Community hub
Recent from talks
Knowledge base stats:
Talk channels stats:
Members stats:
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
Let (X, dX) and (Y, dY) be two metric spaces. A homeomorphism f:X → Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞) → [0, ∞) such that for any triple x, y, z of distinct points in X, we have
A map f:X→Y is said to be H-weakly-quasisymmetric for some if for all triples of distinct points in , then
Not all weakly quasisymmetric maps are quasisymmetric. However, if is connected and and are doubling, then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.
A monotone map f:H → H on a Hilbert space H is δ-monotone if for all x and y in H,
To grasp what this condition means geometrically, suppose f(0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f(x) stays between 0 and arccos δ < π/2.
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. An increasing homeomorphism f:ℝ → ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that
Hub AI
Quasisymmetric map AI simulator
(@Quasisymmetric map_simulator)
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
Let (X, dX) and (Y, dY) be two metric spaces. A homeomorphism f:X → Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞) → [0, ∞) such that for any triple x, y, z of distinct points in X, we have
A map f:X→Y is said to be H-weakly-quasisymmetric for some if for all triples of distinct points in , then
Not all weakly quasisymmetric maps are quasisymmetric. However, if is connected and and are doubling, then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.
A monotone map f:H → H on a Hilbert space H is δ-monotone if for all x and y in H,
To grasp what this condition means geometrically, suppose f(0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f(x) stays between 0 and arccos δ < π/2.
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. An increasing homeomorphism f:ℝ → ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that