# Difference between revisions of "Max-decreasing trajectory"

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

− | is a monotone decreasing function. | + | is a monotone decreasing function. (The function defined above is termed the [[timewise-max function]] for <math>u</math>). |

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

## Latest revision as of 19:48, 18 May 2008

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

## 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 function defined above is termed the timewise-max function for ).

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