Newton's second law, Schrödinger's equation and Maxwell's equations are all theories composed of at most second-time derivatives. Indeed, it is not often we need to take the time derivative of the acceleration. So why are we not seeing more higher-order derivative theories? Although several studies present higher derivatives' usefulness in quadratic gravity and scalar-field theories, one will eventually encounter a problem. In 1850, the physicist Mikhail Ostrogradsky presented a theorem that stated that a non-degenerate Lagrangian composed of finite higher-order time derivatives results in a Hamiltonian unbounded from below. Explicitly, it was shown that the Hamiltonian of such a system includes linearity in physical momenta, often referred to as the ''Ostrogradsky ghost''. This thesis studies how one can avoid the Ostrogradsky ghost by considering degenerate Lagrangians to put constraints on the momenta. The study begins by showing the existence of the ghost and later cover the essential Hamiltonian formalism needed to conduct Hamiltonian constraint analyses of second-order time derivative systems, both single-variable and systems coupled to a regular one. Ultimately, the degenerate second-order Lagrangians successfully eliminate the Ostrogradsky ghost by generating secondary constraints restricting the physical momenta. Moreover, an outline of a Hamiltonian analysis of a general higher-order Lagrangian is presented at the end.

En employant des méthodes de la théorie de Chern-Weil, Reznikov produit une condition suffisante qui assure la non-trivialité de la projectivisation \( \mathbb{P}(E) \) d'un fibré vectoriel complexe en tant que fibré Hamiltonien. Dans le contexte de la quantification géométrique, Savelyev et Shelukhin introduisent un nouvel invariant des fibrés Hamiltoniens avec valeurs dans la K-théorie et étendent le résultat de Reznikov. Cet invariant est donné par l'indice d'Atiyah-Singer d'une famille d'opérateurs \( \text{Spin}^{c} \) de Dirac. Dans ce mémoire, on s'intéresse à des fibrés Hamiltoniens résultant d'un produit fibré et d'un produit cartésien d'une collection de fibrés projectifs complexes \( \mathbb{P}(E_1), \cdots, \mathbb{P}(E_r) \). En usant des mêmes méthodes que Shelukhin et Savelyev, on définit une famille d'opérateurs \( \text{Spin}^{c} \) de Dirac qui agissent sur les sections d'un fibré de Dirac canonique à valeurs dans un fibré pré-quantique. L'indice de famille produit un invariant de fibrés Hamiltoniens avec fibres données par un produit d'espaces projectifs complexes et permet de construire des exemples de fibrés Hamiltoniens non-triviaux. / Using methods of Chern-Weil Theory, Reznikov provides a sufficient condition for the non-triviality of the projectivization \( \mathbb{P}(E) \) of a complex vector bundle \( E \) as a Hamiltonian fibration. In the setting of geometric quantization, Savelyev and Shelukhin introduce a new invariant of Hamiltonian fibrations and a K-theoretic lift of Reznikov's result. This invariant is given by the Atiyah-Singer index of a family of \( \text{Spin}^{c} \)-Dirac operators. In this thesis, we consider Hamiltonian fibrations given by the Cartesian product and the fiber product of a collection of complex projective bundles \( \mathbb{P}(E_1), \cdots, \mathbb{P}(E_r) \). Using the same methods as Savelyev and Shelukhin, we define a family of \( \text{Spin}^{c} \)-Dirac operators acting on sections of a canonical Dirac bundle with values in a suitable prequantum fibration. The family index gives then an invariant of Hamiltonian fibrations with fibers given by a product of complex projective spaces and allows to construct examples of non-trivial Hamiltonian fibrations.

Many systems in nature, e.g. atoms, molecules and planetary motion, can be described as Hamiltonian systems. In such systems, the transport between different regions of phase space determines some of their most important properties like the stability of the solar system and the rate of chemical reactions. While the transport in lower-dimensional systems with two degrees of freedom is well understood, much less is known for the higher-dimensional case. A central new feature in higher-dimensional systems are transport phenomena due to resonance channels. In this thesis, we clarify the complex geometry of resonance channels in phase space and identify a turnstile mechanism that dominates the transport out of such channels. To this end, we consider the coupled standard map for numerical investigations as it is a generic example for 4D symplectic maps. At first, we visualize resonance channels in phase space revealing their highly non-trivial geometry. Secondly, we study the transport away from such channels. This is governed by families of hyperbolic 1D-tori and their stable and unstable manifolds. We provide an approach to measure the volume of a turnstile in higher dimensions as well as the corresponding transport. From the very good agreement of the two measurements we conclude that these structures are a suitable generalization of the well-known 2D turnstile mechanism to higher dimensions. / Viele Systeme in der Natur, z.B. Atome, Moleküle und Planetenbewegungen, können als Hamilton'sche Systeme beschrieben werden. In solchen Systemen bestimmt der Transport zwischen verschiedenen Regionen des Phasenraums einige ihrer wichtigsten Eigenschaften wie die Stabilität des Sonnensystems und die Geschwindigkeit chemischer Reaktionen. Während der Transport in niedrigdimensionalen Systemen mit zwei Freiheitsgraden gut verstanden ist, ist für den höherdimensionalen Fall deutlich weniger bekannt. Eine zentrales neues Merkmal von höherdimensionalen Systemen sind Transportphänomene aufgrund von Resonanzkanälen. In dieser Arbeit verdeutlichen wir die komplexe Geometrie von Resonanzkanälen im Phasenraum und identifizieren einen Drehkreuzmechanismus, der den Transport aus einem solchen Kanal heraus dominiert. Zu diesem Zweck betrachten wir die gekoppelte Standardabbildung für numerische Untersuchungen, da sie ein generisches Beispiel für 4D symplektische Abbildungen ist. Zuerst visualisieren wir Resonanzkanäle im Phasenraum und zeigen ihre höchst nicht-triviale Geometrie. Zweitens untersuchen wir den Transport weg von solchen Kanälen. Dieser wird durch Familien von hyperbolischen 1D-Tori sowie deren stabile und instabile Mannigfaltigkeiten bestimmt. Wir stellen einen Ansatz zur Messung sowohl des eingeschlossenen Volumens in höheren Dimensionen als auch des entsprechenden Transports vor. Aus der sehr guten Übereinstimmung der beiden Messungen schließen wir, dass diese Strukturen eine geeignete Verallgemeinerung des bekannten 2D Drehkreuzmechanismus in höheren Dimensionen sind.

<p>Turniri su najvi&scaron;e istraživana klasa orijentisanih grafova. U tezi su prezentovana dva tipa rezultata. Prvi se odnosi na tzv. neizbežne podgrafove. Obuhvata Hamiltonove bajpase, podgrafove C(<em>n, i</em>) i alternativne Hamiltonove konture. Drugi se bavi problemima frekvencija skorova u običnim, bipartitnim i 3-partitnim turnirima.</p> / <p>Tournaments are the most investigated class of oriented graphs. Two type of results are presented in the thesis. First one is related to so called unavoidable subgraphs. It discusses Hamiltonian bypasses, subgraphs C(n, i) and antidirected Hamiltonian cycles. The second deals with problems of score frequencies in ordinary, bipartite and 3-partite tournaments.</p>

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Floquet’s theorem, a temporal analogue of Bloch’s theorem, is used for studying a time-dependent potential. With applications in cold atoms on optical lattices, quantum dots and more, there is a growing interest in Floquet engineering exotic materials and phases. By solving the time-dependent Schrödinger equation scattering amplitudes are derived from which the transmission spectrum are generated. The driving field induces Floquet sidebands into which the particle can inelastically scatter. Fano resonances are observed when the incoming particle and a bound state of the staticpotential differ by some energy quanta. This process is mediated by the driving field. The scattering matrix and transmission spectra are reproduced from previous work on electron gas, graphene, and a semi-metal admitting a point of quadratic band touching (QBT). The QBT system is extended with linear tilting along the potential, which proves to be another good quantum number for tunable control.

In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses forming a central configuration. An expression of four masses is established depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically it is proved that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central. The singularities of simultaneous binary collisions in collinear four-body problem is regularized by explicitly constructing new coordinates and time transformation in section 3. The motion in the new coordinates and time scale across simultaneous binary collision is at least C^2. Furthermore, the behavior of the motion closing, across and after the simultaneous binary collision, is also studied. Many different types of periodic solutions involving single binary collisions and simultaneous binary collisions are constructed. In section 4, the linear stability is studied for the Kepler orbits of the rhombus four-body problem. We show that, for given four proper masses, there exists a family of periodic solutions for which each body with the proper mass is at the vertex of a rhombus and travels along an elliptic Kepler orbit. Instead of studying the 8 degrees of freedom Hamilton system for planar four-body problem, we reduce this number by means of some symmetry to derive a two degrees of freedom system which then can be used to determine the linear instability of the periodic solutions. After making a clever change of coordinates, a two dimensional ordinary differential equation system is obtained, which governs the linear instability of the periodic solutions. The system is surprisingly simple and depends only on the length of the sides of the rhombus and the eccentricity e of the Kepler orbit. In section 5, index theory for symplectic paths introduced by Y.Long is applied to study the stability of a periodic solution x for a Hamiltonian system. We establish a necessary and sufficient condition for stability of the periodic solution x in two and four dimension.

Infrared divergences arising in theories with massless gauge bosons have been shown to cancel in scattering amplitudes when using dressed states constructed from the Faddeev- Kulish approach to the asymptotic states. It has been established that these states are closely related to asymptotic symmetries of the theory, that is, non-vanishing gauge trans- formations at the asymptotic boundary. In this thesis, we review both of these aspects for QED and non-Abelian gauge theories. We also investigate the expectation value of the non-Abelian field strength tensor using dressed states. We then present a novel con- struction of the dressing operator for non-Abelian gauge theories using Wilson lines. We demonstrate, to order O(g2), that each term of the dressing operator is reproduced in the presented Wilson line approach, along with additional terms that warrant a more thorough understanding. This work extends previous results that pertained to QED and gravity.

Mémoire numérisé par la Direction des bibliothèques de l’Université de Montréal / Ce mémoire se présente comme étant une poursuite de l'étude de la superintégrabilité classique et quantique dans un espace euclidien en deux dimensions avec une intégrale d'ordre trois. La classification de tous les Hamiltoniens séparables en coordonnées carté­siennes qui admettent une constante du mouvement d'ordre trois en les impulsions ayant déjà été complétée, nous proposons une poursuite de ces recherches dans le cas où le système se sépare en coordonnées polaires. Premièrement, nous dérivons les équations qui déterminent complètement le potentiel en ces coordonnées et tentons ensuite de les solutionner selon les différentes simplifications que nous pouvons accomplir sur l'inté­grale par l'action du groupe eulidien E(2). Finalement, nous présentons les équations qui caractérisent entièrement l'intégrabilité euclidienne cubique en coordonnées parabo­liques. / This thesis is a contribution to the study of classical and quantum superintegrabi­lity in a two-dimensional Euclidean space involving a third order integral of motion. A classification of Hamiltonian systems separable in cartesian coordinates that allow a third order invariant in the momenta has already been performed. We propose an ex­tension of this work and investigate Hamiltonians that admit separation of variables in polar coordinates and allow the existence of a third order constant of motion. We deter­mine the equations that characterize the potential in these coordinates and then attempt to solve them while simplifying the integral through the action of Euclidean group E(2). Futhermore, the equations which describe the classical and quantum cubic Euclidean in­tegrability are established in parabolic coordinates.

This document is a Master Thesis report in financial mathematics for KTH. This Master thesis is the product of an internship conducted at Nexialog Consulting, in Paris. This document is about the innovative use of Constrained Gaussian process regression in order to build an arbitrage free swaption cube. The methodology introduced in the document is used on a data set of European Swaptions Out of the Money. / Det här dokumentet är en magisteruppsats i finansiel matematik på KTH. Detta examensarbete är resultatet av en praktik som ufördes på Nexialog Consulting i Paris.Detta dokument handlar om den innovativa användningen av regression för gaussiska processer med bivillkor för att bygga en arbitragefri swaption kub. Den metodik som introduceras i dokumentet används på en datamängd av europeiska swaptions som är "Out of the Money".