12/17/2009, 08:33 AM
It seems to me, that
Sinusoid is only a small angle approximation of any pendullum, top , elastica ( which are all governed by the same equations) . So sinusoidal, Fourier representation of phenomena lacks most of its real properties at high power, high amplitude oscillations.
If the signal pulse front is fast increasing so that its NOT Fourier decomposable in linear superposition of infinite range monohromatic modes ( that is , increasing faster than exponential , e.g. like t^t instead of e^t) than such signal is NOT decomposable in linear superposition of Fourier modes and hence, dispersion relation of such physical phenomena (energy/impulse) may be totally different than usual linear one. And quite rich variety of them.
That is where I think the usefulness of tetration etc is, to study such ultra fast physical phenomena.
I still do not know what differential equation has a solution f(t) = t^t?
Best regards,
Ivars
Sinusoid is only a small angle approximation of any pendullum, top , elastica ( which are all governed by the same equations) . So sinusoidal, Fourier representation of phenomena lacks most of its real properties at high power, high amplitude oscillations.
If the signal pulse front is fast increasing so that its NOT Fourier decomposable in linear superposition of infinite range monohromatic modes ( that is , increasing faster than exponential , e.g. like t^t instead of e^t) than such signal is NOT decomposable in linear superposition of Fourier modes and hence, dispersion relation of such physical phenomena (energy/impulse) may be totally different than usual linear one. And quite rich variety of them.
That is where I think the usefulness of tetration etc is, to study such ultra fast physical phenomena.
I still do not know what differential equation has a solution f(t) = t^t?
Best regards,
Ivars