overgangsfunktion

Overgangsfunktion, or transition function, is a mathematical concept used to describe how a system moves from one state to another as time, input, or conditions change. It is a general idea that appears in several disciplines, with specific definitions that reflect the nature of the system being modeled. In many contexts the transition function is a map that assigns a successor state to every admissible combination of current state and input.

In differential geometry and topology, the transition function is a key component of an atlas on a manifold. For two coordinate charts (U_i, φ_i) and (U_j, φ_j), the transition function is the map φ_j ∘ φ_i^{-1} defined on φ_i(U_i ∩ U_j), taking points in one coordinate representation to another. For smooth manifolds, these transition functions are required to be smooth (and invertible with a smooth inverse), ensuring consistency across overlapping charts and enabling well-defined differentiable structure.

In automata theory and computer science, a transition function describes how a machine changes state in response to inputs. For a deterministic finite automaton, δ: Q × Σ → Q specifies the next state for each current state and input symbol. Non-deterministic automata use a transition function to map to a set of possible next states. The transition function encodes the computational behavior of the system.

In stochastic processes, the transition function (or transition probability function) defines the likelihood of moving from one state to another in a single step. In Markov chains, this is expressed as a transition matrix P with P(i, j) representing the probability of transitioning from state i to state j, with row sums equal to one.

Across these contexts, transition functions formalize the evolution of systems under changing conditions, inputs, or time.