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Synthetic Nanostructures as Quantum Control Systems
Published in Günter Mahler, Volkhard May, Michael Schreiber, Molecular Electronics, 2020
For I = 2 there are n = 2N eigenstates, each directly connectable to c = N others, resulting in a total of e = cn/2 allowed connections (“edges”) out of n(n − 1) possible ones; in this sense the networks are sparsely connected. In Fig. 5 we see how, under these constraints, the structural network translates into a state space network: for N = 2 we get a ring, for N = 3 a connected double ring (as a projection of a 3-dimensional cube on a plane), for N = 4 a torus (as a projection of a 4-cube), for N = 5 a concentrical double torus (or 5-cube) etc. We also indicate to what extent the transitions can selectively be addressed in frequency space only, depending on local parameter variations (“disorder”) and/or interactions. In Fig. 5c,f the transitions from any given node are distinct in frequency space, while in Fig. 5b,e this is not the case. The remaining degeneracies could be lifted if other types of interactions according to Eqs. (6) or (7) were included.
Mesh Parameterization
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
Clearly, not all surfaces admit a parameterization. For instance, a sphere cannot be mapped to the plane without cutting it or poking a hole (see Figure 6.4, left). Therefore, it is a necessary condition that the surface has a boundary. However, it is not a sufficient condition. Consider the surface in Figure 6.4 (right): due to the handle, it is impossible to unfold it to 2D without intersections. Therefore, it is necessary to be able to determine whether a given mesh has a handle or not. The number of handles of a surface is referred to as its genus. A sphere (see Figure 6.5 A, B), has genus 0, a torus (see Figure 6.5 C), has genus 1, a double torus (see Figure 6.5 E), has genus 2.
Design in the service of topology and the classification theorem
Published in International Journal of Mathematical Education in Science and Technology, 2023
Klein bottle and Möbius strip cylinders are popular examples with rectangle presentation of this type of classification. Some surfaces, such as the sphere, can also be constructed by connecting 2 pairs, which creates 2 longer edges instead of 4. One example of octagon representation is the double torus, which has 8 directed edges.