In dieser Arbeit setzen wir uns mit der Ph.D. Thesis von Luis Silvestre auseinander, in welcher er das Hindernisproblem für den gebrochenen Laplace Operator behandelt hat. Das Ziel war es seine Arbeit nachzuvollziehen und seine Beweise vollständig auszuformulieren. Dabei haben wir uns auf die Existenz der Lösung und erste Regularitätsresultate beschränkt.

Der Laplace-Operator auf kompakten Riemannschen Mannigfaltigkeiten besitzt eine natürliche Verallgemeinerung auf kompakte Riemannsche Orbifolds und das Spektrum des so gewonnenen Operators besteht ausschließlich aus Eigenwerten endlicher Vielfachheit. Die Feststellung, dass das Spektrum Informationen über die Geometrie einer Mannigfaltigkeit (oder, allgemeiner, einer Orbifold) enthält, begründete ein ganzes Teilgebiet der Mathematik. Es ist eine offene Frage der sogenannten Spektralgeometrie, ob eine Mannigfaltigkeit und eine singuläre Orbifold isospektral sein (d.h., dasselbe Spektrum mitsamt den Vielfachheiten der Eigenwerte besitzen) können. Angesichts diverser Obstruktionen zur Existenz eines solchen Beispiels für die bekannten Beispiele isospektraler guter Orbifolds, soll diese Arbeit die Spektralgeometrie schlechter Orbifolds erhellen. Zu diesem Zweck geben wir die ersten Beispiele für isospektrale Metriken auf schlechten Orbifolds an. Diese basieren auf bestimmten gewichteten projektiven Räumen, auf denen wir mittels einer Verallgemeinerung von Schüths Version der Torus-Methode nicht-trivial isospektrale Metriken konstruieren. / The Laplace Operator on compact Riemannian manifolds naturally generalizes to compact Riemannian orbifolds and the spectrum of the resulting operator consists only of eigenvalues with finite multiplicities. The observation that the spectrum contains information about the geometry of a manifold (and, more generally, an orbifold) gave rise to a whole field of mathematics. It is an open question of so-called spectral geometry, whether a manifold and a singular orbifold can be isospectral (i.e., have the same spectrum with the same multiplicities of the eigenvalues). Given the various obstructions to the existence of such an example for the known examples of isospectral good orbifolds, this work is an attempt to shed light on the spectral geometry of bad orbifolds by giving the first examples of isospectral Riemannian metrics on bad orbifolds. In our case these are particular fixed weighted projective spaces equipped with non-trivially isospectral metrics obtained by a generalization of Schüth''s version of the torus method.

Orientadores: Yuri Dimitrov Bozhkov, Antonio Carlos Gilli Martins / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-09-24T19:39:04Z (GMT). No. of bitstreams: 1 Freire_IgorLeite_D.pdf: 977261 bytes, checksum: b8ba44493aeac3de0d37cdfff2fc581b (MD5) Previous issue date: 2008 / Resumo: Neste trabalho provamos um teorema que faz a classificacão completa dos grupos de simetrias de Lie da equação semilinear de Kohn - Laplace no grupo de Heisenberg tridimensional. Uma vez que tal equação possui estrutura variacional, determinamos quais são suas simetrias de Noether e a partir delas estabelecemos suas respectivas leis de conservação / Abstract: In this work, we carry out a complete group classification of Lie point symmetries of semilinear Kohn - Laplace equations on the three-dimensional Heisenberg group. Since this equation has variational structure, we determine which of its symmetries are Noether's symmetries. Then we establish their respectives conservation laws / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice. / In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.

Les discrétisations adaptatives sont importantes dans les problèmes de fluxcompressible/incompressible puisqu'il est souvent nécessaire de résoudre desdétails sur plusieurs niveaux, en permettant de modéliser de grandes régionsd'espace en utilisant un nombre réduit de degrés de liberté (et en réduisant letemps de calcul).Il existe une grande variété de méthodes de discrétisation adaptative, maisles grilles cartésiennes sont les plus efficaces, grâce à leurs stencilsnumériques simples et précis et à leurs performances parallèles supérieures.Et telles performance et simplicité sont généralement obtenues en appliquant unschéma de différences finies pour la résolution des problèmes, mais cetteapproche de discrétisation ne présente pas, au contraire, un chemin faciled'adaptation.Dans un schéma de volumes finis, en revanche, nous pouvons incorporer différentstypes de maillages, plus appropriées aux raffinements adaptatifs, en augmentantla complexité sur les stencils et en obtenant une plus grande flexibilité.L'opérateur de Laplace est un élément essentiel des équations de Navier-Stokes,un modèle qui gouverne les écoulements de fluides, mais il se produit égalementdans des équations différentielles qui décrivent de nombreux autres phénomènesphysiques, tels que les potentiels électriques et gravitationnels. Il s'agitdonc d'un opérateur différentiel très important, et toutes les études qui ontété effectuées sur celui-ci, prouvent sa pertinence.Dans ce travail seront présentés des approches de différences finies et devolumes finis 2D pour résoudre l'opérateur laplacien, en appliquant des patchsde grilles superposées où un niveau plus fin est nécessaire, en laissant desmaillages plus grossiers dans le reste du domaine de calcul.Ces grilles superposées auront des formes quadrilatérales génériques.Plus précisément, les sujets abordés seront les suivants:1) introduction à la méthode des différences finies, méthode des volumes finis,partitionnement des domaines, approximation de la solution;2) récapitulatif des différents types de maillages pour représenter de façondiscrète la géométrie impliquée dans un problème, avec un focussur la structure de données octree, présentant PABLO et PABLitO. Le premier estune bibliothèque externe utilisée pour gérer la création de chaque grille,l'équilibrage de charge et les communications internes, tandis que la secondeest l'API Python de cette bibliothèque, écrite ad hoc pour le projet en cours;3) la présentation de l'algorithme utilisé pour communiquer les données entreles maillages (en ignorant chacune l'existence de l'autre) en utilisant lesintercommunicateurs MPI et la clarification de l'approche monolithique appliquéeà la construction finale de la matrice pour résoudre le système, en tenantcompte des blocs diagonaux, de restriction et de prolongement;4) la présentation de certains résultats; conclusions, références.Il est important de souligner que tout est fait sous Python comme framework deprogrammation, en utilisant Cython pour l'écriture de PABLitO, MPI4Py pour lescommunications entre grilles, PETSc4py pour les parties assemblage et résolutiondu système d'inconnues, NumPy pour les objets à mémoire continue.Le choix de ce langage de programmation a été fait car Python, facile àapprendre et à comprendre, est aujourd'hui un concurrent significatif pourl'informatique numérique et l'écosystème HPC, grâce à son style épuré, sespackages, ses compilateurs et pourquoi pas ses versions optimisées pour desarchitectures spécifiques. / Adaptive discretizations are important in compressible/incompressible flow problems since it is often necessary to resolve details on multiple levels,allowing large regions of space to be modeled using a reduced number of degrees of freedom (reducing the computational time).There are a wide variety of methods for adaptively discretizing space, but Cartesian grids have often outperformed them even at high resolutions due totheir simple and accurate numerical stencils and their superior parallel performances.Such performance and simplicity are in general obtained applying afinite-difference scheme for the resolution of the problems involved, but this discretization approach does not present, by contrast, an easy adapting path.In a finite-volume scheme, instead, we can incorporate different types of grids,more suitable for adaptive refinements, increasing the complexity on thestencils and getting a greater flexibility.The Laplace operator is an essential building block of the Navier-Stokes equations, a model that governs fluid flows, but it occurs also in differential equations that describe many other physical phenomena, such as electric and gravitational potentials, and quantum mechanics. So, it is a very importantdifferential operator, and all the studies carried out on it, prove itsrelevance.In this work will be presented 2D finite-difference and finite-volume approaches to solve the Laplacian operator, applying patches of overlapping grids where amore fined level is needed, leaving coarser meshes in the rest of the computational domain.These overlapping grids will have generic quadrilateral shapes.Specifically, the topics covered will be:1) introduction to the finite difference method, finite volume method, domainpartitioning, solution approximation;2) overview of different types of meshes to represent in a discrete way thegeometry involved in a problem, with a focuson the octree data structure, presenting PABLO and PABLitO. The first one is anexternal library used to manage each single grid’s creation, load balancing and internal communications, while the second one is the Python API ofthat library written ad hoc for the current project;3) presentation of the algorithm used to communicate data between meshes (beingall of them unaware of each other’s existence) using MPI inter-communicators and clarification of the monolithic approach applied building the finalmatrix for the system to solve, taking into account diagonal, restriction and prolongation blocks;4) presentation of some results; conclusions, references.It is important to underline that everything is done under Python as programmingframework, using Cython for the writing of PABLitO, MPI4Py for the communications between grids, PETSc4py for the assembling and resolution partsof the system of unknowns, NumPy for contiguous memory buffer objects.The choice of this programming language has been made because Python, easy to learn and understand, is today a significant contender for the numerical computing and HPC ecosystem, thanks to its clean style, its packages, its compilers and, why not, its specific architecture optimized versions.

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I denna avhandling, som omfattar fyra forskningsartiklar, betraktas två operatorer inom den matematiska fysiken. De båda tidigare artiklarna innehåller resultat för Schrödingeroperatorn med Aharonov-Bohm-magnetfält. I artikel I beräknas spektrum och egenfunktioner till denna operator i R2 explicit i ett antal fall då en radialsymmetrisk skalärvärd potential eller ett konstant magnetfält läggs till. I flera av de studerade fallen kan den skarpa konstanten i Lieb-Thirrings olikhet beräknas för γ = 0 och γ ≥ 1. I artikel II bevisas semiklassiska uppskattningar för moment av egenvärdena i begränsade tvådimensionella områden. Vidare presenteras ett exempel då den generaliserade diamagnetiska olikheten, framlagd som en förmodan av Erdős, Loss och Vougalter, är falsk. Numeriska studier kompletterar dessa resultat. De båda senare artiklarna innehåller ett flertal spektrumuppskattningar för Heisenberg-Laplace-operatorn. I artikel III bevisas skarpa olikheter för spektret till Dirichletproblemet i (2n + 1)-dimensionella områden med ändligt mått. Låt λk och μk beteckna egenvärdena till Dirichlet- respektive Neumannproblemet i ett område med ändligt mått. N. D. Filonov har bevisat olikheten μk+1 < λk för den euklidiska Laplaceoperatorn. I artikel IV visas detta resultat för Heisenberg-Laplaceoperatorn i tredimensionella områden som uppfyller vissa geometriska villkor. / In this thesis, which comprises four research papers, two operators in mathe- matical physics are considered. The former two papers contain results for the Schrödinger operator with an Aharonov-Bohm magnetic field. In Paper I we explicitly compute the spectrum and eigenfunctions of this operator in R2 in a number of cases where a radial scalar potential and/or a constant magnetic field are superimposed. In some of the studied cases we calculate the sharp constants in the Lieb-Thirring inequality for γ = 0 and γ ≥ 1. In Paper II we prove semi-classical estimates on moments of the eigenvalues in bounded two-dimensional domains. We moreover present an example where the generalised diamagnetic inequality, conjectured by Erdős, Loss and Vougalter, fails. Numerical studies complement these results. The latter two papers contain several spectral estimates for the Heisenberg Laplacian. In Paper III we obtain sharp inequalities for the spectrum of the Dirichlet problem in (2n + 1)-dimensional domains of finite measure. Let λk and μk denote the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure. N. D. Filonov has proved that the inequality μk+1 < λk holds for the Euclidean Laplacian. In Paper IV we extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfil certain geometric conditions. / QC 20100712

Στην παρούσα διδακτορική διατριβή μελετάμε τρία Προβλήματα που αναφέρονται στην Ψευδο-Ευκλείδεια Γεωμετρία. Στα δύο πρώτα Κεφάλαια, Κεφάλαιο 1 και Κεφάλαιο 2 αναφέρουμε γνωστά αποτελέσματα και περιγράφουμε βασικές έννοιες της Ρημάννιας και Ψευδό - Ρημάννιας Γεωμετρίας. Στο Κεφάλαιο 3 μελετάμε επιφάνειες εκ περιστροφής στον τρισδιάστατο Lorentz - Minkowski χώρο ικανοποιώντας δοσμένη γεωμετρική συνθήκη. Στο Κεφάλαιο 4 βρίσκουμε όλες τις κανονικές μορφές του τελεστή σχήματος των τρισδιάστατων υπερεπιφανειών τύπου (-, +, -) του τετρασδιάστατου Ψευδο - Ευκλείδειου χώρου τύπου (-, +, -, +). Τέλος, στο Κεφάλαιο 5 μελετάμε τη σχέση που υπάρχει μεταξύ των διαρμονικών και ελαχιστικών υπερεπιφανειών που αναφέρθηκαν στο Κεφάλαιο 4, χρησιμοποιώντας τον τελεστή σχήματός τους. Ειδικότερα, αποδεικνύουμε ότι κάθε τέτοια διαρμονική υπερεπιφάνεια είναι ελαχιστική. / In the present PH.D. thesis we study three problems referred in the pseudo-Euclidean geometry. In the first two chapters, Chapter 1 and Chapter 2, we review known results and describe the basic notions of the Riemannian and Pseudo-Riemannian geometry. In Chapter 3, we study surfaces of revolution of the three dimensional Lorentz-Minkowski space satisfying given geometric condition. In Chapter 4, we find all the canonical forms of the shape operator of the three dimensional hypersurfaces of signature (-, +, -) of the four dimensional pseudo-Euclidean space of signature (-, +, -, +). Finally, in Chapter 5, we study the relation which exists between the biharmonic and minimal hypersurfaces referred in Chapter 4, by using their shape operator. Precisely, we prove that every such biharmonic hypersurface is minimal.

This document deals with experimental verifying to use machine learning algorithms AdaBoost or WaldBoost to make classifier, that is able to find point in the second picture that matches original point in the first picture. This work also depicts finding points of interest in image as a first step of finding correspondence. Next there are described some descriptors of points of interest. Corresponding points could be useful for 3D modeling of shooted scene.

This master's thesis contains introduction to image filtration problems, especially to theoretical outlets, whose origin lies in linear systems theory and mathematical function analysis. There are described some approaches and methods which are used to image smoothing and for edge detection in an image. Mainly Sobel operator, Laplace operator and median filter are covered. The main contents of this project is discussion of some approaches of hardware acceleration of image filtering and design of time effective software and hardware implementations of filters in a form of program functions and combinational circuits using theoretical knowledges about time complexity of algorithms. Hardware and software implementation of named filters was done too. For every filter, time of filtration was measured and results were compared and analyzed.