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Dissertations / Theses on the topic 'Geometry of metric spaces'
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Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.
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A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the
space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
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Persson, Nicklas. "Shortest paths and geodesics in metric spaces." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-66732.
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This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.
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Li, Xining. "Preservation of bounded geometry under transformations metric spaces." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722.
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Lopez, Marcos D. "Discrete Approximations of Metric Measure Spaces with Controlled Geometry." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529.
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Alekseevsky, Dmitri, Andreas Kriegl, Mark Losik, Peter W. Michor, and Peter Michor@esi ac at. "The Riemannian Geometry of Orbit Spaces. The Metric, Geodesics, and." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi997.ps.
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StClair, Jessica Lindsey. "Geometry of Spaces of Planar Quadrilaterals." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/26887.
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The purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals.
The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar
quadrilaterals with fixed side lengths) has been well-studied [5], [8], [10]. The symplectic
geometry of these spaces has been studied by Kapovich and Millson [6], but the Riemannian
geometry of these spaces has not been thoroughly examined. We study paths in the moduli
space and the pre-moduli space. We compare intraplanar paths between points in the moduli
space to extraplanar paths between those same points. We give conditions on side lengths
to guarantee that intraplanar motion is shorter between some points. Direct applications of
this result could be applied to motion-planning of a robot arm. We show that horizontal lifts
to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine
exactly which collections of side lengths allow holonomy.<br>Ph. D.
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Le, Brigant Alice. "Probability on the spaces of curves and the associated metric spaces via information geometry; radar applications." Thesis, Bordeaux, 2017. http://www.theses.fr/2017BORD0640/document.
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Nous nous intéressons à la comparaison de formes de courbes lisses prenant leurs valeurs dans une variété riemannienne M. Dans ce but, nous introduisons une métrique riemannienne invariante par reparamétrisations sur la variété de dimension infinie des immersions lisses dans M. L’équation géodésique est donnée et les géodésiques entre deux courbes sont construites par tir géodésique. La structure quotient induite par l’action du groupe des reparamétrisations sur l’espace des courbes est étudiée. À l’aide d’une décomposition canonique d’un chemin dans un fibré principal, nous proposons un algorithme qui construit la géodésique horizontale entre deux courbes et qui fournit un matching optimal. Dans un deuxième temps, nous introduisons une discrétisation de notre modèle qui est elle-même une structure riemannienne sur la variété de dimension finie Mn+1 des "courbes discrètes" définies par n + 1 points, où M est de courbure sectionnelle constante. Nous montrons la convergence du modèle discret vers le modèle continu, et nous étudions la géométrie induite. Des résultats de simulations dans la sphère, le plan et le demi-plan hyperbolique sont donnés. Enfin, nous donnons le contexte mathématique nécessaire à l’application de l’étude de formes dans une variété au traitement statistique du signal radar, où des signaux radars localement stationnaires sont représentés par des courbes dans le polydisque de Poincaré via la géométrie de l’information<br>We are concerned with the comparison of the shapes of open smooth curves that take their values in a Riemannian manifold M. To this end, we introduce a reparameterization invariant Riemannian metric on the infinite-dimensional manifold of these curves, modeled by smooth immersions in M. We derive the geodesic equation and solve the boundary value problem using geodesic shooting. The quotient structure induced by the action of the reparametrization group on the space of curves is studied. Using a canonical decomposition of a path in a principal bundle, we propose an algorithm that computes the horizontal geodesic between two curves and yields an optimal matching. In a second step, restricting to base manifolds of constant sectional curvature, we introduce a detailed discretization of the Riemannian structure on the space of smooth curves, which is itself a Riemannian metric on the finite-dimensional manifold Mn+1 of "discrete curves" given by n + 1 points. We show the convergence of the discrete model to the continuous model, and study the induced geometry. We show results of simulations in the sphere, the plane, and the hyperbolic halfplane. Finally, we give the necessary framework to apply shape analysis of manifold-valued curves to radar signal processing, where locally stationary radar signals are represented by curves in the Poincaré polydisk using information geometry
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Lesser, Alice. "Optimal and Hereditarily Optimal Realizations of Metric Spaces." Doctoral thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8297.
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<p>This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an <i>optimal realization</i> of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique.</p><p>It has been conjectured that <i>extremally weighted</i> optimal realizations may be found as subgraphs of the <i>hereditarily optimal realization</i> Γ<sub>d</sub>, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the <i>tight span</i> of the metric.</p><p>In Paper I, we prove that the graph Γ<sub>d</sub> is equivalent to the 1-skeleton of the tight span precisely when the metric considered is <i>totally split-decomposable</i>. For the subset of totally split-decomposable metrics known as <i>consistent</i> metrics this implies that Γ<sub>d</sub> is isomorphic to the easily constructed <i>Buneman graph</i>.</p><p>In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γ<sub>d</sub>.</p><p>In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γ<sub>d</sub>. However, for these examples there also exists at least one optimal realization which is a subgraph.</p><p>Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γ<sub>d</sub>? Defining <i>extremal</i> optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γ<sub>d</sub> is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span</p>
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Tran, Anh Tuyet. "1p spaces." CSUSB ScholarWorks, 2002. https://scholarworks.lib.csusb.edu/etd-project/2238.
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Hume, David S. "Embeddings of infinite groups into Banach spaces." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:e38f58ec-484c-4088-bb44-1556bc647cde.
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In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the Assouad-Nagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed three-manifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed three-manifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a tree-graded quasi-isometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the Bestvina-Bromberg-Fujiwara quasi-trees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite Assouad-Nagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite Assouad-Nagata dimension if and only if each peripheral subgroup does.
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Antonakoudis, Stergios M. "The complex geometry of Teichmüller space." Thesis, Harvard University, 2014. http://dissertations.umi.com/gsas.harvard:11637.
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We study isometric maps between Teichmüller spaces and bounded symmetric domains in their Kobayashi metric. We prove that every totally geodesic isometry from a disk to Teichmüller space is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. However, we prove that in dimensions two or more there are no holomorphic isometric immersions between Teichmüller spaces and bounded symmetric domains and also prove a similar result for isometric submersions.<br>Mathematics
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Tamanini, Luca. "Analysis and Geometry of RCD spaces via the Schrödinger problem." Thesis, Paris 10, 2017. http://www.theses.fr/2017PA100082/document.
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Le but principal de ce manuscrit est celui de présenter une nouvelle méthode d'interpolation entre des probabilités inspirée du problème de Schrödinger, problème de minimisation entropique ayant des liens très forts avec le transport optimal. À l'aide de solutions au problème de Schrödinger, nous obtenons un schéma d'approximation robuste jusqu'au deuxième ordre et différent de Brenier-McCann qui permet d'établir la formule de dérivation du deuxième ordre le long des géodésiques Wasserstein dans le cadre de espaces RCD* de dimension finie. Cette formule était inconnue même dans le cadre des espaces d'Alexandrov et nous en donnerons quelques applications. La démonstration utilise un ensemble remarquable de nouvelles propriétés pour les solutions au problème de Schrödinger dynamique :- une borne uniforme des densités le long des interpolations entropiques ;- la lipschitzianité uniforme des potentiels de Schrödinger ;- un contrôle L2 uniforme des accélérations. Ces outils sont indispensables pour explorer les informations géométriques encodées par les interpolations entropiques. Les techniques utilisées peuvent aussi être employées pour montrer que la solution visqueuse de l'équation d'Hamilton-Jacobi peut être récupérée à travers une méthode de « vanishing viscosity », comme dans le cas lisse.Dans tout le manuscrit, plusieurs remarques sur l'interprétation physique du problème de Schrödinger seront mises en lumière. Cela pourra aider le lecteur à mieux comprendre les motivations probabilistes et physiques du problème, ainsi qu'à les connecter avec la nature analytique et géométrique de la dissertation<br>Main aim of this manuscript is to present a new interpolation technique for probability measures, which is strongly inspired by the Schrödinger problem, an entropy minimization problem deeply related to optimal transport. By means of the solutions to the Schrödinger problem, we build an efficient approximation scheme, robust up to the second order and different from Brenier-McCann's classical one. Such scheme allows us to prove the second order differentiation formula along geodesics in finite-dimensional RCD* spaces. This formula is new even in the context of Alexandrov spaces and we provide some applications.The proof relies on new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:- equiboundedness of the densities along the entropic interpolations,- equi-Lipschitz continuity of the Schrödinger potentials,- a uniform weighted L2 control of the Hessian of such potentials. These tools are very useful in the investigation of the geometric information encoded in entropic interpolations. The techniques used in this work can be also used to show that the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. Throughout the whole manuscript, several remarks on the physical interpretation of the Schrödinger problem are pointed out. Hopefully, this will allow the reader to better understand the physical and probabilistic motivations of the problem as well as to connect them with the analytical and geometric nature of the dissertation
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Camelo, Botero Miguel Hernando. "A geometric routing scheme in word-metric spaces for data networks." Doctoral thesis, Universitat de Girona, 2014. http://hdl.handle.net/10803/283749.
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This research work explores the use of the Greedy Geometric Routing (GGR) schemes to solve the scalability problem of the routing systems in Internet-like networks and several families of Data Center architectures. We propose a novel and simple embedding of any connected finite graph into a Word-Metric space, i.e., a metric space generated by algebraic groups. Then, built on top of this greedy embedding, we propose three GGR schemes and we prove the theoretical upper bounds of the Routing Table size, vertex label size and stretch. The first scheme works for any kind of graph and the other two are specialized for Internet-like and several families of DC topologies<br>Este trabajo de investigación explora el uso de esquemas de Enrutamiento Geométrico Greedy (Greedy Geometric Routing o GGR) para resolver el problema de escalabilidad de los sistemas de encaminamiento de redes tipo Internet y de varias arquitecturas para Centros de Datos (Data Centers o DCs). Nosotros proponemos un nuevo y simple método de incrustación (embedding) de cualquier grafo finito y conectado en un espacio métrico de palabras (Word-Metric space), es decir, un espacio métrico generado por grupos algebraicos. Luego, construidos sobre esta incrustación, proponemos tres esquemas de GGR y derivamos los límites superiores teóricos de sus tablas de encaminamiento (Routing Table o RT), las etiquetas de los vértices y el stretch. El primer esquema trabaja sobre cualquier tipo de grafo y los otros dos son especializados para topologías tipo Internet y varias familias de arquitecturas de DCs
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Becker, Christian. "On the Riemannian geometry of Seiberg-Witten moduli spaces." Phd thesis, [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975744771.
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Averkov, Gennadiy. "Metrical Properties of Convex Bodies in Minkowski Spaces." Doctoral thesis, Universitätsbibliothek Chemnitz, 2004. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200401537.
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The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found<br>Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet
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Frost, George. "The projective parabolic geometry of Riemannian, Kähler and quaternion-Kähler metrics." Thesis, University of Bath, 2016. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.690742.
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We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.
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Weilandt, Martin. "Isospectral metrics on weighted projective spaces." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2010. http://dx.doi.org/10.18452/16169.
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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.<br>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.
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Arnt, Sylvain. "Large scale geometry and isometric affine actions on Banach spaces." Thesis, Orléans, 2014. http://www.theses.fr/2014ORLE2021/document.
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Dans le premier chapitre, nous définissons la notion d’espaces à partitions pondérées qui généralise la structure d’espaces à murs mesurés et qui fournit un cadre géométrique à l’étude des actions isométriques affines sur des espaces de Banach pour les groupes localement compacts à base dénombrable. Dans un premier temps, nous caractérisons les actions isométriques affines propres sur des espaces de Banach en termes d’actions propres par automorphismes sur des espaces à partitions pondérées. Puis, nous nous intéressons aux structures de partitions pondérées naturelles pour les actions de certaines constructions de groupes : somme directe ; produit semi-directe ; produit en couronne et produit libre. Nous établissons ainsi des résultats de stabilité de la propriété PLp par ces constructions. Notamment, nous généralisons un résultat de Cornulier, Stalder et Valette de la façon suivante : le produit en couronne d’un groupe ayant la propriété PLp par un groupe ayant la propriété de Haagerup possède la propriété PLp. Dans le deuxième chapitre, nous nous intéressons aux espaces métriques quasi-médians - une généralisation des espaces hyperboliques à la Gromov et des espaces médians - et à leurs propriétés. Après l’étude de quelques exemples, nous démontrons qu’un espace δ-médian est δ′-médian pour tout δ′ ≥ δ. Ce résultat nous permet par la suite d’établir la stabilité par produit directe et par produit libre d’espaces métriques - notion que nous développons par la même occasion. Le troisième chapitre est consacré à la définition et l’étude d’une distance propre, invariante à gauche et qui engendre la topologie explicite sur les groupes localement compacts, compactement engendrés. Après avoir montré les propriétés précédentes, nous prouvons que cette distance est quasi-isométrique à la distance des mots sur le groupe et que la croissance du volume des boules est contrôlée exponentiellement<br>In the first chapter, we define the notion of spaces with labelled partitions which generalizes the structure of spaces with measured walls : it provides a geometric setting to study isometric affine actions on Banach spaces of second countable locally compact groups. First, we characterise isometric affine actions on Banach spaces in terms of proper actions by automorphisms on spaces with labelled partitions. Then, we focus on natural structures of labelled partitions for actions of some group constructions : direct sum ; semi-direct product ; wreath product and free product. We establish stability results for property PLp by these constructions. Especially, we generalize a result of Cornulier, Stalder and Valette in the following way : the wreath product of a group having property PLp by a Haagerup group has property PLp. In the second chapter, we focus on the notion of quasi-median metric spaces - a generalization of both Gromov hyperbolic spaces and median spaces - and its properties. After the study of some examples, we show that a δ-median space is δ′-median for all δ′ ≥ δ. This result gives us a way to establish the stability of the quasi-median property by direct product and by free product of metric spaces - notion that we develop at the same time. The third chapter is devoted to the definition and the study of an explicit proper, left-invariant metric which generates the topology on locally compact, compactly generated groups. Having showed these properties, we prove that this metric is quasi-isometric to the word metric and that the volume growth of the balls is exponentially controlled
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Davtyan, Ashot. "Measure generation in the spaces of planes und lines in R^3." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2009. http://nbn-resolving.de/urn:nbn:de:swb:105-7072226.
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Das Ziel der Arbeit besteht darin, einen Beitrag zur Entwicklung der kombinatorischen Integralgeometrie zu leisten. In der Arbeit werden Bewertungen (Valuation) in den Räumen der Geraden und Ebenen im $\R^3$ betrachtet, die von Flagfunktionen abhängen. Unter geeigneten Glattheitsvoraussetzungen an die Flagfunktionen werden notwendige und hinreichende Bedingungen gegeben, die die Fortsetzung der entsprechender Bewertung zu einem signierten Maß sichern. Diese integralgeometrischen Untersuchungen führten zu einer Anzahl von interessanten Ergebnissen, speziell bei der Beschreibung von Metriken im Sinne von Hilberts viertem Problem.
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at, Andreas Cap@esi ac. "Parabolic Geometries, CR-Tractors, and the Fefferman Construction." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1084.ps.
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Campbell, Newton Henry Jr. "Algorithmic Foundations of Heuristic Search using Higher-Order Polygon Inequalities." NSUWorks, 2016. http://nsuworks.nova.edu/gscis_etd/374.
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The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark paradigm. The new heuristic’s novel feature is that it computes and stores a reduced amount of preprocessed information (in comparison to previous landmark-based algorithms) while enabling more informed search decisions. We demonstrate that domination of this heuristic over its predecessor depends on landmark selection and that, in general, the denser the landmark set, the better heuristic performs. Due to the reduced memory requirement, this new heuristic admits much denser landmark sets.
We conduct experiments to characterize the impact that landmark configurations have on this new heuristic, demonstrating that centrality-based landmark selection has the best tradeoff between preprocessing and runtime. Using a developed graph library and static information from benchmark road map datasets, the algorithm is compared experimentally with previous landmark-based shortest path techniques in a fixed-memory environment to demonstrate a reduction in overall computational time and memory requirements. Experimental results are evaluated to detail the significance of landmark selection and density, the tradeoffs of performing preprocessing, and the practical use cases of the algorithm.
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Ståhl, Fredrik. "At the edge of space and time : exploring the b-boundary in general relativity." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-96694.
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This thesis is about the structure of the boundary of the universe, i.e., points where the geometric structures of spacetime cannot be continued. In particular, we study the structure of the b-boundary by B. Schmidt. It has been known for some time that the b-boundary construction has several drawbacks, perhaps the most severe being that it is often not Hausdorff separated from interior points in spacetime. In other words, the topology makes it impossible to distinguish which points in spacetime are near the singularity and which points are ‘far’ from it. The non-Hausdorffness of the b-completion is closely related to the concept of fibre degeneracy of the fibre in the frame bundle over a b-boundary point, the fibre being smaller than the whole structure group in a specific sense. Fibre degeneracy is to be expected for many realistic spacetimes, as was proved by C. J. S. Clarke. His proofs contain some errors however, and the purpose of paper I is to reestablish the results of Clarke, under somewhat different conditions. It is found that under some conditions on the Riemann curvature tensor, the boundary fibre must be totally degenerate (i.e., a single point). The conditions are essentially that the components of the Riemann tensor and its first derivative, expressed in a parallel frame along a curve ending at the singularity, diverge sufficiently fast. We also demonstrate the applicability of the conditions by verifying them for a number of well known spacetimes. In paper II we take a different view of the b-boundary and the b-length functional, and study the Riemannian geometry of the frame bundle. We calculate the curvature Rof the frame bundle, which allows us to draw two conclusions. Firstly, if some component of the curvature of spacetime diverges along a horizontal curve ending at a singularity, R must tend to — oo. Secondly, if the frame bundle is extendible through a totally degenerate boundary fibre, the spacetime must be a conformally flat Einstein space asymptotically at the corresponding b-boundary point. We also obtain some basic results on the isometries and the geodesics of the frame bundle, in relation to the corresponding structures on spacetime. The first part of paper III is concerned with imprisoned curves. In Lorentzian geometry, the situation is qualitatively different from Riemannian geometry in that there may be incomplete endless curves totally or partially imprisoned in a compact subset of spacetime. It was shown by B. Schmidt that a totally imprisoned curve must have a null geodesic cluster curve. We generalise this result to partially imprisoned incomplete endless curves. We also show that the conditions for the fibre degeneracy theorem in paper I does not apply to imprisoned curves. The second part of paper III is concerned with the properties of the b-length functional. The b-length concept is important in general relativity because the presence of endless curves with finite b-length is usually taken as the definition of a singular spacetime. It is also closely related to the b-boundary definition. We study the structure of b-neighbourhoods, i.e., the set of points reachable from a fixed point in spacetime on (horizontal) curves with b-length less than some fixed number e > 0. This can then be used to understand how the geometry of spacetime is encoded in the frame bundle geometry, and as a tool when studying the structure of the b-boundary. We also give a result linking the b-length of a general curve in the frame bundle with the b-length of the corresponding horizontal curve.<br>digitalisering@umu
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Sánchez, Luis Florial Espinoza. "Surfaces in 4-space from the affine differential geometry viewpoint." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-23032015-142340/.
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In this thesis, we study locally strictly convex surfaces from the affine differential viewpoint and generalize some tools for locally strictly submanifolds of codimension 2. We introduce a family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally atrictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal to the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes. More generally, by using the metric of the transversal vector field on M we introduce the affine normal plane and the families of the affine distance and height functions on M. We show that the singularities of the family of the affine height functions appear at directions on the affine normal plane and the singularities of the family of the affine distance functions appear at points on the affine normal plane and the affine focal points correspond as degenerate singularities of the family of affine distance functions. Moreover we show that if M is immersed in a locally strictly convex hypersurface then the affine normal plane contains the affine normal vector to the hypersurface. Finally, we conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.<br>Nesta tese estudamos as superfícies localmente estritamente convexas desde o ponto de vista da geometria diferencial afim e generalizamos algumas ferramentas para subvariedades localmente estritamente convexas de codimensão 2. Introduzimos uma família de métricas afins sobre uma superfície localmente estritamente convexa M no 4-espaço afim. Então, definimos os planos equiafins simétrico e antissimétrico associados com alguma métrica. Mostramos que se M é imersa em uma hiperquádrica localmente estritamente convexa, então os planos simétrico e assimétrico são iguais e contêm o campo vetorial normal afim à hiperquádrica. Em particular, qualquer superfície imersa em uma hiperquádrica localmente estritamente convexa é semiumbílica afim com relação ao plano equiafim simétrico ou antissimétrico. Mais geralmente, usando a métrica do campo transversal sobre M introduzimos o plano normal afim e as famílias de funções distância e altura afim sobre M. Provamos que as singularidades da família de funções altura afim aparecem como direções do plano normal afim e as singularidades da família de funções distância afim aparecem como pontos sobre o plano normal afim e os pontos focais correspondem às singularidades degeneradas da família de funções distância afim. Também provamos que se M é uma superfície imersa em uma hipersuperfície localmente estritamente convexa, então o plano normal afim contém o vetor normal afim à hipersuperfície. Finalmente, concluímos que qualquer superfície imersa em uma hiperesfera localmente estritamente convexa é semiumbílica afim.
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Bonnet, Benoît. "Optimal control in Wasserstein spaces." Electronic Thesis or Diss., Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0442.
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Une vaste quantité d'outils mathématiques permettant la modélisation et l'analyse des problèmes multi-agents ont récemment été développés dans le cadre de la théorie du transport optimal. Dans cette thèse, nous étendons pour la première fois plusieurs de ces concepts à des problématiques issues de la théorie du contrôle. Nous démontrons plusieurs résultats sur ce sujet, notamment des conditions nécessaires d'optimalité de type Pontryagin dans les espaces de Wasserstein, des conditions assurant la régularité intrinsèque de solutions optimales, des conditions suffisantes pour l'émergence de différents motifs, ainsi qu'un résultat auxiliaire à propos des arrangements de certaines singularités en géométrie sous-Riemannienne<br>A wealth of mathematical tools allowing to model and analyse multi-agent systems has been brought forth as a consequence of recent developments in optimal transport theory. In this thesis, we extend for the first time several of these concepts to the framework of control theory. We prove several results on this topic, including Pontryagin optimality necessary conditions in Wasserstein spaces, intrinsic regularity properties of optimal solutions, sufficient conditions for different kinds of pattern formation, and an auxiliary result pertaining to singularity arrangements in Sub-Riemannian geometry
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Carvalho, Luis Carlos de. "Análise da organização didática da geometria espacial métrica nos livros didáticos." Pontifícia Universidade Católica de São Paulo, 2008. https://tede2.pucsp.br/handle/handle/11334.
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Made available in DSpace on 2016-04-27T16:58:45Z (GMT). No. of bitstreams: 1
Luis Carlos de Carvalho.pdf: 2327001 bytes, checksum: 2f70a04711cd873ebfd4e6fc2bab144a (MD5)
Previous issue date: 2008-10-07<br>Secretaria da Educação do Estado de São Paulo<br>This work had as objective to investigate which the organization that the
didactic books of Mathematics destined to 2ª series of Average Education make,
referring to the subject Metric Space Geometry, and if this organization favors the
construction of the geometric thought. For in such a way, the subject was searched in
available literature, official documents, dissertations and theses defended in Brazil
and articles of national and international congresses. They had been analyzed three
of eight books of Mathematics sent for 2ª series of Average Education for the
National Program of the Book for Average Education, 2006 of the Ministry of the
Education and Culture, to the schools of the Direction of Education of São Bernardo
do Campo - SP, according to resulted of the research of: Duval (1995), Robert (1998)
and Parsysz (2000). The analysis of didactic books had as purpose to verify if the
activities for them proposals provide and favor the construction of the knowledge on
the part of the pupils, considering activities with use of material concrete,
constructions with instruments as ruler and compass or softwares that they facilitate
the visualization and they develop the space geometric thought. One concluded that
the analyzed didactic books take care of partially to the construction of the geometric
thought, because the results of the research indicated to little exploration on the part
of the authors of activities that develop the visualization, the representation in the
plan of the three-dimensional figures are not stimulated. A balance with regard to the
number of considered exercises that demand the levels technician and mobilized and
a discrepancy with regard to the available level and to the lack of activities was
evidenced that can be developed by educational software, contributing in this way for
the diffusion of a vision maken a mistake of the teacher on the education of Metric
Space Geometry<br>Este trabalho teve como objetivo investigar qual a organização que os livros didáticos
de Matemática destinados à 2ª série do Ensino Médio fazem referente ao tema Geometria
Espacial Métrica, e se essa organização favorece a construção do pensamento geométrico.
Para tanto, o tema foi pesquisado na literatura disponível, em documentos oficiais, em
dissertações e teses defendidas no Brasil e em artigos de congressos nacionais e
internacionais. Três dos oito livros de Matemática enviados para a 2ª série do Ensino Médio
pelo Programa Nacional do Livro para o Ensino Médio, 2006, do Ministério da Educação e
Cultura, às escolas da Diretoria de Ensino de São Bernardo do Campo SP foram
analisados, segundo os resultados das pesquisas de: Duval (1995); Robert (1998) e Parsysz
(2000). A análise dos livros didáticos teve como finalidade verificar se as atividades por eles
propostas proporcionam e favorecem a construção do conhecimento por parte dos alunos,
propondo atividades com uso de material concreto, construções com instrumentos, como
régua e compasso ou softwares que facilitam a visualização e desenvolvem o pensamento
geométrico espacial. Concluiu-se que os livros didáticos analisados atendem parcialmente à
construção do pensamento geométrico espacial, porque os resultados da pesquisa
indicaram a pouca exploração por parte dos autores de atividades que desenvolvem a
visualização, observou-se que a representação no plano das figuras tridimensionais não é
estimulada. Constatou-se um equilíbrio com relação ao número de exercícios propostos que
exigem os níveis técnicos e mobilizáveis e uma discrepância com relação ao nível disponível
e à falta de atividades que possam ser desenvolvidas por software educacional, contribuindo
desta maneira para a difusão de uma visão equivocada do professor sobre o ensino da
Geometria Espacial Métrica
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García, Pérez Guillermo. "A geometric approach to the structure of complex networks." Doctoral thesis, Universitat de Barcelona, 2018. http://hdl.handle.net/10803/665120.
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Complex networks are mathematical representations of the interaction patterns of complex systems. During the last 20 years of Network Science, it has been recognized that networks from utterly different domains exhibit certain universal properties. In particular, real complex networks present heterogeneous, and usually scale-free, degree distributions, a large amount of triangles, or high clustering coefficient, a very short diameter, and a clear community structure. Among the vast set of models proposed to explain the structure of real networks, geometric models have proven to be particularly promising.
This thesis is developed in the framework of hidden metric spaces, in which the high level of clustering observed in real networks emerges from underlying geometric spaces encoding the similarity between nodes. Besides providing an intuitive explanation to the observed clustering coefficient, geometric models succeed at reproducing the structure of complex networks with high accuracy. Furthermore, they can be used to obtain embeddings of networks, that is, maps of real systems enabling their geometric analysis and efficient navigation.
This work introduces the main concepts in the hidden metric spaces approach and presents a thorough description of the main models and embedding procedures. We generalize these models to generate networks with soft communities, that is, with correlated positions of nodes in the underlying metric space. We also explore one of the models in higher similarity-space dimensions, and show that the maximum clustering coefficient attainable decreases with the dimension, which allows us to conclude that real-world networks must have low-dimensional similarity spaces as a consequence of their high clustering coefficient.
The thesis also includes a detailed geometric analysis of the international trade system. After reconstructing a yearly sequence of world trade networks covering 14 decades, we embed them into hyperbolic space to obtain a series of maps, which we named The World Trade Atlas 1870-2013. In these maps, the likelihood for two countries to be connected by a significant trade channel depends on the distance among them in the underlying space, which encodes the different factors influencing trade interactions. Our analysis of the networks and their maps reveals that the world is being shaped by three different forces acting simultaneously: globalization, localization and hierarchization.
The hidden metric spaces approach can be exploited beyond network metrics. We show that similarity space defines a notion of scale in real-world networks. We present a Geometric Renormalization Group transformation that unveils a previously unknown self-similarity of real networks. Remarkably, the phenomenon is explained by the congruency of real systems with our model. This renormalization transformation provides us with two immediate applications: a method to construct high-fidelity smaller-scale replicas of real networks and a multiscale navigation protocol in hyperbolic space that outperforms single-scale versions.
The geometric origin of real networks is not restricted to their binary structure, but it affects their weighted organization as well. We provide empirical evidence for this claim and propose a geometric model with the capability to reproduce the weighted features of real systems from many different domains. We also present a method to infer the level of coupling of real networks with the underlying metric space, which is generally found to be high in real systems.<br>Les xarxes complexes representen els patrons d’interacció dels sistemes complexos. S’ha observat repetidament que xarxes d’àmbits molt diferents comparteixen certes propietats, com l’heterogeneïtat del nombre de veïns o el clustering elevat (alta presència de triangles), entre d’altres. Tot i que s’han proposat molts models per explicar aquesta universalitat, els models geomètrics han demostrat ser particularment prometedors.
Aquesta tesi es desenvolupa en el context dels espais mètrics ocults, en el qual la natura del clustering s’explica geomètricament en termes de similitud entre nodes. Els models basats en aquesta assumpció no només poden reproduir l’estructura de les xarxes reals amb molta precisió, sinó que permeten obtenir mapes de xarxes reals.
En aquest treball, introduïm els conceptes bàsics dels espais mètrics ocults, els seus models principals i els mètodes d’obtenció de mapes. També generalitzem aquests models al règim amb correlacions geomètriques entre nodes, i explorem la qüestió de la dimensió de l’espai de similitud. La nostra anàlisi ens permet concloure que l’espai de similitud de les xarxes reals ha de tenir dimensionalitat baixa.
Incloem una anàlisi geomètrica detallada de l’evolució del sistema de comerç internacional basada en els mapes a l’espai hiperbòlic de les xarxes corresponents, al llarg de 14 dècades. En aquests mapes, la proximitat entre pa¨ısos representa la probabilitat d’interaccionar comercialment. L’anàlisi mostra que el món evoluciona d’acord amb tres forces que actuen simultàniament: la globalització, la localització i la jerarquització.
Els espais de similitud defineixen una noció d’escala en xarxes reals. Proposem una transformació de renormalització que revela una auto-similitud de sistemes reals anteriorment desconeguda. A més, proposem dues aplicacions d’aquesta transformació: un mètode per a obtenir versions reduïdes de xarxes reals i un mètode multiescalar per a navegar-les.
Finalment, mostrem que les estructures pesades dels sistemes reals també tenen un origen geomètric i proposem un model capaç de reproduir-les amb precisió. Desenvolupem un mètode per a inferir el nivell d’acoblament de les xarxes reals amb els espais mètrics subjacents i trobem que aquest és generalment elevat.
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Razafindrakoto, Ando Desire. "Hyperconvex metric spaces." Thesis, Stellenbosch : University of Stellenbosch, 2010. http://hdl.handle.net/10019.1/4106.
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Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.<br>ENGLISH ABSTRACT: One of the early results that we encounter in Analysis is that every metric space admits
a completion, that is a complete metric space in which it can be densely embedded. We
present in this work a new construction which appears to be more general and yet has
nice properties. These spaces subsequently called hyperconvex spaces allow one to extend
nonexpansive mappings, that is mappings that do not increase distances, disregarding the
properties of the spaces in which they are defined. In particular, theorems of Hahn-Banach
type can be deduced for normed spaces and some subsidiary results such as fixed point
theorems can be observed. Our main purpose is to look at the structures of this new type
of “completion”. We will see in particular that the class of hyperconvex spaces is as large
as that of complete metric spaces.<br>AFRIKAANSE OPSOMMING: Een van die eerste resultate wat in die Analise teegekom word is dat enige metriese ruimte
’n vervollediging het, oftewel dat daar ’n volledige metriese ruimte bestaan waarin die
betrokke metriese ruimte dig bevat word. In hierdie werkstuk beskryf ons sogenaamde
hiperkonvekse ruimtes. Dit gee ’n konstruksie wat blyk om meer algemeen te wees,
maar steeds gunstige eienskappe het. Hiermee kan nie-uitbreidende, oftewel afbeeldings
wat nie afstande rek nie, uitgebrei word sodanig dat die eienskappe van die ruimte
waarop dit gedefinieer is nie ’n rol speel nie. In die besonder kan stellings van die Hahn-
Banach-tipe afgelei word vir genormeerde ruimtes en sekere addisionele ressultate ondere
vastepuntstellings kan bewys word. Ons hoofdoel is om hiperkonvekse ruimtes te ondersoek.
In die besonder toon ons aan dat die klas van alle hiperkonvekse ruimtes net so groot
soos die klas van alle metriese ruimtes is.
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Al-Harbi, Sami. "Clustering in metric spaces." Thesis, University of East Anglia, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396604.
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Herry, Ronan. "Contributions to functional inequalities and limit theorems on the configuration space." Thesis, Paris Est, 2018. http://www.theses.fr/2018PESC1134/document.
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Nous présentons des inégalités fonctionnelles pour les processus ponctuels. Nous prouvons une inégalité de Sobolev logarithmique modifiée, une inégalité de Stein et un théorème du moment quatrième sans terme de reste pour une classe de processus ponctuels qui contient les processus binomiaux et les processus de Poisson. Les preuves reposent sur des techniques inspirées de l'approche de Malliavin-Stein et du calcul avec l'opérateur $Gamma$ de Bakry-Émery. Pour mettre en œuvre ces techniques nous développons une analyse stochastique pour les processus ponctuels. Plus généralement, nous mettons au point une théorie d'analyse stochastique sans hypothèse de diffusion. Dans le cadre des processus de Poisson ponctuels, l'inégalité de Stein est généralisée pour étudier la convergence stable vers des limites conditionnellement gaussiennes. Nous appliquons ces résultats pour approcher des processus Gaussiens par des processus de Poisson composés et pour étudier des graphes aléatoires. Nous discutons d'inégalités de transport et de leur conséquence en termes de concentration de la mesure pour les processus binomiaux dont la taille de l'échantillon est aléatoire. Sur un espace métrique mesuré quelconque, nous présentons un développement de la concentration de la mesure qui prend en compte l'agrandissement parallèle d'ensembles disjoints. Cette concentration améliorée donne un contrôle de toutes les valeurs propres du Laplacien métrique. Nous discutons des liens de cette nouvelle notion avec une version de la courbure de Ricci qui fait intervenir le transport à plusieurs marginales<br>We present functional inequalities and limit theorems for point processes. We prove a modified logarithmic Sobolev inequalities, a Stein inequality and a exact fourth moment theorem for a large class of point processes including mixed binomial processes and Poisson point processes. The proofs of these inequalities are inspired by the Malliavin-Stein approach and the $Gamma$-calculus of Bakry-Emery. The implementation of these techniques requires a development of a stochastic analysis for point processes. As point processes are essentially discrete, we design a theory to study non-diffusive random objects. For Poisson point processes, we extend the Stein inequality to study stable convergence with respect to limits that are conditionally Gaussian. Applications to Poisson approximations of Gaussian processes and random geometry are given. We discuss transport inequalities for mixed binomial processes and their consequences in terms of concentration of measure. On a generic metric measured space, we present a refinement of the notion of concentration of measure that takes into account the parallel enlargement of distinct sets. We link this notion of improved concentration with the eigenvalues of the metric Laplacian and with a version of the Ricci curvature based on multi-marginal optimal transport
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Färm, David. "Upper gradients and Sobolev spaces on metric spaces." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5816.
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<p>The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.</p><p>All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.</p><p>Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.</p><p>This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.</p>
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van, Staden Wernd Jakobus. "Metric aspects of noncommutative geometry." Diss., University of Pretoria, 2019. http://hdl.handle.net/2263/77893.
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We study noncommutative geometry from a metric point of view by
constructing examples of spectral triples and explicitly calculating Connes's
spectral distance between certain associated pure states. After considering
instructive nite-dimensional spectral triples, the noncommutative geometry
of the in nite-dimensional Moyal plane is studied. The corresponding
spectral triple is based on the Moyal deformation of the algebra of Schwartz
functions on the Euclidean plane.<br>Dissertation (MSc)--University of Pretoria, 2019.<br>Physics<br>MSc<br>Unrestricted
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Coffey, Michael R. "Ricci flow and metric geometry." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67924/.
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This thesis considers two separate problems in the field of Ricci flow on surfaces. Firstly, we examine the situation of the Ricci flow on Alexandrov surfaces, which are a class of metric spaces equipped with a notion of curvature. We extend the existence and uniqueness results of Thomas Richard in the closed case to the setting of non-compact Alexandrov surfaces that are uniformly non-collapsed. We complement these results with an extensive survey that collects together, for the first time, the essential topics in the metric geometry of Alexandrov spaces due to a variety of authors. Secondly, we consider a problem in the well-posedness theory of the Ricci flow on surfaces. We show that given an appropriate initial Riemannian surface, we may construct a smooth, complete, immortal Ricci flow that takes on the initial surface in a geometric sense, in contrast to the traditional analytic notions of initial condition. In this way, we challenge the contemporary understanding of well-posedness for geometric equations.
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Lemaire-Beaucage, Jonathan. "Voronoi Diagrams in Metric Spaces." Thesis, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/20736.
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In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.
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Lee, Seunghwan Han. "Probabilistic reasoning on metric spaces." [Bloomington, Ind.] : Indiana University, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3380096.
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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics and Cognitive Science, 2009.<br>Title from PDF t.p. (viewed on Jul 19, 2010). Source: Dissertation Abstracts International, Volume: 70-12, Section: B, page: 7604. Adviser: Lawrence S. Moss.
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Otafudu, Olivier Olela. "Convexity in quasi-metric spaces." Doctoral thesis, University of Cape Town, 2012. http://hdl.handle.net/11427/10950.
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Includes abstract.<br>Includes bibliographical references.<br>The principal aim of this thesis is to investigate the existence of an injective hull in the categories of T-quasi-metric spaces and of T-ultra-quasi-metric spaces with nonexpansive maps.
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Jägrell, Linus. "Geometry of the Lunin-Maldacena metric." Thesis, KTH, Teoretisk fysik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-153502.
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Calisti, Matteo. "Differential calculus in metric measure spaces." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21781/.
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L'obbiettivo di questa tesi è la definizione del calcolo differenziale e dell'operatore di Laplace in spazi metrici di misura. Nel primo capitolo vengono introdotte le definizioni e proprietà principali degli spazi metrici di misura mentre nel secondo quelle riguardanti le funzioni lipschitziane e la derivata metrica di curve assolutamente continue. Nel terzo capitolo quindi viene definito il concetto di p-supergradiente debole e di conseguenza la classe di Sobolev S^p. Nel quarto capitolo viene poi studiata la generalizzazione del concetto di differenziale di f applicato al gradiente di g che da luogo a due funzioni che in generale risultano diverse, ma se coincidono lo spazio verrà detto q-infinitesimamente strettamente convesso.
Vengono quindi dimostrate alcune regole della catena per per queste due funzioni attraverso la dualità fra lo spazio S^p e un opportuno spazio di misure dette q-piani test. In particolare mediante l'introduzione del funzionale energia di Cheeger e il suo flusso-gradiente sarà possibile associare un piano di trasporto al gradiente di una funzione in S^p. Nel quinto capitolo viene definito il p-laplaciano e le regole di calcolo provate precedentemente saranno usate per provare quelle per il laplaciano. Verranno poi definiti gli spazi infitesimamente di Hilbert: in questo caso il laplaciano assume un solo valore e risulta linearmente dipendente da g e si dimostra un'identificazione tra differenziali e gradienti. Nell'ultima parte del quinto capitolo infine viene mostrata un'applicazione del calcolo differenziale in spazi metrici di misura al gruppo di Heisenberg, considerandolo uno spazio metrico di misura munito della metrica di Korany e la misura di Lebesgue. Nella prima parte si mostra che il laplaciano metrico coincide con quello subriemanniano. Viene poi considerata nella seconda parte la sottovarietà {x=0} e si dimostra come il laplaciano metrico sia diverso da quello differenziale.
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Amato, Giuseppe. "Approximate similarity search in metric spaces." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=964997347.
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Kilbane, James. "Finite metric subsets of Banach spaces." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/288272.
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The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of $\ell_2$ isometrically. We say that a Banach space has the Finite Isometric Krivine Property (FIKP) if whenever $\ell_p$ is finitely representable in the space then it contains every subset of $\ell_p$ isometrically. We show that every infinite-dimensional Banach space \emph{nearly} has FIDP and every Banach space nearly has FIKP. We then use convexity arguments to demonstrate that not every Banach space has FIKP, and thus we can exhibit classes of Banach spaces that fail to have Property AI-I. The methods used break down when one attempts to prove that there is a Banach space without FIDP and we conjecture that every infinite-dimensional Banach space has Property FIDP.
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Milicevic, Luka. "Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/273375.
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Farnana, Zohra. "The Double Obstacle Problem on Metric Spaces." Licentiate thesis, Linköping : Linköpings universitet, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10621.
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Stares, Ian S. "Extension of functions and generalised metric spaces." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386678.
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Bellachehab, Anass. "Pairwise gossip in CAT(k) metric spaces." Thesis, Evry, Institut national des télécommunications, 2017. http://www.theses.fr/2017TELE0017/document.
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Cette thèse adresse le problème du consensus dans les réseaux. On étudie des réseaux composés d'agents identiques capables de communiquer entre eux, qui ont une mémoire et des capacités de calcul. Le réseau ne possède pas de nœud central de fusion. Chaque agent stocke une valeur qui n'est pas initialement connue par les autres agents. L'objectif est d'atteindre le consensus, i.e. tous les agents ont la même valeur, d'une manière distribuée. De plus, seul les agents voisins peuvent communiquer entre eux. Ce problème a une longue et riche histoire. Si toutes les valeurs appartiennent à un espace vectoriel, il existe plusieurs protocoles pour résoudre le problème. Une des solutions connues est l'algorithme du gossip qui atteint le consensus de manière asymptotique. C'est un protocole itératif qui consiste à choisir deux nœuds adjacents à chaque itération et de les moyenner. La spécificité de cette thèse est dans le fait que les données stockées par les agents n'appartiennent pas nécessairement à un espace vectoriel, mais à un espace métrique. Par exemple, chaque agent stocke une direction (l'espace métrique est l'espace projectif) ou une position dans un graphe métrique (l'espace métrique est le graphe sous-jacent). Là, les protocoles de gossip mentionnés plus haut n'ont plus de sens car l'addition qui n'est plus disponibles dans les espaces métriques. Cependant, dans les espaces métriques les points milieu ont du sens dans certains cas. Et là ils peuvent se substituer aux moyennes arithmétiques. Dans ce travail, on a compris que la convergence du gossip avec les points milieu dépend de la courbure. On s'est focalisés sur le cas où l'espace des données appartient à une classe d'espaces métriques appelés les espaces CAT(k). Et on a pu démontrer que si les données initiales sont suffisamment "proches" dans un sens bien précis, alors le gossip avec les points milieu - qu'on a appelé le Random Parwise Midpoints- converge asymptotiquement vers un consensus<br>This thesis deals with the problem of consensus on networks. Networks under study consists of identical agents that can communicate with each other, have memory and computational capacity. The network has no central node. Each agent stores a value that, initially, is not known by other agents. The goal is to achieve consensus, i.e. all agents having the same value, in a fully distributed way. Hence, only neighboring agents can have direct communication. This problem has a long and fruitful history. If all values belong to some vector space, several protocols are known to solve this problem. A well-known solution is the pairwise gossip protocol that achieves consensus asymptotically. It is an iterative protocol that consists in choosing two adjacent nodes at each iteration and average them. The specificity of this Ph.D. thesis lies in the fact that the data stored by the agents does not necessarily belong to a vector space, but some metric space. For instance, each agent stores a direction (the metric space is the projective space) or position on a sphere (the metric space is a sphere) or even a position on a metric graph (the metric space is the underlying graph). Then the mentioned pairwise gossip protocols makes no sense since averaging implies additions and multiplications that are not available in metric spaces: what is the average of two directions, for instance? However, in metric spaces midpoints sometimes make sense and when they do, they can advantageously replace averages. In this work, we realized that, if one wants midpoints to converge, curvature matters. We focused on the case where the data space belongs to some special class of metric spaces called CAT(k) spaces. And we were able to show that, provided initial data is "close enough" is some precise meaning, midpoints-based gossip algorithm – that we refer to as Random Pairwise Midpoints - does converge to consensus asymptotically. Our generalization allows to treat new cases of data spaces such as positive definite matrices, the rotations group and metamorphic systems
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Eriksson-Bique, Sylvester David. "Quantitative Embeddability and Connectivity in Metric Spaces." Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10261097.
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<p> This thesis studies three analytic and quantitative questions on doubling metric (measure) spaces. These results are largely independent and will be presented in separate chapters.</p><p> The first question concerns representing metric spaces arising from complete Riemannian manifolds in Euclidean space. More precisely, we find bi-Lipschitz embeddings ƒ for subsets <i>A</i> of complete Riemannian manifolds <i> M</i> of dimension <i>n,</i> where <i>N</i> could depend on a bound on the curvature and diameter of <i>A.</i> The main difficulty here is to control the distortion of such embeddings in terms of the curvature of the manifold. In constructing the embeddings, we will study the collapsing theory of manifolds in detail and at multiple scales. Similar techniques give embeddings for subsets of complete Riemannian orbifolds and quotient metric spaces.</p><p> The second part of the thesis answers a question about finding quantitative and weak conditions that ensure large families of rectifiable curves connecting pairs of points. These families of rectifiable curves are quantified in terms of Poincaré inequalities. We identify a new quantitative connectivity condition in terms of curve fragments, which is equivalent to possessing a Poincaré inequality with some exponent. The connectivity condition arises naturally in three different contexts, and we present methods to find Poincaré inequalities for the spaces involved. In particular, we prove such inequalities for spaces with weak curvature bounds and thus resolve a question of Tapio Rajala.</p><p> In the final part of the thesis we study the local geometry of spaces admitting differentiation of Lipschitz functions with certain Banach space targets. The main result shows that such spaces can be characterized in terms of Poincaré inequalities and doubling conditions. In fact, such spaces can be covered by countably many pieces, each of which is an isometric subset of a doubling metric measure space admitting a Poincaré inequality. In proving this, we will find a new way to use hyperbolic fillings to enlarge certain sub-sets into spaces admitting Poincaré inequalities.</p><p>
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Farnana, Zohra. "The Double Obstacle Problem on Metric Spaces." Doctoral thesis, Linköpings universitet, Tillämpad matematik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51588.
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In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We minimize the p-energy integral among all functions which have prescribed boundary values and lie between two given obstacles. This is a generalization of the Dirichlet problem for p-harmonic functions, in which case the obstacles are —∞ and ∞. We show the existence and uniqueness of solutions, and their continuity when the obstacles are continuous. Moreover we show that the continuous solution is p-harmonic in the open set where it does not touch the continuous obstacles. If the obstacles are not continuous, but satisfy a Wiener type regularity condition, we prove that the solution is still continuous. The Hölder continuity for solutions is shown, when the obstacles are Hölder continuous. Boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles and the boundary values vary and show the convergence of the solutions. We also consider generalized solutions for insoluble obstacle problems, using the convergence results. Moreover we show that for soluble obstacle problems the generalized solution coincides, locally, with the standard solution. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.
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Elkins, Benjamin Joseph. "An investigation of ultrametric spaces." Thesis, Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/28863.
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Hermini, Helba Alexandra [UNESP]. "Algumas observações sobre continuidade de funções." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/150088.
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Previous issue date: 2017-03-09<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Este trabalho consiste em estudar a continuidade de funções do ponto de vista topológico. Além disso, exploramos as diferentes métricas em R² e através de transformações geométricas neste espaço analisamos qual tipo de ação exerce em bolas abertas usando métricas diferentes no domínio e contradomínio.<br>In this work we study the continuity of maps from the topological point of view. In addition, we explore di erent metrics in R² and by using geometric transformations we analyze what kind of action carries in open balls using di erent metrics in the domain and in the codomain.
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Paulik, Gustav. "Gluing spaces and analysis." Bonn : Mathematisches Institut der Universität, 2005. http://catalog.hathitrust.org/api/volumes/oclc/62770010.html.
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CAMFIELD, CHRISTOPHER SCOTT. "Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces." University of Cincinnati / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1211551579.
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Melin, Erik. "Digital Geometry and Khalimsky Spaces." Doctoral thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8419.
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<p>Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young.</p><p>Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s.</p><p>Suppose that a continuous function is defined on a subspace of an <i>n-</i>dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved.</p><p>We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.</p><p>The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane.</p><p>In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. </p><p>The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to <i>n-</i>dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.</p>