Positivity-preserving trajectory: Difference between revisions

Latest revision as of 19:50, 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 positivity-preserving if the function:

$t \mapsto {inf}_{x \in M} u (t, x)$

has the property that once it crosses zero, it never becomes negative again. In other words, if there is a $t_{0}$ such that $u (t_{0}, x) \geq 0$ for all $x \in M$ , then $u (t, x) \geq 0$ for all $t \geq t_{0}, x \in M$ .

@@ Line 10: / Line 10: @@
 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 '''positivity-increasing''' if the function:
+Then, <math>u</math> is said to be '''positivity-preserving''' if the function:
 <math>t \mapsto \inf_{x \in M} u(t,x)</math>
 has the property that once it crosses zero, it never becomes negative again. In other words, if there is a <math>t_0</math> such that <math>u(t_0,x) \ge 0</math> for all <math>x \in M</math>, then <math>u(t,x) \ge 0</math> for all <math>t \ge t_0, x \in M</math>.