Natural pseudodistance


Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb{R})\ , (N,\psi:N\to \mathbb{R})\ is the value \inf_h \|\varphi-\psi\circ h\|_\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to the manifold N\ and \|\cdot\|_\infty\ is the supremum norm. If M\ and N\ are not homeomorphic, then the natural pseudodistance is defined to be \infty\ . It is usually assumed that M\ , N\ are C^1\ closed manifolds and the measuring functions \varphi,\psi\ are C^1\ . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from M\ to N\ .

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function \varphi\ takes values in \mathbb{R}^m\ [1].


Main properties

It can be proved [2] that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the same measuring function) divided by a suitable positive integer k\ . If M\ and N\ are surfaces, the number k\ can be assumed to be 1\ , 2\ or 3\ [3]. If M\ and N\ are curves, the number k\ can be assumed to be 1\ or 2\ [4]. If an optimal homeomorphism \bar h\ exists (i.e., \|\varphi-\psi\circ \bar h\|_\infty=\inf_h \|\varphi-\psi\circ h\|_\infty\ ), then k\ can be assumed to be 1\ [2].

References

  1. ^ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society - Simon Stevin, 6:455-464, 1999.
  2. ^ a b Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
  3. ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
  4. ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.

See also


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Size function — Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half plane to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and… …   Wikipedia

  • List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia

  • Size pair — In size theory, a size pair is a pair (M,varphi), where M is a compact topological space and varphi:M o mathbb{R}^k is a continuous function Patrizio Frosini, Claudia Landi, Size theory as a topological tool for computer vision , Pattern… …   Wikipedia

  • Size homotopy group — The concept of size homotopy group is the anologous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,varphi) is given, where M is a closed manifold of class C^0 and… …   Wikipedia

  • Matching distance — In mathematics, the matching distance[1][2] is a metric on the space of size functions. Example: The matching distance between …   Wikipedia

  • Measuring function — In size theory, a measuring function is a continuous function from a topological space M to mathbb{R}^k . Patrizio Frosini, Claudia Landi, Size theory as a topological tool for computer vision , Pattern Recognition And Image Analysis, 9(4):596… …   Wikipedia