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.

Functions satisfying holonomic q-differential equations

In a similar manner as in some previous papers, where explicit algorithms for finding the differential equations satisfied by holonomic functions were given, in this paper we deal with the space of the q-holonomic functions which are the solutions of linear q-differential equations with polynomial coefficients. The sum, product and the composition with power functions of q-holonomic functions are also q-holonomic and the resulting q-differential equations can be computed algorithmically.

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.

Total differential scattering cross section of {Ar^+}-Ar at 15 to 400 keV

We report on the measurement of the total differential scattering cross section of {Ar^+}-Ar at laboratory energies between 15 and 400 keV. Using an ab initio relativistic molecular program which calculates the interatomic potential energy curve with high accuracy, we are able to reproduce the detailed structure found in the experiment.

An introduction to ordinary differential equations by computer algebra-systems

We report on an elementary course in ordinary differential equations (odes) for students in engineering sciences. The course is also intended to become a self-study package for odes and is is based on several interactive computer lessons using REDUCE and MATHEMATICA . The aim of the course is not to do Computer Algebra (CA) by example or to use it for doing classroom examples. The aim ist to teach and to learn mathematics by using CA-systems.

Root parametrized differential equations

The present thesis is about the inverse problem in differential Galois Theory. Given a differential field, the inverse  problem asks which linear algebraic groups can be realized as differential Galois groups of Picard-Vessiot extensions of this field.   In this thesis we will concentrate on the realization of the classical groups as differential Galois groups. We introduce a method for a very general realization of these groups. This means that we present for the classical groups of Lie rank $l$ explicit linear differential equations where the coefficients are differential polynomials in $l$ differential indeterminates over an algebraically closed field of constants $C$, i.e. our differential ground field is purely differential transcendental over the constants.   For the groups of type $A_l$, $B_l$, $C_l$, $D_l$ and $G_2$ we managed to do these realizations at the same time in terms of Abhyankar's program 'Nice Equations for Nice Groups'. Here the choice of the defining matrix is important. We found out that an educated choice of $l$ negative roots for the parametrization together with the positive simple roots leads to a nice differential equation and at the same time defines a sufficiently general element of the Lie algebra. Unfortunately for the groups of type $F_4$ and $E_6$ the linear differential equations for such elements are of enormous length. Therefore we keep in the case of $F_4$ and $E_6$ the defining matrix differential equation which has also an easy and nice shape.   The basic idea for the realization is the application of an upper and lower bound criterion for the differential Galois group to our parameter equations and to show that both bounds coincide. An upper and lower bound criterion can be found in literature. Here we will only use the upper bound, since for the application of the lower bound criterion an important condition has to be satisfied. If the differential ground field is $C_1$, e.g., $C(z)$ with standard derivation, this condition is automatically satisfied. Since our differential ground field is purely differential transcendental over $C$, we have no information whether this condition holds or not.   The main part of this thesis is the development of an alternative lower bound criterion and its application. We introduce the specialization bound. It states that the differential Galois group of a specialization of the parameter equation is contained in the differential Galois group of the parameter equation. Thus for its application we need a differential equation over $C(z)$ with given differential Galois group. A modification of a result from Mitschi and Singer yields such an equation over $C(z)$ up to differential conjugation, i.e. up to transformation to the required shape. The transformation of their equation to a specialization of our parameter equation is done for each of the above groups in the respective transformation lemma.

Effects of red brick production on land use, household income, and greenhouse gas emissions in Khartoum, Sudan

In Khartoum (Sudan) a particular factor shaping urban land use is the rapid expansion of red brick making (BM) for the construction of houses which occurs on the most fertile agricultural Gerif soils along the Nile banks. The objectives of this study were to assess the profitability of BM, to explore the income distribution among farmers and kiln owners, to measure the dry matter (DM), nitrogen (N), phosphorus (P), potassium (K) and organic carbon (C_org) in cow dung used for BM, and to estimate the greenhouse gas (GHG) emissions from burned biomass fuel (cow dung and fuel wood). About 49 kiln owners were interviewed in 2009 using a semi-structured questionnaire that allowed to record socio-economic and variable cost data for budget calculations, and determination of Gini coefficients. Samples of cow dung were collected directly from the kilns and analyzed for their nutrients concentrations. To estimate GHG emissions a modified approach of the Intergovernmental Panel on Climate Change (IPCC) was used. The land rental value from red brick kilns was estimated at 5-fold the rental value from agriculture and the land rent to total cost ratio was 29% for urban farms compared to 6% for BM. The Gini coefficients indicated that income distribution among kiln owners was more equal than among urban farmers. Using IPCC default values the 475, 381, and 36 t DM of loose dung, compacted dung, and fuel wood used for BM emit annually 688, 548, and 60 t of GHGs, respectively.

Employment and Income of Workers on Indonesian Oil Palm Plantations: Food Crisis at the Micro Level

The importance of oil palm sector for Indonesia is inevitable as the country currently serves as the world’s largest producer of crude palm oil. This paper focuses on the situation of workers on Indonesian oil palm plantations. It attempts to investigate whether the remarkable development of the sector is followed by employment opportunities and income generation for workers. This question is posed within the theoretical framework on the link between trade liberalisation and labour rights, particularly in a labour-intensive and low-skilled sector. Based on extensive field research in Riau, this paper confirms that despite the rapid development of the oil palm plantation sector in Indonesia, the situations of workers in the sector remain deplorable, particularly their employment status and income. This also attests that trade liberalisation in the sector adversely affects labour rights. The poor working conditions also have ramifications for food security at the micro level.

Poverty targeting and income impact of subsidised credit on accessed households in the Northern Mountainous Region of Vietnam

This paper uses the data of 1338 rural households in the Northern Mountainous Region of Vietnam to examine the extent to which subsidised credit targets the poor and its impacts. Principal Component Analysis and Propensity Score Matching were used to evaluate the depth of outreach and the income impact of credit. To address the problem of model uncertainty, the approach of Bayesian Model Average applied to the probit model was used. Results showed that subsidised credit successfully targeted the poor households with 24.10% and 69.20% of clients falling into the poorest group and the three bottom groups respectively. Moreover, those who received subsidised credit make up 83% of ethnic minority households. These results indicate that governmental subsidies are necessary to reach the poor and low income households, who need capital but are normally bypassed by commercial banks. Analyses also showed that ethnicity and age of household heads, number of helpers, savings, as well as how affected households are by shocks were all factors that further explained the probability at which subsidised credit has been assessed. Furthermore, recipients obtained a 2.61% higher total income and a 5.93% higher farm income compared to non-recipients. However, these small magnitudes of effects are statistically insignificant at a 5% level. Although the subsidised credit is insufficient to significantly improve the income of the poor households, it possibly prevents these households of becoming even poorer.