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.

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.

Error estimates for approximate approximations on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in- tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed. In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

Gröbnerbasen in Ore-Algebren

In dieser Arbeit werden grundlegende Algorithmen für Ore-Algebren in Mathematica realisiert. Dabei entsteht eine Plattform um die speziellen Beschränkungen und Möglichkeiten dieser Algebren insbesondere im Zusammenhang mit Gröbnerbasen an praktischen Beispielen auszuloten. Im Gegensatz zu den existierenden Paketen wird dabei explizit die Struktur der Ore-Algebra benutzt. Kandri-Rody und Weispfenning untersuchten 1990 Verallgemeinerungen von Gröbnerbasen auf Algebren ordnungserhaltender Art (``algebras of solvable type''). Diese verhalten sich so, dass Buchbergers Algorithmus stets eine Gröbnerbasis findet. Es wird ein Beispiel gezeigt, an dem klar wird, dass es mehr Ore-Algebren ordnungserhaltender Art gibt als die in der Literatur stets betrachteten Operator-Algebren. Für Ore-Algebren ordnungserhaltender Art werden Algorithmen zu Gröbnerbasen implementiert. Anschließend wird der Gröbner-Walk für Ore-Algebren untersucht. Der Gröbner-Walk im kommutativen Fall wird mit einem instruktiven Beispiel vorgestellt. Dann wird zum nichtkommutativen Fall übergegangen. Es wird gezeigt, dass die Eigenschaft ordnungserhaltender Art zu sein, auf der Strecke zwischen zwei Ordnungen erhalten bleibt. Eine leichte Modifikation des Walks für Ore-Algebren wird implementiert, die im Erfolgsfall die Basis konvertiert und ansonsten abbricht. Es werden Beispiele angegeben, in denen der modifizierte Walk funktioniert sowie ein Beispiel analysiert, in dem er versagt.

Digital image analysis as a tool to estimate legume contributions in legume-grass swards

Summary: Productivity and forage quality of legume-grass swards are important factors for successful arable farming in both organic and conventional farming systems. For these objectives the botanical composition of the swards is of particular importance, especially, the content of legumes due to their ability to fix airborne nitrogen. As it can vary considerably within a field, a non-destructive detection method while doing other tasks would facilitate a more targeted sward management and could predict the nitrogen supply of the soil for the subsequent crop. This study was undertaken to explore the potential of digital image analysis (DIA) for a non destructive prediction of legume dry matter (DM) contribution of legume-grass mixtures. For this purpose an experiment was conducted in a greenhouse, comprising a sample size of 64 experimental swards such as pure swards of red clover (Trifolium pratense L.), white clover (Trifolium repens L.) and lucerne (Medicago sativa L.) as well as binary mixtures of each legume with perennial ryegrass (Lolium perenne L.). Growth stages ranged from tillering to heading and the proportion of legumes from 0 to 80 %. Based on digital sward images three steps were considered in order to estimate the legume contribution (% of DM): i) The development of a digital image analysis (DIA) procedure in order to estimate legume coverage (% of area). ii) The description of the relationship between legume coverage (% area) and legume contribution (% of DM) derived from digital analysis of legume coverage related to the green area in a digital image. iii) The estimation of the legume DM contribution with the findings of i) and ii). i) In order to evaluate the most suitable approach for the estimation of legume coverage by means of DIA different tools were tested. Morphological operators such as erode and dilate support the differentiation of objects of different shape by shrinking and dilating objects (Soille, 1999). When applied to digital images of legume-grass mixtures thin grass leaves were removed whereas rounder clover leaves were left. After this process legume leaves were identified by threshold segmentation. The segmentation of greyscale images turned out to be not applicable since the segmentation between legumes and bare soil failed. The advanced procedure comprising morphological operators and HSL colour information could determine bare soil areas in young and open swards very accurately. Also legume specific HSL thresholds allowed for precise estimations of legume coverage across a wide range from 11.8 - 72.4 %. Based on this legume specific DIA procedure estimated legume coverage showed good correlations with the measured values across the whole range of sward ages (R2 0.96, SE 4.7 %). A wide range of form parameters (i.e. size, breadth, rectangularity, and circularity of areas) was tested across all sward types, but none did improve prediction accuracy of legume coverage significantly. ii) Using measured reference data of legume coverage and contribution, in a first approach a common relationship based on all three legumes and sward ages of 35, 49 and 63 days was found with R2 0.90. This relationship was improved by a legume-specific approach of only 49- and 63-d old swards (R2 0.94, 0.96 and 0.97 for red clover, white clover, and lucerne, respectively) since differing structural attributes of the legume species influence the relationship between these two parameters. In a second approach biomass was included in the model in order to allow for different structures of swards of different ages. Hence, a model was developed, providing a close look on the relationship between legume coverage in binary legume-ryegrass communities and the legume contribution: At the same level of legume coverage, legume contribution decreased with increased total biomass. This phenomenon may be caused by more non-leguminous biomass covered by legume leaves at high levels of total biomass. Additionally, values of legume contribution and coverage were transformed to the logit-scale in order to avoid problems with heteroscedasticity and negative predictions. The resulting relationships between the measured legume contribution and the calculated legume contribution indicated a high model accuracy for all legume species (R2 0.93, 0.97, 0.98 with SE 4.81, 3.22, 3.07 % of DM for red clover, white clover, and lucerne swards, respectively). The validation of the model by using digital images collected over field grown swards with biomass ranges considering the scope of the model shows, that the model is able to predict legume contribution for most common legume-grass swards (Frame, 1992; Ledgard and Steele, 1992; Loges, 1998). iii) An advanced procedure for the determination of legume DM contribution by DIA is suggested, which comprises the inclusion of morphological operators and HSL colour information in the analysis of images and which applies an advanced function to predict legume DM contribution from legume coverage by considering total sward biomass. Low residuals between measured and calculated values of legume dry matter contribution were found for the separate legume species (R2 0.90, 0.94, 0.93 with SE 5.89, 4.31, 5.52 % of DM for red clover, white clover, and lucerne swards, respectively). The introduced DIA procedure provides a rapid and precise estimation of legume DM contribution for different legume species across a wide range of sward ages. Further research is needed in order to adapt the procedure to field scale, dealing with differing light effects and potentially higher swards. The integration of total biomass into the model for determining legume contribution does not necessarily reduce its applicability in practice as a combined estimation of total biomass and legume coverage by field spectroscopy (Biewer et al. 2009) and DIA, respectively, may allow for an accurate prediction of the legume contribution in legume-grass mixtures.

Modulsymbole und Bruhat-Tits-Gebäude der PGL(3) über Funktionenkörpern

Thema der vorliegenden Arbeit ist die Bestimmung von Basen von Räumen spezieller harmonischer 2-Koketten auf Bruhat-Tits-Gebäuden der PGL(3) über Funktionenkörpern. Hierzu wird der Raum der speziellen harmonischen 2-Koketten auf dem Bruhat-Tits-Gebäude der PGL(3) zunächst mit gewissen komplexen Linearkombinationen von 2-Simplizes des Quotientenkomplexes, sogenannten geschlossenen Flächen, identifiziert und anschließend durch verallgemeinerte Modulsymbole beschrieben. Die Darstellung der Gruppe der Modulsymbole durch Erzeuger und Relationen ermöglicht die Bestimmung einer endlichen Basis des Raums der speziellen harmonischen 2-Koketten. Die so gewonnenen Erkenntnisse können zur Untersuchung von Hecke-Operatoren auf speziellen harmonischen 2-Koketten genutzt werden. Mithilfe des hergeleiteten Isomorphismus zwischen dem Raum der speziellen harmonischen 2-Koketten und dem Raum der geschlossenen Flächen wird die Theorie der Hecke-Operatoren auf den Raum der geschlossenen Flächen übertragen. Dies ermöglicht die Berechnung von Abbildungsmatrizen der Hecke-Operatoren auf dem Raum der harmonischen 2-Koketten durch die Auswertung auf den geschlossenen Flächen.