The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean n--space, and are specific cases of the properties of a general topological space.

<p> A class of Banach algebras which generalize the idea of the Lipschitz algebra on a metric space is studied. It is shown that homomorphisms of these algebras correspond to mappings of the underlying space which satisfy certain moduli of continuity. The relation is expressed in categorical terms, and application is made to the theory of quasiconformal mapping. </p> / Thesis / Doctor of Philosophy (PhD)

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This doctoral thesis is devoted to investigate the problem of establishing connections between Domain Theory and the theory of fuzzy metric spaces, in the sense of Kramosil and Michalek, by means of the notion of a formal ball, and then constructing topological and computational models for (complete) fuzzy metric spaces. The antecedents of this research are mainly the well-known articles of A. Edalat and R. Heckmann [A computational model for metric spaces, Theoret- ical Computer Science 193 (1998), 53-73], and R. Heckmann [Approximation of metric spaces by partial metric spaces, Applied Categorical Structures 7 (1999), 71-83], where the authors obtained nice and direct links between Do- main Theory and the theory of metric spaces - two crucial tools in the study of denotational semantics - by using formal balls. Since every metric induces a fuzzy metric (the so-called standard fuzzy metric), the problem of extending Edalat and Heckmann's works to the fuzzy framework arises in a natural way. In our study we essentially propose two di erent approaches. For the rst one, valid for those fuzzy metric spaces whose continuous t-norm is the minimum, we introduce a new notion of fuzzy metric completeness (the so-called standard completeness) that allows us to construct a (topological) model that includes the classical theory as a special case. The second one, valid for those fuzzy metric spaces whose continuous t-norm is greater or equal than the Lukasiewicz t-norm, allows us to construct, among other satisfactory results, a fuzzy quasi-metric on the continuous domain of formal balls whose restriction to the set of maximal elements is isometric to the given fuzzy metric. Thus we obtain a computational model for complete fuzzy metric spaces. We also prove some new xed point theorems in complete fuzzy metric spaces with versions to the intuitionistic case and the ordered case, respec- tively. Finally, we discuss the problem of extending the obtained results to the asymmetric framework. / Ricarte Moreno, L. (2013). Topological and Computational Models for Fuzzy Metric Spaces via Domain Theory [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34670

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.

Problems of population genetics are magnified by problems of big data. My dissertation spans the disciplines of computer science and population genetics, leveraging computational approaches to biological problems to address issues in genomics research. In this dissertation I develop more efficient metric search algorithms. I also show that vast majority of the genomes of great apes are impacted by the forces of natural selection. Finally, I introduce a heuristic to identify neutralomes—regions that are evolving with minimal selective pressures—and use these neutralomes for inferences on effective population size in great apes. We begin with a formal and far-reaching problem that impacts a broad array of disciplines including biology and computer science; the 𝑘-nearest neighbors problem in generalized metric spaces. The 𝑘-nearest neighbors (𝑘-NN) problem is deceptively simple. The problem is as follows: given a query q and dataset D of size 𝑛, find the 𝑘-closest points to q. This problem can be easily solved by algorithms that compute 𝑘th order statistics in O(𝑛) time and space. It follows that if D can be ordered, then it is perhaps possible to solve 𝑘-NN queries in sublinear time. While this is not possible for an arbitrary distance function on the points in D, I show that if the points are constrained by the triangle inequality (such as with metric spaces), then the dataset can be properly organized into a dispersion tree (Appendix A). Dispersion trees are a hierarchical data structure that is built around a large dispersed set of points. Dispersion trees have construction times that are sub-quadratic (O(𝑛¹·⁵ log⁡ 𝑛)) and use O(𝑛) space, and they use a provably optimal search strategy that minimizes the number of times the distance function is invoked. While all metric data structures have worst-case O(𝑛) search times, dispersion trees have average-case search times that are substantially faster than a large sampling of comparable data structures in the vast majority of spaces sampled. Exceptions to this include extremely high dimensional space (d>20) which devolve into near-linear scans of the dataset, and unstructured low-dimensional (d<6) Euclidean spaces. Dispersion trees have empirical search times that appear to scale as O(𝑛ᶜ) for 0<c<1. As solutions to the 𝑘-NN problem are in general too slow to be used effectively in the arena of big data in genomics, it is my hope that dispersion trees may help lift this barrier. With source-code that is freely available for academic use, dispersion trees may be useful for nearest neighbor classification problems in machine learning, fast read-mapping against a reference genome, and as a general computational tool for problems such clustering. Next, I turn to problems in population genomics. Genomic patterns of diversity are a complex function of the interplay between demographics, natural selection and mechanistic forces. A central tenet of population genetics is the neutral theory of molecular evolution which states the vast majority of changes at the molecular level are (relatively) selectively neutral; that is, they do not effect fitness. A corollary of the neutral theory is that the frequency of most alleles in populations are dictated by neutral processes and not selective processes. The forces of natural selection impact not just the site of selection, but linked neutral sites as well. I proposed an empirical assessment of the extents of linked selection in the human genome (Appendix B). Recombination decouples sites of selection from the genomic background, thus it serves to mitigate the effects of linked selection. I use two metrics on recombination, both the minimum genetic distance to genes and local rates of recombination, to parse the effects of linked selection into selection from genic and nongenic sources in the human genome. My empirical assessment shows profound linked selective effects from nongenic sources, with these effects being greater than that of genic sources on the autosomes, as well as generally greater effects on the X chromosome than on the autosomes. I quantify these trends using multiple linear regression, and then I model the effects of linked selection to conserved elements across the whole of the genome. Places predicted to be neutral by my model do not, unlike the vast majority of the genome, show these linked selective effects. This demonstrates that linkage to these regulatory elements, and not some other mechanistic force, accounts for our findings. Further, neutrally evolving regions are extremely rare (~1%) in the genome, and despite generally larger linked selective effects on the X chromosome, the size of this “neutralome” is proportionally larger on the X chromosome than on the autosomes. To account for this and to extend my findings to other great apes I improve on my procedure to find neutralomes, and apply this procedure to the genome of humans, Nigerian chimpanzees, bonobos, and western lowland gorillas (Appendix C). In doing so I show that like humans, these other apes are also enormously impacted by linked selection, with their neutralomes being substantially smaller than the neutralomes of humans. I then use my genomic predictions on neutrality to see how the landscape of linked selection changes across the X chromosome and the autosomes in regions close to, and far from, genes. While I had previously demonstrated the linked selective forces near genes are stronger on the X chromosome than on the autosomes in these taxa, I show that regions far from genes show the opposite; regions far from genes show more selection from noncoding targets on the autosomes than on the X chromosome. This finding is replicated across our great ape samples. Further, inferences on the relative effective population size of the X chromosome and the autosomes both near and far from genes can be biased as a result.

Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking in proofs found the book. Although an elementary knowledge of topology is assumed throughout the paper, references are given for basic definitions and theorems used in developing some of the proofs. The following symbols and notation will be used throughout the paper. X will denote a metric space with metric p, S a topological space, I the set of positive integers, A, B, C... sets of points or elements. Small letters, such as a, b, c, x, y, z... will designate elements or points of sets. U and V will denote open sets Sr(x) a spherical neighborhood of x with radius r. A' denotes the set of limit points of A. A- the set of closure points of A/ U, N, C will denote union, intersection, and set inclusion respectively. The symbol E will mean "is an element of". 0 denotes the empty set. S - A is the set of points in S which are not in A.

<p>The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.</p><p>All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.</p><p>Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.</p><p>This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.</p>

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.

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.