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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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Capolli, Marco. "Selected Topics in Analysis in Metric Measure Spaces." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/288526.
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The thesis is composed by three sections, each devoted to the study of a specific problem in the setting of PI spaces. The problem analyzed are: a C^m Lusin approximation result for horizontal curves in the Heisenberg group, a limit result in the spirit of Burgain-Brezis-Mironescu for Orlicz-Sobolev spaces in Carnot groups and the differentiability of Lipschitz functions in Laakso spaces.
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Tirado, Peláez Pedro. "Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces." Doctoral thesis, Universitat Politècnica de València, 2008. http://hdl.handle.net/10251/2961.
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En los últimos años se ha desarrollado una teoría matemática con propiedades robustas con el fin de fundamentar la Ciencia de la Computación. En este sentido, un avance significativo lo constituye el establecimiento de modelos matemáticos que miden la "distancia" entre programas y entre algoritmos, analizados según su complejidad computacional.
En 1995, M. Schellekens inició el desarrollo de un modelo matemático para el análisis de la complejidad algorítmica basado en la construcción de una casi-métrica definida en el espacio de las funciones de complejidad, proporcionando una interpretación computacional adecuada del hecho de que un programa o algoritmo sea más eficiente que otro en todos su "inputs". Esta información puede extraerse en virtud del carácter asimétrico del modelo. Sin embargo, esta estructura no es aplicable al análisis de algoritmos cuya complejidad depende de dos parámetros. Por tanto, en esta tesis introduciremos un nuevo espacio casi-métrico de complejidad que proporcionará un modelo útil para el análisis de este tipo de algoritmos. Por otra parte, el espacio casi-métrico de complejidad no da una interpretación computacional del hecho de que un programa o algoritmo sea "sólo" asintóticamente más eficiente que otro. Los espacios casi-métricos difusos aportan un parámetro "t", cuya adecuada utilización puede originar una información extra sobre el proceso computacional a estudiar; por ello introduciremos la noción de casi-métrica difusa de complejidad, que proporciona un modelo satisfactorio para interpretar la eficiencia asintótica de las funciones de complejidad.
En este contexto extenderemos los principales teoremas de punto fijo en espacios métricos difusos , utilizando una determinada noción de completitud, y obtendremos otros nuevos. Algunos de estos teoremas también se establecerán en el contexto general de los espacios casi-métricos difusos intuicionistas, de lo que resultarán condiciones de contracción menos fuertes.
Los resultados obtTirado Peláez, P. (2008). Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/2961
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Chowdhury, Samir. "Metric and Topological Approaches to Network Data Analysis." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114.
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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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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’informationWe 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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Peske, Wendy Ann. "A topological approach to nonlinear analysis." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2779.
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A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.
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Malý, Lukáš. "Newtonian Spaces Based on Quasi-Banach Function Lattices." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-79166.
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The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted. The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established. The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.
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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 dissertationMain 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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Don, Sebastiano. "Functions of bounded variation in Carnot-Carathéodory spaces." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3426813.
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We study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H^n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u_j) of functions with
bounded variation in metric spaces (X, d_j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2.Analizziamo alcune proprietà di funzioni a variazione limitata in spazi di Carnot-Carathéodory. Nel Capitolo 2 dimostriamo che esse sono approssimativamente differenziabili quasi ovunque, esaminiamo il loro insieme di discontinuità approssimata e la decomposizione della loro derivata distribuzionale. Assumendo un'ipotesi addizionale sullo spazio, che chiamiamo proprietà R, mostriamo che quasi tutti i punti di discontinuità approssimata sono di salto e studiamo una formula per la parte di salto della derivata. Nel Capitolo 3 dimostriamo un teorema di rango uno à la G. Alberti per la derivata distribuzionale di funzioni vettoriali a variazione limitata in una classe di gruppi di Carnot che contiene tutti i gruppi di Heisenberg H^n con n ≥ 2. Uno strumento chiave nella dimostrazione è costituito da alcune proprietà che legano le derivate orizzontali di una funzione a variazione limitata con il suo sottografico. Nel Capitolo 4 dimostriamo un risultato di compattezza per succesioni (u_j) equi-limitate in spazi metrici (X, d_j) quando lo spazio X è fissato ma la metrica può variare con j. Mostriamo inoltre un'applicazione agli spazi di Carnot-Carathéodory. I risultati del Capitolo 4 sono fondamentali per la dimostrazione di alcuni fatti contenuti nel Capitolo 2.
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Joanpere, Salvadó Meritxell. "Fractals and Computer Graphics." Thesis, Linköpings universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-68876.
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Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature. The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.
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Medwid, Mark Edward. "Rigidity of Quasiconformal Maps on Carnot Groups." Bowling Green State University / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104.
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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-RiemannienneA 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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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.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.
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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Xia, Xiaolin. "A Comparison Study on a Set of Space Syntax based Methods : Applying metric, topological and angular analysis to natural streets, axial lines and axial segments." Thesis, Högskolan i Gävle, Avdelningen för Industriell utveckling, IT och Samhällsbyggnad, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:hig:diva-15524.
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Recently, there has been an increasing interest in looking at urban environment as a complex system. More and more researchers are paying attention to the study of the configuration of urban space as well as human social activities within it. It has been found that correlation exists between the morphological properties of urban street network and observed human social movement patterns. This correlation implies that the influence of urban configurations on human social movements is no longer only revealed from the sense of metric distance, but also revealed from topological and geometrical perspectives. Metric distances, topological relationships and angular changes between streets should be considered when applying space syntax analysis to an urban street network. This thesis is mainly focused on the comparison among metric, topological and angular analyses based on three kinds of urban street representation models: natural streets, axial lines and axial segments. Four study areas (London, Paris, Manhattan and San Francisco) were picked up for empirical study. In the study, space syntax measures were calculated for different combinations of analytical methods and street models. These theoretical space syntax accessibility measures (connectivity, integration and choice) were correlated to the corresponding practical human movement to evaluate the correlations. Then the correlation results were compared in terms of analytical methods and street representation models respectively. In the end, the comparison of results show that (1) natural-street based model is the optimal street model for carrying out space syntax analysis followed by axial lines and axial segments; (2) angular analysis and topological analysis are more advanced than metric analysis; and (3) connectivity, integration and local integration (two-step) are more suitable for predicting human movements in space syntax. Furthermore, it can be hypothesized that topological analysis method with natural-street based model is the best combination for the prediction of human movements in space syntax, for the integration of topological and geometrical thinking.
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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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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.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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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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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.
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.
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.
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.
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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.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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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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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.Includes bibliographical references.
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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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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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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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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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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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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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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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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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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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.
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 A of complete Riemannian manifolds M of dimension n, where N could depend on a bound on the curvature and diameter of A. 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.
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.
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.
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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 consensusThis 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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Huou, Benoit. "Inégalités isopérimétriques produit pour les élargissements euclidien et uniforme : symétrisation et inégalités fonctionnelles." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30239/document.
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Le problème isopérimétrique consiste, dans un espace métrique mesuré, à trouver les ensembles qui, à volume fixé, ont la plus petite mesure de surface. Il peut être formulé dans de nombreux cadres (espaces métriques mesurés généraux, variétés riemanniennes à poids, parties de l'espace euclidien...). Deux questions se dégagent de ce problème : - Quels sont les ensembles solutions, c'est-à-dire ayant la plus petite mesure de surface ? (Il faut noter que ces ensembles n'existent pas toujours). - Que vaut la plus petite mesure de surface ? La solution à la deuxième question peut être formulée sous la forme d'une fonction, appelée profil isopérimétrique, qui, à une valeur de volume (pondéré) donnée, associe la plus petite mesure de surface correspondante. La notion de mesure de surface, quant à elle, peut être définie de plusieurs manières (contenu de Minkowski, périmètre géométrique...), toutes dépendant étroitement à la fois de la distance et de la mesure ambiantes. L'objet principal de cette thèse est l'étude du problème isopérimétrique dans des espaces produits, que ce soit pour transférer des inégalités isopérimétriques d'espaces facteurs vers ces produits, ou pour comparer le profil isopérimétrique de l'espace produit à ceux des facteurs. La thèse se découpe en quatre parties : - Étude de l'opération de symétrisation (pour les ensembles) et de réarrangement (pour les fonctions), notions analogues, du point de vue de la théorie de la mesure géométrique et des fonctions à variations bornée. Ces opérations agissent de sorte à ce que n'augmente pas la mesure de surface (pour les ensembles), ou la variation (pour les fonctions). Nous introduisons notamment une nouvelle classe d'espaces modèles, pour lesquels nous obtenons des résultats qualitativement similaires à ceux obtenus pour les espaces modèles classiques : inégalités isopérimétriques transférées aux produits, comparaison d'énergies (pour des fonctionnelles convexes). - Détail d'un argument de minoration du profil isopérimétrique d'un espace métrique produit XxY par une fonction dépendant des profils de X et Y, pour une large classe de distances produits sur XxY. L'étude de ce problème est faite via la minimisation d'une fonctionnelle sur la classe des mesures de Radon. - Étude du problème isopérimétrique dans un espace métrique mesuré produit (le produit d'ordre quelconque du même espace métrique mesuré), muni de la combinaison uniforme de sa distance (élargissement uniforme). Nous donnons un critère pour que tous les profils isopérimétriques (quel que soit l'ordre d'itération du produit) soient minorés par un multiple du minorant du profil isopérimétrique de l'espace originel. Ceci est fait en utilisant notamment des méthodes ayant trait aux inégalités fonctionnelles. Nous appliquons ensuite les résultats aux influences géométriques. - Étude d'inégalités fonctionnelles dites isopérimétriques, permettant d'appréhender le comportement isopérimétrique dans l'espace produit correspondant d'ordre quelconque. Nous résumons l'état des connaissances à propos des inégalités de ce type et proposons une autre méthode qui pourrait aboutir à prouver une telle inégalité dans le cas de mesures réelles particulières, pour lesquelles le problème est ouvertThe isoperimetric problem in a metric measured space consists in finding the sets having minimal boundary measure, with prescribed volume. It can be formulated in various settings (general metric measured spaces, Riemannian manifolds, submanifolds of the Euclidean space, ...). At this point, two questions arise : - What are the optimal sets, namely the sets having smallest boundary measure (it has to be said that they do not always exist) ? - What is the smallest boundary measure ? The solution to the second answer can be expressed by a function called the isoperimetric profile. This function maps a value of (prescribed) measure onto the corresponding smallest boundary measure. As for the precise notion of boundary measure, it can be defined in different ways (Minkowski content, geometric perimeter, ...), all of them closely linked to the ambient distance and measure. The main object of this thesis is the study of the isoperimetric problem in product spaces, in order to transfer isoperimetric inequalities from factor spaces to the product spaces, or to compare their isoperimetric profiles. The thesis is divided into four parts : - Study of the symmetrization operation (for sets) and the rearrangement operation (for functions), analogous notions, from the point of view of Geometric Measure Theory and Bounded Variation functions. These operations cause the boundary measure to decrease (for sets), or the variation (for functions). We introduce a new class of model spaces, for which we obtain similar results to those concerning classic model spaces : transfer of isoperimetric inequalities to the product spaces, energy comparison (for convex functionals). - Detailed proof of an argument of minorization of the isoperimetric profile of a metric measured product space XxY by a function depending on the profiles of X and Y, for a wide class of product distances over XxY. The study of this problem uses the minimization of a functional defined on Radon measures class. - Study of the isoperimetric problem in a metric measured space (n times the same space) equipped with the uniform combination of its distance (uniform enlargement). We give a condition under which every isoperimetric profile (whatever the order of iteration might be) is bounded from below by a quantity which is proportional to the isoperimetric profile of the underlying space. We then apply the result to geometric influences. - Study of isoperimetric functional inequalities, which give information about the isoperimetric behavior of the product spaces. We give an overview of the results about this kind of inequalities, and suggest a method to prove such an inequality in a particular case of real measures for which the problem reamins open
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Lavenant, Hugo. "Courbes et applications optimales à valeurs dans l'espace de Wasserstein." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS112/document.
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L'espace de Wasserstein est l'ensemble des mesures de probabilité définies sur un domaine fixé et muni de la distance de Wasserstein quadratique. Dans ce travail, nous étudions des problèmes variationnels dans lesquels les inconnues sont des applications à valeurs dans l'espace de Wasserstein.Quand l'espace de départ est un segment, c'est-à-dire quand les inconnues sont des courbes à valeurs dans l'espace de Wasserstein, nous nous intéressons à des modèles où, en plus de l'action des courbes, des termes pénalisant les configurations de congestion sont présents. Nous développons des techniques permettant d'extraire de la régularité à partir de l'interaction entre l'évolution optimale de la densité (minimisation de l'action) et la pénalisation de la congestion, et nous les appliquons à l'étude des jeux à champ moyen et de la formulation variationelle des équations d'Euler.Quand l'espace de départ n'est plus seulement un segment mais un domaine de l'espace euclidien, nous considérons seulement le problème de Dirichlet, c'est-à-dire la minimisation de l'action (qui peut être appelée l'énergie de Dirichlet) parmi toutes les applications dont les valeurs sur le bord du domaine de départ sont fixées. Les solutions sont appelées les applications harmoniques à valeurs dans l'espace de Wasserstein. Nous montrons que les différentes définitions de l'énergie de Dirichlet présentes dans la littérature sont en fait équivalentes; que le problème de Dirichlet est bien posé sous des hypothèses assez faibles; que le principe de superposition est mis en échec lorsque l'espace de départ n'est pas un segment; que l'on peut formuler une sorte de principe du maximum; et nous proposons une méthode numérique pour calculer ces applications harmoniquesThe Wasserstein space is the space of probability measures over a given domain endowed with the quadratic Wasserstein distance. In this work, we study variational problems where the unknowns are mappings valued in the Wasserstein space. When the source space is a segment, i.e. when the unknowns are curves valued in the Wasserstein space, we are interested in models where, in addition to the action of the curves, there are some terms which penalize congested configurations. We develop techniques to extract regularity from the minimizers thanks to the interplay between optimal density evolution (minimization of the action) and penalization of congestion, and we apply them to the study of Mean Field Games and the variational formulation of the Euler equations. When the source space is no longer a segment but a domain of a Euclidean space, we consider only the Dirichlet problem, i.e. the minimization of the action (which can be called the Dirichlet energy) among mappings sharing a fixed value on the boundary of the source space. The solutions are called harmonic mappings valued in the Wasserstein space. We prove that the different definitions of the Dirichlet energy in the literature turn out to be equivalent; that the Dirichlet problem is well-posed under mild assumptions; that the superposition principle fails if the source space is no longer a segment; that a sort of maximum principle holds; and we provide a numerical method to compute these harmonic mappings
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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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Kim, Woojin. "The Persistent Topology of Dynamic Data." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587503336988272.
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Christensen, Andreas. "Capacity estimates and Poincaré inequalities for the weighted bow-tie." Thesis, Linköpings universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-138111.
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We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn, and some modern generalizations of these concepts to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the assumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verification of the sharpness of a parameter. At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rn if and only if it holds on the corresponding bow-tie Finally, we consider a concrete weight function, show that it is a Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].
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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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Celik, Cengiz. "New approaches to similarity searching in metric spaces." College Park, Md. : University of Maryland, 2006. http://hdl.handle.net/1903/3454.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2006.Thesis research directed by: Computer Science. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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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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This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an optimal realization 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.
It has been conjectured that extremally weighted optimal realizations may be found as subgraphs of the hereditarily optimal realization Γd, 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 tight span of the metric.
In Paper I, we prove that the graph Γd is equivalent to the 1-skeleton of the tight span precisely when the metric considered is totally split-decomposable. For the subset of totally split-decomposable metrics known as consistent metrics this implies that Γd is isomorphic to the easily constructed Buneman graph.
In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γd.
In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γd. However, for these examples there also exists at least one optimal realization which is a subgraph.
Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γd? Defining extremal 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 Γd is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span
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Suzuki, Kohei. "Convergence of stochastic processes on varying metric spaces." 京都大学 (Kyoto University), 2016. http://hdl.handle.net/2433/215281.
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Shchur, Vladimir. "Quasi-isometries between hyperbolic metric spaces, quantitative aspects." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00867709.
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In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ''quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form $dt^2+\sum_ie^{2\mu_it}dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).
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Putwain, Rosemary Johanna. "Partial translation algebras for certain discrete metric spaces." Thesis, University of Southampton, 2010. https://eprints.soton.ac.uk/170227/.
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The notion of a partial translation algebra was introduced by Brodzki, Niblo and Wright in [11] to provide an analogue of the reduced group C*-algebra for metric spaces. Such an algebra is constructed from a partial translation structure, a structure which any bounded geometry uniformly discrete metric space admits; we prove that these structures restrict to subspaces and are preserved by uniform bijections, leading to a new proof of an existing theorem. We examine a number of examples of partial translation structures and the algebras they give rise to in detail, in particular studying cases where two different algebras may be associated with the same metric space. We introduce the notion of a map between partial translation structures and use this to describe when a map of metric spaces gives rise to a homomorphism of related partial translation algebras. Using this homomorphism, we construct a C*-algebra extension for subspaces of groups, which we employ to compute K-theory for the algebra arising from a particular subspace of the integers. We also examine a way to form a groupoid from a partial translation structure, and prove that in the case of a discrete group the associated C*-algebra is the same as the reduced group C*-algebra. In addition to this we present several subsidiary results relating to partial translations and cotranslations and the operators these give rise to.
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Mushaandja, Zechariah. "A quasi-pseudometrizability problem for ordered metric spaces." Doctoral thesis, University of Cape Town, 2009. http://hdl.handle.net/11427/4914.
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Includes abstract.Includes bibliographical references (leaves 83-88).
In this dissertation we obtain several results in the setting of ordered topological spaces related to the Hanai-Morita-Stone Theorem. The latter says that if f is a closed continuous map of a metric space X onto a topological space Y then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) For each y 2 Y, f−1{y} has a compact boundary in X; (iii) Y is metrizable. A partial analogue of the above theorem for ordered topological spaces is herein obtained.
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Agyingi, Collins Amburo. "Hyperconvex hulls in catergories of quasi-metric spaces." Doctoral thesis, University of Cape Town, 2014. http://hdl.handle.net/11427/12708.
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Includes bibliographical references.Isbell showed that every metric space has an injective hull, that is, every metric space has a “minimal” hyperconvex metric superspace. Dress then showed that the hyperconvex hull is a tight extension. In analogy to Isbell’s theory Kemajou et al. proved that each T₀-quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T₀-quasi-metric space q-hyperconvex if and only if it is injective in the category of T₀-quasi-metric spaces and non-expansive maps. Agyingi et al. generalized results due to Dress on tight extensions of metric spaces to the category of T₀-quasi-metric spaces and non-expansive maps. In this dissertation, we shall study tight extensions (called uq-tight extensions in the following) in the categories of T₀-quasi-metric spaces and T₀-ultra-quasimetric spaces. We show in particular that most of the results stay the same as we move from T₀-quasi-metric spaces to T₀-ultra-quasi-metric spaces. We shall show that these extensions are maximal among the uq-tight extensions of the space in question. In the second part of the dissertation we shall study the q-hyperconvex hull by viewing it as a space of minimal function pairs. We will also consider supseparability of the space of minimal function pairs. Furthermore we study a special subcollection of bicomplete supseparable quasi-metric spaces: bicomplete supseparable ultra-quasi-metric spaces. We will show the existence and uniqueness (up to isometry) of a Urysohn Γ-ultra-quasi-metric space, for an arbitrary countable set Γ of non-negative real numbers including 0.
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Haihambo, Paulus. "Hyperconvexity and endpoints in T₀-quasi-metric spaces." Master's thesis, University of Cape Town, 2013. http://hdl.handle.net/11427/6617.
Full textAbstract:
Over the last decades much progress has been made in the investigation of hyperconvexity in metric spaces. Recently Kemajou and others have published an article concerning hyperconvexity in T₀-quasi-metric spaces. In 1964 Isbell introduced and studied the concept of an endpoint of a metric space. The aim of this dissertation is to begin an investigation into hyperconvexity and endpoints of T₀-quasi-metric spaces. It starts off with basic definitions and some well-known properties of quasi-pseudometric spaces. We conclude by commencing an investigation into hyperconvexity and endpoints of T₀-quasi-metric spaces. In this dissertation several results obtained for hyperconvexity and endpoints in metric spaces are generalized to T₀-quasi-metric spaces, and some original results for hyperconvexity and endpoints of T₀-quasi-metric spaces are presented. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and the q-hyperconvex hull of its natural T₀-quasi-metric space.
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Birch, Thomas. "Algorithmic randomness on computable metric spaces and hyperspaces." Master's thesis, University of Cape Town, 2012. http://hdl.handle.net/11427/22093.
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In this text we shall be focusing on generalizing Martin-Löf randomness to computable metric spaces with arbitrary measure (for examples of this type of generalization see Gács [14], Rojas and Hoyrup [15]. The aim of this generalization is to define algorithmic randomness on the hyperspace of non-empty compact subsets of a computable metric space, the study of which was first proposed by Barmpalias et al. [16] at the University of Florida in their work on the random closed subsets of the Cantor space. Much work has been done in the study of random sets with authors such as Diamondstone and Kjos-Hanssen [17] continuing the Florida approach, whilst others such as Axon [18] and Cenzer and Broadhead [19] have been studying the use of capacities to define hyperspace measures for use in randomness tests. Lastly in section 6.4 we shall be looking at the work done by Hertling and Weihrauch [13] on universal randomness tests in effective topological measure spaces and relate their results to randomness on computable metric measure spaces and in particular to the randomness of compact sets in the hyperspace of non-empty compact subsets of computable metric spaces.
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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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Babus, Octavian Vladut. "Generalised distributivity and the logic of metric spaces." Thesis, University of Leicester, 2016. http://hdl.handle.net/2381/37701.
Full textAbstract:
The aim of the thesis is to work towards a many-valued logic over a commutative unital quantale and, at the same time, towards a generalisation of coalgebraic logic enriched over a commutative unital quantale Ω. This is done by noticing that the contravariant powerset adjunction can be generalised to categories enriched over a commutative unital quantale. From here we define categorical algebras for the monad generated by this adjunction. We finish by showing that these categorical algebras are algebras over Set with operations and equations, and show that in some cases we can restrict the arity of those operations to be finite.
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PALMISANO, Vincenzo. "Topics in calculus and geometry on metric spaces." Doctoral thesis, Università degli Studi di Palermo, 2022. https://hdl.handle.net/10447/554772.
Full textAbstract:
In this thesis we present an overview of some important known facts related to
topology, geometry and calculus on metric spaces. We discuss the well known
problem of the existence of a lipschitz equivalent metric to a given quasiultrametric,
revisiting known results and counterexamples and providing
some new theorems, in an unified approach. Also, in the general setting of
a quasi-metric doubling space, suitable partition of unity lemmas allows us
to obtain, in step two Carnot groups, the well known Whitney’s extension
theorem for a given real function of class C^m defined on a closed subset of
the whole space: this result relies on relevant properties of the symmetrized
Taylor’s polynomial recently introduced in this setting. Finally, some first
interesting investigations on Menger convexity in the setting of a general
metric spaces concludes this work.
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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.
Full textAbstract:
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 marginalesWe 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