Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table

Almost every graph is vertex-oblique

AbstractThe type of a vertex v of a graph G is the ordered degree sequence of the vertices adjacent to v. The graph G is called vertex-oblique if it contains no two vertices of the same type. We will show that almost every graph G∈G(n,p) is vertex-oblique, if the probability p for each edge to appear in G is within certain bounds

Discussion of a physical optics method and its application to absorbing smooth and slightly rough hexagonal prisms

Three different mathematical solutions of a physical optics model for far field diffraction by an aperture due to Karczewski and Wolf are discussed. Only one of them properly describes diffraction by an aperture and can, by applying Babinet's principle, be used to model diffraction by the corresponding plane obstacle, and by further approximation, diffraction by a particle. Studying absorbing scatterers allows a closer investigation of the external diffraction component because transmission is negligible. The physical optics model has been improved on two aspects: (i) To apply the diffraction model based on two-dimensional apertures more accurately to three-dimensional objects, a size parameter dependent volume obliquity factor is introduced, thus reducing the slightly overestimated side scattering computed for three-dimensional objects. (ii) To compensate simplifications in the underlying physical optics diffraction model for two-dimensional apertures [26] a size parameter dependent cross polarisation factor is implemented. It improves cross polarisation for diffraction and reflection by small particle facets. 2D patterns of P 11, –P 12/P 11 and P 22/P 11 and their azimuthal averages for slightly rough absorbing hexagonal prisms in fixed orientation are obtained and compared with results from the discrete dipole approximation. For particle orientations where shadowing is not negligible, improved phase functions are obtained by using a new method where the incident beam is divided into sub-beams with small triangular cross sections. The intersection points of the three sub-beam edges with the prism define the vertices of a triangle, which is treated by the beam tracer as an incidence-facing facet. This ensures that incident facing but shadowed crystal facets or regions thereof do not contribute to the phase functions. The method captures much of the fine detail contained in 2D scattering patterns obtained with DDA. This is important as speckle can be used for characterizing the size and roughness of small particles such as ice crystals.Peer reviewedFinal Accepted Versio

Liftings and stresses for planar periodic frameworks

We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201

Modelling light scattering by absorbing smooth and slightly rough facetted particles

A method for approximating light scattering properties of strongly absorbing facetted particles which are large compared to the wavelength is presented. It consists in adding the approximated external diffraction and reflection far fields and is demonstrated for a smooth hexagonal prism. This computationally fast method is extended towards prisms with slightly rough surfaces by introducing a surface scaling factor in order to account for edge effects on subfacets forming the rough surface. These effects become more pronounced with decreasing subfacet dimension to wavelength ratio. Azimuthally resolved light scattering patterns, phase functions and degree of linear polarisation obtained by this method and by the Discrete Dipole Approximation are compared for hexagonal prisms with smooth and slightly rough surfaces, respectively.Peer reviewedSubmitted Versio

Exploring local regularities for 3D object recognition

In order to find better simplicity measurements for 3D object recognition, a new set of local regularities is developed and tested in a stepwise 3D reconstruction method, including localized minimizing standard deviation of angles(L-MSDA), localized minimizing standard deviation of segment magnitudes(L-MSDSM), localized minimum standard deviation of areas of child faces (L-MSDAF), localized minimum sum of segment magnitudes of common edges (L-MSSM), and localized minimum sum of areas of child face (L-MSAF). Based on their effectiveness measurements in terms of form and size distortions, it is found that when two local regularities: L-MSDA and L-MSDSM are combined together, they can produce better performance. In addition, the best weightings for them to work together are identified as 10% for L-MSDSM and 90% for L-MSDA. The test results show that the combined usage of L-MSDA and L-MSDSM with identified weightings has a potential to be applied in other optimization based 3D recognition methods to improve their efficacy and robustness