Local map

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Figure 1.193. Locality map of the hot spring-type deposits in Kyushu.

For comparison the same fatty acid data have been visualized by Sammon s NLM with parameter p set to 1 which makes a local mapping preserving well the small distances between objects (Figure 3.25, left). The NLM result shows a compact cluster of the fresh reference samples (class 6), and somewhat separated the desert sample (5). The horizontal map coordinate separates fresh samples from the high mountain samples (4) and the other samples (glacier samples, 2 lake samples, 3 and iceman samples, 1). The clustering has been indicated by manually drawn ellipses it is very similar as obtained with the scores of the PCI and PC2. Furthermore, Figure 3.25... 

Let us remark incidentally that the van der Waals, solvent-accessible and solvent-excluded molecular surfaces commonly used in apparent surface charge calculations, can be discretized without resorting to a polyhedral approximation. Indeed, these surfaces are made of pieces of spheres and tori and it is therefore possible to mesh and compute integrals on the molecular surfaces since analytical local maps are available [19],... 

These local mapping transformations, relating AIM electron populations (charges) to bond lengths, can be easily generalized into relations involving collective electron-population- and/or nuclear-position-displacements, e.g., PNM and nuclear normal modes. For example, the bond-stretching normal vibrations, 2b, defined by the fb principal directions, O = d lb/8Rb,... 

Begley TJ, Rosenbach AS, Ideker T, Samson LD (2004) Hot spots for modulating toxicity identified by genomic phenotyping and localization mapping. Mol Cell 16(1) 117-125. doi 10.1016/j. molcel.2004.09.005... 

Figure 7.1 Locality map for populations of Liothyrella neozelanica from Doubtful Sound, New Zealand, and L uva from Signy Island, South Orkney Islands.

What has all of this to do with coupled map lattices Suppose we take a CML model where the local map at a site leads to cycling among three states a period-3 map. An example is... 

A = a (see Figure 12 for a graphical representation). The local map has the same discrete period-3 behavior as the local dynamics of the resonantly forced system. The discrete time dynamics of the map breaks the time translation symmetry of the system just like the resonant forcing of the local chemical dynamics. We can now make a CML model of the resonantly forced 3 1 reaction-diffusion system by coupling an array of such period-3 maps ... 

In particular, for any qroup the group ring 21") is torsion-free, so that by Proposition 3.6.4 ii) the localization maps... 

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