Cantor construction

Consider the following Cantor construction. A unit section is divided into three equal parts and the middle segment removed. Next, the remaining sections are divided into three equal parts and their middle segments removed again. The procedure, repeated an infinite number of times, yields the fractal Cantor set, see Fig. 91, in which the first few steps of the construction are shown. 

Fig. 2.1 First six steps of the construction of the triadic Cantor Set. The infinite-time limit point set has fractal dimension Dfractai = In 2/In 3.

Thus, whenever the set A has a manifest self-similarity, so that, like the Cantor set, it can be defined by a recursive geometric construction, Dfractal oan be easily calculated from this relation. The Koch Curve, for example, the first three steps in the construction of which are shown in figure 2.2, has a length L which scales as... 

What does the orbit look like for Ooo It is an infinite (and therefore aperiodic) self-similar point set with fractal dimensionality, Dfractai 0.5388 [grass86c]. Figure 4.5 shows the first six stages in the Cantor-set like construction. 

Zhang, A., Gonzalez, S.M., Cantor, E.J., and Chong, S. (2001) Construction of a mini-intein fusion system to allow both direct monitoring of soluble protein expression and rapid purification of target proteins. Gene 275, 241-252. 

Niemeyer, C M, Sano, T, Smith, C L, and Cantor, C R (1994) Oligo-nucleotide-directed self-assembly of proteins semisynthetic DNA-strepta-vidin hybrid molecules as connectors for the generation of macroscopic arrays and the construction of supramolecular bioconjugates. Nucleic Acids Res. 22, 5330-5339. 

Conformational changes in biopolymers are commonly described by a model that has been derived by an application of the one-dimensional Ising model to the problem of cooperative transitions from random coil states into ordered mostly helical conformations of (homo)biopolymers (see e.g. Cantor and Schimmel, 1980). Although the threshold is mostly of the cooperative transition type, landscapes can be constructed for which the threshold corresponds to a first order phase transition. 

Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403 339-342... 

Figure 50. Schematic picture of mg(r) dependence, f,- are the time moments of the interaction that construct in time a fractal Cantor set with dimension df = In 2/ In 3 = 0.63. (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

Fig. 2.2. Sketch of the first three stages in the construction of Cantor s middle thirds set C.

According to (2.3.2), there will be no probability left in the unit interval in the limit of n 00. Therefore, we say that a point xq in A ionizes with probabiUty 1 according to the ionization process defined by the tent map T. But is there really nothing left in A after the application of n -i- 00 mappings The successive steps in the ionization process defined by repeated application of T (see Fig. 2.8) remind us strongly of the construction scheme of Cantor s middle thirds set C (Fig. 2.2) which was introduced and briefiy discussed in Section 2.1. And indeed, there is a whole infinity of points left in A, even in the limit of an infinite number of applications of T. What kind of infinity We can easily answer this question with the tools developed in Section 2.1. Let us introduce a notation for the set of points in A that never ionize. We call this set A" ". For the tent map T we have A" " = C. 

The Cantor set C, as well as the fractal shown in Fig. 2.10, are selfsimilar fractals with simple construction rules. Many fractals encountered in physical and mathematical apphcations are not at all that simple. In order to compute their dimensions, one has to use numerical methods. The following are two frequently employed numerical methods for computing fractal dimensions. 

In the construction of the standard Cantor set the total length of the intervals surviving after the -th step is (2/3)" corresponding to a survival probability... 

This relation also holds for Cantor sets defined by exclusion of other fractions than one third from the intervals in each step. The factor log 2 on the right of Eq. (3) has its origin in the fact that each step in the construction of the Cantor set involves a doubling of the number of surviving intervals. 

Motivated by the remarkable discovery of quasicrystalline ordering in solids in 1984 [1], wave propagation in deterministic non periodic media has been an area of intense research. Following the successful experimental realisation of a multitude of such structures through modem technologies, such as molecular beam epitaxy and laser ablation [2], their interest has increased ever since. The most widely known examples are quasi-periodic structures obtained by substitution rules, such as Fibonacci- or Thue-Morse-chains [3,4], Much less has been published on quasi-periodic chains constructed according to a Cantor-set algorithm, which are the subject of this note. 

This argument (devised by Cantor) is called the diagonal argument, because r is constructed by changing the diagonal entries x in the matrix of digits [x,j]. 

Solution Recall that the Cantor set is covered by each of the sets S used in its construction (Figure 11.2.1). Each consists of 2" intervals of length (1/3)", so if we pick = (1/3)", we need all 2" of these intervals to cover the Cantor set. Hence... 

Cantor set has measure zero) Here s another way to show that the Cantor set has zero total length. In the first stage of construction of the Cantor set, we removed an interval of length from the unit interval [0,1]. At the next stage we re-... 

Middle-halves Cantor set) Construct a new kind of Cantor set by removing the middle half of each sub-interval, rather than the middle third. 

Generalization of even-fifths Cantor set) The even-sevenths Cantor set is constructed as follows divide [0,1] into seven equal pieces delete pieces 2, 4, and 6 and repeat on sub-intervals. 

Figure 5 Construction process of the structure considered in section 4 a one-dimensional line with a distribution of reactive centres localized on a Cantor middle-third set. The two graphs labelled (A ) are copies of the structure at iteration n, while the graph labelled (0) is a one-dimensional chain at iteration n with no reactive centres.

To give a simple example, let us take Q equal to the one-dimensional line (this example corresponds to the classical slab-like model for a cylindrical pore), and let Qr be a Cantor middle-third set, da = log 2/log 3 = 0.631. Figure 5 shows the construction process of the structure applied in the renormalization process. The graph of the structure at... 

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