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- Prime_end abstract "In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding a circle in an appropriate way.The concept of prime ends was introduced by Constantin Carathéodory to describe the boundary behavior of conformal maps in the complex plane in geometric terms. The theory has been generalized to more general open sets, too.Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:If ƒ maps the unit disk conformally and one-to-one onto the domain B, it induces a one-to-one mapping between the points on the unit circle and the prime ends of B.The set of prime ends of the domain B is the set of equivalence classes of chains of arcs converging to a point on the boundary of B.In this way, a point in the boundary may correspond to many points in the prime ends of B, and conversely, many points in the boundary may correspond to a point in the prime ends of B. (for a more precise definition of \"chains of arcs\" and their equivalence classes see the references)The expository paper of Epstein provides a good account of this theory with complete proofs. It also introduces a definition which make sense in any open set and dimension. See also Milnor for a very accessible introduction in the context of complex dynamical systems.".
- Prime_end wikiPageID "23404066".
- Prime_end wikiPageLength "2231".
- Prime_end wikiPageOutDegree "10".
- Prime_end wikiPageRevisionID "678436343".
- Prime_end wikiPageWikiLink Ball_(mathematics).
- Prime_end wikiPageWikiLink Category:Compactification.
- Prime_end wikiPageWikiLink Compactification_(mathematics).
- Prime_end wikiPageWikiLink Complex_plane.
- Prime_end wikiPageWikiLink Conformal_map.
- Prime_end wikiPageWikiLink Constantin_Carathéodory.
- Prime_end wikiPageWikiLink Injective_function.
- Prime_end wikiPageWikiLink Mathematics.
- Prime_end wikiPageWikiLink Unit_circle.
- Prime_end wikiPageWikiLink Unit_disk.
- Prime_end wikiPageWikiLinkText "prime end".
- Prime_end id "l/l058860".
- Prime_end title "Limit elements".
- Prime_end wikiPageUsesTemplate Template:Citizendium.
- Prime_end wikiPageUsesTemplate Template:More_footnotes.
- Prime_end wikiPageUsesTemplate Template:Springer.
- Prime_end wikiPageUsesTemplate Template:Topology-stub.
- Prime_end subject Category:Compactification.
- Prime_end hypernym Method.
- Prime_end type Software.
- Prime_end type Source.
- Prime_end comment "In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding a circle in an appropriate way.The concept of prime ends was introduced by Constantin Carathéodory to describe the boundary behavior of conformal maps in the complex plane in geometric terms.".
- Prime_end label "Prime end".
- Prime_end sameAs Q7243367.
- Prime_end sameAs m.06wb1xb.
- Prime_end sameAs Q7243367.
- Prime_end wasDerivedFrom Prime_end?oldid=678436343.
- Prime_end isPrimaryTopicOf Prime_end.