Bound-narrowing trajectory

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This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold

Definition

Let M be a manifold and u = u(t,x) be a function \R \times M \to \R, where:

  • t denotes the time parameter, and varies in \R
  • x denotes the spatial parameter, and varies in M

In other words, u is a trajectory (or path) in the space of all functions from M to \R.

Then, u is said to be bound-narrowing if it is both max-decreasing and min-increasing or equivalently if it satisfies the following condition:

C_1 \le u(0,x) \le C_2 \forall x \in M \implies C_1 \le u(t,x) \le C_2 \forall x \in M