Related papers: Property A and asymptotic dimension

We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not…

We construct a countable p-local group with a proper invariant metric whose Assouad-Nagata dimension is strictly greater than the asymptotic dimension with linear control. This solves Problem 8.6 from the list [Dr]. We study asymptotic…

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}\alpha_k(x), \text{ where } x=\sum^{\infty}_{k=1}\alpha_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of…

In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains…

Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an aperiodic shift of finite type $(\Sigma_{A}^{+},\S)$. Let $\Delta_{n}$ be the union of cylinders in $\Sigma_{A}^{+}$ corresponding to the points…

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A…

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space…

A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…

Let (X,d) be a metric space and m\in X. Suppose that \phi:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{\phi,m} on X which is equivalent to d. If d^{\phi,m} is totally bounded, its completion is a…

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…

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates…

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale…

In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve…

We define spacetimes that are asymptotically flat, except for a deficit solid angle $\alpha$, and present a definition of their ``ADM'' mass, which is finite for this class of spacetimes, and, in particular, coincides with the value of the…

Let $\left(X_n, d_n\right)$ be a sequence of metric spaces and let $\mathcal{F}=\left\{f_n\right\}_{n \in \mathbb{Z}}$ be a sequence of continuous and onto maps $f_n: X_n \rightarrow X_{n+1}, n \in \mathbb{Z}_{+}$. In this paper, we prove…

Suppose $X$ is a compact connected metric space and $f: X \to X$ is a metric coarse expanding conformal map in the sense of Ha\"issinsky-Pilgrim. We show that if $X$ contains a homeomorphic copy of the letter "Y", then the Hausdorff…

Let $X\subset A^{Z^d}$ be a $2$-dimensional subshift of finite type. We prove that any $2$-dimensional multidimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general…

We show that $(n+1)$-dimensional Myers-Perry metrics, $n\geq4$, have a conformal completion at spacelike infinity of $C^{n-3,1}$ differentiability class, and that the result is optimal in even spacetime dimensions. The associated asymptotic…

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in…

We expose a class of discrete metric spaces, for which bounded geometry is equivalent to the property A of G. Yu. This class includes the coarse disjoint union of $(\mathbb Z/2\mathbb Z)^n$, $n\in\mathbb N$, and consists of spaces of simple…