transition function

English Wikipedia has an article on:

transition function

Wikipedia

This entry needs pronunciation information. If you are familiar with the IPA or enPR then please add some!

transition function (plural transition functions)

(computing theory) A function from (state, input symbol) to state describing what state to move to on receiving a given input in a given state.
(differential geometry) A homeomorphism that bijects between the subsets of the images of two overlapping coordinate charts that are shared in the preimage: For $(U_{1},\varphi _{1})$ $(U_{1},\varphi _{1})$ and $(U_{2},\varphi _{2})$ $(U_{2},\varphi _{2})$ coordinate charts with $U_{1}\cap U_{2}\neq \emptyset$ $U_{1}\cap U_{2}\neq \emptyset$ , the transition functions $\varphi _{12}=\varphi _{2}\circ \varphi _{1}^{-1}$ $\varphi _{12}=\varphi _{2}\circ \varphi _{1}^{-1}$ and its inverse $\varphi _{21}=\varphi _{1}\circ \varphi _{2}^{-1}$ $\varphi _{21}=\varphi _{1}\circ \varphi _{2}^{-1}$ may be constructed.
Synonym: transition map
- 2007, Dominic D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry (Oxford graduate texts in mathematics)‎, Oxford University Press, →ISBN, →LCCN, page 76:
  If $(U_{1},\psi _{1})$ and $(U_{2},\psi _{2})$ are two complex charts, then the transition function is $\psi _{12}:\psi _{1}(U_{1}\cap U_{2})\rightarrow \psi _{2}(U_{1}\cap U_{2})$ , given by $\psi _{12}=\psi _{2}\circ \psi _{1}^{-1}$ . We say $M$ is a complex manifold if it has an atlas of complex charts $(U,\psi )$ , such that all the transition functions are holomorphic, as maps from $\mathbb {C} ^{m}$ to itself.