Difference between revisions of "Max-decreasing trajectory"

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{{trajectory property}}
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==Definition==
 
==Definition==
  
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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-reducing''' if the function:
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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>
  
is a monotone decreasing function.
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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 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 max-decreasing if the function:

t \mapsto \sup_{x \in M} u(t,x)

is a monotone decreasing function. (The function defined above is termed the timewise-max function for u).

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