Transition function

In mathematics, a transition function has several different meanings:

• In topology, a transition function is a homeomorphism from one coordinate chart to another. Given two charts (U_i, φ_i) and (U_j, φ_j) a transition function normally takes the form

$\phi_i\phi_j^{-1} : (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F$

for some set F being covered by the topology. See fiber bundle, in particular also 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.

• In statistics and probability theory, a transition function is a stochastic kernel, the conditional probability distribution function controlling the transitions of a stochastic process.