Dissertations / Theses on the topic 'Metric spaces'

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.

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.

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.

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.

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.

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.

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.

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.

2012 - 2013
The main topics of this thesis are local proximity spaces jointly with some bornological convergences naturally related to them, and ωµ −metric spaces, in particular those which are Atsuji spaces (or UC spaces), jointly with their hyperstructures. Local proximities spaces carry with them two particular features: proximity [48] and boundedness [37], [40]. Proximities allow us to deal with a concept of nearness even though not providing a metric. Proximity spaces are located between topological and metric spaces. Boundedness is a natural generalization of the metric boundedness. When trying to refer macroscopic phenomena to local structures, local proximity spaces appear as a very attractive option. For that, jointly with Prof. A. Di Concilio, in a first step we displayed a uniform procedure as an exhaustive method of generating all local proximity spaces starting from unform spaces and suitable bornologies. After that, we looked at suitable topologies for the hyperspace of a local proximity space. In contrast with the proximity case, in which there is no canonical way of equipping the hyperspaces with a uniformity, the same with a proximity, the local proximity case is simpler. Apparently, at the beginning, we have three natural different ways to topologize the hyperspace CL(X) of all closed non-empty subsets of X: we can think at a local Fell hypertopology or a kind of hit and far-miss topology or also a particular uniform bornological topology. We proved that they match. In the light of the previous local proximity results, we looked for necessary and sufficient conditions of uniform nature for two different uniform bornological convergences to match. This led us to focus on a special class of uniformities: those with a linearly ordered base. They are connected with an interesting generalization of metric spaces, ωµ −metric spaces. These spaces are endowed with special distances valued in ordered abelian additive groups. Furthermore, in relation with ωµ−metric spaces, we looked at generalizations of well known hyperspace convergences, as Hausdorff and Kuratowski convergences obtaining analogue results with respect to the standard case, [28]. Finally, we dealt with Atsuji spaces.We were interested in the problem of constructing a dense extension Y of a given topological space X, which is Atsuji and in which X is topologically embedded. When such an extension there exists, we say that the space X is Atsuji extendable. Atsuji spaces play an important role above all because they allow us to deal with a very nice structure when we concentrate on the most significant part of the space, that is the derived set. Moreover, we know that each continuous function between metric or uniform spaces is uniformly continuous on compact sets. It is possible to have an analogous property on a larger class of topological spaces, Atsuji spaces. They are situated between complete metric spaces and compact ones. We proved a necessary and sufficient condition for a metrizable spaceX to be Atsuji extendable.Moreover we looked at conditions under which a continuous function f X 􀀀 R can be continuously extended to the Atsuji extension Y of X. UC metric spaces admit a very long list of equivalent formulations. We extended many of these to the class of ωµ−metric spaces. The results are contained in [29]. Finally it is presented the idea about the work done jointly with Professor J.F. Peters ( University of Manitoba , Canada). Our research involved the study of more general proximities leading to a kind of strong farness, [52]. Strong proximities are associated with Lodato proximities and the Efremoviˇc property.We say that A and B are −strongly far, where is a Lodato proximity, and we write ~ if and only if A ~ B and there exists a subset C of X such that A ~ X C and C ~ B, that is the Efremoviˇc property holds on A and B. Related to this idea we defined also a new concept of strong nearness, [53]. Starting by these new kinds of proximities we introduced also new kinds of hit-and-miss hypertopologies, concepts of strongly proximal continuity and strong connectedness. Finally we looked at some applicaii tions that in our opinion might reveal interesting.
XII n.s.

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.

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.

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.

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
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

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
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

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.

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 Γ_d, 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 Γ_d 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 Γ_d 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 Γ_d.

In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γ_d. 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 Γ_d? 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 Γ_d is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

Thesis (Ph.D.)--University of Cincinnati, 2008.
Committee/Advisors: David Herron PhD (Committee Chair), David Minda PhD (Committee Member), Nageswari Shanmugalingam PhD (Committee Member). Title from electronic thesis title page (viewed Sep.3, 2008). Keywords: conformal density; uniform spaces; Loewner; quasisymmetry; quasiconofrmal. Includes abstract. Includes bibliographical references.

[EN] Fuzzy mathematics has constituted a wide field of research, since L. A. Zadeh introduced in 1965 the concept of fuzzy set. In particular, the problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors. In 1994, George and Veeramani introduced and studied a notion of fuzzy metric space that constituted a modification of the one given by Kramosil and Michalek. Several authors have contributed to the study of this kind of fuzzy metrics, from the mathematical point of view and for their applications. In this thesis we have contributed to develop the study of these fuzzy metrics, from the mathematical point of view, and we approached the problem of measuring perceptual colour-difference between samples of colour using one of these fuzzy metrics. The contributions of the study carried out in this thesis is summarized as follows: \begin{enumerate} \item[(i)] We have made a detailed study of the fuzzy metric space $(X,M,\cdot)$ where $M$ is given on $X=[0,\infty[$ by $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ and others related to it. As a consequence we have introduced five questions in fuzzy metrics related to continuity, extension, contractivity and completion. \item[(ii)] We have answered an open question constructing a fuzzy metric space $(X,M,\ast)$ in which the assignment $f(t)=\lim_n M(a_n,b_n,t)$, where $\{a_n\}$ and $\{b_n\}$ are $M$-Cauchy sequences in $X$, is not a continuous function on $t$. The response to this question has allowed us to characterize the class of completable strong fuzzy metric spaces. \item[(iii)] We have introduced and studied a stronger concept than convergence of sequences in fuzzy metric spaces, which we call $s$-convergence. In our study, we have gotten a characterization of those spaces in which every convergent sequence is $s$-convergent and we have given a classification of fuzzy metrics attending to the behaviour of the fuzzy metric with respect to the different types of convergence. \item[(iv)] We have studied, in the context of fuzzy metric spaces, when certain families of open balls centered at a point are local bases for this point. \item[(v)] We have answered two open questions related to standard convergence, a stronger concept than convergence of sequences in fuzzy metric spaces, introduced in a natural way attending to the concept of standard Cauchy sequence (introduced in \cite{adomain}). These responses have led us to establish conditions under which Cauchyness and convergence should be considered \textit{compatible}. \item[(vi)] As a practical application, we have shown that a certain fuzzy metric is useful for measuring perceptual colour-differences between colour samples. \end{enumerate}
[ES] La matemática fuzzy ha constituido un amplio campo en la investigación, desde que en 1965 L. A. Zadeh introdujo el concepto de conjunto fuzzy. En particular, la construcción de una teoría satisfactoria de espacios métricos fuzzy ha sido un problema investigado por muchos autores. En 1994, George y Veeramani introdujeron y estudiaron una noción de espacio métrico fuzzy que constituía una modificación de la anteriormente dada por Kramosil y Michalek. Muchos autores han contribuido al estudio de este tipo de métricas fuzzy, desde el punto de vista matemático y de sus aplicaciones. En esta tesis hemos contribuido al desarrollo del estudio de estas métricas fuzzy, desde el punto de vista matemático, y hemos abordado el problema de la medida de la diferencia perceptual de color utilizando una de estas métricas. Las contribuciones que aportamos en esta tesis a dicho estudio, se resumen a continuación: \begin{enumerate} \item[(i)] Hemos hecho un estudio detallado del espacio métrico fuzzy $(X,M,\cdot)$ donde $M$ está dada sobre $[0,\infty[$ por la expresión $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ y de otros espacios métricos fuzzy relacionados con el. Como consecuencia de este estudio hemos introducido cinco cuestiones en la teoría de las métricas fuzzy relacionadas con continuidad, extensión, contractividad y completación. \item[(ii)] Hemos respondido a una cuestión abierta construyendo un espacio métrico fuzzy $(X,M,\ast)$ en el cual la asignación $f(t)=\lim_n M(a_n,b_n,t)$, donde $\{a_n\}$ y $\{b_n\}$ son sucesiones $M$-Cauchy, no es una función continua sobre $t$. La respuesta a esta cuestión nos ha permitido caracterizar la clase de los espacios métricos fuzzy strong completables. \item[(iii)] Hemos introducido y estudiado un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, al que hemos llamado $s$-convergencia. En nuestro estudio hemos conseguido una caracterización de aquellos espacios métricos fuzzy en los cuales toda sucesión convergente es $s$-convergente y hemos dado una clasificación de los espacios métricos fuzzy atendiendo a su comportamiento con respecto a los diferentes tipos de convergencia que se da en él. \item[(iv)] Hemos estudiado, en el contexto de los espacios métricos fuzzy, cuando ciertas familias de bolas abiertas centradas en un punto son base local de este punto. \item[(v)] Hemos respondido a dos cuestiones abiertas relacionadas con la convergencia standard, un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, introducido de forma natural a partir del concepto de sucesión de Cauchy standard (introducido en \cite{adomain}). Estas respuestas nos han llevado a establecer unas condiciones bajo las cuales un concepto relacionado con el concepto de sucesión de Cauchy y un concepto relacionado con el de convergencia deberían satisfacer para ser consideradas \textsl{compatibles}. \item[(vi)] Como aplicación práctica, hemos mostrado que una cierta métrica fuzzy es útil para medir diferencia perceptual de color entre muestras de color. \end{enumerate}
[CAT] La matemàtica fuzzy ha constituït un ampli camp en la investigació, des que el 1965 L. A. Zadeh va introduir el concepte de conjunt fuzzy. En particular, la construcció d'una teoria satisfactòria d'espais mètrics fuzzy ha estat un problema investigat per molts autors. El 1994, George i Veeramani introduiren i estudiaren una noció d'espai mètric fuzzy que constituïa una modificació de la donada per Kramosil i Michalek anteriorment. Molts autors han contribuït a l'estudi d'aquest tipus de mètriques fuzzy, des del punt de vista matemàtic i de les seves aplicacions. En aquesta tesi hem contribuït al desenvolupament de l'estudi d'aquestes mètriques fuzzy, des del punt de vista matemàtic, i hem abordat el problema de la mesura de la diferència perceptiva de color utilitzant aquestes mètriques. Les contribucions que aportem en aquesta tesi a tal estudi es resumeixen a continuació: \begin{enumerate} \item[(i)] Hem fet un estudi detallat de l'espai mètric fuzzy $(X,M,\cdot)$ on $M$ està donada sobre $[0,\infty[$ per l'expressió $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ i d'altres espais mètrics fuzzy relacionats amb ell. Com a conseqüència d'aquest estudi hem introduït cinc qüestions en la teoria de les mètriques fuzzy relacionades amb continuïtat, extensió, contractividad i completació. \item[(ii)] Hem respost a una qüestió oberta construint un espai mètric fuzzy $ (X, M, \ast) $ en el qual l'assignació $ f (t) = \lim_n M (a_n, b_n, t) $, on $ \{a_n\} $ i $ \{b_n \} $ són successions $ M $-Cauchy, no és una funció contínua sobre $ t $. La resposta a aquesta qüestió ens ha permès caracteritzar la classe dels espais mètrics fuzzy strong completables. \item[(iii)] Hem introduït i estudiat un concepte més fort que el de convergència de successions en espais mètrics fuzzy, al qual hem anomenat $ s $-Convergència. En el nostre estudi hem aconseguit una caracterització d'aquells espais mètrics fuzzy en els quals tota successió convergent és $ s $-convergente i hem donat una classificació dels espais mètrics fuzzy atenent al seu comportament respecte als diferents tipus de convergència que es dóna en ell. \item[(iv)] Hem estudiat, en el context dels espais mètrics fuzzy, quan certes famílies de boles obertes centrades en un punt són base local d'aquest punt. \item[(v)] Hem respost a dues qüestions obertes relacionades amb la convergència estàndard, un concepte més fort que el de convergència de successions en espais mètrics fuzzy, introduït de forma natural a partir del concepte de successió de Cauchy estàndard (introduït en \cite{adomain}). Aquestes respostes ens han portat a establir unes condicions sota les quals un concepte relacionat amb el concepte de successió de Cauchy i un concepte relacionat amb el de convergència haurien de satisfer per a ser considerats \textsl{compatibles}. \item[(vi)] Com a aplicació pràctica, hem mostrat que una certa mètrica fuzzy és útil per mesurar la diferència perceptiva de color entre mostres de color. \end{enumerate}
Miñana Prats, JJ. (2015). Fuzzy metric spaces and applications to perceptual colour-differences [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/50612
TESIS

La complexité des données contenues dans les grandes bases de données a augmenté considérablement. Par conséquent, des opérations plus élaborées que les requêtes traditionnelles sont indispensable pour extraire toutes les informations requises de la base de données. L'intérêt de la communauté de base de données a particulièrement augmenté dans les recherches basées sur la similarité. Deux sortes de recherche de similarité bien connues sont la requête par intervalle (Rq) et par k-plus proches voisins (kNNq). Ces deux techniques, comme les requêtes traditionnelles, peuvent être accélérées par des structures d'indexation des Systèmes de Gestion de Base de Données (SGBDs).Une autre façon d'accélérer les requêtes est d'exécuter le procédé d'optimisation des requêtes. Dans ce procédé les données métriques sont recueillies et utilisées afin d'ajuster les paramètres des algorithmes de recherche lors de chaque exécution de la requête. Cependant, bien que l'intégration de la recherche de similarités dans le SGBD ait commencé à être étudiée en profondeur récemment, le procédé d'optimisation des requêtes a été développé et utilisé pour répondre à des requêtes traditionnelles. L'exécution des requêtes de similarité a tendance à présenter un coût informatique plus important que l'exécution des requêtes traditionnelles et ce même en utilisant des structures d'indexation efficaces. Deux stratégies peuvent être appliquées pour accélérer l'execution de quelques requêtes, et peuvent également être employées pour répondre aux requêtes de similarité. La première stratégie est la réécriture de requêtes basées sur les propriétés algébriques et les fonctions de coût. La deuxième stratégie est l'utilisation des facteurs externes de la requête, tels que la sémantique attendue par les usagers, pour réduire le nombre des résultats potentiels. Cette thèse vise à contribuer au développement des techniques afin d'améliorer le procédé d'optimisation des requêtes de similarité, tout en exploitant les propriétés algébriques et les restrictions sémantiques pour affiner les requêtes
The complexity of data stored in large databases has increased at very fast paces. Hence, operations more elaborated than traditional queries are essential in order to extract all required information from the database. Therefore, the interest of the database community in similarity search has increased significantly. Two of the well-known types of similarity search are the Range (Rq) and the k-Nearest Neighbor (kNNq) queries, which, as any of the traditional ones, can be sped up by indexing structures of the Database Management System (DBMS). Another way of speeding up queries is to perform query optimization. In this process, metrics about data are collected and employed to adjust the parameters of the search algorithms in each query execution. However, although the integration of similarity search into DBMS has begun to be deeply studied more recently, the query optimization has been developed and employed just to answer traditional queries.The execution of similarity queries, even using efficient indexing structures, tends to present higher computational cost than the execution of traditional ones. Two strategies can be applied to speed up the execution of any query, and thus they are worth to employ to answer also similarity queries. The first strategy is query rewriting based on algebraic properties and cost functions. The second technique is when external query factors are applied, such as employing the semantic expected by the user, to prune the answer space. This thesis aims at contributing to the development of novel techniques to improve the similarity-based query optimization processing, exploiting both algebraic properties and semantic restrictions as query refinements
A complexidade dos dados armazenados em grandes bases de dados tem aumentadosempre, criando a necessidade de novas operaoes de consulta. Uma classe de operações de crescente interesse são as consultas por similaridade, das quais as mais conhecidas sãoas consultas por abrangência (Rq) e por k-vizinhos mais próximos (kNNq). Qualquerconsulta é agilizada pelas estruturas de indexaçãodos Sistemas de Gerenciamento deBases de Dados (SGBDs). Outro modo de agilizar as operações de busca é a manutençãode métricas sobre os dados, que são utilizadas para ajustar parâmetros dos algoritmos debusca em cada consulta, num processo conhecido como otimização de consultas. Comoas buscas por similaridade começaram a ser estudadas seriamente para integração emSGBDs muito mais recentemente do que as buscas tradicionais, a otimização de consultas,por enquanto, é um recurso que tem sido utilizado para responder apenas a consultastradicionais.Mesmo utilizando as melhores estruturas existentes, a execução de consultas por similaridadetende a ser mais custosa do que as operações tradicionais. Assim, duas estratégiaspodem ser utilizadas para agilizar a execução de qualquer consulta e, assim, podem serempregadas também para responder às consultas por similaridade. A primeira estratégiaé a reescrita de consultas baseada em propriedades algébricas e em funções de custo. Asegunda técnica faz uso de fatores externos à consulta, tais como a semântica esperadapelo usuário, para restringir o espaço das respostas. Esta tese pretende contribuir parao desenvolvimento de técnicas que melhorem o processo de otimização de consultas porsimilaridade, explorando propriedades algébricas e restrições semânticas como refinamentode consultas

In this thesis we present an axiomatic approach to an invariant Harnack inequality for non homogeneous PDEs in the setting of doubling quasi-metric spaces. We adapt the abstract procedure developed by Di Fazio, Gutiérrez and Lanconelli, for homogeneous PDEs taking into account the right hand side of the equation. In particular we adapt the notions of double ball property and critical density property: these notions arise from Krylov-Safonov technique for uniformly elliptic operators and they imply Harnack inequality. Then we apply the axiomatic procedure to subelliptic equations in non divergence form involving Grushin vector fields and to X-elliptic operators in divergence form.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

Desde que L.A. Zadeh presentó la teoría de conjuntos difusos en 1965, esta se ha usado en una amplia serie de áreas de las matemáticas y se ha aplicado en una gran variedad de escenarios de la vida real. Estos escenarios cubren procesos complejos sin modelo matemático sencillo tales como dispositivos de control industrial, reconocimiento de patrones o sistemas que gestionen información imprecisa o altamente impredecible. La topología difusa es un importante ejemplo de uso de la teoría de L.A. Zadeh. Durante años, los autores de este campo han buscado obtener la definición de un espacio métrico difuso para medir la distancia entre elementos según grados de proximidad. El presente trabajo trata acerca de la bicompletación de espacios casi-métricos difusos en el sentido de Kramosil y Michalek. Sherwood probó que todo espacio métrico difuso admite completación que es única excepto por isometría basándose en propiedades de la métrica de Lévy. Probamos aquí que todo espacio casi-métrico difuso tiene bicompletación usando directamente el supremo de conjuntos en [0,1] y límites inferiores de secuencias en [0,1] en lugar de usar la métrica de Lévy. Aprovechamos tanto la bicompletitud y bicompletación de espacios casi-métricos difusos como las propiedades de los espacios métricos difusos y difusos intuicionistas para presentar varias aplicaciones a problemas del campo de la informática. Así estudiamos la existencia y unicidad de solución para las ecuaciones de recurrencia asociadas a ciertos algoritmos formados por dos procedimientos recursivos. Para analizar su complejidad aplicamos el principio de contracción de Banach tanto en un producto de casi-métricas no-Arquimedianas en el dominio de las palabras como en la casi-métrica producto de dos espacios de complejidad casi-métricos de Schellekens. Estudiamos también una aplicación de espacios métricos difusos a sistemas de información basados en localidad de accesos.
Castro Company, F. (2010). Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8420
Palancia

In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.