Related papers: Quasihomomorphisms from the integers into Hamming …

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in…

Within a category $\mathtt{C}$, having objects $\mathtt{C}_0$, it may be instructive to know not only that two objects are non-isomorphic, but also how far from being isomorphic they are. We introduce pseudo-metrics $d:\mathtt{C}_0 \times…

We determine the structure of finitely generated groups which are quasi-isometric to symmetric spaces of noncompact type, allowing Euclidean de Rham factors If $X$ is a symmetric space of noncompact type with no Euclidean de Rham factor,…

Given a proper map f : M $\rightarrow$ Q, having cell-like point-inverses, from a manifold-without-boundary M onto an ANR Q, it is a much-studied problem to find when f is approximable by homeomorphisms, i.e., when the decomposition of M…

Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many…

In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich,…

Given a ring $R$ with center $Z(R)$, we say a linear map $f:R\rightarrow R$ is commuting if $[f(x),x]=0$ for all $x\in R$. Such a map has a standard form if there exists $\lambda\in R$ and additive $\mu:R\rightarrow Z(R)$ such that…

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show…

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the…

In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism,…

S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$ for all…

A survey of properties of the adjunction involving a semisymmetrization functor, which was suggested by J.D.H. Smith, and which maps the category of quasigroups with homotopies to the category of semisymmetric quasigroups with…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC…

For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$ Our main result is that $M(X)$ is an inverse semigroup,…

In this paper we determine all singular endomorphisms of the Hamming graph and other related graphs. The Hamming graph has vertices $\mathbb{Z}^{m}_n$ where two vertices are adjacent, if their Hamming distance is $1$. We show that its…

Given an analytic function $f=u+iv$ in the unit disk $\mathbb{D}$, Zygmund's theorem gives the minimal growth restriction on $u$ which ensures that $v$ is in the Hardy space $h^1$. This need not be true if $f$ is a complex-valued harmonic…

In this paper, we prove Huang et al.'s conjecture stated that if $f$ is a holomorphic function on $\Delta^+:=\{z\in \mathbb C \colon |z|<1,~\mathrm{Im}(z)>0\}$ with $\mathcal{C}^\infty$-smooth extension up to $(-1,1)$ such that $f$ maps…

A code $C$ in a generalised quadrangle ${\mathcal Q}$ is defined to be a subset of the vertex set of the point-line incidence graph $\varGamma$ of ${\mathcal Q}$. The minimum distance $\delta$ of $C$ is the smallest distance between a pair…

We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of…

Chung and Graham define quasirandom subsets of $\mathbb{Z}_n$ to be those with any one of a large collection of equivalent random-like properties. We weaken their definition and call a subset of $\mathbb{Z}_n$ $\epsilon$-balanced if its…