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 Physical model of (infinite) tetration=(NON-isotropic) turbulence Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/17/2009, 08:52 PM (This post was last modified: 03/18/2009, 06:49 PM by Ivars.) I will give it a try: Let us take first order homogenous delay differential equation without dependence of derivative of solution $y' (t)$ on $y(t)$ at the same moment of time $t$: $y'(t) +\alpha y(t-T)=0$ Here T is delay, which shows how values of solution $y(t)$ T moments in past ( or future if T<0) impact derivative of solution $y'(t)$ at current moment $t$. Let us introduce new function $z(t)$ such that : $\ln z(t) = -y(t)$ then $z(t) = e^{-y(t)}$ As is known from literature, solution of such delay equation is given by : $y(t) = \sum_{k=-\infty}^{\infty} C_k *e^{{1/T}*W_k(-\alpha T) t}$ So $z(t) = e^{-y(t)}= e^{\sum_{k=-\infty}^{\infty} -C_k *e^{{1/T}*W_k(-\alpha T) t}}$ Where $W_k$ is a k-th branch of Lambert W function. To reach the solution I am looking for, of form: $e^{-iwt*e^{iwt}}$ Coefficients $C_k$ have to be: $C_k= {1/T}*W_k(-\alpha T) t$ so $z(t) = e^{\sum_{k=-\infty}^{\infty} -{1/T}*W_k(-\alpha T) t *e^{{1/T}*W_k(-\alpha T) t}}$ This can be expressed as infinite product, with each term of sum as exponent of one multiplier $e$ in this infinite product. From here, obviously: $-iwt = -{1/T}*W_k(-\alpha T) t$ so ${1/T}= I$ $T=1/I=-I$ $w= W_k (-\alpha/ I)$ Now if we look back to original delay equation, this means that solution of a form: $e^{-Iwt*e^{Iwt}$ is one of particular solutions corresponding to particular branch of W function of a homogenous logarithmic first order differential equation with IMAGINARY DELAY $T=-I$. $e^{Iwt*e^{-Iwt}}$ corresponds to Imaginary delay $T=1/I$. We can generalize this to imaginary delay $T= +-1/(I\tau)$. (Not sure if + sign is OK). Then we have particular solutions of : $- (\ln z(t))' - \alpha \ln z(t+-1/(I\tau) )=0$ In form: $z_k(t)= e^{+-w_k t/(I\tau) )*e^{-+w_k t/I\tau }$ Where $w_k = W_k (-+\alpha/ (I \tau) )$ I hope there are not too many mistakes I think I could not choose coefficients $C_k$ so arbitrary, it must have an impact on initial conditions or so called preshape function $\phi(t)$ for times $ln z(t) =\phi (t)$ $t [0; +- I\tau]$ which I do not know how to calculate. Next question is how these logarithmic differential delay equations (and thus processes they represent) with imaginary delay have to be nested to produce tetration and , perhaps, turbulence. In case of turbulence, the statistical character of processes will have to be added, somewhere, so that we deal with mean values , distributions and structure functions. (probably, In projective space as that is the space of turbulent time). Ivars « Next Oldest | Next Newest »

 Messages In This Thread Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/13/2009, 07:27 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/18/2009, 07:29 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/20/2009, 10:31 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/25/2009, 10:27 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by bo198214 - 02/26/2009, 12:01 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/26/2009, 06:53 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 03/17/2009, 08:51 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 03/17/2009, 08:52 PM

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