The discrete logarithmic Minkowski problem

Existence of solution of the logarithmic Minkowski problem is proved for the case where the discrete measures on the unit sphere satisfy the subspace concentration condition with respect to some special proper subspaces. In order to understand how optimal this condition is, we discuss certain conditions that any cone volume measure satisfies.Comment: arXiv admin note: text overlap with arXiv:1406.750

Homogeneous linear matrix difference equations of higher order: Singular case

In this article, we study the singular case of an homogeneous generalized discrete time system with given initial conditions. We consider the matrix pencil singular and provide necessary and sufficient conditions for existence and uniqueness of solutions of the initial value problem.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0566

Discrete spacetime and its applications

We survey some results about the asymptotic behavior of discrete spacetime models, which appeared in diverse settings in the physics and math literature. We then discuss some recent applications, including scheduling in disk drives and analysis of airplane boarding strategies

Numerical method for Darcy flow derived using Discrete Exterior Calculus

We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is also included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus.Comment: Added numerical experiment for flow on a surface. Other small changes in meshing related comment

Electrical networks and Stephenson's conjecture

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome

Solutions of Higher Order Homogeneous Linear Matrix Differential Equations: Singular Case

The main objective of this talk is to develop a matrix pencil approach for the study of an initial value problem of a class of singular linear matrix differential equations whose coefficients are constant matrices. By using matrix pencil theory we study the cases of non square matrices and of square matrices with an identically zero matrix pencil. Furthermore we will give necessary and sufficient conditions for existence and uniqueness of solutions and we will see when the uniqueness of solutions is not valid. Moreover we provide a numerical example

A FEM for an optimal control problem of fractional powers of elliptic operators

We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement

The Lp Minkowski problem for polytopes for 0 < p < 1

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.Comment: 19 page

Computing Invariants of Simplicial Manifolds

This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure

Comparing Linear Width Parameters for Directed Graphs

In this paper we introduce the linear clique-width, linear NLC-width, neighbourhood-width, and linear rank-width for directed graphs. We compare these parameters with each other as well as with the previously defined parameters directed path-width and directed cut-width. It turns out that the parameters directed linear clique-width, directed linear NLC-width, directed neighbourhood-width, and directed linear rank-width are equivalent in that sense, that all of these parameters can be upper bounded by each of the others. For the restriction to digraphs of bounded vertex degree directed path-width and directed cut-width are equivalent. Further for the restriction to semicomplete digraphs of bounded vertex degree all six mentioned width parameters are equivalent. We also show close relations of the measures to their undirected versions of the underlying undirected graphs, which allow us to show the hardness of computing the considered linear width parameters for directed graphs. Further we give first characterizations for directed graphs defined by parameters of small width.Comment: 23 page