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Generalized Scale Invariance (GSI):

The parameters &alpha, C1, & H determine how the statistics of the multifractal vary as a function of resolution/scale ratio l. A multifractal at a given resolution is thus defined by averaging the field over all scales smaller than this. In the simplest case the scale can be taken to be the usual distance between two points; so that we average over circles. If the multifractal is scale invariant with this isotropic scale, then the multifractal is "self-similar" ; blowups of small sections give qualitatively the same-looking multifractals. But, what is the scale of rain, or of a cloud?

The usual scale notion is Euclidean distance. It is an a priori, not a physical notion of scale. A physical scale notion is a measure of size of a "typical" structure. The physics determines the appropriate scale notion - but if the structures determine the scale, how to determine the structures? Statistics can define average structures:

Shown in black: isolines of: (changing q does not change the shape; the "< ..>" symbol means statistical averaging)

..and the average structures can define scale changing operators. But, this is too simple without the help of a symmetry principle: scale invariance. (see the zooming movies in the movies section)

Generalized Scale Invariance (see GSI in the glossary for mathematical details) unifies these ideas. It allows the straightforward generalization of "self-affinity"; in this case the scale is defined by various powers of the distance from each of two orthogonal axes. The resulting structures are thus invariant under zooms combined with "compression" along one of the axes. Since the amount of compression is a power law function of the scale, the result is differential (scale dependent) stratification.

GSI also allows the more general case involving nonorthogonal axes (which can possibly rotate as functions of scale; the "rotation dominant" cases). Finally, GSI can be nonlinear, being defined by a position dependent vector field, the generator; however, around each point, linear GSI is approximately valid.

Linear GSI in 2-D:

Mathematically, in linear GSI, the scale function is defined by a dXd matrix; in 2-D:

d: This is the exponent which gives an overall (isotropic) change of scale.

c,f: In a polar representation c = r cos q, f = r sin q, we find that only r is important for the stratification; q gives the overall angle of rotation. Most of the similations below therefore take f=0 (and r=c); the other values of f are obtained by rotation.

e: This gives the contribution to differential (i.e. scale by scale) rotation of structures. The fundamental behaviour depends on the parameter a defined by . When a is real, we have stratification dominance, when a is imaginary, rotation dominance.

Balls and trajectories in linear GSI(position independent):

(The contour lines define all the vectors with equal scale)

Generalized scale reduction of sets:

A generalized blow-down with increasing of the acronym “NVAG”. If , we would have obtained a standard reduction, with all the copies uniformly reduced converging to the centre of the reduction. Each successive reduction is by 28% and

With nonlinear GSI, the anisotropy can depend on both position and scale:

Example of scale functions in Nonlinear GSI (position dependent; each box shows lines of constant scale for strucutres centred at the centre of the box)

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