This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distri...

Abstract. The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An...

A multiple-valued function f : X → QQ(Y ) is essentially a rule assigning Q unordered and non necessarily distinct elements of Y to each element of X. We study the Lipschitz extension problem in this context by using two general Lipschitz extension theorems recently proved by U. Lang and T. Schlichenmaier. We prove that the pair ` X,QQ(Y ) ́ has the Lipschitz extension property if Y is a Banach ...

We prove the existence of a (random) Lipschitz function F : Z → Z such that, for every x ∈ Z, the site (x, F (x)) is open in a site percolation process on Z. The Lipschitz constant may be taken to be 1 when the parameter p of the percolation model is sufficiently close to 1.

The aim of this paper is to introduce and study a new concept ofstrong double $(A)_ {Delta}$-convergent sequence offuzzy numbers with respect to an Orlicz function and also someproperties of the resulting sequence spaces of fuzzy   numbers areexamined. In addition, we define the double$(A,Delta)$-statistical convergence of fuzzy  numbers andestablish some connections between the spaces of stron...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

We prove that, if a compact subspace of a metric space can be covered by a family C of bi–Lipschitz local collars, then it admits also a bi–Lipschitz global collar, whose complexity, measured by the so–called weak bi–Lipschitz constant, depends only on C. If the compactness condition is dropped, a slightly weaker version of this result remains valid, provided C is uniform in a suitable natural ...

The aim of this lecture is to outline the gluing of embeddings at different scales described in James Lee’s paper Distance scales, embeddings, and metrics of negative type from SODA 2005 [1]. We begin by recalling some definitions. A map f : X → Y of metric spaces (X, dX ) and (Y, dY ) is said to be C-Lipschitz if dY (f(x), f(y)) ≤ CdX(x, y) for all x, y ∈ X. The infimum of all C such that f is...

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the CuckerSmale model, adding noise to the behavior of individuals. The difficu...