# Difference between revisions of "Max-decreasing trajectory"

From Diffgeom

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

− | Then, <math>u</math> is said to be '''max- | + | Then, <math>u</math> is said to be '''max-decreasing''' if the function: |

<math>t \mapsto \sup_{x \in M} u(t,x)</math> | <math>t \mapsto \sup_{x \in M} u(t,x)</math> |

## Revision as of 04:38, 8 April 2007

## Definition

Let be a manifold and be a function , where:

- denotes the time parameter, and varies in
- denotes the spatial parameter, and varies in

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

Then, is said to be **max-decreasing** if the function:

is a monotone decreasing function.

The corresponding notion is of a **min-increasing trajectory** -- viz a trajectory where the minimum (or infimum) keeps increasing.