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A Primer on Laplacians
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
In order to mimic this construction in the discrete setting, one requires a bit more structure than just an arbitrary graph. To this end, consider a simplicial manifold, such as a triangulated surface. We keep referring to this manifold as M. As in the smooth case, for simplicity, suppose that M is orientable and has no boundary.
Geometric theory of topological defects: methodological developments and new trends
Published in Liquid Crystals Reviews, 2021
Sébastien Fumeron, Bertrand Berche, Fernando Moraes
The 2 + 1 gravity model can therefore be experimentally investigated from a network of parallel disclinations lines in a 3D nematic sample. Geometry of disclinations networks has been theoretically investigated in the literature, sometimes allowing for analytical expressions for the metric tensor [145–147]. Several authors have shown the possibility of designing arrays of topological linear defects from photopatterning techniques [148–152] and even to manipulate them [153,154]. If this last point opens the possibility of emulating collisions between particles in the 2 + 1 model, it is even more interesting for the extension of the Deser, Jackiw and 't Hooft model to 3 + 1 dimensions [69]: matter particles are represented by a gas of piecewise straight string segments that are likely to collide with a higher frequency. The strings display both positive and negative mass densities, i.e. they are associated with and Frank angles, which makes liquid-crystal-based experiments particularly promising to investigate such models. This model may also have deep connections with Regge calculus in quantum gravity, where the smooth curved spacetime is replaced by a piecewise-flat simplicial manifold. This is like the triangulation of a surface in 3D where the local curvature is described by the dihedral angle between adjacent triangles (the triangle is a 2D simplex). The effect of gluing the edges of the simplexes generates a network of cone-like singularities (Regge cones) which are analogs to wedge disclinations [155,156] (see Figure 6).