Fold catastrophe

The number of elementary catastrophe types depends upon q. It has been shown that only for values of g < 5 is the number of catastrophe types finite. Thom has classified these types by their co-rank k and co-dimension q for values of g < 4. The concept of an unfolding and the accompanying definitions are illustrated first in terms of the simplest of all catastrophe types, the so-called fold catastrophe for which both the co-rank and co-dimension equal one. 

The analytical description of this fold catastrophe is obtained by replacing p(r) by a function of the single behaviour coordinate and parametrized by a single control parameter v, denoting displacements from the catastrophe point along the reaction coordinate. The topological behaviour of p(r) in the... 

For all molecules the cis conformation is the absolute minimum, whereas the trans conformation is either a secondary minimum (oxalyl fluoride) or a maximum (oxalyl chloride and oxalyl bromide). Oxalyl bromide presents a shallow minimum corresponding to a gauche conformer. Substituting F by Cl yields a cusp catastrophe which changes the two maxima at tt/2 and the minimum at 0 into a maximum at 0. The substitution of Cl by Br is responsible for a dual-fold catastrophe in which two wandering points near 27t/3 give rise to a new minimum (gauche conformation) and a new maximum. 

As the dihedral angle is increased from cf) = 0° to the critical value beyond which the bond path vanishes (27°), the H-H BCP and the corresponding 6-MR RCP come closer and closer. The annihilation of the H-H BCP at ca. 27° entails a sudden change in structure , a catastrophe [130], known as the fold catastrophe [63] (see change in structure in Fig. 6). As the BCP and the RCP approach one another, the property densities evaluated at each CP assume closer and closer values until the two points eventually merge and annihilate each other [57]. The plot in Fig. 7a shows the change in the distance... 

We shall begin the investigation of Thom potential functions with the family of functions V(x a) = x3 + ax corresponding to the fold catastrophe or A2, see Table 2.2. The catastrophe manifold corresponding to this potential function is obtained from equation (2.13), c = (a),... 

Fig. 17. Catastrophe surface M2, singularity set I2 and bifurcation set B2 of the fold catastrophe (.42).

The function S(b /) = S,(b), playing the role of a potential function of the system, thus corresponds to the potential function of an elementary fold catastrophe (A2), b plays the role of a state variable and l is a control parameter. 

Detailed analysis of a number and types of critical points (attractors) of the ELF field enables a characterisation of quantitative changes of ELF, as well as the electronic structure of the N-Ol bond, within the catastrophe theory. For F01N02, three catastrophes observed during dissociation of the N-Ol bond, one cusp and two fold catastrophes, are shown in Fig. 19.2. The V(N,01) attractor is annihilated in the cusp catastrophe and the protocovalent bond is created by two attractors, V(N) and V(01). Subsequently the V(N) and V(01) attractors are annihilated in the fold catastrophes. 

We now turn to obtaining estimates for the expected crystal thickness, that is the solution of Eq. (3.97), for various values of C, where C, = 0 for / < lmi and increases with l otherwise. The case C, = constant can be used to show that the finite probability of folding is sufficient to obtain a finite thickness at all supercoolings thus avoiding the SI catastrophe, which was demonstrated in Sect. 3.7.1. This case is unphysical and was only considered because of its mathematical simplicity. It leads to the prediction that the thickness, though finite, increases with AT. 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

Therefore, by (34), there will only be a fold in the rs-catastrophe surface if T satisfies ... 

The solution of this set of equations gives the non-isothermal induction period x (8) as a function of non-isothermal shear rate 8 for different values of the parameters P and Fig. 2.32 shows the results of calculations for a wide range of dimensionless shear rates from 0.01 up to 100. The parameter P is equal to 0.03 and % varies from 0 to 1. At high shear rates, the decrease in the induction period is proportional to 8 1. This means that a 100-fold increase in shear rate results in an almost 100-fold reduction in the induction period, which could well be catastrophic for material processing if the process rate is increased. The influence of the parameter on x (8) is significant only for high shear rates. 

With increasingly networked, distributed computer systems the risk of deliberate malicious interactions, using software-based tools, became a serious threat. Many-fold related issues like data protection, privacy, integrity, authenticity, and denial of service attacks, viruses, worms etc. lead to a separate community to be established, which is nowadays in the main focus of the public as was safety some time ago (and still is—but only after catastrophic events). This community developed separate standards, methods, taxonomy and ways of thinking. 

There is one last way to plot the results, which may appeal to you if you like to picture things in three dimensions. This method of presentation contains all of the others as cross sections or projections. If we plot the fixed points jt above the (r,/z) plane, we get the cusp catastrophe surface shown in Figure 3.6.5. The surface folds over on itself in certain places. The projection of these folds onto the (r,h) plane yields the bifurcation curves shown in Figure 3.6.2. A cross section at fixed h yields Figure 3.6.3, and a cross section at fixed r yields Figure 3.6.4. 

The aminotransferases, AST and ALT, are enzymes located in the cytoplasm of hepatocytes and their levels will be elevated with hepatocellular injury. The degree of elevation of the aminotransferases is helpful in suggesting possible etiologies. The highest levels (>20-fold increase above normal) are typically seen in acute viral, drug-induced, or ischemic events associated with circulatory catastrophes. Alcoholic liver disease rarely presents with ALT values >500 units/L, and higher values should alert the clinician to complicating problems... 

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