Cantor

Cantor C R and Schimmel P R 1980 Biophysical Chemistry, Part II Techniques for the Study of Biological Structure and Function (San Francisco Freeman)... 

S. M. Complex, ia B. Cantor, ed.. Proceedings of the 3rd International Conference on Rapidly Quenched Metals Vol. 1, The Metals Society, London, 1978, p. 

C. R. Cantor and P. R. Schimmel, Biophysical Chemistry, Part II, Techniques for the Study ofiBiological Structure andFunction, W. H. Ereeman, New York, 1980. 

CR Cantor, PR Schimmel. Biophysical Chemistry. New York WH Ereeman, 1980. 

Clark, R. M. (1990). Unit process research for removing volatile organic chemicals from drinking water An overview. In Significance and Treatment of Volatile Organic Compounds in Water Supplies, (N. M. Ram, R. F. Christman, and K. P. Cantor, eds.), Lewis Publishers, Chelsea, ML... 

Adapted from Cantor, C., and Schimmel, P., 1980. Biophysical Chemistry.. San Franci.sco W.H. Freeman, and Tanford, C., 1968. Protein denatnration. Advances in Protein Chemistry 23 121-282. 

Cantor, G. (1991). Michael Faraday Sandemanian and Scientist A Study of Science and Religion in the Mineteenth Ceiitmy London Macmillan. 

Although it is not the only such measure, the fractal dimension docs quantify the intuitive belief that the Cantor set is somewhere in-between a point and a line. We will consider generalizations of fractal needed in later chapters,... 

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... 

If we give E the discrete topology and T the product topology, then T becomes a compact metric space homeomorphic to the Cantor set under the metric... 

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. 

Fig. 4.5 Sets of stable attractors for the first six critical values of a. Note the self-similarity between the boxed subpattern for oe and the entire pattern for 04 appearing two lines above. A Cantor-set-like fractal pattern appears in the limit an-+oo-...

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

This Cantor set may be explicitly visualized in the n-dimensional euclidean space 7 " by defining a mapping xp F —> 7 " ([grass83] and [packl]). The coordinate of the resulting vector xpid) is given by ... 

The formalism for computing Lyajmnov exponents for continuous dynamical systems that was introduced in the last section can also be used, with only minor modifications, for determining exponents for CA as well. The major modification involves replacing the Euclidean norm, V t) - used for measuring the divergence of two nearby trajectories (see equation 4.60) - by the Cantor-set metric, d t) ... 

The (finite time) limit set Df may be considered to be a Cantor Set, with dimension S given by (see section 4.5.2) ... 

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