skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • Home
  • Search Results
  • Page 1 of 1

Search for: All records

Award ID contains: 2154918

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. ; ; (, Revista Matemática Iberoamericana)
    In this paper, we generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by bi-Lipschitz curves. 
    more » « less
    Full Text Available 
  2. ; (, Journal d'Analyse Mathématique)
    The Decomposition Problem in the class $$LIP(\S^2)$$ is to decompose any bi-Lipschitz map $$f:\S^2 \to \S^2$$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set $$X$$ of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface $$\S^2 \setminus X$$. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that $$X \subset \S^2$$ is uniformly disconnected if and only if the Riemann surface $$\S^2 \setminus X$$ has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest. 
    more » « less
    Full Text Available 
  3. ; (, Journal of Logic and Analysis)
    The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of $$\R^N$$ is contained on a curve of finite length. We show that for finite sets, the algorithm constructed in \cite{Schul-Hilbert,BNV} that solves the Analyst's Traveling Salesman Problem has polynomial time complexity and we determine the sharp exponent. 
    more » « less
    Full Text Available 
  4. ; (, Annales Fennici Mathematici)
    An infinite iterated function system (IIFS) is a countable collection of contraction maps on a compact metric space. In this paper we study the conditions under which the attractor of such a system admits a parameterization by a continuous or Hölder continuous map of the unit interval. 
    more » « less
    Full Text Available