Lorentzian area measures and the Christoffel problem

We introduce a particular class of unbounded closed convex sets of $\R^{d+1}$, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature $(+,...,+,-)$ instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space \H^d. Important examples are embeddings of the universal cover of so-called globally hyperbolic maximal flat Lorentzian manifolds. Basic tools are first derived, similarly to the classical study of convex bodies. For example, F-convex sets are determined by their support function, which is defined on \H^d. Then the area measures of order $i$, $0\leq i\leq d$ are defined. As in the convex bodies case, they are the coefficients of the polynomial in $\epsilon$ which is the volume of an $\epsilon$ approximation of the convex set. Here the area measures are defined with respect to the Lorentzian structure. Then we focus on the area measure of order one. Finding necessary and sufficient conditions for a measure (here on \H^d) to be the first area measure of a F-convex set is the Christoffel Problem. We derive many results about this problem. If we restrict to "Fuchsian" F-convex set (those who are invariant under linear isometries acting cocompactly on \H^d), then the problem is totally solved, analogously to the case of convex bodies. In this case the measure can be given on a compact hyperbolic manifold. Particular attention is given on the smooth and polyhedral cases. In those cases, the Christoffel problem is equivalent to prescribing the mean radius of curvature and the edge lengths respectively

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Tropical varieties, maps and gossip

Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group

A distributed solution to the adjustable robust economic dispatch problem

The problem of maintaining balance between consumption and production of electric energy in the presence of a high share of intermittent power sources in a transmission grid is addressed. A distributed, asynchronous optimization algorithm, based on the ideas of cutting-plane approximations and adjustable robust counterparts, is presented to compute economically optimal adjustable dispatch strategies. These strategies guarantee satisfaction of the power balancing constraint as well as of the operational constraints for all possible realizations of the uncertain power generation or demand. The communication and computational effort of the proposed distributed algorithm increases for each computational unit only slowly with the number of participants, making it well suited for large scale networks. A distributed implementation of the algorithm and a numerical study are presented, which show the performance in asynchronous networks and its robustness against packet loss