Scale Relativity, Non-differentiability and Fractal Space-time

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With Galileo, time becomes a primary variable and velocity is derived from a ratio of space over time, which are now considered on the same footing, in terms of a space-time (which remains, however, degenerated, since the speed limit C is implicitly in nite there) This involves the vectorial character of velocity and its local aspect ( nally implemented by its de nition like the derivative of the position with respect to time) The same reversal can be applied to scales The scale dimension itself becomes a primary variable, treated on the same footing as space and time, and the resolutions are therefore de ned as derivatives from the fractal coordinate and (ie as a scale-velocity ): V = ln = d ln L/d (1418)

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This new and fundamental meaning given to the scale exponent = D DT , now treated like a variable, makes it necessary to allot a new name to it Henceforth, we will call it djinn (in preceding articles, we had proposed the word zoom, but this already applies more naturally to the scales transformation themselves, ln( / )) This will lead us to work in terms of a generalized 5D space, the space-time-djinn In analogy with the vectorial character of velocity, the vectorial character of the zoom (ie, of the scale transformations) is then apparent because the four spatio-temporal resolutions can now be de ned starting from the four coordinates of space-time and of the djinn: v i = dxi /dt ln = d ln L /d (1419)

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Note however that, in more recent works, a new generalization of the physical nature of the resolutions is introduced, which attributes a tensorial nature to them, analogous to that of a variance-covariance error matrix [NOT 06, NOT 08] We could object to this reversal of meaning of the scale variables, that, from the point of view of the measurements, it is only through L and that we have access to the djinn , which is deduced from them However, we notice that it is the same for the time variable, which, though being a primary variable, is always measured in an indirect way (through changes of position or state in space) A nal advantage of this inversion will appear later on in the attempts to construct a generalized scale relativity It allows the de nition of a new concept, ie that of scale-acceleration = d2 ln L /d 2 which is necessary for the passage to non-linear scale laws and to a scale dynamics The introduction of this concept makes it possible to further reinforce the identi cation of fractals of constant fractal dimension with scale inertia Indeed,

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Scaling, Fractals and Wavelets

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the free scale equation can be written (in one dimension to simplify the writing): = d2 ln L/d 2 = 0 It integrates as: d ln L/d = ln = constant (1421) (1420)

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The constancy of resolution means here that it is independent of the djinn The solution therefore takes the awaited form L = L0 ( / ) More generally, we can then make the assumption that the scale laws can be constructed from a least action principle A scale Lagrange function, L(ln L, V, ), with V = ln( / ) is introduced, and then a scale action:

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L(ln L, V, ) d

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(1422)

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The principle of stationary action then leads to Euler-Lagrange scale equations: L d L = d V ln L 1455 Scale dynamics and scale force The simplest possible form of these equations corresponds to a cancellation of the second member (absence of scale force), and to the case where the Lagrange function takes the Newtonian form L V 2 We once again recover, in this other way, the scale inertia power law behavior Indeed, the Lagrange equation becomes in this case: dV =0 d V = constant (1424) (1423)

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The constancy of V = ln( / ) means here, as we have already noticed, that it is independent of Equation (1423) can therefore be integrated under the usual fractal form L = L0 ( / ) However, the principal advantage of this representation is that it makes it possible to pass to the following order, ie, to non-linear scale dynamic behaviors We consider that the resolution can now become a function of the djinn The fact of having

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