Locally topologically equivalent

This pattern of dots and gaps is a topological Cantor set. Since each dot corresponds to one layer of the complex, our model of the Rdssler attractor is a Cantor set of suifaces. More precisely, the attractor is locally topologically equivalent to the Cartesian product of a ribbon and a Cantor set. This is precisely the structure we would expect, based on our earlier work with... 

This is a key result since it links the moments of the eigenspectrum directly to the local topology of the molecule. Let us consider the first five moments, 0, filf 2, and respectively. The moment, ju0 gives the total number of states in the eigenspectrum or, equivalently, the total number of atomic orbitals, since from eqn (4.46)... 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

Contrary to a periodic lattice, the symmetry of which is entirely characterized by its space group, the description of an amorphous structure faces the problem of no long-range order and of a local coordination that varies from site to site. The polycluster description of amorphous solids is a classification based on a set of Coordination Polyhedra (CP) for each atom determined by its nearest neighbors. Topologically equivalent polyhedra form classes and are referred to as... 

Fig. 16.5, Phylogenetic trees derived on the basis of distance metrics derived from differences in various local structural parameters at topologically equivalent positions in a family of homologous cytochrome structures...

Definition 8.2. A system X [continuous or discrete) is said to have a modulus if in some subspace B of the space of dynamical systems, where X G B, a continuous, locally non-constant functional h is defined such that if X and X are topologically equivalent, then h[X) = h[X). 

The most successful theoretical framework in which the accumulating data has been understood is the tube model of de Gennes, Doi and Edwards [2]. We visit the model in more detail in Sect. 2, but the fundamental assumption is simple to state the topological constraints by which contingent chains may not cross each other, which act in reality as complex many-body interactions, are assumed to be equivalent for each chain to a tube of width a surrounding and coarse-graining its own contour (Fig. 2). So, motions perpendicular to the tube contour are confined while those curvilinear to it are permitted. The theory then resembles a dynamic version of rubber elasticity with local dissipation, and with the additional assumption of the tube constraints. 

The local equivalence that we have just proved implies that the predictive contents of the Maxwell s theory and of this topological model are exactly the same when referred to local experiments, as most of them are. Accordingly, it is not possible to discern between the two by viewing locally. This is the operative meaning of local equivalence. 

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