Scale Relativity, Non-differentiability and Fractal Space-time

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identi ed the resolution logarithm with a scale-velocity , V = ln( / ), then results naturally in de ning a scale acceleration: = d2 ln L/d 2 = d ln( / )/d (1425)

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The introduction of a scale force then makes it possible to write a scale analog of Newton s dynamic equation (which is simply the preceding Lagrange equation (1423)): d2 ln L (1426) d 2 where is a scale-mass which measures how the system resists scale force F = = 14551 Constant scale force Let us rst consider the case of a constant scale-force Continuing with the analogy with motion laws, such a force derives from a scale-potential = F ln L We can write equation (1426) in the form: d2 ln L =G d 2 (1427)

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where G = F/ = constant This is the scalar equivalent to parabolic motion in constant gravity Its solution is a parabolic behavior: V = V0 + G , ln L = ln L0 + V0 + 1 G 2 2 (1428)

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The physical meaning of this result is not clear in this form Indeed, from the experimental point of view, ln L and possibly are functions of V = ln( / ) After rede nition of the integration constants, this solution is therefore expressed in the form: L 1 1 ln , ln ln2 (1429) = = G L0 2G Thus, fractal dimension, usually constant, becomes a linear function of the log-resolution and the logarithm of length now no longer varies linearly, but in a parabolic way This result is potentially applicable to many situations, in all the elds where fractal analysis prevails (physics, chemistry, biology, medicine, geography, etc) Frequently, after careful examination of scale dependence for a given quantity, the power law model is rejected because of the variation of the slope in the plane (ln L, ln ) In such a case, the conclusion that the phenomenon considered is not fractal could appear premature It could, on the contrary, be a non-linear fractal behavior relevant to scale dynamics, in which case the identi cation and the study of scale force responsible for the distortion would be most interesting

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Scaling, Fractals and Wavelets

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14552 Scale harmonic oscillator Another interesting case of scale potential is that of the harmonic oscillator In the case where it is attractive , the scale equation is written as: ln L + 2 ln L = 0 (1430)

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where the notation indicates the second derivative with respect to the variable Setting = ln( / ), the solution is written as: L ln = L0 ln2 ( / ) 1 2 ln ( / )

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(1431)

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Thus, there is a minimal or maximal scale for the considered system, whereas the slope d ln L/d ln (which can no longer be identi ed with the djinn in this non-linear situation) varies between zero and in nity in the eld of resolutions allowed between and More interesting still is the repulsive case, corresponding to a potential which we can write as = (ln L/ 0 )2 /2 The solution is written as: ln L = 0 L0 ln2 ln2 (1432)

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This solution is more general than that given in previous publications, where we 1 had considered only the case ln( / ) = 0 The interest of this solution is that it again yields asymptotic behavior of very large or very small scales ( or ) the standard solution L = L0 ( / ) 0 , of constant fractal dimension D = 1 + 0 On the other hand, this behavior is faced with increasing distortions when the resolution approaches a maximum scale max = , for which the slope (which we can identify with an effective fractal dimension minus the topological dimension) becomes in nite In physics, we suggested that such a behavior could shed new light on the quarks con nement: indeed, within the reinterpretive framework of gauge symmetries as symmetries on the spatio-temporal resolutions (see below), the gauge group of quantum chromodynamics is SU(3), which is precisely the dynamic symmetry group of the harmonic oscillator Solutions of this type could also be of interest in the biological eld, because we can interpret the existence of a maximum scale where the effective fractal dimension becomes in nite, like that of a wall, which could provide models, for example, of cell walls With scales lower than this maximum scale (for small components which evolve inside the system considered), we tend either towards scale-independence (zero slope) in the rst case, or towards free fractal behavior with constant slope in the second case, which is still in agreement with this interpretation

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