boundarymaintained

Boundarymaintained is a term used to describe a class of methods and properties in mathematics, computer science, and engineering in which the boundary of a domain is kept invariant under a transformation or process. The concept is often important when the behavior at the boundary is critical for safety, stability, or correctness, such as physical containment, numerical stability, or boundary-value problems.

In formal terms, let D be a domain in R^n with boundary ∂D. A mapping f: D → D is boundarymaintained if f(∂D) ⊆ ∂D and, typically, f(D) ⊆ D. In many formulations, interior points map to interior points, making the boundary a limit set or invariant boundary. In discrete systems, a state-transition relation preserves boundary states if, whenever a boundary state is present, successors remain on or inside the boundary.

Examples include a reflecting boundary in a physical simulation, where particles bounce at ∂D rather than leaving D; a mesh parametrization that keeps edge vertices on the boundary; and control policies that prevent boundary states from being crossed by enforcing constraint adherence.

Applications of boundarymaintained concepts appear in safety verification, computational fluid dynamics, robotics, and network security, where maintaining the boundary corresponds to respecting defined limits and invariants. The idea is closely related to invariant sets and boundary conditions, providing a framework for ensuring systems behave within prescribed constraints. See also invariant set, boundary conditions, preserving maps, and containment.