducted on two adjacent fields at the Agricultural Research Station, Rumais (23°41’15” N, 57°59’1” E) in the South of Al-Batinah Plain in Oman from October 2007 to July 2009. In this experiment, the effects of 24 soil and fertilizer treatments on the growth and productivity of Musa AAA cv. 'Malindi' were evaluated. The treatments consisted of two soil types (saline and amended non-saline), two fertilizer application methods (mixed and ring applied), six fertilizer amendments (1: fresh dairy manure, 2: composted dairy manure, 3: composted dairy manure and 10% date palm straw, 4: composted dairy manure and 30% date palm straw, 5: only NPK, and 6: NPK and micronutrients). Sandy loam soil was imported from another part of Oman to amended the soil in the planting holes and create non-saline conditions in the root-zone. The results indicate that replacing the saline soil in the root zone by non-saline soil improved plant growth and yield more than fertilizer amendments or application methods. Particularly those plants on amended soil where NPK was applied using the ring method and which received micronutrients grew significantly faster to harvest (339 days), had a higher average bunch weight (9.5 kg/bunch) and were consequently more productive (10.6 tonnes/hectare/cycle) compared to the other treatments.

Entwicklung und Anwendung einer Kraftwerks- und Speichereinsatzoptimierung für die Untersuchung von Energieversorgungsszenarien mit hohem Anteil erneuerbarer Energien in Deutschland

In dieser Arbeit wurde ein gemischt-ganzzahliges lineares Einsatzoptimierungsmodell für Kraftwerke und Speicher aufgebaut und für die Untersuchung der Energieversorgung Deutschlands im Jahre 2050 gemäß den Leitstudie-Szenarien 2050 A und 2050 C ([Nitsch und Andere, 2012]) verwendet, in denen erneuerbare Energien einen Anteil von über 85 % an der Stromerzeugung haben und die Wind- und Solarenergie starke Schwankungen der durch steuerbare Kraftwerke und Speicher zu deckenden residualen Stromnachfrage (Residuallast) verursachen. In Szenario 2050 A sind 67 TWh Wasserstoff, die elektrolytisch aus erneuerbarem Strom zu erzeugen sind, für den Verkehr vorgesehen. In Szenario 2050 C ist kein Wasserstoff für den Verkehr vorgesehen und die effizientere Elektromobilität hat einen Anteil von 100% am Individualverkehr. Daher wird weniger erneuerbarer Strom zur Erreichung desselben erneuerbaren Anteils im Verkehrssektor benötigt. Da desweiteren Elektrofahrzeuge Lastmanagementpotentiale bieten, weisen die Residuallasten der Szenarien eine unterschiedliche zeitliche Charakteristik und Jahressumme auf. Der Schwerpunkt der Betrachtung lag auf der Ermittlung der Auslastung und Fahrweise des in den Szenarien unterstellten ’Kraftwerks’-parks bestehend aus Kraftwerken zur reinen Stromerzeugung, Kraft-Wärme-Kopplungskraftwerken, die mit Wärmespeichern, elektrischen Heizstäben und Gas-Backupkesseln ausgestattet sind, Stromspeichern und Wärmepumpen, die durch Wärmespeicher zum Lastmanagment eingesetzt werden können. Der Fahrplan dieser Komponenten wurde auf minimale variable Gesamtkosten der Strom- und Wärmeerzeugung über einen Planungshorizont von jeweils vier Tagen hin optimiert. Das Optimierungsproblem wurde mit dem linearen Branch-and-Cut-Solver der software CPLEX gelöst. Mittels sogenannter rollierender Planung wurde durch Zusammensetzen der Planungsergebnisse für überlappende Planungsperioden der Kraftwerks- und Speichereinsatz für die kompletten Szenariojahre erhalten. Es wurde gezeigt, dass der KWK-Anteil an der Wärmelastdeckung gering ist. Dies wurde begründet durch die zeitliche Struktur der Stromresiduallast, die wärmeseitige Dimensionierung der Anlagen und die Tatsache, dass nur eine kurzfristige Speicherung von Wärme vorgesehen war. Die wärmeseitige Dimensionierung der KWK stellte eine Begrenzung des Deckungsanteils dar, da im Winter bei hoher Stromresiduallast nur wenig freie Leistung zur Beladung der Speicher zur Verfügung stand. In den Berechnungen für das Szenario 2050 A und C lag der mittlere Deckungsanteil der KWK an der Wärmenachfrage von ca. 100 TWh_th bei 40 bzw. 60 %, obwohl die Auslegung der KWK einen theoretischen Anteil von über 97 % an der Wärmelastdeckung erlaubt hätte, gäbe es die Beschränkungen durch die Stromseite nicht. Desweiteren wurde die CO2-Vermeidungswirkung der KWK-Wärmespeicher und des Lastmanagements mit Wärmepumpen untersucht. In Szenario 2050 A ergab sich keine signifikante CO2-Vermeidungswirkung der KWK-Wärmespeicher, in Szenario 2050 C hingegen ergab sich eine geringe aber signifikante CO2-Einsparung in Höhe von 1,6 % der Gesamtemissionen der Stromerzeugung und KWK-gebundenen Wärmeversorgung. Das Lastmanagement mit Wärmepumpen vermied Emissionen von 110 Tausend Tonnen CO2 (0,4 % der Gesamtemissionen) in Szenario A und 213 Tausend Tonnen in Szenario C (0,8 % der Gesamtemissionen). Es wurden darüber hinaus Betrachtungen zur Konkurrenz zwischen solarthermischer Nahwärme und KWK bei Einspeisung in dieselben Wärmenetze vorgenommen. Eine weitere Einschränkung der KWK-Erzeugung durch den Einspeisevorrang der Solarthermie wurde festgestellt. Ferner wurde eine untere Grenze von 6,5 bzw. 8,8 TWh_th für die in den Szenarien mindestens benötigte Wasserstoff-Speicherkapazität ermittelt. Die Ergebnisse dieser Arbeit legen nahe, das technisch-ökonomische Potential von Langzeitwärmespeichern für eine bessere Integration von KWK ins System zu ermitteln bzw. generell nach geeigneteren Wärmesektorszenarien zu suchen, da deutlich wurde, dass für die öffentliche Wärmeversorgung die KWK in Kombination mit Kurzzeitwärmespeicherung, Gaskesseln und elektrischen Heizern keine sehr effektive CO2 -Reduktion in den Szenarien erreicht. Es sollte dabei z.B. untersucht werden, ob ein multivalentes System aus KWK, Wärmespeichern und Wärmepumpen eine ökonomisch darstellbare Alternative sein könnte und im Anschluss eine Betrachtung der optimalen Anteile von KWK, Wärmepumpen und Solarthermie im Wärmemarkt vorgenommen werden.

Utilisation of Giant African snail (Achatina fulica) meal as protein source by laying hens

A 12-week experiment was carried out to investigate the effects of substituting Giant African snail meal for fish meal in laying hens diet. Four diets were formulated to contain snail meal as replacement for fish meal at 0 (control), 33, 67 and 100%. A total of 120 Shaver Brown pullets aged 18 weeks were allocated to the dietary treatments in a randomised design. Each treatment consisted of three replicates and ten birds per replicate. Feed intake increased only for the 33% treatment as compared to the 67% replacement diet but did not differ from the other treatments. There were no significant treatment effects on egg performance parameters observed (egg production, egg weight, total egg mass, feed conversion ratio and percent shell). The overall feed cost of egg production reduced on the snail meal-based diets. The organoleptic evaluation of boiled eggs revealed no difference between the treatments. Based on these results it was concluded that total replacement of fish meal with cooked snail meat meal does not compromise laying performance or egg quality. The substitution is beneficial in terms of production cost reduction and the reduction of snails will have a beneficial impact especially where these snails are a serious agricultural pest. The manual collection and processing of snails can also become a source of rural income.