相关论文: On the distance between homotopy classes in $W^{1/…

We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special…

We introduce an equivalence relation on the space $W^{1,1}(\Omega;{\mathbb S}^1)$ which classifies maps according to their "topological singularities". We establish sharp bounds for the distances (in the usual sense and in the Hausdorff…

Let $V$ be an inner product space, and $x, y \in V$; the conjecture is made that, for any $p \in [1, \infty]$, the function $d_p(x, y):=\|x-y\|/(\|x\|^p+ \|y\|^p)^{1/p}$ is a distance on $V$.

Homotopic distance $\D$ as introduced in \cite{MVML} can be realized as a pseudometric on $\mathrm{Map}(X,Y)$. In this paper, we study the topology induced by the pseudometric $\D$. In particular, we consider the space…

We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our…

The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the…

For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $\gamma_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with…

Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of…

We examine the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w(x)=\operatorname{dist}(x,E)^{-\alpha}$ belongs to the Muckenhoupt class $A_1$, for some $\alpha>0$, if and only if…

In this note, we show how the results of Mazowiecka--Schikorra, combined with those of Bourgain--Brezis--Mironescu, imply the existence of minimal maps of degree one in $ W^{\frac{1}{p},p}(\mathbb{S}^1,\mathbb{S}^1) $ for $ p \in [p', 2] $,…

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$,…

In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a…

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that…

Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such…

Kusner asked if $n+1$ points is the maximum number of points in $\mathbb{R}^n$ such that the $\ell_p$ distance $(1<p<\infty)$ between any two points is $1$. We present an improvement to the best known upper bound when $p$ is large in terms…

For the unit sphere S^d in Euclidean space R^(d+1), we show that for d-1<s<d and any N>1, discrete N-point minimal Riesz s-energy configurations are well separated in the sense that the minimal distance between any pair of distinct points…

We consider a differential inclusion on a manifold, defined by a field of open half-spaces whose boundary in each tangent space is the kernel of a one-form. We make the assumption that the corank one distribution associated to the kernel is…

We provide a local theory for the optimization of the Hausdorff distance between a polytope and a zonotope. To do this, we compute explicit local formulae for the Hausdorff function $d(P, -) : Z_n \to \mathbb{R}$, where $P$ is a fixed…

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The…

Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If ${H^p}\left( \mathbb{D}…