Meaning of transition function | Babel Free

Definitions

1. A function from (state, input symbol) to state describing what state to move to on receiving a given input in a given state.
2. A homeomorphism that bijects between the subsets of the images of two overlapping coordinate charts that are shared in the preimage: For (U_1,φ₁) and (U_2,φ₂) coordinate charts with U_1∩U_2≠∅, the transition functions φ₁₂=φ₂∘φ₁⁻¹ and its inverse φ₂₁=φ₁∘φ₂⁻¹ may be constructed.

Equivalents

Examples

“If (U#95;1,#92;psi#95;1) and (U#95;2,#92;psi#95;2) are two complex charts, then the transition function is #92;psi#95;#123;12#125;#58;#92;psi#95;1(U#95;1#92;capU#95;2)#92;rightarrow#92;psi#95;2(U#95;1#92;capU#95;2), given by #92;psi#95;#123;12#125;#61;#92;psi#95;2#92;circ#92;psi#95;1#123;-1#125;. We say M is a complex manifold if it has an atlas of complex charts (U,#92;psi), such that all the transition functions are holomorphic, as maps from #92;mathbb#123;C#125;ᵐ to itself.”

CEFR level