Critical point index

A point K of M where the gradient of E(K) vanishes [where the tangent hyperplane to E(K) is "horizontal"], is a point where the force of deformation is zero, i.e., point K represents an equilibrium configuration. Such a point is called a critical point, and is denoted by K(A,i). Here, the first derivatives being zero, the second partial derivatives of the energy hypersurface are used to characterize the critical points. The first quantity in the parentheses, X, is the critical point index (and not the "order of critical point" as it is sometimes incorrectly called). The index A, of a critical point is defined as the number of negative eigenvalues of the Hessian matrix H(K(A,i)), defined by the elements... 

TABLE 7. Calculated relative energies at the Cl level (in kcalmol )21S (ATM = absolute true minima, TM = true minima, SP = saddle point, CP2 = critical point of index 2)... 

Most remarkable is the immense increase of the polydispersity index x x and of the ratio x/x which both increase with x This distribution follows asymptotically a power law of wfx)°cx with r 2.5, when the critical point of gelation is approached. Figure 20 shows some of these distributions for various a, or different x ... 

This shows that the Hessian of / is positive definite (resp. negative definite) on (resp. N ). Therefore / is non-degenerate in the sense of Bott, i.e. the set of critical points is a disjoint union of submanifolds of X, and the Hessian is non-degenerate in the normal direction at any critical point. We put = dim N = 2 dime N which is the index of / at the critical manifold Cj,. Note that the index is always even in this case. 

Proof. We only need to calculate the index of the critical point Z. By the choice of our Morse function, this is equal to the sum of dimensions of the weight spaces which satisfy either of the following conditions,... 

From Eq. (5.7), we find an = (0/9vxv0)112 = N112/3 at the critical point where N = (f>0/viv0 is the effective polymerization index. To approach this critical point avoiding the spinodal decomposition in the perpendicular directions, we require... 

E. Cornec 16 gave 1-3507 for the index Of refraction of the acid soln. of sp. gr. 1-1056. J. H. Gladstone found for the refractive index of soln. of sp. gr. 1-180 at 7-5°, 1-3584, 1-3630, and 1-3746 respectively for the A-, D-, and ff-lines. W. J. Pope gave 21-6 for the refraction equivalent of the H2P04-radicle. C. Fery said that in the progressive neutralization of phosphoric acid by sodium hydroxide there are three critical points corresponding with the appearance of the primary,... 

When the critical point must be a vertex of the hyperrectangle 0(F), the simplest approach to calculating the flexibility index F is to maximize s in each vertex direction 61 (Fig. 5) by the following (N)LP (Swaney and Grossmann, 1985a) ... 

Step 6. Apply the active constraint strategy to the flexibility index (F) at the stage of structure (without the energy recovery constraint). The form of this flexibility index problem is described in a later section, (a) If F a 1, then the HEN is operable in the specified uncertainty range. Stop, (b) If F< 1, then add the critical point for operability as another period of operation and return to step 5. 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

While the law with index 3.4 for viscosity is valid in the whole region above Mc, the dependence of terminal relaxation time is different for weakly and strongly entangled systems (Ferry 1980) and determines the second critical point M ... 

Wheeler (1936) uses a different method of determining inflammability. Having determined whether a given dust is explosive, he next proceeds to add known percentages of inert dust such as fuller s earth (200 I.M.M.-mesh) until the dust is no longer explosive. This critical point is called the index of inflammability." If the amount of fuller s earth used is denoted by / (in percent), the index of inflammability I may be written... 

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