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 Physical model of (infinite) tetration=(NON-isotropic) turbulence Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/13/2009, 07:27 AM (This post was last modified: 02/19/2009, 09:52 AM by Ivars.) Hello, I am still around, but have been looking for physical models of tetration just to have SOME intuitive understanding what it may represent. Here is how far I have got: I truly think that Nature has more than logarithms to give us for improving computation speed,since it ITSELF somputes much faster. If we take a truly continuous medium based system which has e.g infinite number of nested phase transitions and within all these phase transition between them the decay time behaves exponentially, if we take them all together, as a whole, we get infinitely iterated exponential, or infinite tetration which will characterize the system. Well, infinite tetration ( and its inverse, self root) exibits enormous speeds and very peculiar properties, like returning purely imaginary or complex results for iteration of purely real arguments ( as analytical continuation). As long as Complex math is true, this means existance of such systems is possible (e.g turbulence of an infinitely continuous dissipationless fluid , or , one which dissipates from real value of parameter into imaginary ( e.g. spinor)). I think turbulence in general would be the right place to use also finite tetration (when number of scales is limited due to existance of quanta/discreteness in the system) and other hyperfast functions, since it involves multiple nested time/space scales. Ivars Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/18/2009, 07:29 PM (This post was last modified: 02/23/2009, 04:33 PM by Ivars.) If we have a process whose observed single appearance (n=1) has Poisson statistics: $P= \lambda*e^{-\lambda}$ Then, it can be either result of simple Poisson process, or, process whose logarithm has Poisson distribution (and some mechanism hidden to us performs logarithm operation on input process) - so called log Poisson process: (such processes have recently been involved in models of turbulence (Extended Self Similarity) and Multiplicative ( chain, cascade) stochastic processes). $ln(W) = \lambda*e^{-\lambda}$ The obvious distribution for process W itself is then (without norming , just to see the functional form) : $W = e^{{\lambda}*e^{-\lambda}}$ if we substiture then $e^{\lambda} = z$ the log-Poisson process has distribution: $W=z^{(1/z)}$ -self root which is so familiar from infinite tetration as it is its inverse! $h(W) = h(z^{(1/z)}) = z$ So we can say that: Self -root represents a distribution for a log-Poisson process W itself (unnormed-I do not know how to do norming) ( process W whose logarithm has Poisson distribution); What is infinite tetration then? Since infinite tetration work on both z^(1/z) and (1/z)^z returning either z or 1/z , we can say that infinite tetration returns a pair of conjugate values linked to log-Poisson process , so that log-Poisson process underlying $W$ distribution is split into 2 parts: $z= e^{\lambda}$ and $1/z= e^{-\lambda}$ From these we can see that these conjugate values lead to a pair of growth/decay process..They are also "memoryless" processes, each of them in opposite direction ( if we consider $\lambda$ as "time" - memoryless towards past and predictionless towards future ( a well known property of exponential process- whereever You are on it, it looks the same). I guess that if we had a process which randomly grows as e^x and has a random instantenous feedback e^-x in each scale (value) x (and vice versa) than it could end up with log-Poisson distribution or distribution characterized by a pair of conjugate values. The above can be extended to $\lambda$ being imaginary and complex, thus leading to a notion that underlying math behind complex mathematics is actually stochastic, random, and e.g. $I=e^{I*pi/2} ,-I= e^{-I*pi/2 }$ are exactly such a pair of growth/decay speed values, more so, the distribution $W$ in this case is invariant relative to order of growth/decay process- any order as long as they oscillate between themselves will lead to the same distribution- which is characterized in real domain by its value, $I^{(1/I)}=(1/I)^I= e^{\pi/2}=4,81...$. Which in turn may explain the true reason behind amazing correlation between 2D turbulence and conformality of functions of complex argument. Ivars Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/20/2009, 10:31 PM (This post was last modified: 02/21/2009, 11:31 AM by Ivars.) I was little wrong about spinors ( though they have a relation , obviously, to this). At least in pure imaginary case , the values obtained by infinite tetration $h( e^{pi/2}) = +- I$are propagators ( of imaginary time) and their further research (further research of analytical properties of tetration) has to be done via Green function, a 2 point function!Yep! I think Green function can be applied sucessfully also to real and complex , may be even quaternion etc . arguments z of h(z), but I am not sure. http://en.wikipedia.org/wiki/Green's_fun...dy_theory) But there is a hitch that these things might involve 2D complex projective space (and not just complex projective plane, plane in projective space is a very misleading name.... Argand plane will not be enough. The "curvature" of complex projective 2D space can be either 1 when its analoque to Argand plane, a real or imaginary constant , or a function of a special kind.) This "curvature" appears in expression for distances /angles for complex projective 2D space, given by F.Klein $distance (or angle) = {1/ln \lambda(\tau)} *ln (crossratio)$ The "area'" "curvature" of complex projective 2D space is then: $c^2= ({1/ln \lambda(\tau)})^2$ Actually, one has to take multiplication of "distance" and angle constants since they both together characterize the space, c^2 =c1*c2 . We can also define another parameter, helicity of complex projective 2D space: $h= c1/c2$ Which is a division of " distance" constant c1 by angle constant c2. From this it is obvious that in general, c1/c2 is a ratio of curvature /torsion of some other space, space of all Complex Projective 2D spaces with different "distance"' and angle constants c1 and c2. Now, if c=1 we get Argand plane. In this case c^2 = 1 , I guess, so it is either c1, c2=1, or c1, c2=-1 ( corresponding to Argand halfplanes) If c=const we get various complex projective 2D spaces with curvature $c^2 = ({1/ln \lambda})^2$ Since in Quantum physics we have operator $\Omega(t)$ which is a multiplication of propagators , it seems that in more general case this operator is a function of "curvature" of corresponding Complex projective 2D space: $\Omega(t) = f(c^2) = f(({1/ln \lambda(\tau)})^2)$ However, if c= function of $\tau$ the curvature becomes quite intricate, but it seems that that has been already solved, since in definition of projective "distance" is not unique, as crossratio can have 24 different values in 6 groups by 4 , or, alternatively 4 groups by 6. So in any arbitrary complex projective 2D space, (not just Argand conjugate halfplanes which is just a degenerate case of general complex projective 2D space) the "distance " between any 2 points has 24 values which may not be equal depending on "curvature" of complex projective 2D space. These 24 different values seem to be adressed by Dedekind Eta function and involved invariants $g2$ and $g3$ which is homogenous of order -4 and -6 respectively. So it not so simple, but it seems that most things have been already adressed. Except few ( like turbulence of time) . Excuse me for little stretching of topic and not being exact ( yet) , but if You can offer any help, please do. The appearance of modular functions in relation to tetration is not at all a surprise to me. This model of curved complex projective space does not involve any other metrical issues as those arising in transformations of various invariants (crossratio is a projective invariant (invariant to Mobius transfromations) , Klein invariant is invariant to Unimodular Mobius transformations etc ( g2, g3 and corresponding Discriminant and eta function) - that means that to work with these, ordering present on real number line is not needed explicitly, at least not as a starting point. Only in case c1=1 and c2=-1 (Argand half Planes) the ordering of numbers becomes the same as on the real number line, but that is so so so specific subcase that to focus on it too much is just misleading. Ivars Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/25/2009, 10:27 PM (This post was last modified: 02/25/2009, 10:31 PM by Ivars.) Here is one example of article going into tetration (iterated exponential - Authors do not notice it explicitly ) in probabilities of critical processes - of course, including turbulence. Amazingly, one of constants in exponent of probability distribution function is $\pi/2$. Universal fluctuations in correlation functions Ivars bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 02/26/2009, 12:01 AM Ivars Wrote:Here is one example of article going into tetration (iterated exponential - Authors do not notice it explicitly ) I only see a two times iterated exponential on page 3745. Do you mean that with tetration??? Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/26/2009, 06:53 PM (This post was last modified: 02/27/2009, 09:34 PM by Ivars.) bo198214 Wrote:I only see a two times iterated exponential on page 3745. Do you mean that with tetration??? Oops. Somehow it looked very promising even though it contains just: $e^x/{e^{e^x}}$ I hoped to see $(e^x)^{(e^{-x})}$ But since Poisson distribution is : $x* e^{-x} = x/e^x$ There could be such pattern for (nested) critical processes: $\ln(\ln(x))-> \ln(x)/x-> x/e^x-> e^x/{e^{e^x}}-> e^{e^x}/e^{e^{e^x}}->$ Also, $e^x$ is just a special value for $x^x=x[4]2$ Still, appearance of iterated exp indicates a new kind of process, and probably the by nesting it one finally arrives at tetration. Not yet though... What should be done to make Area of $(e^x)^{(e^{-x})}$=1? Ivars Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/08/2009, 10:35 PM (This post was last modified: 03/11/2009, 07:59 PM by Ivars.) If there is a statistic process whose distribution w function logarithm $\ln w$ is Poisson distribution: $\ln w_k=T_k(z)*e^{-T_k(z)}$ with mean value $<\ln w_k>= T_k(z)$. Here $T(z)=\sum_{n=1}^{\infty}{n^{n-1}*z^n/n!$ Is the Euler tree function. k denotes branches. $w_k$ is a probability of state k. Then the process w istself probability distribution (unnormed) is : $w_k = e^{T_k(z)*e^{-T_k(z)}}$ with, mean value $=e^{T_k(z)}$ If there is a physical (or mathematical process ) whose probality distribution (distribution function) is: $w_k$ then its entropy S is defined as mean value of logarithm of logarithm of its distribution function: $S= -<\ln w_k> = -<\ln (e^{T_k(z)*e^{-T_k(z)}})>$ For w_k as log Poisson process entropy is then: $S= -$ If we assume that negative sign can (?) be moved inside entropy mean ,than $S= <-T_k(z)*e^{-T_k(z)}>$ But $T_k(z) = -W_k(-z)$ and where $W_k (-z)$ is a branch of Lambert function. $W_0(-z) = \sum_{n=1}^{\infty}{(-n)^{n-1}*(-z)^n/n!$ $S = -z_k=$ But on the right we now have Poisson process 1 event with mean $W_k (-z)$ . Since entropy is also mean value <>, than $S = W_k(-z)$ For example, if $z=\pi/2$ the $S= W_k(-\pi/2) = I*\pi/2$ if $z=-1$ then $S= W_k(1) = \Omega$ if $z=\ln2/2$, then $S=W_k(-\ln2/2) =\ln 2$ if $z=1/e$ , then $S=W_k(-1/e) = 1$ Such interpretation of entropy allows negative, positive and complex values of entropy $S=W_k(-z)$. When is entropy of such process 0 ? if $z=0$ then $S = W_k(0) = 0$ Based on the above, every complex number w can be viewed as entropy of some log Poisson process and $w=S= W_k(-z)$. Obviously, the fluctuations around mean value in such log Poisson process depend on fluctuations of z. Are there any inherent fluctuations of complex numbers?. A process whose mean is $-\Omega$ I described Here The result was: $-\Omega$ $f(x) = \ln(x) \text{ if } x>0$ $f(x) = \ln(-x) \text{ if }x<0$ $\lim_{n\to\infty}\frac{\sum_{n=1}^\infty f^{\circ n}(x)}{n}= -\Omega=-0.567143..=\ln(\Omega)$ if we took $1/x$ instead of x, mean was $\Omega$. From above , $\Omega$ then is entropy S of some log-Poisson process, while $-\Omega$ is entropy of negative log Poisson process such that $S= - W(-z) = T(z)$ If some process has entropy $S=\Omega=-<\ln\Omega>$ then from above average value of such process is $ = 0,567143 = \Omega$ So in this specific case (iteration of logarithm of absolute value of 1/x) , mean value of process and entropy are the same. Would that mean that this iteration of logarithm of absolute value of (x) has log Poisson distribution? and in general iterations could be looked upon as statistical mathematical processes? Each try n would give a random value of iteration and the mean is then the same as limit of sum of iteration values as n->oo divided by n as it goes to infinity. It would be interesting to find higher moments and also the structure functions which involve statistical deviations between 2 values of iteration as a function of "distance" n2-n1 between them. Since log Poisson processes are involved in intermittency of turbulence, might be that Tree and Lambert functions are as well. Ivars Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/17/2009, 08:51 AM (This post was last modified: 03/22/2009, 05:07 PM by Ivars.) Based on what I have been writing all along I have a tiny idea that something like that expressions: $e^{iwt*e^{-iwt}}$ in case of e.g. $I^{1/I} = e^{pi/2}$ must be a "time" oscillator. Obviously non-linear. When You try to solve any differential equation for a linear component $w$ of Fourier spectrum, You always get: Advanced solution : $e^{-Iwt}$ Retarded solution: $e^{Iwt}$ We know, $e^{iwt} = \cos wt + I*sin wt$ , is a wave solution of a single infinite Fourier component with frequency w. Replacing t with -t just changes direction of time, so usually if there is no damping, and system is symmetric in time, one of solutions is enough. Usually the full solution of the oscillator differential equation is the linear combination of both particular solutions: $A*e^{-Iwt} + B*e^{Iwt}$. The combination of particular solutions $A*e^{Iwt}*B*e^{-Iwt} =A*B$ is not very useful since it is independent of time. However, we can proceed further: if each of $A*e^{Iwt}, B*e^{-Iwt}$ are solutions of some differential equation, what is the physical process and what is its differential or functional equation whose solution is exponential combination of particular solutions: $A*e^{{Iwt}* B*e^{-Iwt}}$ or $B*e^{{-Iwt}* A*e^{Iwt}}$ which is also at the same time log Poisson type expression? Here we exponate advanced time solution in retarded, or vice versa: $a^{(b*a^-b}$ or $a^{-b*a^b}$ Without doubt, it is an nonlinear oscillator since if we write the full Euler formula: $e^{Iwt*e^{-Iwt}} = (\cos wt+I*\sin wt)^{(\cos wt-I*\sin wt)}$ It is a power combination of 2 complex helical sinusoidal waves of same frequency. Question: Solution of WHAT physical process and corresponding differential or functional equation it is? I have no doubt it is related to delay differential equations, but I have to learn about them more. Of course, this plugs DIRECTLY into tetration since it is a small power tower which can be extended. Also obviously we can use instead of $iwt$ imaginary part of tree function $T(z)$. It than connects this solution to infinite tetration. Also, as another approach, we can create new variable that is somehow normed vs. infinite tetration so that: $p(z, a) = (z[4]a)/{h(z)}$ Also, we can create new function $(W_k(-ln z)* T_k(ln z)) / -(ln_k ( z)) ^2 = (h_k(z))^2$ In this case, the conjugate values of h(z) then would be 2 values of square root, a geometric mean between 2 expressions for h: $h_k(z)= +- sqrt((W_k(-ln z)* T_k(ln z))) / I*ln_k ( z)$ Also, using Wright $\omega$ function one can similarly define $\tau(z) = T_{K(z)}(e^z)$ I wonder if all k are the same? Corless has written that $W_k(z)$ is an analytic ( in most part of k-plane) function of k where k can be any complex number. Is then $T_k(z)$ also analytic in the same k region? Logarithm? $(h_k(z))^2$? More later when I have understood what I have written Best regards, 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 »

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