The vertical profile of embedded trees

Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is $$\frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}},$$ with n_{l-1}=n_{r+1}=0. The sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the tree. The vertical profile of a uniform random tree of size n is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in Z. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa i, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page

The representation of the symmetric group on m-Tamari intervals

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journa

Description of the unsteady flow pattern from peak efficiency to near surge in subsonic centrifugal compressor stage

This paper aims to describe the flow structure modifications when the operating point moves from peak efficiency to near stall condition in a moderate pressure ratio centrifugal compressor stage consisted of a splittered unshrouded impeller and a vaned diffuser. The investigations are based on three-dimensional U-RANS simulation results. The flow is described in the impeller and in the vaned diffuser through time-averaged flow quantities and unsteady fluctuations. Results show that at low mass flow rate, the effects of secondary flow in the impeller are more pronounced, inducing both, high time-averaged values and temporal fluctuations of the flow angle near the shroud at the diffuser inlet, leading to vane suction side boundary layer separation. Pressure waves due to impeller diffuser interaction spread through the vaned diffuser generating unsteadiness which intensifies at near surge condition

Les valeurs afférentes à la Charte canadienne des droits et libertés dans le discours judiciaire : utilisations et sources

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.Dans les décisions où entre en jeu la Charte canadienne des droits et libertés, les juges construisent leurs raisonnements à partir des droits et libertés énumérés dans son libellé, mais également à partir de valeurs. Parmi ces valeurs afférentes à la Charte se retrouvent notamment la protection des groupes vulnérables, la protection de la vie privée et de la réputation de la personne, le respect de la dignité inhérente de l'être humain, la promotion de la justice et de l'égalité sociale de même que l'encouragement à l'épanouissement personnel. Contrairement aux droits et libertés, lesquels furent directement inscrits par le constituant dans le libellé de la Charte, les valeurs représentent une création de l'appareil judiciaire. Malgré la composante subjective que l'on attribue généralement aux valeurs, nous constatons que la sécurité du droit ne se trouve pas fragilisée par l'incorporation d'une dimension axiologique dans le raisonnement judiciaire en matière de Charte. Au contraire, le recours aux valeurs favorise la constance et la prévisibilité dans l'application et l'interprétation de cet instrument de protection des droits et libertés. D'une part, les valeurs afférentes à la Charte sont utilisées de façon rigoureuse dans le contrôle judiciaire de la constitutionnalité des lois, dans l'adaptation des règles de common law et dans l'interprétation des lois. Les juges invoquent fréquemment les mêmes valeurs, à quelques variantes près, assurant ainsi une certaine stabilité dans le traitement des composantes axiologiques de la Charte. D'autre part, les juges sont largement capables de dissocier de leurs convictions personnelles les valeurs qu'ils invoquent dans leurs décisions. Les valeurs tirées des comportements sociaux actuels, du droit international et du droit comparé, de même que des ouvrages des philosophes politiques et des théoriciens du droit, sont des valeurs qui ont peu à voir avec les sentiments, les opinions ou les intérêts personnels des juges.In cases involving the Canadian Charter of Rights and Freedoms, judges include in their reasoning not only rights and freedoms, but also values. Among these Charter values we find protection of vulnerable groups, respect for the inherent dignity of human beings, cultivation oUndividual self-fulfillment and human flourishing, commitment to social justice and protection of privacy rights and reputation. Contrary to rights and freedoms, which are specifically enumerated in the Charter, values represent a creation of the judicial apparatus. Notwithstanding their inherent subjective component, values do not weaken the law's predictability nor do they endanger judicial reasoning concerning the Charter. On the contrary, applying values brings constancy and certainty in the interpretation of the Charter. On the one hand, Charter values are used in rigorous ways in the judicial review of constitutionality, in the adaptation of common law mIes and in the interpretation of laws and by-Iaws. Judges have recourse to more or less the same values, thus bringing stability in the Charter's analysis. On the other hand, judges are able to dissociate their personal beliefs from the values they caU upon in their judgments. Values drawn from contemporary social behaviors, international law and comparative law, as weU as political and legal philosophers' works, are values quite distant from judges' feelings, opinions and personal interests

Asymptotic enumeration of constellations and related families of maps on orientable surfaces

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps and m-constellations -- for example, large maps of genus g with even face degrees are bipartite with probability tending to 1/2^{2g}. A special case of our results implies former conjectures of Gao.Comment: 39 pages, 9 figure

Tamari lattices and parking functions: proof of a conjecture of F. Bergeron

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^(m), which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in T_n^(m) is $$\frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.$$ This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees. Here, we prove another conjecture of Bergeron, which deals with the number of labelled, intervals. An interval [P,Q] of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in T_n^(m) is $${(m+1)^n(mn+1)^{n-2}}.$$ The form of these numbers suggests a connection with parking functions, but our proof is non-bijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper. Finding a bijective proof remains an open problem.Comment: 21 pp. This paper is now subsumed by arXiv:1202.5925, and will not be submitted to any journa

Analysis of the Unsteady Flow Field in a Centrifugal Compressor from Peak Efficiency to Near Stall with Full-Annulus Simulations

This study concerns a 2.5 pressure ratio centrifugal compressor stage consisting of a splittered unshrouded impeller and a vaned diffuser. The aim of this paper is to investigate the modifications of the flow structure when the operating point moves from peak efficiency to near stall. The investigations are based on the results of unsteady three-dimensional simulations, in a calculation domain comprising all the blade. A detailed analysis is given in the impeller inducer and in the vaned diffuser entry region through time-averaged and unsteady flow field. In the impeller inducer, this study demonstrates that the mass flow reduction from peak efficiency to near stall leads to intensification of the secondary flow effects. The low momentum fluid accumulated near the shroud interacts with the main flow through a shear layer zone. At near stall condition, the interface between the two flow structures becomes unstable leading to vortices development. In the diffuser entry region, by reducing the mass flow, the high incidence angle from the impeller exit induces a separation on the diffuser vane suction side. At near stall operating point, vorticity from the separation is shed into vortex cores which are periodically formed and convected downstream along the suction side

Déformation et découpe interactive de solides à géométrie complexe

Cette thèse consiste à explorer une nouvelle approche pour la simulation d'objets flexibles par la mécanique des milieux continus, dans le cadre d'applications graphiques interactives telles que le jeu vidéo ou l'entraînement aux gestes chirurgicaux. Elle s'inscrit en continuité d'un stage de M2-R sur ce même sujet. Il est important de pouvoir régler simplement un compromis entre précision et temps de calcul suivant la nature de l'application. Les approches actuelles de simulation utilisent principalement la méthode des éléments finis. Celle-ci repose sur un maillage volumique des objets qu'il est souvent difficile d'adapter dynamiquement aux besoins de l'application. La nouveauté introduite par cette thèse est d'utiliser des repères déformables comme primitives cinématiques, avec des champs de déplacements inspirés des méthodes de 'skinning' utilisées en informatique graphique. Le but est d'éviter ainsi les difficultés liées au maillage volumique, ainsi que de faciliter le raffinement et la simplification adaptatives par simple ajout ou suppression de repère déformable là où c'est souhaitable. Ce travail est financé par le projet européen 'Passport for Virtual Surgery', dont le but est de créer automatiquement des modèles physiques pour l'entraînement aux gestes de chirurgie hépatique, à partir de données médicales et anatomiques personnalisées. Dans ce contexte, Guillaume, en collaboration avec d'autres membres du projet, mettra en place les outils nécessaires pour construire la scène physique à partir d'images médicales segmentées et de connaissances anatomiques génériques. Le foie sera dans un premier temps représenté par des modèles physiques précédemment développés à EVASION et étendus aux opérations de découpe. Par la suite, il y appliquera son nouveau modèle mécanique basé sur des repères déformables. The aim of this thesis is to develop a new approach for the simulation of flexible objects based on the continous middle method, related with interactive graphics applications such as video games or training in surgery. It is a continuity of the M2 research internship on the same topic. It is important to simply settle a compromise between accuracy and time computing according to the application. Current simulation approaches mainly use the finite element method, which is based on a volumetric mesh of the simulated objects. It is often difficult to dynamically adapt the needs to the application. The novelty of this thesis is to use deformable reference frames as kinematic primitives, with displacement fields based on 'skinning' methods used in computer graphics. The aim is to avoid the difficulties associated with volumetric mesh, and make the refinement and the adaptive simplification easier by adding or deleting deformable reference frames if necessary. This work is funded by the European project 'Passport for Virtual Surgery', which aims to automatically create models for physical training in gestures of liver surgery, from medical and anatomical custom data. In this context, Guillaume, in collaboration with other members of the project, will develop the tools necessary to build the physical scene from segmented medical images and generic anatomical knowledge. The liver will initially be represented by physical models previously developed in the EVASION team and then extended to cutting operations. Thereafter, Guillaume will apply his new mechanical model based on deformable reference frames.Physically based deformable models have become ubiquitous in computer graphics. It allow to synthetize real behaviors, based on the physical laws from continuum mechanics. In this thesis, we focus on interactive simulations such as to video games or surgical simulators. The majority of the existing works focused up to here on the animation of objects made of homogeneous materials. Nevertheless, plenty of real objects, for instance like the biological structures, consist of multiple imbricated materials. Their decomposition in homogeneous zones requires a high-resolution spatial discretization to solve the variations of the material properties, which requires prohibitive computation time. In this context, we present new real time simulation techniques for deformable objects which can be cut. First of all, we present a real time method for cutting deformable objects in which, contrary to the previous methods, the object deforms on the cutting tool contact and cuts occur only when the pressure reaches a certain level. The independence of the physical, collision and visual models makes the topological changes easier. The GPU computing and local modifications enable fast execution. Then, a dynamic meshless method is described, which uses reference frames as control nodes instead of using points, with a displacement field formulation similar to skinning. It allows to easily tune the weights and benefits from the rigor of physical methods as the finite elements. The introduction of integration points, reducing the samples number by a least squares approximation, speeds up the spatial integrations. Other pre-computations are proposed in order to speed up the simulation time. Finally, new anisotropic shape functions are defined to encode the variations of material properties thanks to the introduction of the compliance distance. These complex shape functions uncouple the material resolution of the displacement functions ones. It allow an extremely sparse nodes sampling. The use of the compliance distance allows an automatic nodes distribution with regard to the material properties.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Numerical Investigation of Kelvin–Helmholtz Instability in a Centrifugal Compressor Operating Near Stall

The present work details the occurrence of the Kelvin–Helmholtz instability in a centrifugal compressor operating near stall. The analysis is based on unsteady three-dimensional simulations performed on a calculation domain covering the full annulus for the impeller and the vaned diffuser. A detailed investigation of the flow structure is presented, together with its evolution consequent to the mass flow reduction. It is demonstrated that this reduction leads to an enlargement of the low-momentum flow region initially induced by the combination of the secondary and leakage flows. When the compressor operates near stall, the shear layer at the interface between the main flow and this low-momentum flow becomes unstable and induces a periodic vortex shedding. The frequency of such an unsteady phenomenon is not correlated with the blade-passing frequency. Its signature is thus easily isolated from the deterministic rotor/stator interaction. Its detection requires full-annulus simulations with an accurate resolution in time and space, which explains why it has never been previously observed in centrifugal compressor

Volumetric modeling and interactive cutting of deformable bodies

Special Issue on Biomechanical Modelling of Soft Tissue MotionInternational audienceA new approach for the interactive simulation of viscoelastic object cutting is presented. Two synchronized geo- metrical models at different resolutions are used, both derived from medical images. In contrast with most previous approaches, the blade deforms the object, and cutting occurs once a contact pressure threshold is exceeded. Moreover, we achieve interactive simulation rates by embedding a high resolution geometry within a regular grid with arbitrary resolution. This allows to trade off accuracy for speed in the computation of deformations. The input data is a high- resolution volumetric model of the objects. The surface model of the object, used for rendering as well as collision detection and response, is a polygonal level set of the volumetric data. It is embedded in the volume model using barycentric coordinates. Cutting is performed by removing voxels at the fine level, and updating the surface and volume models accordingly. We introduce a new data structure, which we call a Dynamic Branched Grid, in order to preserve the fine level topology at the coarse level. When an element of the coarse volumetric model is cut, it is replaced by a number of superimposed elements with the same size and at the same rest position as the original one. Each new element is assigned a part of material contained in the original one, and the mass and stiffness are recomputed accordingly. The well-known problem of creating small, ill-shaped finite elements while remeshing is thus completely avoided