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We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [7]. We also bound the number of generic measures such a subshift can support based on its…
We study mappings defined in the domain of a metric space that distort the modulus of families of paths by the type of the inverse Poletskii inequality. Under certain conditions, it is proved that such mappings have a continuous extension…
We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the…
General equations of the unified field theory, obtained using the curved and torsional space-time, are presented. They contain only independent geometrical parameters (metric and connections) of the metric-affine space, and describe the…
Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be…
Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces $X$ and $Y$ are weakly similar if there exists a weak similarity $\Phi\colon X\to Y$. We find a structural characteristic of finite…
This article investigates the topological pressure of isotropic axial products of Markov subshifts on the $d$-tree. We show that the quantity increases with dimension $d$. To achieve this, we introduce the pattern distribution vectors and…
In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted…
We first identify (up to linear isomorphism) the Lipschitz free spaces of quasiarcs. By decomposing quasiconformal trees into quasiarcs as done in an article of David, Eriksson-Bique, and Vellis, we then identify the Lipschitz free spaces…
We give a short and elementary proof of the fact that every metric space of finite asymptotic dimension can be embedded into a finite product of trees.
We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric cpaces with Busemann convex pretangent spaces is also…
We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$.…
The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$…
This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel--Lizorkin-type spaces $\dot B^{s,\tau}_{p,q}(W)$ and $\dot F^{s,\tau}_{p,q}(W)$. In this article, the authors establish the molecular…
In this paper, we give an interesting extension of the partial S-metric space which was introduced [4] to the M_s-metric space. Also, we prove the existence and uniqueness of a fixed point for a self mapping on an Ms-metric space under…
Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion…
Fractons and other subdimensional particles are an exotic class of emergent quasi-particle excitations with severely restricted mobility. A wide class of models featuring these quasi-particles have a natural description in the language of…
We find universal spaces for Alexandroff and finite spaces and explore some of its topological properties as well as their description as inverse limits of finite spaces and Alexandroff extensions. They can be used as a natural environment…
We prove that Sobolev spaces on Cartesian and warped products of metric spaces tensorize, only requiring that one of the factors is a doubling space supporting a Poincar\'e inequality.
For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as $\sum_{i=1}^d…