Stationary points classification

Point Stationary point ( i, 2) m Hessian matrix eigenvalues Classification... 

List stationary points and their classification (maximum, minimum, saddle point) of... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

The properties of a nondegenerate stationary point, and its stability, derive from the properties of nondegenerate critical points of the potential. Note that the classification of stationary points described in Section 5.1 applies directly to a nondegenerate stationary point of a gradient system. Hence, in the case of gradient systems, the requirement of lack of degeneracy of a critical point constitutes the criterion of applicability of this classification. 

States of a system may be classified into three groups (a) the state beyond a stationary point, F(x, y) 0 (b) nondegenerate stationary state, F(x, y) = 0, det(dFi/dxj) 0 (c) degenerate stationary state, F(x, y) = 0, det(dFJdxj) = 0 analogously with the classification of gradient systems. 

Haken has considered the applicability of "Rohrschneider/ McReynolds constants" for the classification of stationary phases for the separation of fatty esters (13). He concluded that the approach was limited since the measurements used to determine the aforementioned "constants" are made at 100°C and most fatty acid methyl ester separations are carried out at about 200°C. He had previously shown significant variation in the, what will now be called, Rohrschneider/McReynolds coefficients, with temperature (14). Polar polysiloxanes such as XF-1150 demonstrated greatest variability in the coefficients and nonpolar types such as SE-30 demonstrated least variation. Supina pointed out that the X factor in the McReynolds coefficients should be indicative of extent of interaction with olefinic substituents (15). Figure 9.5 demonstrates the utility of this approach the 18 3 and 20 0 methyl esters are used as markers for the consideration of... 

Although the orthoester method has not been investigated kinetically, it probably can be considered as an activated monomer synthesis, a classification introduced by Bamford (131) and elaborated by Szwarc (132). Some implications of Szwarc s analysis may be pertinent to the orthoester polymerization. Szwarc has pointed out regarding a related polymerization of Leuch s anhydrides Increasing the concentration of initiator has a dual effect on the rate of such a polymerization. It increases the stationary concentration rrf growing species — a trivial effect expected in... 

Based on the above-mentioned six key properties of reversed phases, the stationary phases can be characterized. A wide variety of literature exists on this subject [9-15]. Of course, the synthesized stationary phases can be subjected to a full physicochemical examination (nitrogen adsorption measurements to determine the specific surface area, the pore volume and the pore size, CHN analysis to determine the surface coverage of the stationary phase, particle size measurements, etc.). However, all these characterizations are not really to the point, because in the end only the chromatographic separation counts. As a result, chromatographic tests for the characterization and classification of reversed phases have established themselves, from which a representative few, without any claim to being exhaustive, are presented here (Figures 4.1—4.3). 

Physical phenomena in science and engineering satisfy partial differential equations (PDEs), which relate changes in measurable quantities, like pressure or velocity, through partial derivatives taken in space and time. ETnlike ordinary differential equations (ODEs), whose integration constants are fixed by specifying values of the function and its derivatives at one or more points, PDEs require, in addition to functional information on curved boundaries, the specification of initial conditions for equations of evolution. Boundaries, we emphasize, may be external or internal, stationary or moving. The exact manner in which auxiliary conditions apply depends on the physical nature of the problem and is reflected in the type classification of the PDE studied. 

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