Topological isomorphism

For the knot plane projection with defined passages, the following Reidemeister theorem is valid [39] different knots (or links) are topologically isomorphic to each other if they can be transformed continuously into one another by means of a sequence of simple local Reidemeister moves of types 1, 2 and 3 (see Fig. 9). Two knots are called regular isotopic if they are isomorphic with respect to the last two types of moves (2 and 3) if they are isomorphic with respect to all types of Reidemeister moves, they are called ambient isotopic. As can be seen from Fig. 9, a Reidemeister move of type 1 leads to the cusp creation on chain projection. At the same time, it is noteworthy that all real 3D-knots (links) are of ambient isotopy. 

We also believe that almost all paths in the ensemble of lines with fractal dimension df = 3 are topologically isomorphic with a trivial knot or at least with a sufficiently simple one. 

There are 2 different realizations of the lattice. The natural physical question here is [3] What is the part Po( ) of unknotted (i.e. topologically isomorphic to trivial ring) paths on the lattice among 2 possible ones ... 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

The question whether ir Ms(G) — Mt(Gt0) is an isomorphism for general G ObC(n)s0 is local in the Zariski topology on S, so we may assume that G is embedded into an abelian scheme A over Sq. Put B = A/G. The result follows from the above and the commutative diagram with exact rows... 

If all phase trajectories by means of continuous deformation can be tranformed into one another, then the summation in Eq. (11) can be extended to all possible paths. However, if the phase space consists of topological non-isomorphic classes C, the summation in Eq. (11) extends only to paths from a given class and the problem of determination of the topological invariant arises. 

It is clear that in the case of MFI, the zeolite pore entrances should not be considered as rigid apertures. Instead, zeolite framework topologies can show flexibility. While the O-Si-0 angle in the tetrahedral unit is rigid (109 + 1 °), the Si-O-Si angle between the units can vary between 145 and 180°. Based on isomorphous substitution of Si by other T-atoms in the framework [18], framework defects [19], cation positions, changes in the water content [16], external forces on the crystalline material [20] and upon adsorption of guest molecules [21] phase transitions can occur that have a dramatic influence in particular cases on the framework atom positions. 

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