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) [Pg.20]

From Eq. (5.7), we find an = (

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) [Pg.591]

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 [Pg.116]

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 [Pg.156]

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. [Pg.53]

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. [Pg.76]

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 [Pg.257]

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. [Pg.167]

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