Cette thèse est divisée en trois parties. Dans la première partie, nous commençons par décrire les théories de Ziglin, Yoshida et Morales-Ramis et les motiver. Dans la deuxième partie, on étudie l’intégrabilité des équations différentielles de Newton à trois degrés de liberté dont les forces sont des polynômes homogènes de degrés trois. En utilisant une analyse du groupe de Galois différentiel des équations aux variations d’ordre supérieur, nous faisons une classification (presque) complète des forces génériques et intégrables. Dans une dernière partie, nous intéressons à l’intégrabilité d’un système d’équations différentielles homogènes d’ordre un (système A). L’application directe de la théorie de Morales-Ramis ne donne des obstructions à l’intégrabilité. En dérivant le système A par rapport au temps, nous obtenons un système différentiel de Newton homogène d’ordre 2 (système B). L’avantage est que ce dernier possède des solutions particulières algébriquement non triviales et le critère classique de Morales-Ramis nous permet d’établir des conditions nécessaires d’intégrabilité. Nous prouvons qu’il existe des relations explicites entre les intégrales premières des deux systèmes et nous introduisons une nouvelle méthode de recherche d’intégrales premières que l’on appelle « Extension tangente double ». Nous appliquons cette méthode à des systèmes planaires homogènes quadratiques. Comme deuxième application, nous montrons que, sous certaines conditions, les racines newtoniennes d’un système différentiel de Newton avec force centrale sont intégrables par quadratures. Nous présentons plusieurs systèmes intégrables avec deux, trois et quatre degrés de liberté. / This thesis is divided into three parts. In the first part we begin by describing the theories of Ziglin, Yoshida and Morales-Ramis and motivating them. In the second part we study the integrability of three-dimensional differential Newton equations with homogeneous polynomial forces of degree three. Using an analysis of differential Galois group of higher order variational equations, we give an almost complete classification of integrable generic forces. The last part is devoted to a study of the integrability of a system of first order homogeneous differential equations (system A ). The direct application of the Morales-Ramis theory does not lead to obstructions to the integrability. If we differentiate the differential system A with respect to time, we obtain a homogeneous Newtonian system (system B). The advantage is that the system B has a non-trivial particular solution and the classical criterion of Morales-Ramis allows us to establish necessary conditions for integrability. We prove that there are explicit relationships between first integrals of the both systems and we introduce a new method for finding first integrals called ``Double tangent extension method''. We apply the obtained results for a detailed analysis of homogeneous planar differential system. Using the double tangent extension method, we formulate some conditions under which the Newtonian roots of Newton's system with central force are integrable by quadratures. Some new cases of integrability with two, three and four degrees of freedom are found.

High power Gallium Nitride (GaN) based field effect transistors are used in many high power applications from RADARs to communications. These devices dissipate a large amount of power and sustain high electric fields during operation. High power dissipation occurs in the form of heat generation through Joule heating which also results in localized hot spot formation that induces thermal stresses. In addition, because GaN is strongly piezoelectric, high electric fields result in large inverse piezoelectric stresses. Combined with residual stresses due to growth conditions, these effects are believed to lead to device degradation and reliability issues. This work focuses on studying these effects in detail through modeling of Heterostructure Field Effect Transistors (HFETs) and metal oxide semiconductor hetero-structure field effect transistor (MOSHFETs) under various operational conditions. The goal is to develop a thorough understanding of device operation in order to better predict device failure and eventually aid in device design through modeling. The first portion of this work covers the development of a continuum scale model which couples temperature and thermal stress to find peak temperatures and stresses in the device. The second portion of this work focuses on development of a micro-scale model which captures phonon-interactions at the device scale and can resolve local perturbations in phonon population due to electron-phonon interactions combined with ballistic transport. This portion also includes development of phonon relaxation times for GaN. The model provides a framework to understand the ballistic diffusive phonon transport near the hotspot in GaN transistors which leads to thermally related degradation in these devices.

Simulationsmodelle sind ein etabliertes Instrument zur Analyse von Auswirkungen exogener Schocks in komplexen Systemen. Die in jüngster Zeit gestiegene verfügbare Rechenleistung und -geschwindigkeit hat die Entwicklung detaillierterer und komplexerer Simulationsmodelle befördert. Dieser Trend hat jedoch Bedenken hinsichtlich der Unsicherheit solcher Modellergebnisse aufgeworfen und daher viele Nutzer von Simulationsmodellen dazu motiviert, Unsicherheiten in ihren Simulationen zu integrieren. Eine Möglichkeit dies systematisch zu tun besteht darin, stochastische Elemente in die Modellgleichungen zu integrieren, wodurch das jeweilige Modell zu einem Problem (mehrfacher) numerischer Integrationen wird. Da es für solche Probleme meist keine analytischen Lösungen gibt, werden numerische Approximationsmethoden genutzt. Die derzeit zur Quantifizierung von Unsicherheiten in Simulationsmodellen genutzt en Techniken, sind entweder rechenaufwändig (Monte Carlo [MC] -basierte Methoden) oder liefern Ergebnisse von heterogener Qualität (Gauß-Quadraturen [GQs]). In Anbetracht der Bedeutung von effizienten Methoden zur Quantifizierung von Unsicherheit im Zeitalter von „big data“ ist es das Ziel dieser Doktorthesis, Methoden zu entwickeln, die die Näherungsfehler von GQs verringern und diese Methoden einer breiteren Forschungsgemeinschaft zugänglich machen. Zu diesem Zweck werden zwei neuartige Methoden zur Quantifizierung von Unsicherheiten entwickelt und in vier verschiedene, große partielle und allgemeine Gleichgewichtsmodelle integriert, die sich mit Agrarumweltfragen befassen. Diese Arbeit liefert methodische Entwicklungen und ist von hoher Relevanz für angewandte Simulationsmodellierer. Obwohl die Methoden in großen Simulationsmodellen für Agrarumweltfragen entwickelt und getestet werden, sind sie nicht durch Modelltyp oder Anwendungsgebiet beschränkt, sondern können ebenso in anderen Zusammenhängen angewandt werden. / Simulation models are an established tool for assessing the impacts of exogenous shocks in complex systems. Recent increases in available computational power and speed have led to simulation models with increased levels of detail and complexity. However, this trend has raised concerns regarding the uncertainty of such model results and therefore motivated many users of simulation models to consider uncertainty in their simulations. One way is to integrate stochastic elements into the model equations, thus turning the model into a problem of (multiple) numerical integration. As, in most cases, such problems do not have analytical solutions, numerical approximation methods are applied. The uncertainty quantification techniques currently used in simulation models are either computational expensive (Monte Carlo [MC]-based methods) or produce results of varying quality (Gaussian quadratures [GQs]). Considering the importance of efficient uncertainty quantification methods in the era of big data, this thesis aims to develop methods that decrease the approximation errors of GQs and make these methods accessible to the wider research community. For this purpose, two novel uncertainty quantification methods are developed and integrated into four different large-scale partial and general equilibrium models addressing agro-environmental issues. This thesis provides method developments and is of high relevance for applied simulation modelers who struggle to apply computationally burdensome stochastic modeling methods. Although the methods are developed and tested in large-scale simulation models addressing agricultural issues, they are not restricted to a model type or field of application.

The first International Congress of Mathematicians (ICM) was held in Zurich in 1897, setting the standards for all future ICMs. Whilst giving an overview of the congress itself, this thesis focuses on the Swiss organisers, who were predominantly university professors and secondary school teachers. As this thesis aims to offer some insight into their lives, it includes their biographies, highlighting their individual contributions to the congress. Furthermore, it explains why Zurich was chosen as the first host city and how the committee proceeded with the congress organisation. Two of the main organisers were the Swiss geometers Carl Friedrich Geiser (1843-1934) and Ferdinand Rudio (1856-1929). In addition to the congress, they also made valuable contributions to mathematical education, and in Rudio's case, the history of mathematics. Therefore, this thesis focuses primarily on these two mathematicians. As for Geiser, the relationship to his great-uncle Jakob Steiner is explained in more detail. Furthermore, his contributions to the administration of the Swiss Federal Institute of Technology are summarised. Due to the overarching theme of mathematical education and collaborations in this thesis, Geiser's schoolbook "Einleitung in die synthetische Geometrie" is considered in more detail and Geiser's methods are highlighted. A selection of Rudio's contributions to the history of mathematics is studied as well. His book "Archimedes, Huygens, Lambert, Legendre" is analysed and compared to E W Hobson's treatise "Squaring the Circle". Furthermore, Rudio's papers relating to the commentary of Simplicius on quadratures by Antiphon and Hippocrates are considered, focusing on Rudio's translation of the commentary and on "Die Möndchen des Hippokrates". The thesis concludes with an analysis of Rudio's popular lectures "Leonhard Euler" and "Über den Antheil der mathematischen Wissenschaften an der Kultur der Renaissance", which are prime examples of his approach to the history of mathematics.