Catastrophe map

Figure 24. The Xiao-Kellman catastrophe map for L = 0. Regions I-IV differ according to the numbers and types of fixed point listed in Table 11. The points on the curved trajectories mark the coordinates of successive polyads for the relevant molecule. Taken from Ref. [14] with permission of the PCCP Owner Societies.

Cooper and Child [14] have given an extensive description of the effects of nonzero angular momentum on the nature of the catastrophe map and the quantum eigenvalue distributions for polyads in its different regions. Here we note that the fixed points and relative equilibria, for nonzero L = L/2J, are given by physical roots of the equation... 

It should be noted that Fig. 25 is not exhaustive, because the catastrophe map for general L in Fig. 2 of Ref. [14] shows a narrow band that is bounded below by the junction between regions III and IV of the L = 0 catastrophe map in Fig. 24 and above by a line of cusps, parametric on L. The nature of the monodromy in this region remains to be investigated. 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

In addition, the catastrophe mapping /7 — a projection of the point (x c) on the control parameters space... 

Recently, another class of MAPs has been identified that lead to depolymerization of MT, stathmin and katanin [11], Stathmin destabilizes MT either by reducing available tubulin dimer or by altering the frequency of catastrophe at MT plus ends [18]. In contrast, katanin is an ATP-ase that severs MT [19]. In neurons, one important role of katanin is to release the MT from the MTOC and allow transport of the assembled MT into axons and dendrites. 

Rgure 1.6. (a) Schematic formulation-composition map. SAD is the surfactant affinity difference it is positive for a lipophilic surfactant and negative for a hydrophilic one. The gray zones are abnormal, (b) Schematic representation of the proposed mechanism for a transitional inversion, (c) Schematic representation of the proposed mechanism for a catastrophic phase inversion. 

This concept has been borrowed from the "catastrophe theory . Nowadays this theory has been extensively developed [220]. Strictly speaking, it is the theory of the peculiarities of differentiable mappings [221]. 

Fig. 3 A The face - chalice ambiguous figure drawn in B as a series of altered figures that map onto a cusp catastrophe. The potential energy of each ( , v) point as projected onto the manifold...

The Thom s catastrophe theory (Thom, 1973) basically describes how, for a given system, a continuous action on the control space (C ), parameterized by c s, provides a suddenly change on its behavior space (/ ), described by variables x s, through the stable singularities of the smooth map... 

FIGURE 10.1 Two maps from Antonio Snider-Pelligrini s (1858) book La Creation et ses Mysteres Devoiles (the maps are numbered 9 and 10 and inserted between pages 63 and 64). He was an ardent Christian with immense foith that the earth was created by a catastrophic event. According to him, the continents of America, South America, Europe and Africa once formed a biock of fused rocks prior to the great flood. The flood cooled the block and a gigantic and rapid rupture then occurred, involving the separation of Americas and the Old World. 

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