Double cusps

The cusps which have just been described are called single cusps in contradistinction to double cusps or points of osculation in which the curves extend to both sides of the point of contact. These are what Cayley calls tacnodes. The differential coefficient has now two or more equal roots and y has at least two equal values. The different branches of the curve have a common tangent. 

At Xi i I he double cusp point coinciding with the double critical point (Figure 3.97, the point A), appears inside the initial binodal at W2a- A new binodal then arises from this point. An increase in x, within x, < X, < Xio causes a splitting of the double cusp point into two cusp points with two arcs of the new binodal between them (Figure 3.98a) (cusp- -also means the moon s horn as two arcs of the binodal look). 

When W i = W2,U Ihe double cusp point splits into two cusp points, and there appears a configuration with two cusp points. The stable critical point is far from the cusp points on the left-hand branch of the CPC (see Figure 3.97) and, apparently, does not play any role. 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

The first reported study of the behaviour of double differential cross sections for positron impact ionization was that of Moxom et al. (1992) these workers conducted a search for electron capture to the continuum (ECC) in positron-argon collisions. In this experiment electrons ejected over a restricted angular range around 0° were energy-analysed to search for evidence of a cusp similar to that found in heavy-particle collisions (e.g. Rodbro and Andersen, 1979 Briggs, 1989, and references therein), which would be the signature of the ECC process. 

Another example is provided by pressure-induced phase transitions in which a cusp catastrophe transforms a double-well of the potential energy curve into a single-well and vice versa. In a wide pressure range, between 10 and 100 GPa, stishovite is the stable modification of silica. At about 100 GPa, a phase transition from PA2/mnm. (rutile) to Pmnm(CaCl2) occurs which corresponds to the twinning of the initial tetragonal cell... 

Figure 3. These high-resolution Galileo images show the pointy cusps of some Cycloids, where the double ridges make sharp turns. The crack in the crust lies between the ridges in each ridge-pair. The two examples to the left are details of cusps that appear in Figure 1.

Table I summarizes the results for Hg. In order to calculate the activation energy the results shown in Table II for the H atom and Hg molecule at the same level of approximation were used. The total energy, of course, improves as the wavefunction is allowed more variation. However, because the cusp in the wavefunction is poorly represented by one or two Gaussians the total energy is still far from the experimental value. In comparison with the Gaussian work of Schwartz and Schaad,i our best total energy, which involves full configuration interaction, is naturally better than their double Gaussian SCF calculation but not as good as their results with larger basis sets.

Tables 1.7 and 1.8 list Huzinaga s (41,45) basis sets for the Is and 2p Slater functions in hydrogen. These basis sets are interesting particularly because they were the first basis sets subjeeted to the double-zeta procedure of Slater theory. This procedure has the effect of reducing the number of terms for variation in calculations, but, more particularly, philosophically, provides for the better representation of details of the atomic radial functions, such as the cusp near the origin in s-functions, since the components subject to optimization separately, can be included in the linear combination.

The CeCuj orders antiferromagnetically at = 3.8 K, showing a structure in the specific heat jump at 4 K. This double transition is also seen in thermal expansion and in magnetic susceptibility measurements at 3.6 and 3.8 K, respectively. The entropy associated with both transitions is approximately Rln2 and 7lt = 50 mJ K /Ceatom (Willis et al. 1987). The temperature dependence of C at T second-order transition, furthermore, the cusp observed at and the strong anomaly in the thermal expansion suggest such a... 

Repairs of a tooth may make problems of cervical lesions more serious. For example, it has been shown that mesial occlusal distal (MOD) cavity preparations result in tooth cusps becoming effectively cantilever beams that deflect when the tooth is subject to occlusal loading [60,95-99]. Deeper cavity preparation causes longer cantilever beams to be created. Because the deflection varies as the length of the beam cubed, it follows that doubling the depth of the cavity causes cuspal deflection to... 

It should be noticed however that the period-doubling behavior is only observed for elastomer substrates without pre-strain. When the compression is achieved through a pre-stretching of the elastomer slab prior deposition of the thin sheet, the nonlinear elasticity of the PDMS mbber comes into play and other morphology characterized by periodic cusps is observed [21]. 

Differential thermal analysis measurements indicated double TgS for solutions with lower polymer concentrations beyond the depicted cusp in Fig. 12.10 [59]. Creep recovery measurements on this system showed that the solvent molecules in the solutions have higher mobilities than the polymer chain segments. A lower temperature and greater crowding is therefore necessary to force the solvent molecules from an equilibrium response. Therefore, it was... 

The pentary critical point results from the junction of four double critical points or of two pairs of cusp points with the critical point (Figure 3.101d). Such a point on the CPC is indistinguishable from the tricritical one in a ternary system. In the general case, the m multiple critical point gets into the CPC when the critical point merges with the (m — l)-multiple cusp point. The even-multiple critical points are always away from the CPC stable branch while an odd-multiple critical point may well appear on the CPC stable branch. 

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