Deformation Quantization by Moyal Star-Product and Stratonovich Chaos

We make a deformation quantization by Moyal star-product on a space of functions endowed with the normalized Wick product and where Stratonovich chaos are well defined

Combinatorial Hopf algebra structure on packed square matrices

We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of $k$-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics.Comment: 35 page

Survey of correlation properties of polyatomic molecules vibrational energy levels using FT analysis

International audienceIn the last few years molecular spectroscopists have begun to study the highly excited vibrational levels of polyatomic molecules. In this high energy regime vibrational quantum numbers can no longer be intrinsically assigned (in contrast to vibrational levels at low energy). One can only characterize these levels by their correlation properties. The authors consider: short range correlations which are characterized by the next neighbor distribution, (NND). These correlations range from a Poisson (random or uncorrelated spectra) to a Wigner distribution (which shows 'level repulsion'); (ii) long range correlations are characterized by the$\mathrm{\Sigma ^{2}}$(L) and$\mathrm{\Delta _{3}}$(L) function. They describe the behavior which ranges from an uncorrelated spectra (Poisson statistic) to a spectra with 'spectral rigidity'

Zero-field anisotropic spin hamiltonians in first-Row rransition metal complexes: theory, models and applications

L'anisotropie magnétique est à l'origine de la lente relaxation de l'aimantation des aimants moléculaires. L'objectif principal de ce travail est de comprendre les facteurs qui gouvernent les anisotropies locales et intersites dans les composés polynucléaires. Des calculs relativistes et corrélés ont été effectués sur des systèmes mono- et bi-nucléaires. Les dégrés de liberté principaux de la méthode ab initio d'interaction d'états en deux étapes ont été optimisés pour obtenir des paramètres d'anisotropie en bon accord avec les résultats spectroscopiques. La théorie des hamiltoniens effectifs procure un procédé universel d'extraction de ces paramètres. Elle a donc été utilisée pour vérifier la validité des modèles usuels et proposer des éventuelles améliorations aux modèles. Enfin, les paramètres d'anisotropies ont été rationalisés dans certains cas à l'aide de la théorie des perturbations quasi-dégénérées.Magnetic anisotropy is responsible for the slow relaxation of the magnetization in single molecule magnets. The main goal of this work is to understand the factors that govern local and intersite anisotropies in polynuclear compounds. For this purpose, correlated relativistic calculations are performed in mono- and bi-nuclear species. The main degrees of freedoms of the two-step state-interaction ab initio method have been optimized in order to obtain anisotropic parameters in good agreement with the spectroscopic data. The effective Hamiltonian theory provides a universal procedure of extraction of these parameters. It has therefore been used to check the accuracy of the standard model and to propose improved models when necessary. Finally, the anisotropy parameters have been rationalized in several cases by using the quasi-degenerate perturbation theory

Algèbres de Hopf combinatoires

This thesis is in the field of algebraic combinatorics. In other words, the idea is to use algebraic structures, in this case of combinatorial Hopf algebras, to better study and understand the combinatorial objects and algorithms for composition and decomposition about these objects. This research is based on the construction and study of algebraic structure of combinatorial objects generalizing permutations. After recalling the background and notations of various objects involved in this research, we propose, in the second part, the study of the Hopf algebra introduced by Aguiar and Orellana based on uniform block permutations. By focusing on a description of these objects via well-known objects, permutations and set partitions, we propose a polynomial realization and an easier study of this algebra. The third section considers a second generalization interpreting permutations as matrices. We define and then study the families of square matrices on which we define algorithms for composition and decomposition. The fourth part deals with alternating sign matrices. Having defined the Hopf algebra of these matrices, we study the statistics and the behavior of the algebraic structure with these statistics. All these chapters rely heavily on computer exploration, and is the subject of an implementation using Sage software. This last chapter is dedicated to the discovery and manipulation of algebraic structures on Sage. We conclude by explaining the improvements to the study of algebraic structure through the Sage softwareCette thèse se situe dans le domaine de la combinatoire algébrique. Autrement dit, l'idée est d'utiliser des structures algébriques, en l'occurence des algèbres de Hopf combinatoires, pour mieux étudier et comprendre les objets combinatoires ainsi que des algorithmes de composition et de décomposition agissant sur ces objets. Ce travail de recherche repose sur la construction et l'étude de structure algébrique sur des objets combinatoires généralisant les permutations. Après avoir rappelé le contexte et les notations des différents objets intervenant dans cette recherche, nous proposons dans la seconde partie l'étude de l'algèbre de Hopf introduite par Aguiar et Orellana indexée par les permutations de blocs uniformes. En se focalisant sur une description de ces objets via d'autres bien connus, les permutations et les partitions d'ensembles, nous proposons une réalisation polynomiale et une étude plus simple de cette algèbre. La troisième partie étudie une deuxième généralisation en interprétant les permutations comme des matrices. Nous définissons et étudions alors des familles de matrices carrées sur lesquelles nous définissons des algorithmes de composition et de décomposition. La quatrième partie traite des matrices à signes alternants. Après avoir définie l'algèbre de Hopf sur ces matrices, nous étudions des statistiques et le comportement de la structure algébrique vis-à-vis de ces statistiques. Tous ces chapitres s'appuient fortement sur l'exploration informatique, et fait l'objet d'une implémentation utilisant le logiciel Sage. Ce dernier chapitre est consacré à la découverte et la manipulation de structures algébriques sur Sage. Nous terminons en expliquant les améliorations apportées pour l'étude de structure algébrique au travers du logiciel Sag

A polynomial realization of the Hopf algebra of uniform block permutations

Poster sessio

The resolution of the weak-exchange limit made rigorous, simple and general in binuclear complexes

The correct interpretation of magnetic properties in the weak-exchange regime has remained a challenging task for several decades. In this regime, the effective exchange interaction between local spins is quite weak, of the same order of magnitude or smaller than the various anisotropic terms, which \textit{in fine} generates a complex set of levels characterized by spin intercalation if not significant spin mixing. Although the model multispin Hamiltonian, \hms{} = \js{} + \da{} +\db{} + \dab{}, is considered good enough to map the experimental energies at zero field and in the strong-exchange limit, theoretical works pointed out limitations of this simple model. This work revives the use of \hms{} from a new theoretical perspective, detailing point-by-point a strategy to correctly map the computational energies and wave functions onto \hms{}, thus validating it regardless of the exchange regime. We will distinguish two cases, based on experimentally characterized dicobalt(II) complexes from the literature. If centrosymmetry imposes alignment of the various rank-2 tensors constitutive of \hms{} in the first case, the absence of any symmetry element prevents such alignment in the second case. In such a context, the strategy provided herein becomes a powerful tool to rationalize the experimental magnetic data, since it is capable of fully and rigorously extracting the multispin model without any assumption on the orientation of its constitutive tensors. Finally, previous theoretical data related to a known dinickel(II) complex is reinterpreted, clarifying initial wanderings regarding the weak-exchange limit

Interplay between Local Anisotropies in Binuclear Complexes

A systematic study has been undertaken to determine how local distortions affect the overall (molecular) magnetic anisotropies in binuclear complexes. For this purpose we have applied a series of distortions to two binuclear Ni(II) model complexes and extracted the magnetic anisotropy parameters of multispin and giant-spin model Hamiltonians. Furthermore, local and molecular magnetic axes frames have been determined. It is shown that certain combinations of local distortions can lead to constructive interference of the local anisotropies and that the largest contribution to the anisotropic exchange does not arise from the second-rank tensor normally included in the multispin Hamiltonian, but rather from a fourth-rank tensor. From the comparison of the extracted parameters, simple rules are obtained to maximize the molecular anisotropy by controlling the local magnetic anisotropy, which opens the way to tune the anisotropy in binuclear or polynuclear complexes

Développement et validation de schémas de calcul à double niveau pour les réacteurs à eau sous pression

Notions de transport neutronique -- Équation du tranport neutronique -- Éléments de calculs neutroniques -- Présentation du code DRAGON, autres codes utilisés -- Présentation des schémas à double niveau -- Généralités -- Résultats et limites du schéma Le Tellier -- Vers une amélioration du schéma Le Tellier et résultats -- Chainage de DRAGON avec le code de coeur COCAGNE -- Présentation de COCAGNE et réalisation du chainage -- Présentation des calculs effectués -- Résultats obtenus sur le cluster

Excited states of polonium(IV): Electron correlation and spin-orbit coupling in the Po^{4+} free ion and in the bare and solvated [PoCl5]^- and [PoCl6]^{2-} complexes

Polonium (Po, Z = 84) is a main-block element with poorly known physico-chemical properties. Not much information has been firmly acquired since its discovery by Marie and Pierre Curie in 1898, especially regarding its speciation in aqueous solution and spectroscopy. In this work, we revisit the absorption properties of two complexes, [PoCl5]^- and [PoCl6]^{2-}, using quantum mechanical calculations. These complexes have the potential to exhibit a maximum absorption at 418 nm in HCl medium (for 0.5 mol/L concentrations and above). Initially, we examine the electronic spectra of the Po^{4+} free ion and of its isoelectronic analogue, Bi^{3+}. In the spin-orbit configuration interaction (SOCI) framework. Our findings demonstrate that the SOCI matrix should be dressed with correlated electronic energies and that the quality of the spectra is largely improved by decontracting the reference states at the complete active space plus singles (CAS+S) level. Subsequently, we investigate the absorption properties of the [PoCl5]^- and [PoCl6]^{2-} complexes in two stages. Firstly, we perform methodological tests at the MP2/def2-TZVP gas phase geometries, indicating that the decontraction of the reference states can there be skipped without compromising the accuracy significantly. Secondly, we study the solution absorption properties by means of single-point calculations performed at the solvated geometries, obtained by an implicit solvation treatment or a combination of implicit and explicit solvation. Our results highlight the importance of saturating the first coordination sphere of the Po^{IV} ion to obtain a qualitatively correct picture. Finally, we conclude that the known-for-decades 418 nm peak could be attributed to a mixture of both the [PoCl5(H2O)]^- and [PoCl6]^{2-} complexes. This finding not only aligns with the behaviour of the analogous Bi^{III} ion under similar conditions but..