We explore the problem of transforming plane triangulations using simultaneous diagonal flips. Wagner showed that any n-vertex plane triangulation can be transformed to any other plane triangulation on equal number of vertices using a finite sequence of diagonal flips. Later on it has been established that O(n) individual flips suffice to complete this transformation. Bose et al. showed that the transformation can also be done in 4 × ( 2 / log 54/53 + 2 / log 6/5 ) logn + 2 ≈ 327.1 log n simultaneous flips. This bound is asymptotically tight. We present two algorithms to improve the leading coefficient of this bound for transforming any plane triangulation into any other. The first of the two algorithms lowers this bound down to 4 × ( 2 / log 12/11 + 2 / log 9/7 ) logn + 2 ≈ 85.8 log n. By processing and preprocessing the interior and exterior of the triangulation’s Hamiltonian cycle parallelly in an interlaced fashion, we make further improvement of the algorithm from ≈ 327.1 log n down to 12 / log 6/5 logn + 2 ≈ 45.6 log n.

Dans le cadre du contrôle quantique bilinéaire, cette thèse étudie la possibilité de retrouver l'Hamiltonien et/ou le moment dipolaire à l'aide de mesures d'observables pour un ensemble grand de contrôles. Si l'implémentation du contrôle fait intervenir des bruits alors les mesures prennent la forme de distributions de probabilité. Nous montrons qu'il y a toujours unicité (à des phases près) des Hamiltoniens de du moment dipolaire retrouvés. Plusieurs modèles de bruit sont étudiés: bruit discrète constant additif et multiplicatif ainsi qu'un modèle de bruit dans les phases sous forme de processus Gaussien. Les résultats théoriques sont illustrés par des implémentations numériques. / The problem of recovering the Hamiltonian and dipole moment, termed inversion, is considered in a bilinear quantum control framework. The process uses as inputs some measurable quantities (observables) for each admissible control. If the implementation of the control is noisy the data available is only in the form of probability laws of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to phase factors). Several models of noise are considered including the discrete noise model, the multiplicative amplitude noise model and a Gaussian process phase model. Both theoretical and numerical results are established.

A Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph is called Hamiltonian if it contains such a cycle. In this thesis we investigate two classes of graphs, defined by local criteria. Graphs in these classes, with a simple set of exceptions K, were proven to be Hamiltonian by Asratian, Broersma, van den Heuvel, and Veldman in 1996 and by Asratian in 2006, respectively. We prove here that in addition to being Hamiltonian, graphs in these classes have stronger cyclic properties. In particular, we prove that if a graph G belongs to one of these classes, then for each vertex x in G there is a sequence of cycles such that each cycle contains the vertex x, and the shortest cycle in the sequence has length at most 5; the longest cycle in the sequence is a Hamilton cycle (unless G belongs to the set of exceptions K, in which case the longest cycle in the sequence contains all but one vertex of G); each cycle in the sequence except the first contains all vertices of the previous cycle, and at most two other vertices. Furthermore, for each edge e in G that does not lie on a triangle, there is a sequence of cycles with the same three properties, such that each cycle in the sequence contains the edge e.

The existence and uniqueness of Lagrangian invariant subspaces of Hamiltonian matrices is studied. Necessary and sufficient conditions are given in terms of the Jordan structure and certain sign characteristics that give uniqueness of these subspaces even in the presence of purely imaginary eigenvalues. These results are applied to obtain in special cases existence and uniqueness results for Hermitian solutions of continuous time algebraic Riccati equations.

On étudie les propriétés de rigidité symplectique des difféomorphismes hamiltoniens en dimension finie et en dimension infinie. En dimension finie, les outils principaux qu'on utilise sont les fonctions génératrices et les capacités symplectiques. En dimension infinie on regarde les flots des équations en dérivées partielles (EDPs) hamiltoniennes et, en particulier, les flots qui peuvent être approchés uniformément par des flots hamiltoniens de dimension finie.Dans la première partie de la thèse on étudie les sélecteurs d'action définies à partir des fonctions génératrices et on construit des invariants hamiltoniens pour les sous-ensembles de $R^{2m}times T^*T^k$. Cela nous permet de démontrer un théorème non-squeezing coisotrope pour les difféomorphismes hamiltoniens à support compact de $R^{2n}$. On montre à continuation que cette propriété apparaisse dans certains cas non compacts. Finalement, on explique comment ce résultat donne aussi l'information sur le problème de rigidité symplectique en dimension intermédiaire. Encore en dimension finie, on démontre qu'on peut utiliser le théorème du chameau symplectique pour produire des sous-ensembles invariants compacts dans des surfaces d'energie.Dans la deuxième partie on étudie les propriétés de rigidité symplectique des flots des EDPs hamiltoniennes. On se place dans le contexte introduit par Kuksin et on étudie une classe particulière de EDPs semi-linéaires qui peuvent être approchées par flots hamiltoniens de dimension finie. D'abord on donne une nouvelle construction de capacité symplectique en dimension infinie à partir des capacités de Viterbo. Puis on démontre l'analogue de la rigidité intermédiaire pour certaines EDPs hamiltoniennes. Cette classe inclue l'équation d'ondes en dimension 1 avec une non-linéarité bornée, comme par exemple l'équation de Sine-Gordon. Dans la dernière partie de la thèse on s'intéresse à un analogue de la conjecture d'Arnold pour l'équation de Schrödinger périodique avec une non linéarité de convolution. / We study symplectic rigidity properties in both finite and infinite dimension. In finite dimension, the main tools that we use are generating functions and symplectic capacities. In infinite dimension we study flows of Hamiltonian partial differential equations (PDEs) and, in particular, flows which can be uniformly approximated by finite dimensional Hamiltonian diffeomorphisms.In the first part of this thesis we study the action selectors defined from generating functions and we build Hamiltonian invariants for subsets of $R^{2m}times T^*T^k$. This allows us to prove a coisotropic non-squeezing theorem for compactly supported Hamiltonian diffeomorphisms of $R^{2n}$. We then extend this result to some non-compact settings. Finally we explain how this result can give information about the middle dimensional symplectic rigidity problem. Still in finite dimensions, we show that it is possible to use the symplectic camel theorem to create energy surfaces with compact invariant subsets.In the second part of the thesis we study symplectic rigidity properties of flows of Hamiltonian PDEs. We work in the context introduced by Kuksin and study a particular class of semi-linear Hamiltonian PDEs that can be approximated by finite dimensional Hamiltonian diffeomorphisms. We first give a new construction of an infinite dimensional capacity using Viterbo's capacities. The main result of this part is the proof of the analogue of the middle dimensional rigidity for certain types of Hamiltonian PDEs. These include nonlinear string equations with bounded nonlinearity such as the Sine-Gordon equation. In the final part of this thesis we study an analogue of Arnold's conjecture for the periodic Schrödinger equations with a convolution nonlinearity.

La première partie de cette thèse concerne l'étude du problème de Cauchy pour l'équation de KdV quasi-linéaire.On établit un théorème d'existence locale obtenu grâce à des propriétés structurelles et des techniques de jauge qui permettent de compenser les pertes de dérivées apparentes dans les estimations a priori.Dans la seconde partie, les propriétés de stabilité orbitale co-périodique et modulationnelle sont explorées numériquement en exploitant des critères algébriques tous établis à partir d'une même intégrale d'action et de ses dérivées secondes. Notre méthode utilise des quadratures numériques suivies de différences finies afin de calculer la matrice hessienne de l'intégrale d'action. Le comportement asymptotique de cette matrice nous pousse à prêter beaucoup d'attention à l'étude des ondes de grande période ou de faible amplitude. Les résultats numériquesprésentés fournissent de nombreuses informations en lien avec des questions ouvertes.On effectue également des simulations directes sur le système d' ÉDP original pour étudier à la fois le comportement des ondes périodiques sous différents types de perturbations, et les solutions de problèmes de Cauchy avec donnée initiale discontinue. Pour ces derniers, on s'attend à observer des chocs dispersifs, dont la compréhension est basée sur le problème de Gurevich-Pitaevskii, où les équations modulées à la Whitham sont utilisées pour approcher la zone oscillante des chocs. On compare des simulations directes aux solutions idéales du problème de Gurevich-Pitaevskii, en commençant par la célèbre équation de KdV / The first part of this manuscript presents a well-posedness result for a quasilinear version of the KdV equation.The proof takes advantage of structural properties and gauge techniques to deal with apparent loss of derivativesin a priori estimates.In the second part, we investigate the modulational and orbital coperiodic stability of periodic waves by computingalgebraic criteria involving the same abbreviated action integral and its second order derivatives. Our methoduses numerical integrations followed by finite differences to compute the Hessian matrix of the action integral.We pay attention to the asymptotic behavior of this matrix in the large period and small amplitude limits. Thenumerical results about stability give some new insight on several analytical open questions.Finally, direct numerical computations are done on the original system of PDEs to study the behavior of periodictraveling waves under various kinds of perturbations and the solutions of Cauchy problem with discontinuousinitial data. For the latter, we expect dispersive shock waves to arise. The building block for understandingdispersive shocks is known as the Gurevich-Pitaevskii problem, in which modulated equations 'a la Whitham'are used as an approximate model for the oscillatory zone. We compare direct numerical simulations to idealizedsolutions of Gurevich-Pitaevskii problems, starting with the famous KdV equation

Ce manuscrit est consacré à l'étude de la mécanique classique et la mécanique quantique en présence d'un champ magnétique. En mécanique classique, nous utilisons un Hamiltonien pour décrire la dynamique d'une particule chargée dans un domaine soumis à un champ magnétique. Nous nous intéressons ici à deux problèmes classiques de physique : le problème de confinement et le problème de scattering. Dans le cas quantique, nous étudions le problème spectral du laplacien magnétique au niveau semi-classique dans des domaines de dimension deux: sur une variété Riemanienne compacte à bord et dans ℝ ². En supposant que le champ magnétique ait un unique minimum strictement positif et non-dégénéré, nous pouvons décrire les fonctions propres par les méthodes WKB. Grâce au théorème spectral, nous pouvons estimer efficacement les vraies fonctions propres et les fonctions propres approchées localement proche du minimum du champ magnétique. Dans ℝ ², sous l'hypothèse additionnelle d'une symétrie radiale du champ magnétique, nous pouvons montrer que les fonctions propres du laplacien magnétique décroissent de manière exponentielle à l'infini avec une vitesse contrôlée par la fonction phase de la procédure WKB. De plus, les fonctions propres sont très bien approchées dans un espace à poids exponentiel. / This manuscript is devoted to classical mechanics and quantum mechanics, especially in the presence of magnetic field. In classical mechanics, we use Hamiltonian dynamics to describe the motion of a charged particle in a domain affected by the magnetic field. We are interested in two classical physical problems: the confinement and the scattering problem. In the quantum case, we study the spectral problem of the magnetic Laplacian at the semi-classical level, in two-dimensional domains: on a compact Riemmanian manifold with boundary and on ℝ ². Under the assumption that the magnetic field has a unique positive and non-degenerate minimum, we can describe the eigenfunctions by WKB methods. Thanks to the spectral theorem, we estimated efficiently the true eigenfunctions and the approximate eigenfunctions locally near the minimum point of the magnetic field. On ℝ ², with the additional assumption that the magnetic field is radially symmetric, we can show that the eigenfunctions of the magnetic Laplacian decay exponentially at infinity and at a rate controlled by the phase function created in WKB procedure. Furthermore, the eigenfunctions are very well approximated in an exponentially weighted space.

Indiana University-Purdue University Indianapolis (IUPUI) / We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.

In many areas of the world, seismic hazards pose a great risk to both human and natural populations. In particular, earthquake-induced tsunamis are especially dangerous to many areas in the Pacific. The study and quantification of these seismic events can both help scientists better understand how these natural hazards occur and help at-risk populations make better preparations for these events. However, many events of interest occurred too long ago to be recorded by modern instruments, so data on these earthquakes are sparse and unreliable. To remedy this, a Bayesian method for reconstructing the source earthquakes for these historical tsunamis based on anecdotal data, called TsunamiBayes, has been developed and used to study historical events that occurred in 1852 and 1820. One drawback of this method is the computational cost to reconstruct posterior distributions on tsunami source parameters. In this work, we improve on the TsunamiBayes method by introducing higher-order MCMC methods, specifically the Hamiltonian Monte Carlo (HMC) method to increase sample acceptance rate and therefore reduce computation time. Unfortunately the exact gradient for this problem is not available, and so we make use of a surrogate gradient via a neural network fitted to the forward model. We examine the effects of this surrogate gradient HMC sampling method on the posterior distribution for an 1852 event in the Banda Sea, compare results to previous results collected usisng random walk, and note the benefits of the surrogate gradient in this context.

Density functional theory calculations were performed to study the dissociative chemisorption of methane over Pt(111) with the idea of finding the minimum energy path for the reaction and its dependence on surface coverage. Two approaches were used to evaluate this problem; first, we used different sizes of supercells (2x2, 3x3, 4x4) in order to decrease surface coverage in the absence of pre-adsorbed H and CH3 fragments to calculate the energy barriers of dissociation. The second approach uses a 4x4 unit cell and surface coverage is simulated by adding pre-absorbed H and CH3 fragments. Results for both approaches show that in general the height of the dissociation barriers increases as the surface coverage increases, although, the first approach yields slightly lower barriers due to the fact that all repeatable images of the incident molecule are approaching the surface simultaneously. Using the reaction path formulation we were able to compute the potential energy surface for CH4 dissociation. Our results suggest that excitation of the symmetric stretch and bend modes will likely increase the probability for reaction.