Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Dual virtual element method for discrete fractures networks

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments

Electron-impact rotational and hyperfine excitation of HCN, HNC, DCN and DNC

Rotational excitation of isotopologues of HCN and HNC by thermal electron-impact is studied using the molecular {\bf R}-matrix method combined with the adiabatic-nuclei-rotation (ANR) approximation. Rate coefficients are obtained for electron temperatures in the range 5$-$6000 K and for transitions among all levels up to J=8. Hyperfine rates are also derived using the infinite-order-sudden (IOS) scaling method. It is shown that the dominant rotational transitions are dipole allowed, that is those for which $\Delta J=1$. The hyperfine propensity rule $\Delta J=\Delta F$ is found to be stronger than in the case of He$-$HCN collisions. For dipole allowed transitions, electron-impact rates are shown to exceed those for excitation of HCN by He atoms by 6 orders of magnitude. As a result, the present rates should be included in any detailed population model of isotopologues of HCN and HNC in sources where the electron fraction is larger than 10$^{-6}$, for example in interstellar shocks and comets.Comment: 12 pages, 4 figures, accepted in MNRAS (2007 september 3