两点伸缩齐性度量空间

$λ$-Scale, a lambda calculus for spaces with dilations

$λ$-Scale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations.

Maps of metric spaces

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into another. See arXiv:1103.6007 for the context.

A metric characterization of snowflakes of Euclidean spaces

We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to $\mathbb R^n$ equipped with a distance $(d_{\rm E})^ε$, for some $n\in \mathbb N_0$ and $ε\in (0,1]$, where $d_{\rm E}$ is the Euclidean distance, if and only if it is locally compact, $2$-point isometrically homogeneous, and admits dilations of any factor.

Sub-riemannian geometry from intrinsic viewpoint

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian geometry may be described by…

具有伸缩的拟度量空间的代数和解析性质

Algebraic and analytic properties of quasimetric spaces with dilations

本文给出了正则次黎曼流形的局部切锥理论和这类空间之间映射的可微性的公理化方法。这种公理化的方法依赖于一个膨胀结构的概念，这是在准度量空间的一般框架中引入的。考虑准度量使我们能够覆盖一般情况下，特别是，最小的光滑性假设的向量场定义的子黎曼结构。值得注意的是，现有的度量空间的理论不能直接推广到拟度量空间。

Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds

We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, it coincides with the Naimark Dilation and only one rotation matrix is computed, when the process is non-stationary, a set of rotation matrices are computed. In particular, the peri…

Regularity properties of spheres in homogeneous groups

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context o…

等距齐性度量空间中Hilbert流形的检测

Detecting Hilbert manifolds among isometrically homogeneous metric spaces

我们在等距齐性度量空间中发现了Hilbert流形，并将所得结果应用于识别G/H型齐性空间中的Hilbert流形，其中G是可度量化的拓扑群，H是G的闭平衡子群.

Coarse classification of abelian groups and amenable shift-homogeneous metric spaces

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

具有非负曲率的齐次多项式

Homogeneous Metrics with nonnegative curvature

给定紧李群H \subset G，研究G/H上具有非负截面曲率的G-不变度量空间.对于H和G之间的一个中间子群K，我们得到了扩大K的李代数在G/H上保持非负曲率的条件.如果（K，H）是对称对，则这种扩大是可能的，这产生了许多非负弯曲齐次度量的新例子。我们提供了其他的例子空间G/H的非负弯曲齐次度量出乎意料的大家庭。

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.

齐型度量子空间上Lipschitz函数的延拓

Extension of Lipschitz Functions Defined on Metric Subspaces of Homogeneous Type

若任意度量空间$M$的度量子空间$M^{o}$带有一个加倍测度$μ$，则存在$M^{o}$上所有巴拿赫空间的Lipschitz函数到$M$上的Lipschitz函数的同时线性扩张.此外，该线性算子的范数由$μ$的加倍常数的对数控制。

On the distribution of distances in homogeneous compact metric spaces

We provide a simple proof that in any homogeneous, compact metric space of diameter $D$, if one finds the average distance $A$ achieved in $X$ with respect to some isometry invariant Borel probability measure, then $$\frac{D}{2} \leq A \leq D.$$ This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds. We then classify the cases where one of the extremes occurs. In particular any homogeneous compact metric space where $A=\frac{D}{2}$ possesses a strict antipodal property which implies in particular that the distribution of distances in…