Unstable direction

Beyond the third jet zone, there is a stagnant zone in which the velocity values are relatively uniform and have an unstable direction. There is reverse flow in zones I through III which is located between the jet boundaries and the cylinder walls. The maximum value of the velocity in the reverse flow is in the cross-section at the end of zone I at the distance Xj. The following equation was derived to calculate the length of the first zone X, ... 

Figure Jh Homoclinic tangle associated with the fixed point at (—a, 0) for case A. Near the fixed point, the solid line gives the unstable direction while the dashed line is the stable direction. The size of Planck s constant h is shown to illustrate that several states can be supported by the single structure. An estimate of the number of states is given by the number of h boxes needed to cover the structure.

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The multibaker map preserves the vertical and horizontal directions, which correspond respectively to the stable and unstable directions. Accordingly, the diffusive modes of the forward semigroup are horizontally smooth but vertically singular. Both directions decouple, and it is possible to write down iterative equations for the cumulative functions of the diffusive modes, which are known as de Rham functions [ 1, 29]... 

Tedmical Observations. The manufacture of benzidine is one of the most important operations of dye chemistry, because this base is used in the preparation of numerous valuable, although generalty unstable, direct dyes. Similar methods are used to prepare the tolidines (ortho and meta) from o- and m-nitrotoluenes, and o-dianisidine from o-nitroanisole. Dyes from m-tolidine will rare go on cotton, but they are interesting wool dyes. 

Attach to each point of the PODS its stable/unstable directions. 

If a system is uniformly hyperbolic, every point in phase space has both stable and unstable directions, and the maximum Lyapunov exponent with respect the maximum entropy measure is positive. The system has the mixing property and is therefore ergodic. The correlation function of observables also shows exponential decay. Uniformly hyperbolicity, which is sometimes rephrased as strong chaos in physical literature, is a well-established class of systems and is controllable by means of many mathematical tools [15]. In hyperbolic systems, there are no sources to make the relaxation process slow. 

This is because it represents an open one-dimensional flow in which fluid continuously escapes along the unstable foliation, that is not included in the model (2.87). The loss of fluid along the unstable direction, however, is balanced by the apparent compressibility of the flow along the x direction, dvx/dx = —A, which is consistent with the loss rate in (2.88). 

Nucleophilic addition to the carbonyl carbon of an a,f3-unsaturated Class II carbonyl compound is called direct addition addition to the j8-carbon is called conjugate addition. Whether direct or conjugate addition occurs depends on the nature of the nucleophile, the structure of the carbonyl compound, and the reaction conditions. Nucleophiles that form unstable direct addition products— halide ions, cyanide ion, thiols, alcohols, and amines— form conjugate addition products. Nucleophiles that form stable addition products—hydride ion and carbaiuons—form direct... 

In chemical terms, normally hyperbolic invariant manifolds play the role of an extension of the concept of transition states. The reason why it is an extension is as follows. As already explained, transition states in the traditional sense are regarded as normally hyperbolic invariant manifolds in phase space. In addition to them, those saddle points with more than two unstable directions can be considered as normally hyperbolic invariant manifolds. Such saddle points are shown to play an important role in the dynamical phase transition of clusters [14]. Furthermore, as is already mentioned, a normally hyperbolic invariant manifold with unstable degrees of freedom along its tangential directions can be constructed as far as instability of its normal directions is stronger than its tangential ones. For either of the above cases, the reaction paths in the phase space correspond to the normal directions of these manifolds and constitute their stable or unstable manifolds. 

Because of the greater potential dependence of the direct charge transfer than of the monoelectron transfer, the first one must always predominate at high overvoltage even if at the equilibrium potential the monoelectron transfer is energetically favored, so that it cannot be considered as barred. Moreover, if the intermediates occurring in multistep mechanisms are relatively unstable, direct charge transfer may become the fastest of the parallel paths even at equilibrium. 

In the general case where there are both stable and unstable characteristic exponents, or stable and unstable multipliers in the spectrum, the local bifurcation problem does not cause any special difficulties, thanks to the reduction onto the center manifold. Consequently, the pictures from Chaps. 9-H will need only some slight modifications where unstable directions replace stable ones, or be added to existing directions in the space. However, the reader must... 

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