This cumulative dissertation is organized as follows: After the introduction, the second chapter, based on “Asymptotic independence of bivariate order statistics” (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived. Chapter 3 is from “Conditional tail independence in archimedean copula models” (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas. “Generalized pareto copulas: A key to multivariate extremes” (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017. The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing. / Diese kumulative Dissertation ist wie folgt aufgebaut: Nach der Einleitung wird im zweiten Kapitel, welches auf “Asymptotic independence of bivariate order statistics” (2017) von Falk und Wisheckel beruht, die asymptotische Abhängigkeitsstruktur von bivariaten Ordnungsstatistiken untersucht. Dazu wird eine explizite Darstellung der bedingten Verteilung einer bivariaten Ordnungsstatistik hergeleitet. Kapitel 3, basierend auf “Conditional tail independence in archimedean copula models” (2019) von Falk, Padoan und Wisheckel, zeigt, dass unter schwachen Anforderungen an den Generator einer Archimedischen Copula die übrigen Komponenten unabhängig werden, wenn man auf eine davon bedingt. Das insbesondere auch dann, wenn die Komponenten ohne die Bedingung abhängig waren. Die theoretischen Erkenntnisse werden anhand von Simulationsergebnissen verdeutlicht. “Generalized pareto copulas: A key to multivariate extremes” (2019) von Falk, Padoan und Wisheckel liefert Kapitel 4. Es wird ein nichtparametrischer Ansatz vorgestellt um die Überschreitungswahrscheinlichkeit eines Zufallsvektors über einen festen, hohen Schwellenwert zu schätzen, wenn dieser einer verallgemeinerten Pareto Copula folgt. Das Verfahren wird in den theoretischen Rahmen eingebettet, anhand einer Simulation validiert und auf Luftverschmutzungsdaten in Mailand, Italien, von 2002 bis 2017 angewendet. Im fünften Kapitel werden einige weitere Ergebnisse gesammelt: es geht um Ableitungen von D-Normen, insbesondere um eine Bedingung, die die Existenz der Richtungsableitungen sicherstellt. Außerdem werden multivariate Spacings thematisiert.

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For larger instances of multi-objective optimisation problems, the exact Pareto frontier can be both difficult and time-consuming to calculate. There is a wide range of methods to find feasible solutions to such problems, but techniques for finding good optimistic bounds to compare the feasible solutions with are missing. In this study, we investigate the use of Lagrangian relaxation to create optimistic bounds to bi-objective optimisation problems with complicating side constraints. The aim is to develop an effective method to produce optimistic bounds that are closer to the Pareto frontier than the commonly used linear programming bounds.  In order to use Lagrangian relaxation on the bi-objective problem, the objectives are combined using the weighted sum method. A Lagrangian dual function is then constructed by relaxing the complicating constraints and the subgradient method is used to optimise the dual problem in order to find an optimistic solution. By solving the single-objective problem for multiple weights, an optimistic bound to the Pareto frontier can be constructed. The subgradient method also includes a heuristic to find feasible solutions. The feasible solutions found by the heuristic form a pessimistic bound to the frontier. The method has been implemented and tested on several instances of a capacitated facility location problem with cost and CO2 emission as objectives. The results indicate that, by using Lagrangian relaxation, an optimistic bound close to the Pareto frontier can be found in a relatively short time. The heuristic used also manages to produce good pessimistic bounds, and hence the Pareto frontier can be tightly enclosed. The optimistic bounds found by Lagrangian relaxation are better and more consistent along the Pareto frontier than the bounds found by linear programming.

<p>This thesis is both an extension and a critique of Roy Hornosty's doctoral dissertation. In " Conceptions of Human Nature in the Sociological Tradition", Hornosty traces the development and career of two distinct concepts of human nature as they are reflected in sociological theory. Hornosty argues that sociology originally emerged with two competing ideas of man, one stressing the logical priority of the individual, and the other, the predominance of the collective.</p> <p>In the course of his study, Hornosty discusses what he refers to as the second generation of European sociologists, comprised suggests of Durkheim, Weber, Simmel that each theorist of this and Pareto. Hornosty generation describes a conception of human nature based on an 'inner dialectic' between the individual, who seeks independent self-actualization, over and against the demands of the collective, which develops according to laws which are often in stark contrast to the dictates of individuality.</p> <p>One chapter of Hornosty's study is devoted to Georg Simmel's sociological thought. In it, Hornosty argues that Simmel views man in terms of an antinomy between social and individual forces, locked in an irreconcilable struggle. For Simmel, Hornosty believes, man is both social and, at the same time, independent of society, although never completely. According to Hornosty's interpretation of Simmel, this dialectic is an ineradicable condition of mankind.</p> <p>The present study focuses entirely on the writings of Georg Simmel. The author argues that while an undeniable dialectic exists between the individual and society, it is by no means irreconcilable as Hornosty and others have suggested. In fact, it will be argued that Simmel saw ways in which the dichotomous relationship between the individual and the collective could be overcome.</p> <p>The author contends that Simmel, influenced by Nietzsche's philosophy, details avenues of consequence removed from the sociological nexus. By focusing on Simmel 's four categories of human experience, two of which are completely removed from the social paradigm, this thesis outlines Simmel's attempt to find a manner in which certain talented individuals could transcend the dichotomy between the individual and society through devotion to objective goals which lie beyond the dialectic.</p> <p>The author suggests that Simmel in fact reserves his highest praise for the creative genius, the individual able to tolerate, and in turn, transcend the tensions of his or her existence in the name of a higher good, whether artistic or intellectual. By outlining Simmel's preoccupation with the creative process, and his search for viable expressions of individuality removed from society, the author attempts to illustrate the limitations of the sociological paradigm.</p> / Thesis / Master of Arts (MA)