The supersymmetric solutions of PT-/non-PT symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schr&Auml / odinger equation with the deformed Morse, Hulth&para / en, P&Auml / oschl-Teller, Hyperbolic Kratzer-like, Screened Coulomb, and Exponential-Cosine Screened Coulomb (ECSC) potentials. The Hamiltonian hi- erarchy method is used to get the real energy eigenvalues and corresponding wave functions.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) The electronic structure and rovibrational spectroscopy of MH2, MHn+2, HMHen+ and MHen+2 (M = Li, Be, Na, Mg, K, Ca; n = 1, 2) have been investigated using correlated ab initio ansatz. In order to determine the efficacy of various electronic structure methods with respect to Group-I and II hydrides and helides, atomic properties of Li, Be,Na, Mg, K and Ca were calculated. Relativistically-corrected UCCSD(T) and ICMRCI(+Q) were deemed to be the most suitable ansatz with respect to both efficiency and accuracy. The lowest 2A1 and 2Σ- states of MH2 were found to be purely repulsive, in agreement with previous predictions. The main factor determining the structure and stability of the excited states of MH2 was the relative orientations and occupations of the valence p atomic orbital of M and the H2 1Ou orbital. The ground states of MHn+2 were found to be the result of the charge-quadrupole interaction between Mn+ and the H2 molecular subunit. The structures of the ground states of HMHe+ were extremely uxional with respect to the central bond angle co-ordinate. The ground state PESs of MHe+2 were also extremely sensitive to the ab initio ansatz by which they are modelled. The respective bonding of the H and He in both HMHe+ and HMHe2+ appeared to be charge-dependent in the case of Be, Mg and Ca. Despite the weak bonding observed for the Group-II hydrohelide and helide monocations, the corresponding dications each exhibit thermodynamically stable equilibria. The solution algorithm of von Nagy-Felsobuki and co-workers was employed in the calculation of vibrational and rovibrational spectra. This algorithm employed an Eckart-Watson Hamiltonian in conjunction with rectilinear normal co-ordinates. Vibrational and rovibrational Hamiltonian matrices were diagonalised using variational methods. This algorithm was extended so that the vibration transition moment integrals, and hence vibrational radiative properties, of linear triatomic molecules could be calculated. A method by which vibration-averaged structures are calculated was also developed and implemented. Analytical potential energy functions (PEFs) and dipole moment functions (DMFs) of (1A1)LiH+2, (1A1)NaH+2, (1A1)BeH2+2,(1A1)MgH2+2, (1Σ+g )BeHe2+2, (2Σ+)HBeHe2+, (1Σ+g )MgHe2+2 and (2Σ+)HMgHe2+ were developed using leastsquare regression techniques in conjunction with discrete ab initio grids. Vibrational structures and spectra of these species were subsequently calculated. In addition, the rovibrational spectra of (1A1)LiH+2, (1A1)NaH+2, (1A1)BeH2+2 and (1A1)MgH2+2 were calculated. For (1A1)LiH+2 and (1A1)LiD+2 , calculated rovibrational transition frequencies for J ≤ 10 and 0 ≤ K ≤ 3 were within ca. 0.1-0.2% of experimental values.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) The electronic structure and rovibrational spectroscopy of MH2, MHn+2, HMHen+ and MHen+2 (M = Li, Be, Na, Mg, K, Ca; n = 1, 2) have been investigated using correlated ab initio ansatz. In order to determine the efficacy of various electronic structure methods with respect to Group-I and II hydrides and helides, atomic properties of Li, Be,Na, Mg, K and Ca were calculated. Relativistically-corrected UCCSD(T) and ICMRCI(+Q) were deemed to be the most suitable ansatz with respect to both efficiency and accuracy. The lowest 2A1 and 2Σ- states of MH2 were found to be purely repulsive, in agreement with previous predictions. The main factor determining the structure and stability of the excited states of MH2 was the relative orientations and occupations of the valence p atomic orbital of M and the H2 1Ou orbital. The ground states of MHn+2 were found to be the result of the charge-quadrupole interaction between Mn+ and the H2 molecular subunit. The structures of the ground states of HMHe+ were extremely uxional with respect to the central bond angle co-ordinate. The ground state PESs of MHe+2 were also extremely sensitive to the ab initio ansatz by which they are modelled. The respective bonding of the H and He in both HMHe+ and HMHe2+ appeared to be charge-dependent in the case of Be, Mg and Ca. Despite the weak bonding observed for the Group-II hydrohelide and helide monocations, the corresponding dications each exhibit thermodynamically stable equilibria. The solution algorithm of von Nagy-Felsobuki and co-workers was employed in the calculation of vibrational and rovibrational spectra. This algorithm employed an Eckart-Watson Hamiltonian in conjunction with rectilinear normal co-ordinates. Vibrational and rovibrational Hamiltonian matrices were diagonalised using variational methods. This algorithm was extended so that the vibration transition moment integrals, and hence vibrational radiative properties, of linear triatomic molecules could be calculated. A method by which vibration-averaged structures are calculated was also developed and implemented. Analytical potential energy functions (PEFs) and dipole moment functions (DMFs) of (1A1)LiH+2, (1A1)NaH+2, (1A1)BeH2+2,(1A1)MgH2+2, (1Σ+g )BeHe2+2, (2Σ+)HBeHe2+, (1Σ+g )MgHe2+2 and (2Σ+)HMgHe2+ were developed using leastsquare regression techniques in conjunction with discrete ab initio grids. Vibrational structures and spectra of these species were subsequently calculated. In addition, the rovibrational spectra of (1A1)LiH+2, (1A1)NaH+2, (1A1)BeH2+2 and (1A1)MgH2+2 were calculated. For (1A1)LiH+2 and (1A1)LiD+2 , calculated rovibrational transition frequencies for J ≤ 10 and 0 ≤ K ≤ 3 were within ca. 0.1-0.2% of experimental values.

The dynamics of Hamiltonian systems (e.g. planetary motion, electron dynamics in nano-structures, molecular dynamics) can be investigated by symplectic maps. While a lot of work has been done for 2D maps, much less is known for higher dimensions. For a generic 4D map regular 2D-tori are organized around a skeleton of families of elliptic 1D-tori, which can be visualized by 3D phase-space slices. An analysis of the different bifurcations of the families of 1D-tori in phase space and in frequency space by computing the involved hyperbolic and elliptic 1D-tori is presented. Applying known results of normal form analysis, both the local and the global structure can be understood: Close to a bifurcation of a 1D-torus, the phase-space structures are surprisingly similar to bifurcations of periodic orbits in 2D maps. Far away the phase-space structures can be explained by remnants of broken resonant 2D-tori. / Die Dynamik Hamilton'scher Syteme (z.B. Planetenbewegung, Elektronenbewegung in Nanostrukturen, Moleküldynamik) kann mit Hilfe symplektischer Abbildungen untersucht werden. Bezüglich 2D Abbildungen wurde bereits umfassende Forschungsarbeit geleistet, doch für Systeme höherer Dimension ist noch vieles unverstanden. In einer generischen 4D Abbildung sind reguläre 2D-Tori um ein Skelett aus Familien von elliptischen 1D-Tori organisiert, was in 3D Phasenraumschnitten visualisiert werden kann. Durch die Berechnung der beteiligten hyperbolischen und elliptischen 1D-Tori werden die verschiedenen Bifurkationen der Familien von 1D-Tori im Phasenraum und im Frequenzraum analysiert. Die Anwendung bekannter Ergebnisse aus Normalformanalysen ermöglicht das Verständnis sowohl des lokalen, als auch des globalen Regimes. Nahe an der Bifurkation eines 1D-Torus sind die Phasenraumstrukturen denen von Bifurkationen periodischer Orbits in 2D Abbildungen überraschend ähnlich. Weit entfernt können die Phasenraumstrukturen als Überreste eines zerplatzten resonanten 2D-Torus erklärt werden.

Nesta tese discutimos sistematicamente os efeitos magnetocalórico e barocalórico em sistemas físicos com transição de fase magnética de primeira e de segunda ordem em sistemas físicos cujo magnetismo tem carater local. Para essa finalidade, utilizamos um modelo de momentos magnéticos locais interagentes, incluindo um termo de acoplamento magnetoelástico. Nossos cálculos mostram que a relação entre a interação de troca e o acoplamento magnetoelástico é responsável pela ordem da transição de fase e pelo aparecimento da histerese térmica e magnética. Os resultados mostram que as grandezas magnetocalóricas exibem grandes valores quando o sistema sofre uma transição de fase de primeira ordem. Além disso, quando existe uma histerese visível as grandezas magnetocalóricas dependem do processo de aquecimento e resfriamento do sistema. Ainda de acordo com nossos resultados, existe um efeito barocalórico normal (sistema aquece sob aumento de pressão) quando a pressão aplicada aumenta a temperatura de ordenamento magnético sem alterar a ordem da transição de fase magnética. O efeito barocalórico inverso (sistema resfria sob aumento de pressão) ocorre quando a pressão aplicada diminui a temperatura crítica sem mudar a ordem da transição de fase. Nossos cálculos mostram, ainda, que um efeito barocalórico anômalo (mudança de sinal nas grandezas barocalóricas) ocorre em casos especiais onde a pressão aplicada muda a natureza da transição de fase magnética do primeira para segunda ordem e vice-versa. / In this thesis we systematically discuss the magnetocaloric and barocaloric effects in physical systems, whose magnetism has a localized character and undergo both second and first order phase transitions. For this purpose, we use a model of interacting localized magnetic moments, including the magnetoelastic coupling. Our calculations show that the ratio between the exchange and magnetoelastic parameters is responsible for the appearance of the first order transition and consequently the thermal and magnetic hysteresis. Our calculations also show that the magnetocaloric quantities exhibit large values when the system undergoes a first order transition. Furthermore, when the hysteresis is not negligible, the magnetocaloric quantities depend on the heating and cooling processes. As far as the barocaloric effect is concerned, our calculations show that the normal barocaloric effect occurs (the system heat up with increasing applied pressure) when the applied pressure increases the magnetic ordering temperature without changing the order of the magnetic phase transition. On the other hand, the inverse barocaloric effect (the system cools down with increasing applied pressure) occurs when the applied pressure reduces the magnetic ordering temperature without changing the order of the magnetic phase transition. Finally, our calculations show that an anomalous barocaloric effect (i.e., change of sign in the barocaloric quantities) occurs in special cases when the applied pressure change the nature of the magnetic phase transition from first to second order and vice-versa.

Tout phénomène, qu’il soit biologique, géologique ou mécanique peut être décrit à l’aide de lois de la physique en termes d’équations différentielles, algébriques ou intégrales, mettant en relation différentes variables physiques. Les objectifs de la thèse sont de montrer comment les systèmes à paramètres distribués peuvent être modélisés par un modèle bond graph, qui est par nature un modèle à paramètres localisés. Deux approches sont possibles : - utiliser une technique d’approximation qui discrétise le modèle initialement sous forme d’équations aux dérivées partielles (EDP) dans le domaine spatial, en supposant que les phénomènes physiques distribués peuvent être considérés comme homogènes dans certaines parties de l’espace, donc localisés. - déterminer la solution des EDP qui dépend du temps et de l’espace, puis à approximer cette solution avec différents outils numériques. Le premier chapitre rappelle quelques méthodes classiques utilisées pour l’approximation des EDP et les modèles bond graphs correspondants.Dans le deuxième chapitre, l’approche port-Hamiltonienne est présentée et son extension aux systèmes à paramètres distribués est proposée. Dans le troisième chapitre, les principaux modèles utilisés pour la représentation des flux de trafic routier sont rappelés et mis en œuvre en simulation. Ceci conduit à des comparaisons, d’une part entre différentes méthodes de résolution numérique et d’autre part entre différents modèles. Dans le quatrième chapitre, une approche originale propose d’étendre la représentation bond graph issue de la méthodologie Computational Fluid Dynamics au flux de trafic, en utilisant un modèle EDP à deux équations proposé par Jiang / Virtually every phenomenon in nature, whether biological, geological, or mechanical, can be described with the aid of the laws of physics, in terms of algebraic, differential, or integral equations relating various quantities of interest. The objectives of the thesis were to show how distributed parameter systems can be modeled using a bond graph model, which is by its nature itself a lumped parameter model. Two ways are possible :- using an approximation technique to discretize the model in the space domain, assuming that physical distributed phenomena can be considered as homogenous in some parts of space, and thus lumped. Different bond graph models can be obtained depending on the technique used.- determining a solution of the PDE depending on space and time, and thus to approximate this solution by means of different kinds of tools.In chapter 1, some classical methods used for approximation of partial differential equations are recalled and the corresponding bond graph model is designed. For each of them advantages and drawbacks are presented.In the second chapter, the port-Hamiltonian approach for distributed parameter system is presented, and a new result is proposed for telegrapher’s equation solving.In the third chapter, the main models used for traffic flow representation are presented and some of them are implemented in simulation. A comparison is done on one hand on different numerical methods applied on the first class of models (1-eq. model) and on the other hand between 1-equation and 2- equation models.In chapter 4, we have proposed an original approach extending Computational Fluid Dynamics bond graph representation to traffic flow, using Jiang’s model

Nous avons développé une nouvelle approche basée sur les méthodes Monte Carlo par chaînes de Markov pour déterminer les distributions de Partons et quantifier leurs incertitudes expérimentales. L’intérêt principal d’une telle étude repose sur la possibilité de remplacer la minimisation standard avec MINUIT de la fonction χ 2 par des procédures fondées sur les méthodes Statistiques et sur l’inférence Bayésienne en particulier,offrant ainsi une meilleure compréhension de la détermination des distributions de partons. Après avoir examiné ces techniques Monte Carlo par chaînes de Markov, nous introduisons l’algorithme que nous avons choisi de mettre en œuvre, à savoir le Monte Carlo hybride (ou Hamiltonien). Cet algorithme, développé initialement pour la chromodynamique quantique sur réseau, s’avère très intéressant lorsqu’il est appliqué à la détermination des distributions de partons par des analyses globales. Nous avons montré qu’il permet de contourner les difficultés techniques dues à la grande dimensionnalité du problème, en particulier celle relative au taux d’acceptation. L’étude de faisabilité réalisée et présentée dans cette thèse indique que la méthode Monte Carlo par chaînes de Markov peut être appliquée avec succès à l’extraction des distributions de partons et à leurs in-certitudes expérimentales. / We have developed a new approach to determine parton distribution functions and quantify their experimental uncertainties, based on Markov Chain Monte Carlo methods.The main interest devoted to such a study is that we can replace the standard χ 2 MINUIT minimization by procedures grounded on Statistical Methods, and on Bayesian inference in particular, thus offering additional insight into the rich field of PDFs determination.After reviewing these Markov chain Monte Carlo techniques, we introduce the algorithm we have chosen to implement – namely Hybrid (or Hamiltonian) Monte Carlo. This algorithm, initially developed for lattice quantum chromodynamique, turns out to be very interesting when applied to parton distribution functions determination by global analyses ; we have shown that it allows to circumvent the technical difficulties due to the high dimensionality of the problem, in particular concerning the acceptance rate. The feasibility study performed and presented in this thesis, indicates that Markov chain Monte Carlo method can successfully be applied to the extraction of PDFs and of their experimental uncertainties.

We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated. / Mathematical Sciences / D. Phil. (Mathematics)

How to evaluate if a spatial model is well ajusted to a problem? How to know if it is the best model between the class of conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models, including homoscedasticity and heteroscedasticity cases? To answer these questions inside Bayesian framework, we propose new ways to apply Bregman divergence, as well as recent information criteria as widely applicable information criterion (WAIC) and leave-one-out cross-validation (LOO). The functional Bregman divergence is a generalized form of the well known Kullback-Leiber (KL) divergence. There is many special cases of it which might be used to identify influential points. All the posterior distributions displayed in this text were estimate by Hamiltonian Monte Carlo (HMC), a optimized version of Metropolis-Hasting algorithm. All ideas showed here were evaluate by both: simulation and real data. / Como avaliar se um modelo espacial está bem ajustado? Como escolher o melhor modelo entre muitos da classe autorregressivo condicional (CAR) e autorregressivo simultâneo (SAR), homoscedásticos e heteroscedásticos? Para responder essas perguntas dentro do paradigma bayesiano, propomos novas formas de aplicar a divergência de Bregman, assim como critérios de informação bastante recentes na literatura, são eles o widely applicable information criterion (WAIC) e validação cruzada leave-one-out (LOO). O funcional de Bregman é uma generalização da famosa divergência de Kullback-Leiber (KL). Há diversos casos particulares dela que podem ser usados para identificar pontos influentes. Todas as distribuições a posteriori apresentadas nesta dissertação foram estimadas usando Monte Carlo Hamiltoniano (HMC), uma versão otimizada do algoritmo Metropolis-Hastings. Todas as ideias apresentadas neste texto foram submetidas a simulações e aplicadas em dados reais.

Nesta tese discutimos sistematicamente os efeitos magnetocalórico e barocalórico em sistemas físicos com transição de fase magnética de primeira e de segunda ordem em sistemas físicos cujo magnetismo tem carater local. Para essa finalidade, utilizamos um modelo de momentos magnéticos locais interagentes, incluindo um termo de acoplamento magnetoelástico. Nossos cálculos mostram que a relação entre a interação de troca e o acoplamento magnetoelástico é responsável pela ordem da transição de fase e pelo aparecimento da histerese térmica e magnética. Os resultados mostram que as grandezas magnetocalóricas exibem grandes valores quando o sistema sofre uma transição de fase de primeira ordem. Além disso, quando existe uma histerese visível as grandezas magnetocalóricas dependem do processo de aquecimento e resfriamento do sistema. Ainda de acordo com nossos resultados, existe um efeito barocalórico normal (sistema aquece sob aumento de pressão) quando a pressão aplicada aumenta a temperatura de ordenamento magnético sem alterar a ordem da transição de fase magnética. O efeito barocalórico inverso (sistema resfria sob aumento de pressão) ocorre quando a pressão aplicada diminui a temperatura crítica sem mudar a ordem da transição de fase. Nossos cálculos mostram, ainda, que um efeito barocalórico anômalo (mudança de sinal nas grandezas barocalóricas) ocorre em casos especiais onde a pressão aplicada muda a natureza da transição de fase magnética do primeira para segunda ordem e vice-versa. / In this thesis we systematically discuss the magnetocaloric and barocaloric effects in physical systems, whose magnetism has a localized character and undergo both second and first order phase transitions. For this purpose, we use a model of interacting localized magnetic moments, including the magnetoelastic coupling. Our calculations show that the ratio between the exchange and magnetoelastic parameters is responsible for the appearance of the first order transition and consequently the thermal and magnetic hysteresis. Our calculations also show that the magnetocaloric quantities exhibit large values when the system undergoes a first order transition. Furthermore, when the hysteresis is not negligible, the magnetocaloric quantities depend on the heating and cooling processes. As far as the barocaloric effect is concerned, our calculations show that the normal barocaloric effect occurs (the system heat up with increasing applied pressure) when the applied pressure increases the magnetic ordering temperature without changing the order of the magnetic phase transition. On the other hand, the inverse barocaloric effect (the system cools down with increasing applied pressure) occurs when the applied pressure reduces the magnetic ordering temperature without changing the order of the magnetic phase transition. Finally, our calculations show that an anomalous barocaloric effect (i.e., change of sign in the barocaloric quantities) occurs in special cases when the applied pressure change the nature of the magnetic phase transition from first to second order and vice-versa.