A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Solution properties of the de Branges differential recurrence equation

In this 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper we study differential recurrence equations equivalent to de Branges' original ones and show that many solutions of these differential recurrence equations don't change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.

Solution of the time dependent Dirac-Fock-Slater equation for many-electron collisions systems using a time window method

We present a new scheme to solve the time dependent Dirac-Fock-Slater equation (TDDFS) for heavy many electron ion-atom collision systems. Up to now time independent self consistent molecular orbitals have been used to expand the time dependent wavefunction and rather complicated potential coupling matrix elements have been neglected. Our idea is to minimize the potential coupling by using the time dependent electronic density to generate molecular basis functions. We present the first results for 16 MeV S{^16+} on Ar.

Effects of plant diversity on bioenergy parameters in grassland biomass

Extensive grassland biomass for bioenergy production has long been subject of scientific research. The possibility of combining nature conservation goals with a profitable management while reducing competition with food production has created a strong interest in this topic. However, the botanical composition will play a key role for solid fuel quality of grassland biomass and will have effects on the combustion process by potentially causing corrosion, emission and slagging. On the other hand, botanical composition will affect anaerobic digestibility and thereby the biogas potential. In this thesis aboveground biomass from the Jena-Experiment plots was harvested in 2008 and 2009 and analysed for the most relevant chemical constituents effecting fuel quality and anaerobic digestibility. Regarding combustion, the following parameters were of main focus: higher heating value (HHV), gross energy yield (GE), ash content, ash softening temperature (AST), K, Ca, Mg, N, Cl and S content. For biogas production the following parameters were investigated: substrate specific methane yield (CH4 sub), area specific methane yield (CH4 area), crude fibre (CF), crude protein (CP), crude lipid (CL) and nitrogen-free extract (NfE). Furthermore, an improvement of the fuel quality was investigated through applying the Integrated generation of solid Fuel and Biogas from Biomass (IFBB) procedure. Through the specific setup of the Jena-Experiment it was possible to outline the changes of these parameters along two diversity gradients: (i) species richness (SR; 1 to 60 species) and (ii) functional group (grasses, legumes, small herbs and tall herbs) presence. This was a novel approach on investigating the bioenergy characteristic of extensive grassland biomass and gave detailed insight in the sward-composition¬ - bioenergy relations such as: (i) the most relevant SR effect was the increase of energy yield for both combustion (annual GE increased by 26% from SR8→16 and by 65% from SR8→60) and anaerobic digestion (annual CH4 area increased by 22% from SR8→16 and by 49% from SR8→60) through a strong interaction of SR with biomass yield; (ii) legumes play a key role for the utilization of grassland biomass for energy production as they increase the energy content of the substrate (HHV and CH4 sub) and the energy yield (GE and CH4 area); (iii) combustion is the conversion technique that will yield the highest energy output but requires an improvement of the solid fuel quality in order to reduce the risk of corrosion, emission and slagging related problems. This was achieved through applying the IFBB-procedure, with reductions in ash (by 23%), N (28%), K (85%), Cl (56%) and S (59%) and equal levels of concentrations along the SR gradient.

Effects of grassland conversion and tillage intensities on soil microbial biomass, residues and community structure

Agricultural intensification has a strong impact on level of soil organic matter (SOM), microbial biomass stocks and microbial community structure in agro-ecosystems. The size of the microbial necromass C pool could be about 40 times that of the living microbial biomass C pool in soils. Due to the specificity, amino sugar analysis gives more important information on the relative contribution of fungal and bacterial residues to C sequestration potential of soils. Meanwhile, the relationship between microbial biomass and microbial necromass in soil and its ecological significance on SOM are not fully understood and likely to be very complex in grassland soils. This thesis focuses on the effects of tillage, grassland conversion intensities and fertilisation on microbial biomass, residues and community structure. The combined analyses of microbial biomass and residue formation of both fungi and bacteria provided a unique opportunity to study the effect of tillage, grassland conversion and fertilisation on soil microbial dynamics. In top soil at 0-30 cm layer, a reduction in tillage intensity by the GRT and NT treatments increased the accumulation of saprotrophic fungi in comparison with the MBT treatment. In contrast, the GRT and NT treatments promoted AMF at the expense of saprotrophic fungi in the bottom soil layer at 30-40 cm depth. The negative relationship between the ergosterol to microbial biomass C ratio and the fungal C to bacterial C ratio points to the importance of the relationship between saprotrophic fungi and biotrophic AMF for tillage-induced changes in microbial turnover of SOC. One-season cultivation of winter wheat with two tillage events led to a significant loss in SOC and microbial biomass C stocks at 0-40 cm depth in comparison with the permanent grassland, even 5 years after the tillage event. However, the tillage induced loss in microbial biomass C was roughly 40% less in the long-term than in the short-term of the current experiment, indicating a recovery process during grassland restoration. In general, mould board tillage and grassland conversion to maize monoculture promoted saprotrophic fungi at the expense of biotrophic AMF and bacteria compared to undisturbed grassland soils. Slurry application promoted bacterial residues as indicated by the decreases in both, the ergosterol to microbial biomass C ratio and the fungal C to bacterial C ratio. In addition, the lost microbial functional diversity due to tillage and maize monoculture was restored by slurry application both in arable and grassland soils. I conclude that the microbial biomass C/S ratio can be used as an additional indicator for a shift in microbial community. The strong relationships between microbial biomass and necromass indices points to the importance of saprotrophic fungi and biotrophic AMF for agricultural management induced effects on microbial turnover and ecosystem C storage. Quantitative information on exact biomass estimates of these two important fungal groups in soil is inevitably necessary to understand their different roles in SOM dynamics.