Min-increasing trajectory - Revision history

Vipul: 4 revisions

2008-05-18T19:49:46Z

4 revisions

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Vipul at 11:04, 8 April 2007

2007-04-08T11:04:10Z

← Older revision		Revision as of 11:04, 8 April 2007
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	is a monotone increasing function. (the function defined above is called the [[timewise-min function]] for <math>u</math>).		is a monotone increasing function. (the function defined above is called the [[timewise-min 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 '''max-decreasing trajectory''' -- viz a trajectory where the maximum (or supremum) keeps decreasing.

Vipul at 10:54, 8 April 2007

2007-04-08T10:54:20Z

← Older revision		Revision as of 10:54, 8 April 2007
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	<math>t \mapsto \inf_{x \in M} u(t,x)</math>		<math>t \mapsto \inf_{x \in M} u(t,x)</math>

	is a monotone increasing function.		is a monotone increasing function. (the function defined above is called the [[timewise-min 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.

Vipul at 04:41, 8 April 2007

2007-04-08T04:41:45Z

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			{{trajectory property}}

	==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:		Then, <math>u</math> is said to be '''min-increasing''' if the function:

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

	is a monotone ~~decreasing~~ function.		is a monotone increasing function.

	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.

Vipul at 04:06, 8 April 2007

2007-04-08T04:06:12Z

New page

==Definition==

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

* <math>t</math> denotes the time parameter, and varies in <math>\R</math>
* <math>x</math> denotes the spatial parameter, and varies in <math>M</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:

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

is a monotone decreasing function.

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