Biophysical Chemistry of Fractal Structures and Processes in by P. Baveye
By P. Baveye
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Additional info for Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems
In other words, the Cantor set has zero length and its topological dimension is zero. Yet it is an uncountable set, containing inﬁnitely many points in any neighborhood of each of its points. Furthermore, one can show that the points PHILIPPE BAVEYE ET AL. 15 of the Cantor set can be put in one-to-one correspondence with those of the initiator [0, 1]; mathematically, P and [0, 1] have the same ‘cardinality’. This raises some very serious questions about the use of traditional dimensions to characterize these two sets, and justiﬁes the labels of ‘monstrous’ and ‘pathological’ that rapidly became associated with the Cantor set.
1). Indeed, he argues that the divider dimension does not have a precise metric meaning in these cases. g. 14 (p. 736), 32 (p. 334)]. e. one determines the spatial coordinates of N0 points uniformly distributed along the curve. Then, one centers a ball Br (x) of radius r at some location x on the curve, and one counts the number N(r, x) of sampled points within this ball. 22) This limit may also be viewed as the measure or ‘mass’ of the curve that is contained in the ball Br (x). 23) and the ratio ln μ(Br (x))/ ln r is termed in this case the Hölder , coarse Hölder  or Lipschitz–Hölder exponent .
A number of alternative dimensions have been proposed to overcome the difﬁculties associated with the traditional box-counting dimension DBC . They include the lower and upper modiﬁed box-counting dimensions  and the packing or Tricot dimension [5, 33, 34]. Unfortunately, these dimensions reintroduce all the difﬁculties of calculation associated with DH , and in some cases are even more awkward to use! One key advantage of the box-counting dimension DBC over the similarity dimension DS is that DBC can be used to evaluate the dimension of self-afﬁne sets.