Parabolic stationary point

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

Finding and characterizing the stationary states of systems with more than two DOFs is an unsolved problem. Isolated stationary points are the best known of these manifolds. In systems with two DOFs, in addition to isolated stationary points, it is also possible to find a closed loop of stationary points [77]. These are associated with parabolic resonances. More complicated manifolds wUl exist in systems having more than two DOFs. In the present discussion we wUl focus on the consequences of the existence of isolated points of stationary flow in phase space. 

The response surfaces in Figs. 2 and 3b are a part of distorted parabolic cylinder, which show a minimum ridge in the experimental domain. The response surface in Figure 3a is an inverted paraboloid (dome), and the corresponding contour plot is elliptical. The stationary point, which is the point at which the slope of the response surface is zero when taken in all directions, on this surface is within the design domain however, it is a minimum point. The response surfaces in Figs. 4a and 4b have a saddle behavior or minimax nature. The stationary point is not a maximum or a minimum point, but a saddle point. 

The case when the eigenvalues are equal to zero is special and must be treated separately. One is tempted to assume that such cases are rare. In fact these cases occur when the manifolds of stationary flow are not isolated points. The simplest of these cases give rise to parabolic resonances [77] however, they are beyond the scope of this review. These resonances have been observed in some of the simplest reactive systems [78]. 

The velocity of a liquid flowing in a capillary varies with the distance from the centre of the tube as shown in Fig. 7.9. The liquid at the surface of the tube is stationary so that the double layer at the interface consists of a stationary and a moving part. It is the relative movement of these two planes of the double layer which gives rise to the streaming potential. The velocity of the liquid at any point on the parabolic front distant x from the wall is given by... 

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