Дисертації з теми "Intrinsic geometry"

An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.

Le probleme du tolerancement des pieces mecaniques est decisif pour I'industrie de pointe. Ses incidences economiques sont importantes pour le secteur manufacturier qui subit des transformations profondes imposees par la globalisation des marches et revolution constante des technologies (CAO, CMM, 3D Scanner, etc.). U est admit aujourd'hui que I'optimisation des performances des produits requiert la prise en compte des variations inherentes aux processus de fabrication, d'ou le controle de la qualite a travers le processus de developpement et de fabrication. Dans le cas des composantes dites 'flexibles' (ou non rigides), par exemple des pieces mecanique a parol minces comme le revetement d'un avion ou d'une auto, I'usage industriel actuel est limite encore a I'utilisation de gabarit de conformite, relativement couteux, qui contraignent la geometric de la piece a un etat qui reflete 1'assemblage. Par la suite, des mesures par contact directe ou par scanner sont effectuees. C'est ainsi I'industrie elimine I'effet des deformations dues a la flexibilite de la piece pour tenter de detecter les defauts dus au procede de fabrication. Le projet propose a pour objectif de faciliter les operations d'inspection dimensionnelle et geometrique des composantes flexibles a partir d'un nuage de points, et ce, sans recours a un gabarit ou des operations secondaires de conformation. Plus specifiquement, nous visons le developpement d'une methodologie qui permettra de localiser et de quantifier les defauts de profil dans le cas des coques minces typique des industries aerospatiales et automobiles. Pour arriver a cet objectif, nous mettons en oeuvre une idee que nous appelons Numerical Inspection Fixture. Nous utilisons les distances geodesiques pour detecter la similarite intrinseque entre une piece a I'etat libre qui inclus les effets de gravite, des contraintes internes et des defauts de fabrication, et la meme piece telle que definie nominalement par un modele CAO. Ce memoire developpe le fondement theorique de cette methode et les algorithmes qu'y sont relies. Nous employons une approche, deja employe dans le domaine de I'imagerie medicale, pour identifier les distances geodesiques minimales (geometric metrique), les statistiques {Multidimensional Scaling - MDS) pour analyser les similarites et dissimilarites entre deux objets, ainsi que la methode d'elements finis (FE) pour aboutir a une approche generate et original pour I'inspection de pieces geometriques non rigides. Deux methodes y sont proposees avec des validations numeriques.

The aim of the present study was to investigate the potential for children to use the turtle methaphor to develop understandings of intrinsic, euclidean and cartesian geometrical ideas. Four aspects of the problem were investigated. a) the nature of the schema children form when they identify with the turtle in order to change its state on the screen; b) whether it is possible for them to use the schema to gain insights into certain basic geometrical principles of the cartesian geometrical system; c) how they might use the schema to form understandings of euclidean geometry developed inductively from specific experiences; d) the criteria they develop for choosing between intrinsic and euclidean ideas. Ten 11 to 12 year - old children participated in the research, previously having had 40 to 50 hours of experience with Turtle geometry. The research involved three case - studies of pairs of children engaging in cooperative activities, each case - study within a geometrical Logo microworld. The data included hard copies of everything that was said, typed and written. Issues a) and b) were investigated by means of the first case - study which involved three pairs of children and a microworld embedding intrinsic and coordinate ideas. A model of the children's intrinsic schema and a model of the coordinate schema which they formed during the study were devised. The analysis shows that the two schemas remained separate in the children's minds with the exception of a limited number of occasions of context specific links between the two. Issue c) was investigated in the second case - study involving one pair of children and a microworld where the turtle was equipped with distance and turn measuring instruments and a facility to mark positions. The analysis illustrates how a turtle geometric environment of a dynamic mathematical nature was generated by the children, who used their intrinsic schema and predominantly engaged in inductive thinking. The geometrical content available to the children within this environment was extended from intrinsic to both intrinsic and euclidean geometry. Issue d) was investigated by means of the third case - study involving a pair of children and a microworld where the children could choose among circle procedures embedding intrinsic and/or euclidean notions in order to construct figures of circle compositions. The analysis shows that the children employed their turtle schema in using both kinds of notions and did not seem to perceive qualitative differences between them. Their decisions on which type of notion to use were influenced by certain broader aspects of the mathematical situations generated in the study.

Line geometry is a foundation of screw algebra in line coordinates that were created by Plücker as ray coordinates taking a line as a ray between two points and axis coordinates taking a line as the intersection of two planes. This Thesis reveals the geometrical meaning and intrinsic relationship between these ray coordinates and axis coordinates, leading to an in-depth understanding of conformability and duality of these two sets of screw coordinates, and their related vector space and dual vector space. Based on the study of screw algebra, the resultant twist of a serial manipulator is presented geometrically by an assembly of unit joint screws with the corresponding velocity amplitudes. This leads to the geometrical interpretation for the resultant twist with its instantaneous screw axis (ISA) that is formulated by a combination of weighted position vectors of joint screws. The screw-based Jacobian is then derived after recognizing the resultant twist of a serial manipulator. The case leads to a revelation for the first time the relationship of a Jacobian matrix acquired by using screw algebra and a derivative Jacobian matrix using differential functions, and to an in-depth investigation of transformation between these two Jacobians. To extend the application of screw algebra and this derivative Jacobian, kinematics analysis of a novel reconfigurable base-integrated parallel mechanism is proposed and its screw-based Jacobian is derived, leading to its equivalent model, the Metamorphic hand with a reconfigurable palm. The method is then applied to the investigation of the Metamorphic Hand, while manipulating an object, based on the product sub-manifolds and the exponential method. Evaluation of the functionality of the Metamorphic hand is further analysed, with the Anthropomorphism Index (AI) and palmar shape modulation as the criteria, evaluating performance enhancement of the Metamorphic hand in comparison to other robotic hands with a fixed palm. The Thesis presents novel discoveries in the intrinsic geometry of screw coordinates and the coherent connection between Jacobian formed by screw algebra and the Jacobian using the derivative method. This intrinsic geometry insight is then used to investigate for the first time the parallel mechanism with a reconfigurable base, paving a way for an in-depth investigation of the Metamorphic hand on its reconfigurability and grasp affordability and for the first time using Anthropomorphic Index to evaluate the Metamorphic hand.

Surface topography is, generally, composed of many length scales starting from its physical geometry, to its microscopic or atomic scales known by roughness. The spatial and geometrical evolution of the roughness topography of engineering surfaces avail comprehensive understanding, and interpretation of many physical and engineering problems such as friction, and wear mechanisms during the mechanical contact between adjoined surfaces. Obviously, the topography of rough surfaces is of random nature. It is composed of irregular hills/valleys being spatially correlated. The relation between their densities and their geometric properties are the fundamental topics that have been developed, in this research study, using the theory of random fields and the integral geometry.An appropriate random field model of a rough surface has been defined by the most significant parameters, whose changes influence the geometry of its excursion. The excursion sets were quantified by functions known as intrinsic volumes. These functions have many physical interpretations, in practice. It is possible by deriving their analytical formula to estimate the parameters of the random field model being applied on the surface, and for statistical analysis investigation of its excursion sets. These subjects have been essentially considered in this thesis. Firstly, the intrinsic volumes of the excursion sets of a class of mixture models defined by the linear combination of Gaussian and t random fields, then for the skew-t random fields are derived analytically. They have been compared and tested on surfaces generated by simulations. In the second stage, these random fields have been applied to real surfaces measured from the UHMWPE component, involved in application of total hip implant, before and after wear simulation process. The primary results showed that the skew-t random field is more adequate, and flexible for modelling the topographic roughness. Following these arguments, a statistical analysis approach, based on the skew-t random field, is then proposed. It aims at estimating, hierarchically, the significant levels including the real hills/valleys among the uncertain measurements. The evolution of the mean area of the hills/valleys and their levels enabled describing the functional behaviour of the UHMWPE surface over wear time, and indicating the predominant wear mechanisms.

Les ``C-arm'' sont des systémes de radiologie interventionnelle fréquemment utilisés en salle d'opération ou au lit du patient. Des images 3D des structures anatomiques internes peuvent être calculées à partir de multiples radiographies acquises sur un ``C-arm mobile'' et isocentrique décrivant une trajectoire généralement circulaire autour du patient. Pour cela, la géométrie conique d'acquisition de chaque radiographie doit être précisément connue. Malheureusement, les C-arm se déforment en général au cours de la trajectoire. De plus leur motorisation engendre des oscillations non reproductibles. Ils doivent donc être calibrés au cours de l'acquisition. Ma thèse concerne la calibration intrinsèque d'un C-arm à partir de la détection de la projection du collimateur de la source dans les radiographies.Nous avons développé une méthode de détection de la projection des bords linéaires du collimateur. Elle surpasse les méthodes classiques comme le filtre de Canny sur données simulées ou réelles. La précision que nous obtenons sur l'angle et la position (phi,s) des droites est de l'ordre de: phi{RMS}=+/- 0.0045 degrees et s{RMS}=+/- 1.67 pixels. Nous avons évalué nos méthodes et les avons comparés à des méthodes classiques de calibration dans le cadre de la reconstruction 3D
Mobile isocentric x-ray C-arm systems are an imaging tool used during a variety of interventional and image guided procedures. Three-dimensional images can be produced from multiple projection images of a patient or object as the C-arm rotates around the isocenter provided the C-arm geometry is known. Due to gravity affects and mechanical instabilities the C-arm source and detector geometry undergo significant non-ideal and possibly non reproducible deformation which requires a process of geometric calibration. This research investigates the use of the projection of the slightly closed x-ray tube collimator edges in the image field of view to provide the online intrinsic calibration of C-arm systems.A method of thick straight edge detection has been developed which outperforms the commonly used Canny filter edge detection technique in both simulation and real data investigations. This edge detection technique has exhibited excellent precision in detection of the edge angles and positions, (phi,s), in the presence of simulated C-arm deformation and image noise: phi{RMS} = +/- 0.0045 degrees and s{RMS} = +/- 1.67 pixels. Following this, the C-arm intrinsic calibration, by means of accurate edge detection, has been evaluated in the framework of 3D image reconstruction

L'ensemble d'excursion EX(u) d'un champ aléatoire réel X sur R^d à un niveau de seuil u ∈ R est le sous-ensemble du domaine R^d où X dépasse u. Ainsi, l'ensemble d'excursion est aléatoire, et sa distribution à un niveau fixe u est déterminée par la distribution de X. Étant des sous-ensembles de R^d, les ensembles d'excursions peuvent être étudiés en termes de leurs propriétés géométriques comme moyen d'obtenir des informations partielles sur les propriétés de distribution des champs aléatoires sous-jacents.Cette thèse examine :(a) comment les mesures géométriques d'un ensemble d'excursion peuvent être déduites à partir d'un échantillon discret de l'ensemble d'excursion, et(b) comment ces mesures peuvent être liées aux propriétés distributionnelles du champ aléatoire à partir duquel l'ensemble d'excursion a été obtenu. Chacun de ces points est examiné en détail dans le Chapitre 1, qui fournit un aperçu global des résultats trouvés tout au long du reste de ce manuscrit. Les mesures géométriques que nous étudions (pour les ensembles d'excursion et les sous-ensembles déterministes de R^d) lors de l'adressage du point (a) sont la mesure de la surface de dimension (d−1), le reach, et le rayon de r-convexité. Chacune de ces quantités peut être liée à la régularité de la frontière de l'ensemble, ce qui est souvent difficile à déduire à partir d'échantillons discrets de points.Pour résoudre ce problème, nous apportons les contributions suivantes au domaine de la géométrie computationnelle :- Dans le Chapitre 2, nous identifions le facteur de biais qui correspond aux algorithmes de comptage local pour calculer la mesure de la surface de dimension (d − 1) des ensembles d'excursion sur une grande classe de pavages de R^d. Le facteur de biais dépend uniquement de la dimension d et non de la géométrie précise du pavage.- Dans le Chapitre 3, nous introduisons un algorithme de comptage pseudo-local pour calculer le périmètre des ensembles d'excursion en deux dimensions. L'algorithme proposé est convergent multigrille (multigrid convergent en anglais) et comporte un hyper paramètre réglable pouvant être choisi automatiquement à partir d'informations accessibles.- Dans le Chapitre 4, nous introduisons le β-reach en tant que généralisation du reach, et l'utilisons pour prouver la cohérence d'un estimateur du reach des sous-ensembles fermés de R^d. De même, nous définissons un estimateur cohérent du rayon de r-convexité des sous-ensembles fermés de R^d. De nouvelles relations théoriques sont établies entre le reach et le rayon de r-convexité. Nous étudions également comment ces mesures géométriques des ensembles d'excursion sont liées à la distribution du champ aléatoire.- Dans le Chapitre 5, nous introduisons l'extremal range : une statistique géométrique locale qui caractérise l'étendue spatiale des dépassements de seuil à un niveau fixe u ∈ R. La distribution de l'extremal range est entièrement déterminée par la distribution de l'ensemble d'excursion au niveau u. Nous montrons comment l'extremal range est liée distributionnellement aux volumes intrinsèques de l'ensemble d'excursion. De plus, le comportement limite de l'extremal range aux grands seuils est étudié en relation avec la stabilité des peaks-over-threshold du champ aléatoire sous-jacent. Enfin, la théorie est appliquée à des données climatiques réelles pour mesurer le degré d'indépendance asymptotique présent et sa variation dans l'espace.Des perspectives sur la manière dont ces résultats peuvent être améliorés et étendus sont fournies dans le Chapitre 6
The excursion set EX(u) of a real-valued random field X on R^d at a threshold level u ∈ R is the subset of the domain R^d on which X exceeds u. Thus, the excursion set is random, and its distribution at a fixed level u is determined by the distribution of X. Being subsets of R^d, excursion sets can be studied in terms of their geometrical properties as a means of obtaining partial information about the distributional properties of the underlying random fields.This thesis investigates(a) how the geometric measures of an excursion set can be inferred from a discrete sample of the excursion set, and(b) how these measures can be related back to the distributional properties of the random field from which the excursion set was obtained.Each of these points are examined in detail in Chapter 1, which provides a broad overview of the results found throughout the remainder of this manuscript. The geometric measures that we study (for both excursion sets and deterministic subsets of R^d) when addressing point (a) are the (d − 1)-dimensional surface area measure, the reach, and the radius of r-convexity. Each of these quantities can be related to the smoothness of the boundary of the set, which is often difficult to infer from discrete samples of points. To address this problem, we make the following contributions to the field of computational geometry:• In Chapter 2, we identify the bias factor in using local counting algorithms for computing the (d − 1)-dimensional surface area of excursion sets over a large class of tessellations of R^d. The bias factor is seen to depend only on the dimension d and not on the precise geometry of the tessellation.• In Chapter 3, we introduce a pseudo-local counting algorithm for computing the perimeter of excursion sets in two-dimensions. The proposed algorithm is multigrid convergent, and features a tunable hyperparameter that can be chosen automatically from accessible information.• In Chapter 4, we introduce the β-reach as a generalization of the reach, and use it to prove the consistency of an estimator for the reach of closed subsets of R^d. Similarly, we define a consistent estimator for the radius of r-convexity of closed subsets of R^d. New theoretical relationships are established between the reach and the radius of r-convexity.We also study how these geometric measures of excursion sets relate to the distribution of the random field.• In Chapter 5, we introduce the extremal range: a local, geometric statistic that characterizes the spatial extent of threshold exceedances at a fixed level threshold u ∈ R. The distribution of the extremal range is completely determined by the distribution of the excursion set at the level u. We show how the extremal range is distributionally related to the intrinsic volumes of the excursion set. Moreover, the limiting behavior of the extremal range at large thresholds is studied in relation to the peaks-over-threshold stability of the underlying random field. Finally, the theory is applied to real climate data to measure the degree of asymptotic independence present, and its variation throughout space.Perspectives on how these results may be improved and expanded upon are provided in Chapter 6

Virtual reality systems are nowadays common part of many research institutes due to its low cost and effective visualization of data. They mostly allow visualization and exploration of virtual worlds, but many lack user interaction. In this paper we suggest multi-camera optical system, which allows effective user interaction, thereby increasing immersion of virtual system. This paper describes the calibration process of multiple cameras using point correspondences.

Reinforced concrete frames infilled with unreinforced masonry are commonly used in structures worldwide. The interaction between the frame and the infill panel is usually ignored in engineering practice, and the masonry infill is not considered as a structural element. However, observations made after the occurrence of strong earthquakes have shown that the bare frame and infill-frame behave differently when subjected to in-plane lateral loads. Extensive research has been conducted on the behaviour of infill-frames when laterally loaded. This research focuses on the analysis of infill-frames using the equivalent strut modelling method, whereby an infill-frame is simplified, and the infill panel is replaced by one or more compressive strut elements. A large number of strut models have been proposed in the literature, but recent studies have demonstrated that it is not possible to apply one strut model to all infill-frame structures. It has been found that changing the properties of an infill-frame can also change the geometric properties of struts, namely width, location and number of struts in an equivalent strut model. For this reason, recent studies have proposed a case-specific strut modelling approach. In the current study, a macro script available in the literature that can be used to generate a detailed finite element (FE) model has been applied to construct and analyse a number of infill-frames with different material and geometric properties. Sensitivity analyses on some of these infill-frames have also been conducted by varying the material properties of the infill, and the amount and distribution of vertical loads on the frame. The results of detailed FE analyses, more specifically contours of the compressive principal stresses, have been used to define the geometric properties of the struts of case-specific strut models for each of the infill-frames. Equivalent strut models were then analysed and compared. Further, the proposed strut models were applied to other infill-frames selected for this study; two strut models from the literature were also applied to these infill-frames. It was concluded that the geometric properties of, and the vertical load on an infill-frame can be related to the geometric properties of its equivalent strut model. In contrast, a variation of up to 25% in the masonry material properties did not have a significant effect on the strut properties. It was shown that casespecific strut modelling is a versatile and generic technique that can adequately replicate the highly nonlinear behaviour of infill-frames regardless of their geometric or material properties. By expanding the current research, it is hoped that a rigorous classification of infill-frames and their relevant equivalent strut models can be developed to assist structural engineers in their everyday design tasks.

Orientador: Francesco Mercuri
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatistica e Computação Cientifica
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Resumo: A tese está dividida em duas partes: Parte 1: Hipersuperfícies conformemente planas com curvatura média constante. Seja Mn uma variedade riemanniana conformemente plana de dimensão n = 3 e f : Mn -IRn+1 uma hipersuperfície com curvatura média constante. Para estas imersões, o único caso não classificado é aquele onde o operador de Weingarten possui três autovalores distintos. Para este caso em aberto, pusemos uma condicão adicional natural: Por todo ponto passa uma linha de curvatura que também é uma geodésica. Esta condição é natural pois ela é sempre verdadeira para dimensão maior do que três. O teorema principal classifica tais hipersuperfícies como sendo um aberto de um cone sobre um toro de Clifford. Parte 2: Hipersuperfícies compactas de cohomogeneidade 1. Seja Mn uma variedade riemanniana compacta de dimensão n = 5, p : G x M M uma ação por isometrias de cohomogeneidade 1 de um grupo compacto e conexo G e f : Mn -lRn+1 uma hipersuperfície. Dizemos que uma isometria g é induzida por uma isometria h do espaço ambiente se f o g = h o f. f é dita padrão se todas as isometrias de G são induzidas por isometrias do ambiente. O teorema principal diz que uma hipersuperfície f nas condições acima é uma imersão padrão.
Abstract: The thesis has two parts: Part 1: Conformally fiat hypersurfaces with constant mean curvature. Let Mn be a conformally fiat manifold with dimension n = 3 and f : Mn -IRn+1 a hypersurface with constant mean curvature. For this immersions, the only unclassified case is the tridimensional one, where the Weingarten operator has three different eigenvalues. In this open case, we have put a natural condition: For every point passes a curvature line that is also a geodesic. This condition is natural because it is always true for dimensions greater than 3. The main theorem classifies those hypersurfaces as a piece of a cone over the Clifford Torus. Part 2: Compact hypersurfaces of cohomogeneity one. Let Mn a compact riemannian manifold with dimension n = 5, p : G x M - M a cohomogeneity one action of a connected and compact isometry group G and f : Mn -lRn+1 a hypersurface. We say that an isometry g ? G is induced by an isometry h of the ambient space if f o g = h o f. f is called a standard immersion if every isometry that belong to G are induced by an ambient space isometry. The main theorem proves that a hypersurface f that satisfies the aboveI conditions is a standart immersion.
Doutorado
Doutor em Matemática

Malgré les améliorations considérables des performances du matériel graphique ainsi que les avancées algorithmiques majeures depuis le début des années 2000, certains phénomènes naturels restent extrêmement coûteux à simuler. Par exemple, plusieurs pistes ont été proposées pour améliorer les performances des simulations de fluides, qui sont animées par la résolution d'équations différentielles partielles (EDP), plus particulièrement les équations hautement non linéaires de Navier-Stokes. Dans cette thèse, nous explorons d'abord l'utilisation de l'apprentissage profond pour créer un espace réduit dans lequel un solveur peut opérer à moindre coût, tout en produisant des solutions de haute qualité. Nous proposons un modèle qui permet la simulation d'écoulements turbulents à une résolution quatre fois supérieure à celle de l'entrée dans chaque dimension, avec des performances d'exécution améliorées par rapport à un solveur haute résolution. Ensuite, nous utilisons les contributions sur les opérateurs intrinsèques pour simuler des fluides sur des surfaces 3D avec des coûts réduits. Nous nous concentrons sur le modèle smoothed-particle hydrodynamics (SPH) que nous adaptons aux surfaces 3D, en rassemblant les voisinages des particules grâce aux géodésiques de plus court chemin, et en déplaçant ces particules de manière intrinsèque sur la surface. Tout ceci est simple à mettre en œuvre sur le GPU, ce qui permet la simulation de dizaines de milliers de particules sur différents maillages triangulaires à une vitesse interactive
Despite tremendous improvements in graphics hardware performance aswell as key algorithmic advancements since the beginning of the years 2000, some natural phenomena remain extremely costly to simulate. For instance, several tracks have been proposed to improve the performance of fluid simulations, that are animated by solving partial differential equations (PDE), more specifically the highlynon-linear Navier-Stokes equations. In this thesis, we first explore the use of deep learning to create a reduced space in which a solver can operate with lower costs, while still out putting high-quality solutions. We propose a model that enables the simulation of turbulent flows at a resolution four times higher than that of the given input in each dimension, with improved runtime performance compared to a high-resolution solver. Secondly, we use the contributions on intrinsic operators for simulating fluids on 3D surfaces with reduced costs. We focus on the smoothed-particle hydrodynamics (SPH) model that we adapt to 3D surfaces, by gathering the particles' neighborhoods thanks to shortest-path geodesics, and by displacing such particles in an intrinsic manner on the surface. All of this is straightforward to implement on the GPU, enablingthe simulation of tens of thousands of particles on various triangle meshes at interactive speed

The main body of this thesis consists of six appended papers. In the  first two, different  cooperative surveillance problems are considered. The second two consider different aspects of the trajectory planning problem, while the last two deal with observer design for mobile robotic and Euler-Lagrange systems respectively.In Papers A and B,  a combinatorial optimization based framework to cooperative surveillance missions using multiple Unmanned Ground Vehicles (UGVs) is proposed. In particular, Paper A  considers the the Minimum Time UGV Surveillance Problem (MTUSP) while Paper B treats the Connectivity Constrained UGV Surveillance Problem (CUSP). The minimum time formulation is the following. Given a set of surveillance UGVs and a polyhedral area, find waypoint-paths for all UGVs such that every point of the area is visible from  a point on a waypoint-path and such that the time for executing the search in parallel is minimized.  The connectivity constrained formulation  extends the MTUSP by additionally requiring the induced information graph to be  kept recurrently connected  at the time instants when the UGVs  perform the surveillance mission.  In these two papers, the NP-hardness of  both these problems are shown and decomposition techniques are proposed that allow us to find an approximative solution efficiently in an algorithmic manner.Paper C addresses the problem of designing a real time, high performance trajectory planner for an aerial vehicle that uses information about terrain and enemy threats, to fly low and avoid radar exposure on the way to a given target. The high-level framework augments Receding Horizon Control (RHC) with a graph based terminal cost that captures the global characteristics of the environment.  An important issue with RHC is to make sure that the greedy, short term optimization does not lead to long term problems, which in our case boils down to two things: not getting into situations where a collision is unavoidable, and making sure that the destination is actually reached. Hence, the main contribution of this paper is to present a trajectory planner with provable safety and task completion properties. Direct methods for trajectory optimization are traditionally based on a priori temporal discretization and collocation methods. In Paper D, the problem of adaptive node distribution is formulated as a constrained optimization problem, which is to be included in the underlying nonlinear mathematical programming problem. The benefits of utilizing the suggested method for  online  trajectory optimization are illustrated by a missile guidance example.In Paper E, the problem of active observer design for an important class of non-uniformly observable systems, namely mobile robotic systems, is considered. The set of feasible configurations and the set of output flow equivalent states are defined. It is shown that the inter-relation between these two sets may serve as the basis for design of active observers. The proposed observer design methodology is illustrated by considering a  unicycle robot model, equipped with a set of range-measuring sensors. Finally, in Paper F, a geometrically intrinsic observer for Euler-Lagrange systems is defined and analyzed. This observer is a generalization of the observer proposed by Aghannan and Rouchon. Their contractivity result is reproduced and complemented  by  a proof  that the region of contraction is infinitely thin. Moreover, assuming a priori bounds on the velocities, convergence of the observer is shown by means of Lyapunov's direct method in the case of configuration manifolds with constant curvature.
QC 20100622
TAIS, AURES

We prove a Harnack inequality for non-negative solutions of a parabolic equation having an anisotropic slow diffusion. We study the propagation of support of solutions, through an iterative technique reminiscent of De Giorgi’s method and through the investigation of particular embeddings in anisotropic Sobolev spaces. At this point, we make an analysis of the natural scaling of the equation to reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution thanks to finite speed of propagation. Then we exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale-invariant Harnack inequality in an anisotropic intrinsic geometry, dictated by the powers of the diffusion coefficients. Finally we show some consequences as H¨older continuity of solutions, Liouville-type theorems and we formulate some open problems.

The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.

This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by k dimensional planes but which are embedded in a D>>k dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.

Dissertation

碩士
國立臺北藝術大學
音樂學系碩(博)士班
106
This thesis aims to illustrate the process and practice of transforming geometric shapes into the acoustic sound of music. In the first chapter, the relationships between graphics and music were discussed. This includes a probe into the question of “shape” in the visual arts and music, in particular the “shape” of the Chinese written characters. The connections between the Chinese calligraphy, the structure of the Chinese written characters, and music were explored in order to support my main thesis. Next chapter focuses on the spatiality of the acoustic sounds in music and goes into how spatial movement is created through the transformations of acoustic sound. The discussion is divided into the following two aspects: the structures of timbres and the structures of acoustic sounds. In the third chapter, I elaborated on how the concepts discussed in the previous chapters became the source of inspiration for my orchestral piece "Skewb Diamond". A detailed account of how I composed the piece was given, starting with a definition of my inspiration and musical ideas to the transformation of the initial concepts and the actual process of composition and musical design. In the final chapter, an analysis of my orchestral piece "Skewb Diamond" was given. I also shared how my observations and perceptions of visual artworks were transformed into acoustic imageries in my music. In this case, the characteristics and structures of diamonds and geometric figures were the main subject matter. The whole purpose of this study is to take a deeper look into my own musical language and explore further the relationships between the domain of visual art and that of music, in particular in regards to visual space and graphic notations in music, with the hope of stimulating more thoughts for my musical creations in the future. Keywords: “geometric”, “spatial”, “music”, “intrinsic sounds”