The intent of this thesis is to explore whether any patterns emerge among families or through graph operations regarding the appearance of Weierstrass vertices on graphs. Currently, patterns have been identified and proven on cycles, complete graphs, complete bipartite graphs, and the house and house-x graphs. A Python program developed as part of this thesis to perform the algorithms used in this analysis confirms these findings. This program also revealed a pattern: if v is a Weierstrass vertex, then the vertex v* added to the graph as a pendant vertex to v is also a Weierstrass vertex. The converse is also true: if v is not a Weierstrass vertex, v* will not be either.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23683号 / 理博第4773号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Yes / Polygon, subdivision, and NURBS are three mainstream modeling techniques widely applied in commercial software packages. They require heavy manual operations, and involve a lot of design variables leading to big data, high storage costs and slow network transmissions. In this paper, we integrate the strengths of boundary-based surface creation and partial differential equation (PDE)-based geometric modeling to obtain the first analytical C continuous 4-sided PDE patches involving sculpting force-based shape creation and manipulation and use them to develop an interactive modeling technique for easy and quick creation of 3D models with small data from vertex-frames. With this modeling technique, a vertex frame is defined by eight vertices, and a C continuous 4-sided PDE patch is created from the vertex-frame through an analytical solution to a vector-valued second-order PDE subjected to the boundary conditions determined by the eight vertices of a vertex-frame. A user-friendly interface is developed from the obtained analytical solution, which enables users to interactively input and modify vertex-frame models easily and create 3D models in real time. Different surface modeling tasks are carried out to test the developed interactive tool and compare our proposed method with polygon and NURBS modeling and Coons surfaces. The results demonstrate the effectiveness of our proposed method and its advantages in reducing design variables, saving storage costs, and effective shape creation and manipulation. / European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 778035; MCIN/ AEI /10.13039/501100011033/ FEDER “Una manera de hacer Europa”

In the last few years, large-scale experiments have generated genome-wide protein interaction networks for many organisms including Saccharomyces cerevisiae (baker's yeast), Caenorhabditis elegans (worm) and Drosophila melanogaster (fruit fly). In this thesis, we examine the vertex and edge betweenness centrality measures of these graphs. These measures capture how "central" a vertex or an edge is in the graph by considering the fraction of shortest paths that pass through that vertex or edge. Our primary observation is that the distribution of the vertex betweenness centrality follows a power law, but the distribution of the edge betweenness centrality has a Poisson-like distribution with a very sharp spike. To investigate this phenomenon, we generated random networks with degree distribution identical to those of the protein interaction networks. To our surprise, we found out that the random networks and the protein interaction networks had almost identical distribution of edge betweenness. We conjecture that the "Poisson-like" distribution of the edge betweenness centrality is the property of any graph whose degree distribution satisfies power law. / Master of Science

Le but de ce travail est d’utiliser les simulations du type vertex afin d’étudier l’effet de l’orientation cristallographique sur la cinétique de la restauration dans l’alliage binaire Al-0,1%Mn. La caractérisation de la cinétique de la restauration dans un polycristal déformé par laminage à froid a été faite à l’aide des mesures de microdureté sur les échantillons recuits à quatre températures pendant différentes durées. Le modèle de restauration développé par Vandermeer et al a été appliqué pour estimer l’énergie d’activation pour la restauration. Plusieurs problèmes ont été rencontrés pendant l’application du modèle, dus à la complexité de ses équations, et également l’amplification de l’incertitude des mesures expérimentales par les termes exponentiels qui figurent dans le modèle. Le simulations vertex de la croissance des sous-grains ont été faites avec les propriétés du matériau trouvées dans la littérature. Les microstructures de déformation des monocristaux orientés Brass et Goss ont été reproduites dans les simulations. Les résultats des simulations étaient en bon accord avec les résultats expérimentaux d’Albou et al. Le faible taux de la croissance des sous-grains dans la microstructure Brass a été à cause de la présence des sous-joints de faible désorientation. La microstructure Goss contient des bandes avec les sous-joints de fortes désorientations, et en conséquent, fortes mobilités, qui entrainent la croissance des sous-grains dans cette structure. Ainsi, il a été établi que l’effet de l’orientation cristallographique est présent dans la microstructure de déformation, les sous-joints ainsi créés et la distribution de désorientations. / The present study aims to investigate the effect of crystallographic orientation on the kinetics of recovery in deformed samples of the Al-0.1%Mn binary alloy using vertex dynamics simulations of subgrain growth. In order to estimate the recovery kinetics experimentally, a polycrystalline bar of the alloy was cold rolled and samples extracted from the rolled bar were subjected to isothermal annealing at four temperatures for different durations. The activation energy for recovery was estimated using microhardness measurements and the recovery kinetics model developed by Vandermeer et al. Several practical difficulties were encountered when using this model due to the complexity of the model equations and the amplification of experimental uncertainty by exponential functions in the model. The activation energy found was therefore not reliable. Using the material properties mentioned in the literature, vertex dynamics simulations of subgrain growth were carried out. Deformation microstructures of monocrystals having Brass and Goss orientations were reproduced in the simulations. Good agreement was achieved with the experimental results of Albou et al. The low speed of recovery in Brass crystals was due to the presence of subgrain boundaries of low disorientations. The Goss deformation microstructure consists of bands with subgrain boundaries of high disorientations. High disorientations lead to high mobilities of subgrain boundaries and this led to subgrain growth in the Goss crystal. Thus the effect of crystallographic orientation is related to the deformation microstructure, the subgrain boundaries and the distribution of their disorientations.

Cette thèse se déroule au sein du projet ToDo (Time versus Optimality in discrete Optimization ANR 09-EMER-010) financé par l'Agence Nationale de la Recherche. Nous nous intéressons à la résolution exacte et approchée de deux problèmes de graphes. Dans un souci de compromis entre la durée d'exécution et la qualité des solutions, nous proposons une nouvelle approche par partition en cliques qui a pour but (1) de résoudre de manière rapide des problèmes exacts et (2) de garantir la qualité des résultats trouvés par des algorithmes d'approximation. Nous avons combiné notre approche avec des techniques de filtrage et une heuristique de liste. Afin de compléter ces travaux théoriques, nous avons implémenté et comparé nos algorithmes avec ceux existant dans la littérature. Dans un premier temps, nous avons traité le problème de l'indépendant dominant de taille minimum. Nous résolvons de manière exacte ce problème et démontrons qu'il existe des graphes particuliers dans lesquels le problème est 2-approximable. Dans un second temps nous résolvons par un algorithme exact et un algorithme d'approximation le problème du vertex cover et du vertex cover connexe. Puis à la fin de cette thèse, nous avons étendu nos travaux aux problèmes proches, dans des graphes comprenant des conflits entre les sommets. / This thesis takes place in the project ToDo 2 funded by the french National Research Agency. We deal with the resolution of two graph problems, by exact and approximation methods. For the sake of compromise between runtime and quality of the solutions, we propose a new approach by partitioning the vertices of the graph into cliques, which aims (1) to solve problems quickly with exact algortihms and (2) to ensure the quality if results with approximation algorithms. We combine our approach with filtering techniques and heuristic list. To complete this theoretical work, we implement our algorithms and compared with those existing in the literature. At the first step, we discuss the problem of independent dominating of minimum size. We solve this problem accurately and prove that there are special graphs where the problem is 2-approximable. In the second step, we solve by an exact algorithm and an approximation algorithm, the vertex cover problem and the connected vertex cover problem. Then at the end of this thesis, we extend our work to the problems in graphs including conflicts between vertices.

Ce travail est consacré à l'étude de faisabilité d'une imagerie par reconstruction de vertex (IRV) pour le contrôle qualité en temps réel de la thérapie par faisceau d'ions carbone. La détection de vertex d'interactions nucléaires repose sur la détection de particules secondaires : grâce à un dispositif de détection spatiale des fragments chargés (tracker), on peut reconstruire les trajectoires des particules émergeant du patient et les extrapoler jusqu'à leur point d'origine (le vertex)... Dans le cadre de notre étude, la position du vertex est déterminée de deux manières différentes : soit en calculant l'intersection de la trajectoire d'un fragment émergent avec celle de l'ion incident (connue grâce à l'utilisation d'un hodoscope de faisceau placé en amont du patient), soit grâce à l'intersection de la trajectoire de deux fragments émergents détectés en coïncidence. Notre étude de faisabilité de la technique repose sur l'outil de simulation GEANT4. La première partie de l'étude a consisté à valider cet outil grâce à plusieurs expériences réalisées au GANIL (Caen) et au GSI (Darmstadt) avec des ions carbone de différentes énergies dans des cibles d'eau ou de PMMA Par la suite, la comparaison des deux modes de détection des particules secondaires a montré que la technique utilisant l'hodoscope est la plus performante. Enfin, après l'optimisation des principaux paramètres de cette technique, une simulation réaliste montre qu'il est possible de mesurer le parcours des ions avec une précision millimétrique à l'échelle d'une tranche en énergie voire à l'échelle d'un voxel unique / This work is devoted to the feasibility study of interaction vertex imaging (IVI) for real-time quality assurance in hadrontherapy treatments with carbon ion beams. Nuclear interaction vertex detection is based on secondary particle detection: a device allows us to spatially detect charged particles (tracker), thus we can reconstruct the trajectories of particles emerging from the patient and then extrapolate their emission point (vertex). In our study, the vertex position is determined by two ways: either by calculating the intersection of the trajectory of an emerging fragment with the trajectory of the incident ion (measured by means of a beam hodoscope upstream of the patient), or with the intersection of two emerging particles trajectories detected in coincidence. Our feasibility study of this technique relies on the GEANT4 simulation tool. The first part of the study aimed to validate this tool with experiments performed at GANIL (Caen) and GSI (Darmstadt) using carbon ion beams at various energies and in various targets (water or PMMA). Secondly, the comparison of two different technics for secondary particles detection showed that the technique using the hodoscope is the most efficient. Finally, after the parameters optimization of this technique, a realistic simulation shows that it is possible to measure ion paths within millimeter precision during each energy slices. A control of each beam spot may also be possible

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

Given a graph G and an initial colouring of its vertices with two colours, say black and white, an irreversible k-threshold conversion process on G is an iterative process in which a white vertex becomes permanently coloured black at time t if at least k of its neighbours are coloured black at time t-1. A set S of vertices is an irreversible k-threshold conversion set (k-conversion set) of G if the initial colouring in which the vertices of S are black and the others are white results in the whole vertex set becoming black eventually. In the case where G is (k+1)-regular, it can be shown that the k-conversion sets coincide with the so-called feedback vertex sets, or decycling sets. In this dissertation we study the size and structure of minimum k-conversion sets in several classes of graphs. We examine conditions that lead to equality and inequality in existing bounds on the minimum size of a k-conversion set of G, for k- and (k+1)-regular graphs G. Furthermore, we derive new sharp lower bounds on this number for regular graphs of degree ranging from k+1 to 2k-1 and for graphs of maximum degree k+1. We determine exact values of the minimum size of a k-conversion set for certain classes of trees. We show that every (k+1)-regular graph has a minimum k-conversion set that avoids certain structures in its induced subgraph. These results lead to new proofs of several known results on colourings and forest partitions of (k+1)-regular graphs and graphs of maximum degree k+1. / Graduate

La hadronthérapie est une méthode de radiothérapie utilisant des ions (ici le carbone) comme faisceau plutôt que des rayons X plus conventionnels pour le traitement des cancers. Étant donné le parcours spécifique des ions dans la matière, ils permettent de traiter des tumeurs profondes dans des zones délicates telles que le cerveau par exemple. Ceci est complémentaire à tout ce qui existe depuis des dizaines d’années (intervention chirurgicale, rayons X, chimiothérapie). Deux futurs centres de traitement et de recherche (ARCHADE à Caen et ETOILE à Lyon) seront opérationnels en France à partir de 2018 en ce qui concerne ARCHADE afin de profiter des avancées récentes et de poursuivre les recherches sur cette méthode. La perte d’énergie des ions carbone dans la matière suit la loi de Bethe-Bloch, le maximum de dépôt d’énergie se situant dans une zone restreinte appelée « pic de Bragg ». En modulant la position et l’énergie du faisceau, il est possible d’irradier l’ensemble du volume de la tumeur. Cependant, les réactions nucléaires de l’ion carbone dans les tissus entrainent la production de fragments plus légers (H, He, Li etc.) qui déposent leur énergie au-delà du pic de Bragg. Les modèles implémentés dans les codes de simulation couramment utilisés en hadronthérapie (FLUKA, GEANT4 etc.) sont incapables de reproduire en même temps les distributions angulaires des fragments générés ainsi que les distributions en énergie. Le fait de ne pas reproduire fidèlement ce phénomène de fragmentation nuit à la précision des systèmes de planification de traitement utilisés cliniquement. En effet, une mauvaise estimation du processus de fragmentation entraine un biais dans le calcul de la dose déposée dans les cellules saines en arrière du pic de Bragg. Ainsi, afin de mieux contraindre les modèles, deux expériences de mesure de sections efficaces de fragmentation du carbone ont été menées. La première en mai 2011 avec un faisceau à 95MeV/n au GANIL à CAEN avec les collaborateurs du LPC Caen et la seconde en août 2011 avec un faisceau à 400 MeV/n au GSI à Darmstadt, avec la collaboration FIRST. L’expérience E600 étudie la fragmentation des ions du faisceau de carbone à 95 MeV/n dans différentes cibles minces (Au, C, , Ti etc.) correspondant aux différents constituants élémentaires du corps humain. Les différents fragments sont détectés à l’aide de cinq télescopes. Chacun d’eux est constitué de 3 étages (2 détecteurs silicium et un scintillateur CsI) afin de faire des mesures de perte d’énergie et d’énergie totale permettant une identification par la méthode du ΔE-E. Ces télescopes étaient disposés sur des raquettes pilotées à distance afin de pouvoir modifier leur position angulaire par rapport à la position de la cible. Ainsi, les taux de production des différents fragments permettent de remonter aux sections efficaces de fragmentation doublement différentielles (en énergie et en angle). [...] / The hadrontherapy is a radiotherapy method using ions (carbon ions here) instead of the more conventional X-rays for cancer treatment. Deep radioresistant tumour areas, as brain carcinoma for example, can be treated thanks to the specific dosedeposition at the end of the ion path. This is an additional method to older classic ones (surgery, X-rays, chemotherapy). Two hadrontherapy centres for treatment and research are planned in France from 2018 (ARCHADE) in order to benefit from the newest progress and to keep improving this method. Carbon ions energy loss in the matter follows the Bethe-Bloch law. The maximum of energy depth is located in a limited area called “Bragg peak”. By adjusting the beam position and energy, the whole volume of the tumor can be irradiated. Nevertheless, nuclear reactions of carbon ion in tissues generate the production of lighter fragments (H, He, Li etc.) that deposit their energy beyond the Bragg peak. Models implemented in hadrontherapy simulation codes (FLUKA, GEANT4 etc.) cannot reproduce angular distributions of the lighter fragments and energy distributions at the same time. These poor estimations affect the treatment planning systems accuracy that are clinically used.Indeed, a bad estimation of fragmentation process induces a bias in the dose calculation concerning healthy cells beyond the Bragg peak. In order to better constraint models, two experiments based on fragmentation cross-sections measurements have been performed. The first one in may 2011 with a beam at 95 MeV/u (GANIL) in collaboration with the LPC Caen and the second one in august 2011 with a beam at 400 MeV/u (GSI) with the FIRST collaboration. E600 experiment is devoted to the study of carbon ions fragmentation at 95 MeV/u in several thin targets (Au, C, , Ti etc.) corresponding to the basic building blocks of human body. Five telescopes are designed for the fragments detection. Each one is a three-stage detector (2 silicon detectors and one CsI scintillator) that allows energy loss and total energy measurements for the ΔE-E identification method.Telescopes were disposed two by two in the reaction chamber with a remote control of the angular position. From the production rate measurements, the double differential fragmentation cross-sections (energy and angle) can be computed.From the experimental data for + reaction at 95 MeV/u on a 250 μm thick carbon target, all cross-sections were deduced.FIRST experiment uses a very different set-up. It is composed of: a beam monitoring, a vertex detector (CMOS), a calorimeter(KENTROS), a magnet (ALADIN), MUSIC (3 ionization chambers and 4 proportional counters) and a TOF-wall. Generated particles trajectory is reconstructed thanks to the vertex detector + TOF-wall for all fragments emitted with an angle lower than 5° and thanks to the vertex detector + KENTROS for higher angles. In the first case, the ALADIN magnet deflects the trajectory of the particles (MUSIC detector ran out). One 8 mm thick target has been used here. Preliminary results concerning production rates of the different charges, angular distributions and reconstruction efficiencies have been obtained. Heavier fragments mass identification is quite difficult because of the non-working MUSIC detector; it degrades the fragments momentumaccuracy.[...]