Related papers: Asymptotic Resemblance

We study a long-recognised but under-appreciated symmetry called "dynamical similarity" and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a…

The cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of…

Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension $n$ containing a bi-infinite geodesic can be coarsely separated by a subset $S$ of asymptotic dimension equal…

This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…

We characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of sets of (almost) farthest points from the unit sphere using the generalized diameter. For this we introduce few…

In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this…

By utilizing the idea of Colombeau's generalized function, we introduce a notion of asymptotic map between arbitrary diffeological spaces. The category consisting of diffeological spaces and asymptotic maps is enriched over the category of…

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large $n$, a networked data…

We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in…

We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the…

We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number.…

The theory of descriptive nearness is usually adopted when dealing with sets that share some common properties even when the sets are not spatially close, i.e., the sets have no members in common. Set description results from the use of…

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns…

A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in $\mathbf{ZF}$ a new characterization of iso-dense spaces in terms of…

We introduce the notion of asymptotic coarse Lipschitz equivalence of metric spaces. We show that it is strictly weaker than coarse Lipschitz equivalence. We study its impact on the asymptotic dimension of metric spaces. Then we focus on…

Roughly speaking, let us say that a map between metric spaces is large scale conformal if it maps packings by large balls to large quasi-balls with limited overlaps. This quasi-isometry invariant notion makes sense for finitely generated…

Symmetries are ubiquitous in modern physics. They not only allow for a more simplified description of physical systems but also, from a more fundamental perspective, can be seen as determining a theory itself. In the present paper, we…

A new line of research uses compression methods to measure the similarity between signals. Two signals are considered similar if one can be compressed significantly when the information of the other is known. The existing compression-based…

In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in…

Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we…