Transition function: Difference between revisions

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In [[mathematics]], a '''transition function''' has several different meanings:

* In topology and in particular in the theory of manifolds, a [[transition map|transition function]] between two charts of an atlas is a map which allows to pass from one chart to the other in the region where they intersect.

* In [[topology]], a '''transition function''' is a [[homeomorphism]] from one coordinate [[Atlas (topology)|chart]] to another. Given two charts (''U''_''i'', φ_''i'') and (''U''_''j'', φ_''j'') a transition [[function (mathematics)|function]] normally takes the form

::<math>\phi_i\phi_j^{-1} : (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F</math>

:for some [[Set (mathematics)|set]] ''F'' being covered by the [[topology]]. See [[fiber bundle]], in particular also [[Vector_bundle#Transition_functions|vector bundle]], and [[atlas (topology)]] for additional details.

* In [[computing]], a '''transition function''' is the function that defines the state transitions of a [[Turing machine]], [[finite-state machine]], or [[cellular automaton]].