Topology torus

O Equation (60.10) states that both J and B must be perpendicular to VP. This means that J and B must lie on a surface with no components perpendicular to it. Then, the question arises What can the shape of the surface to which the J and B are bound be. The answer is a consequence of the Poincare-Hopf theorem, namely, that the surface must have the form of a topological torus, where the field lines lie on nested surfaces, and there are no field lines connecting one surface with the other. 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Here, the topological nature of the tori will be discussed briefly. Figure 1 shows the five possible prototypes of toroidal forms that are considered to be related to fullerenes. These structures are classified by the ratios of the inner and outer diameters r, and r, and the height of the torus, h. (Note that is larger than / ,) As depicted in Fig. 1, if r, = r, and h r, and h = — r,) then the toroidal forms are of type... 

Exercise 4.10 Consider the (topological) two-torus in and outer radius R. An equation for this two-torus is... 

We can also use link polynomials to prove that certain unoriented links are topologically chiral. For example, let L denote the (4,2)-torus link which is illustrated on the left in Figure 12. This is called a torus link because it can be embedded on a torus (i.e. the surface of a doughnut) without any self-intersections. It is a (4,2)-torus link, because, when it lies on the torus, it twists four times around the torus in one direction, while wrapping two times around the torus the other way. Let L denote the oriented link that we get by putting an arbitrary orientation on each component of the (4,2)-torus link, for example, as we have done in Figure 12. Now the P-polynomial of L is P(L ) = r5m l - r3m x + ml 5 -m3r + 3m r3. 

We can use this same approach to prove that other molecular knots and links are topologically chiral. For example, consider the molecular link illustrated in Figure 18. This catenane was synthesized by Nierengarten et al. [12]. For this molecule the set T(G) consists of many unlinks together with many copies of the (4,2)-torus link, illustrated as L in Figure 12. However we saw earlier that this unoriented link is topologically chiral. Therefore, the molecular (4,2)-torus link is topologically chiral as well. 

In a similar way we can prove that the embedded cell complex of the molecular (4,2)-torus link (see Figure 18) is topologically chiral. Also, by adding appropriate labels we can similarly prove the topological chirality of the oriented embedded cell complex of the molecular Hopf link (see Figure 19). 

Figure 13.12 A SOM-based pharmacophore road map. Different sets of ligands were projected onto a SOM that was generated by using the complete COBRA library. Black areas indicate the characteristic distributions of the compounds. Crosses indicate empty neurons in the map, i.e., areas of pharmacophore space that are not populated by the respective compound class. All molecules were encoded by a topological pharmacophore descriptor (CATS) [4], Note that each map forms a torus.

An example of topological analysis is shown in figure 13. A rectangular rubber sheet is transformed into a torus by the identification of pairs of opposite sides, first to form an open cylinder which transforms into a torus by joining the two open ends. In technical language, the Euclidean plane is... 

A surface is closed if it has no boundary curves. By this definition surfaces of a sphere and a torus are closed, whilst the surfaces of a hollow cylinder and of a disc are open. Boundary curves of two-sided surfaces are curves which separate one side of the surface from the other, for example the edges of a piece of thin paper. A completely open cylinder has two boundary curves. A cylinder which is half-open has only one boundary curve, and is continuously deformable into, and therefore topologically equivalent to a disc. Similarly, the removal of a disc from the surface of a sphere leaves an... 

Sheaf in fpqc topology 117 Smooth group scheme 88 Solvable group scheme 73 Spec A 41 Split torus 56 Strictly upper triangular 62 Subcomodule 23 Symplectic group 99... 

A properly folded chain molecule can form a cavity with a shape globally complementing the shape of an enclosed, quasi-spherical molecule. If, however, the enclosed molecule has a toroidal or more complicated topology, where the hole of the torus is small so the surrounding chain molecule cannot enter this hole, then an approximate global complementarity by a chain molecule is unlikely to reflect the correct topology of the enclosed molecule. 

Open-ended question for the topologically minded) Does Theorem 6.8.2 hold for surfaces other than the plane Check its validity for various types of closed orbits on a torus, cylinder, and sphere. 

Using the frequency map analysis we visualized (Lega and Froeschle 1996) the predicted result on the topology of the neighborhood of noble tori for values of the perturbing parameter well above the ones allowed by the mathematical demonstration. Moreover, we have measured the size of the complementary set of tori, showing that the size of islands and chaotic zones decreases exponentially when the distance to the noble torus goes to zero. 

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