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.

Eros and Thanatos in Ernest Hemingway's "The Short Happy Life of Francis Macomber" and Angela Carter's "Master"

According to Hemingway all good prose has the dignity and draught of an iceberg. This is especially true of Hemingway's short stories. "Francis Macomber" counts among the best composed short stories in English. Interpretation sways between Hemingway's idealisation of the male code and its deconstruction. Is the White Hunter a British scourge of American values or is Margot the tragic victim of a newly founded male friendship? Is the open ending rather a hunting accident or the mean murder of an unloved spouse? Hemingway gives hitherto uninterpreted clues by mentioning strategies of big game hunting and ballistics. Carter's magic story "Master"; deals with the regression of a White Hunter who has become 'his own negative'. An Englishman dissolves into the Amazonian jungle and is killed by his own victim. The story can be read as a macabre crime of sexuality and murder or as a dystopian warning of the future of mankind if we go on exploiting and destroying our planet.

Gone with the internet - How the old education was lost

Despite its young history, Computer Science Education has seen a number of "revolutions". Being a veteran in the field, the author reflects on the many changes he has seen in computing and its teaching. The intent of this personal collection is to point out that most revolutions came unforeseen and that many of the new learning initiatives, despite high financial input, ultimately failed. The author then considers the current revolution (MOOC, inverted lectures, peer instruction, game design) and, based on the lessons learned earlier, argues why video recording is so successful. Given the fact that this is the decade we lost print (papers, printed books, book shops, libraries), the author then conjectures that the impact of the Internet will make this revolution different from previous ones in that most of the changes are irreversible. As a consequence he warns against storming ahead blindly and suggests to conserve - while it is still possible - valuable components of what might soon be called the antebellum age of education.

Institutions, Behavior and Evolution

The three articles constituting this thesis are for reasons of content or method related to the following three fields in economics: Behavioral Economics, Evolutionary Game Theory and Formal Institutional Economics. A core element of these fields is the concept of individual preferences. Preferences are of central importance for the conceptional framework to analyze human behavior. They form the foundation for the theory of rational choice which is defined by the determination of the choice set and the selection of the most preferred alternative according to some consistency requirements. The theory of rational choice is based on a very simplified description of the problem of choice (object function and constraints). However, that choices depend on many more factors is for instance propagated by psychological theories and is supported by many empirical and experimental studies. This thesis adds to a better understanding of individual behavior to the extent that the evolution of certain characteristics of preferences and their consequences on human behavior forms the overarching theme of the dissertation. The long-term effect of evolutionary forces on a particular characteristic of importance in the theoretical, empirical and experimental economic literature, the concept of inequality aversion, is subject of the article “The evolution of inequality aversion in a simplified game of life” (Chapter 4). The contribution of the article is the overcoming of a restriction of former approaches to analyze the evolution of preferences in very simple environments. By classifying human interaction into three central economic games, the article provides a first step towards a simplified and sufficiently complete description of the interaction environment. Within such an environment the article characterizes the evolutionary stable preference distribution. One result shows, that the interaction of the aforementioned three classes can stabilize a preference of inequality aversion in the subpopulation which is favored in the problem of redistribution. The two remaining articles are concerned with social norms, which dissemination is determined by medium-run forces of cultural evolution. The article “The impact of market innovations on the evolution of social norms: the sustainability case.“ (Chapter 2) studies the interrelation between product innovations which are relevant from a sustainability perspective and an according social norm in consumption. This relation is based on a conformity bias in consumption and the attempt to avoid cognitive dissonances resulting from non-compliant consumption. Among others, it is shown that a conformity bias on the consumption side can lead to multiple equilibria on the side of norm adoption. The article “Evolution of cooperation in social dilemmas: signaling internalized norms.” (Chapter 3) studies the emergence of cooperation in social dilemmas based on the signaling of social norms. The article provides a potential explanation of cooperative behavior, which does not rely on the assumption of structured populations or on the unmotivated ability of social norms to restrict individual actions or strategy spaces. A comprehensive result of the single articles is the explanation of the phenomenon of partial norm adaption or dissemination of preferences. The plurality of the applied approaches with respect to the proximity to the rational choice approach and regarding the underlying evolutionary mechanics is a particular strength of the thesis. It shows the equality of these approaches in their potential to explain the phenomenon of cooperation in environments that provide material incentives for defective behavior. This also points to the need of a unified framework considering the biological and cultural coevolution of preference patterns.