Non-leading manifold

Here the fixed point is asymptotically stable. All trajectories apart belonging to the non-leading manifold x = 0 enter O along the leading direction... 

If Lk < 0, then for the original multi-dimensional map (10.4.1), the fixed point is also a stable focus. Moreover, its leading manifold coincides with the center manifold. This means that all positive semi-trajectories, excluding those in the non-leading manifold tend to O along spirals which are... 

Remark. In the multi-dimensional case where besides the central coordinates there are also the stable ones, the unstable set consists of three curves, whereas the stable set is a bimch consisting of three semi-planes intersecting along the non-leading manifold as shown in Fig. 10.5.4, for the three-dimensional example. 

Recall that the non-leading manifold is (n — l)-dimensional. It partitions into two components. If the loop F lies in then a small perturbation may make it miss so that it enters the saddle from either component of We will show (Subsec. 13.6.2) that when the loop is moved from one component to the other, it is accompanied by a change in the sign of the separatrix value A. 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

The manifold intermediates in homogeneous transition-metal catalysis are certainly metal complexes and therefore show a behaviour like ordinary coordination compounds associations of phosphorus donors open up multifarious additional controls. Both, substrates and P ligands are Lewis bases that we have to consider and that compete at the coordination centers of the metal, leading to competitive, non-competitive or uncompetitive activation or inhibition processes in analogy to the terminology of enzyme chemistry... 

In the schematic diagram of the vacuum line (Fig. 2.1) a U-bend is shown between the main tap T, and the manifold, which has two functions (a) It lends an extra element of mechanical flexibility to the system by absorbing small movements at the traps or along the main manifold (WL or WL ) which might otherwise lead to fracture and (b) it acts as a sink for non-volatile residues in the line and for grease which may be washed away from the taps 7, 7 and 7. ... 

In the Sharf and Fischer treatment the real manifold is subdivided into n idealized submanifolds where each one conforms with the Bixon-Jortner model. By taking suitable linear combinations of the n idealized submanifolds the problem may be reduced to that of a discrete state and two manifolds, both of which carry intensity but only one of which is associated with a non-zero interaction. The Bixon-Jortner procedure applied to this situation leads to a Fano lineshape superposed on a constant background absorption. 

Causes of fatty liver are manifold, and combinations of causes quite common. Acquired causes are by far the most frequent, but there are also rare causes, e.g. coeliac disease (9, 25), parenteral nutrition. (28, 29) Congenital metabolic disorders can also lead to the development of a fatty liver, as in the case of a rare thesaurismosis. It is of considerable therapeutic and prognostic importance to differentiate between an alcoholic fatty liver (AFL) and alcoholic steatohepatitis (ASH) (s. pp 529, 531) as well as between non-alcoholic fatty liver (NAFLD) and non-alcoholic steatohepatitis (NASH). (2, 20, 24, 36) (s. tabs. 31.5-31.7)... 

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