Convergence of Cauchy sequences for the covariant Gromov–Hausdorff propinquity

The covariant Gromov–Hausdorff propinquity is a distance on Lipschitz dynamical systems over quantum compact metric spaces, up to equivariant full quantum isometry. It is built from the dual Gromov–Hausdorff propinquity which, as its classical counterpart, is complete. We prove in this paper several sufficient conditions for convergence of Cauchy sequences for the covariant propinquity and apply it to show that many natural classes of dynamical systems are complete for this metric.     

Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces

In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive self-mapping T of a hyperconvex metric space (or a CAT(0) space) X, we look for a solution to the problem of existence of z∈E⊂X such that

     

Hereditary tree growth and Lévy forests

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space $\mathbb{T}$ of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in $\mathbb{T}$ and limit theorems for growing families of trees. We apply these results to Galton–Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton–Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Lévy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Lévy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton–Watson trees, including the super-critical cases.     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

完备凸度量空间中(次)相容和集值广义非扩张映象的公共不动点定理

本文在完备凸度量空间中,利用集值和单值映象(次)相容的一些条件,建立了数值广义非扩张映象存在公共不动点的一个充要条件和一个充分条件.我们的结果改进、扩充和发展了文[2～7]中的主要结果.     

On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet‐Urysohn space. Although, in its absence even ? may fail to be a sequential space. Our goal in this paper is to discuss under which set‐theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ?, are classes of Fréchet‐Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ? if and only if the axiom of countable choice holds for families of subsets of ?, and every metric space has a unique ‐completion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

We present a new and self‐contained theory for mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld's diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel we also study the corresponding mapping properties and derive explicit solutions together with certain regularity results.     

Berezin Quantization and Reproducing Kernels on Complex Domains

Let be a non-compact complex manifold of dimension , a Kähler form on , and the reproducing kernel for the Bergman space of all analytic functions on square-integrable against the measure . Under the condition

F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just and a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as . This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in . Along the way, we also fix two gaps in Berezin's original paper, and discuss, for a domain in , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure .

     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.