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...

The Jacobi polynomials induce a translation operator on function spaces on the interval [−1, 1]. For any homogeneous Banach space B w.r.t. this translation, we can study the according little and big Lipschitz spaces, lipB(λ) and LipB(λ), respectively. The big Lipschitz spaces are not homogeneous themselves. Therefore we introduce semihomogeneous Banach spaces w.r.t. Jacobi translation, of which...

using a bessel generalized translation, we obtain an analog of titchmarsh's theorem for the bessel transform for functions satisfying the lipschitz condition in the space $mathrm{l}_{p,alpha}(mathbb{r}_{+})$, where $alpha>-frac{1}{2}$ and $1

using a generalized spherical mean operator, we obtain the generalizationof titchmarsh's theorem for the dunkl transform for functions satisfyingthe lipschitz condition in l2(rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

The goal of this article is to develop a framework for large margin classification in metric spaces. We want to find a generalization of linear decision functions for metric spaces and define a corresponding notion of margin such that the decision function separates the training points with a large margin. It will turn out that using Lipschitz functions as decision functions, the inverse of the...

In this note we consider the computability of the solution of the initial-value problem for ordinary differential equations with continuous right-hand side. We present algorithms for the computation of the solution using the “thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of a differential equa...

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in (Cn, 0). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs (X1, 0...

In this paper, we propose the Lipschitz margin ratio and a new metric learning framework for classification through maximizing the ratio. This framework enables the integration of both the inter-class margin and the intra-class dispersion, as well as the enhancement of the generalization ability of a classifier. To introduce the Lipschitz margin ratio and its associated learning bound, we elabo...