Künstliche Randbedingungen und Algebraische Äquivalenzen in der Elastizitätstheorie

In dieser Arbeit werden zwei Aspekte bei Randwertproblemen der linearen Elastizitätstheorie untersucht: die Approximation von Lösungen auf unbeschränkten Gebieten und die Änderung von Symmetrieklassen unter speziellen Transformationen. Ausgangspunkt der Dissertation ist das von Specovius-Neugebauer und Nazarov in "Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type"(Math. Meth. Appl. Sci, 2004; 27) eingeführte Verfahren zur Untersuchung von Petrovsky-Systemen zweiter Ordnung in Außenraumgebieten und Gebieten mit konischen Ausgängen mit Hilfe der Methode der künstlichen Randbedingungen. Dabei werden für die Ermittlung von Lösungen der Randwertprobleme die unbeschränkten Gebiete durch das Abschneiden mit einer Kugel beschränkt, und es wird eine künstliche Randbedingung konstruiert, um die Lösung des Problems möglichst gut zu approximieren. Das Verfahren wird dahingehend verändert, dass das abschneidende Gebiet ein Polyeder ist, da es für die Lösung des Approximationsproblems mit üblichen Finite-Element-Diskretisierungen von Vorteil sei, wenn das zu triangulierende Gebiet einen polygonalen Rand besitzt. Zu Beginn der Arbeit werden die wichtigsten funktionalanalytischen Begriffe und Ergebnisse der Theorie elliptischer Differentialoperatoren vorgestellt. Danach folgt der Hauptteil der Arbeit, der sich in drei Bereiche untergliedert. Als erstes wird für abschneidende Polyedergebiete eine formale Konstruktion der künstlichen Randbedingungen angegeben. Danach folgt der Nachweis der Existenz und Eindeutigkeit der Lösung des approximativen Randwertproblems auf dem abgeschnittenen Gebiet und im Anschluss wird eine Abschätzung für den resultierenden Abschneidefehler geliefert. An die theoretischen Ausführungen schließt sich die Betrachtung von Anwendungsbereiche an. Hier werden ebene Rissprobleme und Polarisationsmatrizen dreidimensionaler Außenraumprobleme der Elastizitätstheorie erläutert. Der letzte Abschnitt behandelt den zweiten Aspekt der Arbeit, den Bereich der Algebraischen Äquivalenzen. Hier geht es um die Transformation von Symmetrieklassen, um die Kenntnis der Fundamentallösung der Elastizitätsprobleme für transversalisotrope Medien auch für Medien zu nutzen, die nicht von transversalisotroper Struktur sind. Eine allgemeine Darstellung aller Klassen konnte hier nicht geliefert werden. Als Beispiel für das Vorgehen wird eine Klasse von orthotropen Medien im dreidimensionalen Fall angegeben, die sich auf den Fall der Transversalisotropie reduzieren lässt.

Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Research through DESIGN through research - a problem statement and a conceptual sketch

This paper re-addresses the issue of a lacking genuine design research paradigm. It tries to sketch an operational model of such a paradigm, based upon a generic design process model, which is derived from basic notions of evolution and learning in different domains of knowing (and turns out to be not very different from existing ones). It does not abandon the scientific paradigm but concludes that the latter has to be embedded into / subordinated under a design paradigm.

The problem of noncharacteristic quasimolecular X-rays in heavy ion collision

Quasi-molecular X-rays observed in heavy ion collisions are interpreted within a relativistic calculation of correlation diagrams using the Dirac-Slater model. A semiquantitative description of noncharacteristic M X rays is given for the system Au-I.

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

The problem of the graphic artist

'The problem of the graphic artist' is a small example of applying elementary mathematics (divisibility of natural numbers) to a real problem which we ourselves have actually experienced. It deals with the possibilities for partitioning a sheet of paper into strips. In this contribution we report on a teaching unit in grade 6 as well as on informal tests with students in school and university. Finally we analyse this example methodologically, summarise our observations with pupils and students, and draw some didactical conclusions.

Cleanliness versus cow comfort – an insolvable problem?

Cubicle should provide good resting comfort as well as clean udders. Dairy cows in cubicle houses often face a restrictive environment with regard to resting behaviour, whereas cleanliness may still be impaired. This study aimed to determine reliable behavioural measures regarding resting comfort applicable in on-farm welfare assessments. Furthermore, relationships between cubicle design, cow sizes, management factors and udder cleanliness (namely teats and teat tips) were investigated. Altogether 15 resting measures were examined in terms of feasibility, inter-observer reliability (IOR) and consistency of results per farm over time. They were recorded during three farm visits on farms in Germany and Austria with cubicle, deep litter and tie stall systems. Seven measures occurred to infrequently to allow reliable recording within a limited observation time. IOR was generally acceptable to excellent except for 'collisions during lying down', which only showed good IOR after improvement of the definition. Only three measures were acceptably repeatable over time: 'duration of lying down', 'percentage of collisions during lying down' and 'percentage of cows lying partly or completely outside lying area'. These measures were evaluated as suitable animal based welfare measures regarding resting behaviour in the framework of an on-farm welfare assessment protocol. The second part of the thesis comprises a cross-sectional study on resting comfort and cow cleanliness including 23 Holstein Friesian dairy herds with very low within-farm variation in cubicle measures. Height at withers, shoulder width and diagonal body length were measured in 79-100 % of the cows (herd size 30 to115 cows). Based on the 25 % largest animals, compliance with recommendations for cubicle measures was calculated. Cleanliness of different body parts, the udder, teats and teat tips was assessed for each cow in the herd prior to morning milking. No significant correlation was found between udder soiling and teat or teat tip soiling on herd level. The final model of a stepwise regression regarding the percentage of dirty teats per farm explained 58.5 % the variance and contained four factors. Teat dipping after milking which might be associated with an overall clean and accurate management style, deep bedded cubicles, increasing cubicle maintenance times and decreasing compliance concerning total cubicle length predicted lower teat soiling. The final model concerning teat tip soiling explained 46.0 % of the variance and contained three factors. Increasing litter height in the rear part of the cubicle and increased alley soiling which is difficult to explain, predicted for less soiled teat tips, whereas increasing compliance concerning resting length was associated with higher percentages of dirty teat tips. The dependent variable ‘duration of lying down’ was analysed using again stepwise regression. The final model explained 54.8 % of the total variance. Lying down duration was significantly shorter in deep bedded cubicles. Further explanatory though not significant factors in the model were neck-rail height, deep bedding or comfort mattresses versus concrete floor or rubber mats and clearance height of side partitions. In the attempt to create a more comprehensive lying down measure, another analysis was carried out with percentage of ‘impaired lying down’ (i.e. events exceeding 6.3 seconds, with collisions or being interrupted) as dependent variable. The explanatory value of this final model was 41.3 %. An increase in partition length, in compliance concerning cubicle width and the presence of straw within bedding predicted a lower proportion of impaired lying down. The effect of partition length is difficult to interpret, but partition length and height were positively correlated on the study farms, possibly leading to a bigger zone of clear space for pelvis freedom. No associations could be found between impaired lying down and teat or teat tip soiling. Altogether, in agreement with earlier studies it was found that cubicle dimensions in practice are often inadequate with regard to the body dimensions of the cows, leading to high proportions of impaired lying down behaviour, whereas teat cleanliness is still unsatisfactory. Connections between cleanliness and cow comfort are far from simplistic. Especially the relationship between cubicle characteristics and lying down behaviour apparently is very complex, so that it is difficult to identify single influential factors that are valid for all farm situations. However, based on the results of the present study the use of deep bedded cubicles can be recommended as well as improved management with special regard to cubicle and litter maintenance in order to achieve both better resting comfort and teat cleanliness.