Dimensionality mixed-phase space

Figure 1. Typical mixed phase space for the two-dimensional mapping. Quasi-periodic and chaotic regions coexist. The motion around tori is sticky and the orbit sometimes takes a long time to get through the sticky region.

The constant (3 can be regarded as a Liapunov exponent, the inverse of which gives a time scale for a system to reach the state of inter-basin mixing. In Fig. 8, we have p 0.17 or ( 1 6.0. And, at time t = 6, it is observed in Fig. 9 that Gnax(6) = 0.05. This means rmax(6) = 0.37 x rmax(0), but since the dimensionality of the cross section of a phase space is 29, the area of the cross section of this reaction tube is already as small as roughly rmax(6)29 3 x 10-13 x rmax(0)29. 

The last property ensures that all the lower-order mixed moments are included in the set. This turns out to be very important since generally neglecting mixed moments leads to abscissas that lie on lower-dimensional subspaces of the M-dimensional phase space. Whenever the dynamics of the investigated problem is not confining the abscissas on lower-dimensional supports, it is appropriate to choose a moment set to define A that is not restricted to generating such behavior. Therefore, it is suggested for many applications that one should use moment sets that treat all the M internal coordinates equally. If we relax this third condition a valid (but not optimal) moment set is instead obtained. 

Single-crystal X-ray analysis revealed that the purple phase [(Cp Ru)2(paracyclophane)] (TCNQ)2 crystallized in the P space group and contained one-dimensional mixed stacks of [(Cp Ru)2(paracyclophane)] and... 

Given the algorithm data flow structure of flgure 1, modeled by the PDG model, the definition of a control flow is performed in two phases. First, the placement of individual domains is done in a three-dimensional common node space. This placement step is performed incrementally, steered by accurate mixed ILP optimization techniques [31]. The result is depicted in figure 3. The extreme points of node spaces 1 and 3 are located at [0,0,2] and [561,0,2] and at [0,-1,0] and [511,-1,0], respectively. This means that both spaces are nicely aligned to each other and connected by 512 dependencies with direction [0,-1,-2]. The extreme points of node space 4 are located at [0,0,1], [511,50,1], [511,0,1] and [0,50,1]. 

For a material system with density and viscosity differences, the 5-parametric relevance list, eq. (5.1), has to be extended by the physical properties of the second mixing component, by the volume ratio of both phases, (p = V2/Vi, and - inevitably - by the weight difference, gAp, due to the prevailing density differences, to a 9-parametric dimensional space [15] ... 

In this CCC system, the column holder revolves around the vertical axis of the centrifuge while it rotates about its horizontal axis at the same angular velocity. This second HSCCC system is based on a hybrid between type L and X synchronous systems (Fig. 5), and it leads to bilateral hydrodynamic distribution of the two phases in an end-closed coaxial multilayer coil as in the type J HSCCC (Fig. 6B). However, in contrast to type J synchronous planetary motion, the centrifugal vectors fluctuate in a three-dimensional space where one component steadily acts across the diameter of the tube to stabilize the retention of the stationary phase. This stabilizing effect becomes greater as the hybrid approaches the type L synchronous system, while the phase-mixing effect is reduced. The optimum column position for... 

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