scholarly journals Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

10.29007/pw5g ◽  
2018 ◽  
Author(s):  
Larry Moss ◽  
Jayampathy Ratnayake ◽  
Robert Rose
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Fractal Sets ◽  
Cauchy Completion ◽  
Sweeping Generalization ◽  
Initial Algebra ◽  
Finite Metric Spaces ◽  
Set Of Points

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offersa sweeping generalization of this result. He is able to represent many of what would be intuitivelycalled "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to thecharacterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutionsis not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors asclosures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].Our second contribution is not completed at this time, but it is a set of results on \emph{metric space}characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our workdevelops (a) and (b) in metric settings while dropping (c).

scholarly journals An Intrinsic Characterization of Five Points in a CAT(0) Space

2020 ◽  
Vol 8 (1) ◽  
pp. 114-165
Author(s):  
Tetsu Toyoda
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
New Approach ◽  
Intrinsic Characterization ◽  
New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

Characterization of Certain Classes of Spaces With Gδ Points as Open Images of Metric Spaces

1975 ◽  
Vol 27 (6) ◽  
pp. 1229-1238
Author(s):  
Kenneth C. Abernethy
Keyword(s):  
Metric Spaces ◽  
Topological Spaces ◽  
Generalized Metric Spaces ◽  
The Past ◽  
Generalized Metric

The study of metrization has led to the development of a number of new topological spaces, called generalized metric spaces, within the past fifteen years. For a survey of results in metrization theory involving many of these spaces, the reader is referred to [13]. Quite a few of these generalized metric spaces have been studied extensively, somewhat independently of their role in metrization theorems. Specifically, we refer here to characterizations of these spaces by various workers as images of metric spaces. Results in this area have been obtained by Alexander [2], Arhangel'skii [3], Burke [5], Heath [10], Michael [15], Nagata [16], and the author [1], to mention a few. Later we will recall specifically some of these results.

scholarly journals Lipschitz-free Spaces on Finite Metric Spaces

2019 ◽  
Vol 72 (3) ◽  
pp. 774-804 ◽  
Author(s):  
Stephen J. Dilworth ◽  
Denka Kutzarova ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Space ◽  
Free Space ◽  
Metric Spaces ◽  
Large Classes ◽  
Complemented Subspace ◽  
Finite Metric Space ◽  
Free Spaces ◽  
Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

scholarly journals Monad Metrizable Space

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1891
Author(s):  
Orhan Göçür
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Real Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Metrizable Spaces

Do the topologies of each dimension have to be same and metrizable for metricization of any space? I show that this is not necessary with monad metrizable spaces. For example, a monad metrizable space may have got any indiscrete topologies, discrete topologies, different metric spaces, or any topological spaces in each different dimension. I compute the distance in real space between such topologies. First, the passing points between different topologies is defined and then a monad metric is defined. Then I provide definitions and some properties about monad metrizable spaces and PAS metric spaces. I show that any PAS metric space is also a monad metrizable space. Moreover, some properties and some examples about them are presented.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

scholarly journals Topological pattern recognition for point cloud data

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 289-368 ◽  
Author(s):  
Gunnar Carlsson
Keyword(s):  
Pattern Recognition ◽  
Algebraic Topology ◽  
Point Cloud ◽  
Metric Spaces ◽  
Persistent Homology ◽  
Topological Spaces ◽  
Data Sets ◽  
Point Cloud Data ◽  
Cloud Data ◽  
Finite Metric Spaces

In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to aspersistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
Author(s):  
TIMOTHY FAVER ◽  
KATELYNN KOCHALSKI ◽  
MATHAV KISHORE MURUGAN ◽  
HEIDI VERHEGGEN ◽  
ELIZABETH WESSON ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Independent Set ◽  
Ultrametric Space ◽  
Ultrametric Spaces ◽  
Negative Type ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Generalized Roundness

AbstractMotivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

scholarly journals The isometric embedding problem for length metric spaces

2019 ◽  
pp. 1-44
Author(s):  
Barry Minemyer
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Energy Functional ◽  
Embedding Problem ◽  
Lorentzian Space ◽  
Set Of Points ◽  
Dense Set

We prove that every proper [Formula: see text]-dimensional length metric space admits an “approximate isometric embedding” into Lorentzian space [Formula: see text]. By an “approximate isometric embedding” we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

scholarly journals Closedness, separation and connectedness in pseudo-quasi-semi metric spaces

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4757-4766
Author(s):  
Tesnim Baran
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
Topological Category ◽  
The Relationship

In this paper, we give the characterization of closed and strongly closed subsets of an extended pseudo-quasi-semi metric space and show that they induce closure operator. Moreover, we characterize each of Ti, i = 0, 1, 2 and connected extended pseudo-quasi-semi metric spaces and investigate the relationship among them. Finally, we introduce the notion of irreducible objects in a topological category and examine the relationship among each of irreducible, Ti,i = 1,2, and connected extended pseudo-quasi-semi metric spaces.

Sign in / Sign up

Export Citation Format

Share Document