Maps Which Preserve Graphs

Periodic Maps Which Preserve a Complex Structure

Mappings which preserve regular dodecahedrons

for all x, y ∈ X . A distance r > 0 is said to be preserved by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X whenever ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is preserved by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → ...

Resolvable Maps Preserve Complete Metrizability

Let X be a Polish space, Y a separable metrizable space, and f : X → Y a continuous surjection. We prove that if the image under f of every open set or every closed set is resolvable, then Y is Polish. This generalizes similar results by Sierpiński, Vainštain, and Ostrovsky.

Quasimöbius maps preserve uniform domains

We show that if a domain Ω in a geodesic metric space is quasimöbius to a uniform domain in some metric space, then Ω is also uniform. Mathematics Subject Classification (2000). 30C65

Characterisation of Graphs Which Underlie Regular Maps on Closed Surfaces

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex Š is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incid...