transition function

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transition function (plural transition functions)

1. (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.
2. (differential geometry) A homeomorphism that bijects between the subsets of the images of two overlapping coordinate charts that are shared in the preimage: For ${\displaystyle (U_{1},\varphi _{1})}$ and ${\displaystyle (U_{2},\varphi _{2})}$ coordinate charts with ${\displaystyle U_{1}\cap U_{2}\neq \emptyset }$, the transition functions ${\displaystyle \varphi _{12}=\varphi _{2}\circ \varphi _{1}^{-1}}$ and its inverse ${\displaystyle \varphi _{21}=\varphi _{1}\circ \varphi _{2}^{-1}}$ may be constructed.

   Synonym: transition map

   • 2007, Dominic D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry (Oxford graduate texts in mathematics)‎^[1], Oxford University Press, →ISBN, →LCCN, page 76:

     If ${\displaystyle (U_{1},\psi _{1})}$ and ${\displaystyle (U_{2},\psi _{2})}$ are two complex charts, then the transition function is ${\displaystyle \psi _{12}:\psi _{1}(U_{1}\cap U_{2})\rightarrow \psi _{2}(U_{1}\cap U_{2})}$, given by ${\displaystyle \psi _{12}=\psi _{2}\circ \psi _{1}^{-1}}$. We say ${\displaystyle M}$ is a complex manifold if it has an atlas of complex charts ${\displaystyle (U,\psi )}$, such that all the transition functions are holomorphic, as maps from ${\displaystyle \mathbb {C} ^{m}}$ to itself.

• state machine
• coordinate chart
• atlas