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Let $x(t)$ a process with a differentiable trajectory. How do you understand $\int_0^t |\dot{x}(s)| \delta_{x(s) = 0} d s$?

megaproba
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    Dirac "functions" (which are not functions) are defined by the fact that $\int_I f(t), \delta_{t_0}(t),dt = f(t_0)$. In your case, the notation is a bit peculiar, because the condition $x(s)=0$ does not define a single point (unless you've hidden some hypotheses). Is there at least a reason for this equation $x(s)=0$ to have a finite number of solutions? or a discrete set of solutions? – PseudoNeo Jul 13 '16 at 09:48
  • According to André Preumont in Random Vibrations and Spectral Analysis, page 190, formula 10.3. Such a formula gives the average number of crossing of $0$ by $x(t)$. I think the book's explanation is a bit handwavy. I am trying to understand that from a rigorous viewpoint. – megaproba Jul 13 '16 at 10:38
  • Try using this. The answer there also mentions a number of scenarios that you may want to check. – stochasticboy321 Jul 13 '16 at 13:01
  • $$\int y(s) \delta_{x(s) = 0} d s=\sum_{s:x(s)=0}\frac{y(s)}{|x'(s)|}$$ – Did Jul 14 '16 at 00:58

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